/ / i c^,/I // c/ /~~ / i y y' 1" ~..c// ^1lil~-r C. -f ^ THE AMERICAN COMMERCIAL ARITHMETIC, FOR THE USE OF COMMERCIAL COLLEGES, Private Students, Schools and Counting-Houses, EMBRACING AN EXTENSIVE COURSE BOTH IN THEORY AND PRACTICE TOGETHER'WITH THE LAWS OF THE UNITED STATES RELATING TO INTEREST. DAMAGES ON BILLS, AND THE COLLECTING OF DEBTS, BY T. A. BRYCE, M.A., LL.D., AUTHOR OF TREATISES ON ALGEBRA AND GEOMETRY. ADOPTED AND USED IN THE ANN ARBOR BUSINESS COLLEGE. ANN ARBOR, MICH.: PUBLISHED BY A. C. PARSON, A.11[., PRESIDENT, ANN ARBOR BUSINESS COLLEGE 1867. ENTERED according to Act or Congress, in the year 1866, by GILBERT Y. BURNS, in the Clerk'sOffice of the District Court of the Northern District of New York. PREFACE. THOUGH elementary works on Arithmetic are in abundance, yet it seems desirable that there should be added to this an extensive treatise on the commercial rules, and commercial laws and usages. It is not enough that the school-boy should be provided with v course suited to his age. There must be supplied to him something higher as he advances in age and progress, and Vears the period when he is to enter on real business life. The Author's aim has, therefore, been to combine these two objects, and to produce a work adequate to carry the learner from the very elements up to the highest rules required by those preparing for business. As the work proceeded, it was found necessary to extend the original programme considerably, and, therefore, also the limits of the book, so as to make it useful to all classes in the community. In carrying out this plan, much care has been taken to unfold the theory of Arithmetic as a SCIENCE in as concise a manner as seemed consistent with clearness, and at the same time to show its; applications as an ART. Every effort has been made to render the: business part so copious and practical as to afford the young student ample information and discipline in all the principles and usages of commercial intercourse. For the same reason some articles on Commercial Law have been introduced, as it was a prominent part of the Author's aim to produce a work which should be found useful, not only in the class-room, and the learner's study, but also on the merchant's table, and the accountant's desk. The Author begs to tender his best thanks to J. Smith Iomans, Esq., New York City, Editor and Proprietor of the "Banker's Magazine and Statistical Register," for the able manner in which he supplied this part of the work. Throughout the work particular care has been taken not to enunciate any rule without explaining the reason of the operation, for, without a knowledge of the principle, the operator is a mert calculating machine that can work but a certain round, and is almost sure to be at fault when any ndvel case arises. The explanations IV. PREFACE. are, of course, more or less the result of reading, but, nevertheless: they are mainly derived from personal study and experience in teaching. The great mass of the exercises are likewise entirely new: though the Author has not scrupled to make selections from some oi the most approved works on the subject; but in doing so, he has confined himself almost entirely to such questions as are to be found in nearly all popular books, and which, therefore, are to be looked upon as the common property of science. Algebraic forms have been avoided as much as possible, as being unsuited to a large proportion of those for whom the book is intended, and to many altogether unintelligible, and besides, those who understand Algebraic modes will have all the less difficulty in understanding the Arithmetical ones. Even in the more purely mathematical parts the subject has been popularized as much as possible. In arranging the subjects it was necessary to follow. a certain logical order, but the intelligent teacher and learner will often find it necessary to depart from that order. (See suggestions to teachers.) Every one will admit that rules and definitions should be expressed in the smallest possible number of words, consistent with perspicuity and accuracy. Great pains have been taken to carry out this principle in every case. Indeed, it might be desirable, if practicable, not to enunciate any rules, but simply to illustrate each case by a few examples, and leave the learner to take the principle into his mind, as his rule, without the encumbrance of words. Copious exercises are appended to each rule, and especially to the most important, such as Fractions, Analysis, Percentage, with its applications, &c. Besides these, there have been introduced extensive collections of mixed exercises throughout the body of the work, besides a large number at the end. The utility of such miscellaneous questions will be readily admitted by all, but the reason why they are of so much importance seems strangely overlooked or misunderstood even by writers on the subject. They are spoken of as review exercises, but their great value depends on something still more important. An illustration will best serve here. A class is working questions on a certain rule, and each member of the. class has just heard the rule enunciated and explained, and therefore readily applies it. So far one important object is attained, viz., freedom of operation. But sofnething more is necessary. The SUGGESTIONS TO TEACHERS. THE author would first refer to the remark made in the Preface that he does not expect that the Teacher will follow the logical order adopted in the book, and even advises that he should not do so in many cases. He knows by experience that the same order does not suit all students any more than the same medical treatment suits all patients. t The coure requires to be varied according to age, ability and acquirements. The greatest difficulties generally present themselves at the earliest stages. What more serious difficulty, for example, has a child to encounter than the learning of the alphabet? Though this is perhaps the extreme case, yet others will be found to be in proportion. For beginners, therefore, we recommend the following course. Let the elementary rules be carefully explained and illustrated by simple examples, and the pupil shown how to work easy exercises; this done, let the whole be reviewed; and exercises of a more difficult kind proposed. The decimal coinage should then be taken up. In explaining this part of the subject the teacher ought to notice carefully that the operations in this case differ in no way from those already gone through in reference to whole numbers, except in the preserving of the mark that separates the cents from the dollars, usually called the decinal point. The next step ought to be the whole subject of denominate numbers, and in illustration and application, the rule of practice. After a thorough review of all the ground now gone over, Simple Proportion may be entered upon, using such questions as do not involve Fractions. Then, after a course of Fractions has been gone through, Proportion should be reviewed, and questions which involve Fractions proposed. After this it will generally be found desirable to study Percentage, with its applications. The order in which the rest of the course shall be taken is comparatively unimportant, as the student has now realized a capital on which he can draw upon for any purpose. The author would, in the strongest manner possible, impress on the minds of teachers the great utility of frequent reviews, and especially of constant exercise in the addition of money columns vi. PREFACE. by that of Decimals. Barter, too, has been passed by, as questions of that class can easily be solved by the Rule of Proportion, which has been fully explained. The subject of Analysis has been gone into at considerable length, and it is hoped that the new manner in which the explanations and solutions are presented, and the extensive collection of exercises appended, will contribute to make this a valuable part of the treatise. The view given of Decimal Fractions seems the only true one, and calculated to give the student clear notions regarding the nature of the notation, as a simple extension of the common Arabic system, and also appropriate to show the convenience and utility of Decimals. The distinction beeween -Decimals and Decimal Fractions has been ignored-as being " A distinction without a difference." Decimals is merely a short way of writing Decimal Fractions; thus,.7 is merely a convenient mode of writing N.' These differ in form only, but otherwise are as perfectly identical as i and j. Tile contracted methods of Multiplication and Division will be found, after some practice, extremely useful and expeditious in Decimals expressed by long lines of figures. The averaging of Accounts and Equations of Payments, Cash Balance and Partnership Settlements, have been introduced as essential parts of a commercial education, and, it is hoped, will form a most important and useful study for those preparing for business, and probably a safe guide to many in business who have not systematically studied the subject. SUGGESTIONS TO TEACHERS. THE author would first refer to the remark made in the Preface that he does not expect that the Teacher will follow the logical order adopted in the book, and even advises that he should not do so in many cases. He knows by experience that the same order does not suit all students any more than the same medical treatment suits all patients. The coure requires to be varied according to age, ability and acquirements. The greatest difficulties generally present themselves at the earliest stages. What more serious difficulty, for example, has a child to encounter than the learning of the alphabet? Though this is perhaps the extreme case, yet others will be found to be in proportion. For beginners, therefore, we recommend the following course. Let the elementary rules be carefully explained and illustrated by simple examples, and the pupil shown how to work easy exercises; this done, let the whole be reviewed; and exercises of a more difficult kind proposed. The decimal coinage should then be taken up. In explaining this part of the subject the teacher ought to notice carefully that the operations in this case differ in no way from those already gone through in reference to whole numbers, except in the preserving of the mark that separates the cents from the dollars, usually called the decinal point. The next step ought to be the whole subject of denominate numbers, and in illustration and application, the rule of practice. After a thorough review of all the ground now gone over, Simple Proportion may be entered upon, using such questions as do not involve Fractions. Then, after a course of Fractions has been gone through, Proportion should be reviewed, and questions which involve Fractions proposed. After this it will generally be found desirable to study Percentage, with its applications. The order in which the rest of the course shall be taken is comparatively unimportant, as the student has now realized a capital on which he can draw upon for any purpose. The author would, in the strongest manner possible, impress on the minds of teachers the great utility of frequent reviews, and especially of constant exercise in the addition of money columns Viii. SUGGESTIONS TO TEACIERS. To make the exercises under each rule of progressive difficulty, as far as possible, has been an object kept constantly in view, as also to give each exercise the semblance of a real question, for all persons, especially the young, take greater interest in exercises that assume the form of reality than in such as are merely abstract; and, besides, this is a preparatory exercise to the application of the rules afterwards. At every stage the greatest care should be taken that the learner thoroughly understands the meaning of each rule, and the conditions of each question and the terms in which it is expressed, before he attempts to solve it. The Teacher should not always be talking or working on the black-board; he should require the pupils to speak a good deal in answer to questions, and also work much on their slates, and each in his turn on the board for illustration to the rest. Finally, it is suggested to every Teacher to keep constantly before his mind both of the two chief works he has to accomplish. First, the developement of the mental powers of his pupil; and, secondly, imparting to him such knowledge as he will require to use when he enters upon life, either as a professional man, or a merchant or clerk. Some seem to consider these two objects incompatible, as if taking up time in mental training left insufficient time for the imparting of actual knowledge. This is a palpable errror, for the more the mental powers are cultivated, the more readily and rapidly will any species of knowledge be apprehended, and the more surely, too, will it be retained when it has been mastered. Mental culture is at once the foundation and the means; the other is the superstructure raised on that foundation and by that means; or it may be compared to a great capital judiciously embarked in trade, and often turned, and therefore yielding good. profits. It frequently happens, however, from the peculiar circumstances of individuals and families, and even communities, that young men require to be hurried into business, so as to be able to support themselves; but even in such cases the desired object will be much more readily and securely attained by such a course than by what is usually and not inappropriately called " Cramming." Every effort has been made to give to this book the character here recommended, especially in the explanatory parts. SUGGESTIONS TO COMMERCIAL STUDENTS. THE foregoing suggestions are addressed directly to the Teacher, but a careful consideration of them by the Student will, it is hoped, be found highly profitable. A few additional hints are subjoined for the benefit of those seeking a liberal and practical commercial education. As in all branches, so in Arithmetic, it is of the utmost consequence to digest the rules of the art thoroughly, and store them in the memory, to be reproduced when required, and applied with accuracy. But this is not enough; something more is needed by the Student. To be an eminent accountant he must acquire rapidity of operation. Accuracy, it is true, should be attained first, especially as it is the direct means of arriving at readiness and rapidity. Accuracy may be called the foundation, readiness and rapidity the two wings of the superstructure. Either of these acquirements is indeed valuable in itself, but it is the combination of them that constitutes real effective skill, and makes the possessor relied upon, and looked up to in mercantile circles. Some one may ask, " How are these to be acquired? " The answer is as simple as it is undeniably true; only by extensive practice, not in the counting-house or warehouse, indeed, though these will improve and mature them, but in the school and college, so that you may take them with you to the business office when you go to your first day's duty. Go prepared is a maxim that all intelligent business men will affirm. Be so prepared that you will not keep your customers waiting restlessly in your office or warehouse while you are puzzling through the account you are to render to him, but strive rather to surprise him by having your bill ready so soon. Another important help to the attaining of this rapidity, as noticed in the note at foot of page 18, is not to use the tongue in calculating but the eye and the mind. Nor should the course of self-discipline end here. To be an expert accountant even, is but one part, though an important one of a qualification for business. Study Commercial geography-commercial and international relations-political economy-tariffs, &c., &c. X. SUGGESTIONS TO COMMERCIAL STUDENTS. Study even politics, not for their own sake but on account of the manner in which they affect trade and commerce. Do not, except in the case of some serious difficulty, indulge in the indolent habit of asking your teacher or fellow student to work the question for you; work it out yourself-rely upon your self, and aim at the freedom and correctness which will give you confidence in yourself, or rather in your powers and acquirements. Another caution will not be out of place. Many students follow the practice of keeping the text book beside them to see what the answer is; this has the same effect as a leading question in an examination, being a guide to the mode by seeing the result. Study and use the mode to come at the result; gain that knowledge of principles and correctness of operation that will inspire the confidence that your answer is cor. rect without knowing what answer the text book or the teacher may assign to it. There are two things of such constant occurrence and requiring such extreme accuracy that they must be specially mentioned,-they are the addition of money columns and the making of Bills of Pareels. Too much care and practice can scarcely be bestowed on these. TABLE OF CONTENTS. PAGE. Ac counts and envoices.................................... 122 Addition........,..............................17 Alligation...................................................... 232 Analysis........................................................ 1ll Annuities....................................,.... 286 Averaging of Accounts........................................... 223 Average, General................................................ 196 Axioms........................................................... 17 Banking...................................................... 167 Bankruptcy.................................. 214 Book-keeping Exercises........................................... 299 Brokerage....................................................... 175 Commercial Paper........................................ 149 Commission................................ 172 Coins, Foreign.......................................... 326 Currency of Canada.............................................. 240 Custom House Business........................................... 199 Decimal Coinage.............................................. 32 Denominate Numbers........................45 Discount........................................................ 165 Division, Simple............................................. 26 Equation of Payments............................................216 Exercises-Set I.....................................55 -- -- -Set II............................................. 107 --- Set III......................................... 318 Exchange...................................................... 241 Arbitration of......................................... 251 - ---- American..................., 242 Britain............................................ 247 Evolution.......................................... 258 Foreign MIoneys, Table........................................... 250 Fractions........................................................ 58 -- Common......................................... 6 61 - Decimal............................................... 71 Greatest Common Measure........................................ 52 Insllance............................................... 179 terest......................................................... 134 Simple............................................. 135 -- -- Compound............................................. 162 Introduction...................................... 13 xii INDEX. Involution...................................... 256 Laws of the United States...................... 328 Least Common Multiple.................................... 54 M ensuration............................................ 304 Money, its Nature and Value..................................... 258 M ultiplication, Simple........................................... 22 N otation..................................................... 15 Numeration................................................ 14 Paper Currency................................................. 239 Partial Payments............................................. 52 Partnership.................................................. 209 - Settlements........................................... 290 Percentage.............................................. 129 Practice........................................ 119 Profit and Loss............................................ 188 Progression.................................................. 270 Properties o' Numbers...........................................297 Proportion................................................. 92 - Simple....................................... 92 Compound........................................... 101 R atio.......................................................... 92 Reduction....................................................... 42 Stocks and Bonds.................................... 203 Storage................................................ 194 Subtraction, Simple............................................. 20 Tables of money; &c........................................ 30 Taxes............................................... 199 ARITHMETIC ARTICLE 1.-ARITHMETIC treats of numbers in theory and practice. In relation to theory it is a science, and in relation to practice it is an art. All computations are made by fixing on a certain quantity, called a unit, or one, and repeating that unit any required number of times. Various units are selected, according to the nature and extent of the quantity or space to be measured. For example, in measuring length or distance, if the extent is small, such as the length of a pane of glass, we select a small unit, called an inch, and repeating that unit any required number of times, say twelve, we say the pane is twelve inches long,-if a more extended space is to be measured, it is convenient to adopt a larger unit,-thus, if we wish to measure the length of a desk, we should probably select a unit called a foot, equal to twelve of the preceding units,-if we wish to measure the length of a room, we should select a still larger unit, called a yard, equal to three of the last,-again, if we wish to measure the length of a field, we should adopt a unit equal to five and a-half of the last, and called a perch or rod,-if we wish to note the distance between New York and Buffalo, we have recourse to a still larger unit, called a mile, and equal to three hundred and twenty of the last,-finally, when astronomers are estimating the distance of any planet, say the earth, from the sun, they generally use a unit equal to a million of the last-mentioned, and they say that the earth is ninety-five millions of miles from the sun, but they simply note the distance as ninety-five; and in the same manner they mark the distance of Venus as sixty-nine, meaning in both cases that each unit is a million of miles. A similar illustration may be applied to every kind or measurement. The symbols or characters now almost universally used to denote quantity or magnitude, are the Arabic figures, or digits, 1, 2, 3, 4, 5, 6, 7, 8,.9, 0. These, by various combinations, can be made to represent any quantity or magnitude whatsoever. The first nine are called significant figures, because they always denote some real quantity,-the last, called nought (often improperly ought,) or cipher, or zero, simply indicates the absence of any significant figure. 14 ARITHMETIC. NU MERATION. 2.-NUMERATION is the mode of marking and reading off any line of figures that has been written down, so as to ascertain its value readily and express that value in words. For this purpose every such line is divided into sets or lots of three figures each, counting from right to left, and each set is called a period,-thus, 888888888 forms three periods by marking the figures in threes from right to left by a character of the same form as the comma in composition,-thus, 888,888,888. The first period is called the period of units, the second the period of thousands, the third the period of millions, and so on,-billions, trillions, quadrillions, &c., &c.: to any required extent, which seldom exceeds millions. The first figure of each period denotes units* of that period, the second tens, and the third hundreds of that period. Thus, in the example given above, the first figure denotes eight units in the period of units, or eight ones, or, as it is usually read, simply eight; so, also, the fourth denotes eight units in the period of thousands; or eight times one thousand, or eight thousands; the seventh figure again denotes eight units in the period of millions, or eight times one million, or eight millions; again, the second, fifth, and eighth figures denote tens in the period of units, thousands and millions, respectively; lastly, the third, sixth and ninth figures denote hundreds in the periods of units, thousands and millions, respectively. Such a line, then, as 888,888,888 is read eight hundred and eigety-eight millions, eight hundred and eighty-eight thousands, eight hundred and eighty-eight. Every period but the last must have three figures. Thus, in the line 43,279,865 the first and second periods have three digits each, units, tens and hundreds, but the third has only two, units and tens, but no hundreds, and therefore is read forty-three millions, two hundred and seventy-nine thousands, eight hundred and sixty-five, RULE FOR NUMERATION. Beginning at the right, count off periods of three digits each till not more than three are left; then read off each period from left to * It is somewhat awkward that the term units is used for two purposes, viz.: as the name of the first period and also as the name of the first figure of each period. Though we cannot well change what usage has so long estab. lished, yet the teacher may obviate the difficulty by varying the expression occasionally, if not habitually, saying, E. G., units in the unity period, or the place of units in the units period. NOTATION. 15 right by naming as many hundreds, tens and units as each contains, and adding at the end of each period its proper name. The name of the unity period is usually omitted. When a cipher occurs no mention is made of that place in the period, but the cipher is counted as a digit; thus, in the line 360,708,091 each cipher is counted a digit, but the reading is three hundred and sixty millions, seven hundred and eight thousands and ninety-one. EXE RCISES Divide into periods and read the following lines: 1.-586729341 2.-976852734 3.-2178427385 4.-92879357485 5.-4638709120 6.-11illill 1111 7.-2822828228288 8.-10904870 9.-101010101011 N.OTATION. 3 — NOTATION is the mode of expressing any quantity or Inagnitude by the combination of conventional symbols or characters. Thus, by the Roman notation, the letter I. stands for one, II. for two, X. for ten, &c.; thus, XII. stands for one ten and two units. By the Arabic notation, any digit standing alone, as 5 in the margin, denotes simply five units, but if another digit (5) be placed to the right of it, then the new 5 denotes units and the other 5 becomes tens, so that appending a second digit makes the first one ten times its original value; again, if 5 another digit (5) be subjoined, it takes the place of units, 55 5 and the 5 next to it becomes teng and the third becomes 5555 hundreds, so that each of them has ten times the value in the third line that it haa in the second; so also, if another digit (5) be added, each of the three to the left of it will have ten times the value that it had in the third line, and so on. Universally, every digit placed to the right makes every one to the left ten times its previous'value. The use of the tenth of the Arabic characters, the cipher (0) will be made more clear by the rule of notation than by numeration. [f I am counting my cash and find that I have eight ten-dollar bills, and eight one-dollar bills, it is plain from Art. 2 that if I write 8 alone this must represent the one-dollar bills, and to represent the ten-dollar bills along with the one-dollar bills I must 16 ARITHMETIC. write 88, for the figure to the left being ten times that to the right, will stand correctly for the ten-dollar bills, just as that to the right, being in the units' place, stands for the one-dollar bills.But if I have no one-dollar bills and write 8, this would stand for only one-dollar bills, and hence the necessity for introducing a non-significant character and writing 80, for though the cipher represents no quantity, yet by being put in the place of units it throws the 8 to be in the place of tens, and therefore the 8 now stands fitly for the eight ten-dollar bills, and is written $80.Again, if I find that I have two one-hundred-dollar bills, six onedollar bills, but no ten-dollar bills, and I write only 26, this would be plainly incorrect, for the 2 would stand for ten-dollar bills only, but by inserting a zero mark between the figures I throw the 2 into the place of hundreds, and $206 represents correctly. that I have two one-hundred dollar bills, and six one-dollar bills, but no ten-dollar bills. The superiority of this simple system over the cumbrous Roman one will be manifest from its simplicity and brevity by writing eighty-eight according to both systems-thus: LXXXVIII. and 88. RULE FOR NOTATION..Write tne significant figures of the first period named in their proper places, filling up any places not named with ciphers, just as if you were writing the units period with nothing to follow; then, to indicate that something is to follow, place a comma to the right, and do the same for every period down to units, inclusive. For example, teacher says: " Write down one hundred and six millions;" pupil writes 106 and pauses; teacher adds, "ninety thousand;' pupil fills up thus: 106,090, and pauses; teacher concludes: "anc eighteen;" pupil completes 106,090,018. If the teacher should say sixteen millions and the pupil write 016, the cipher woulc be manifestly superfluous, as it has no effect on figures placed to the right of it, but only on those placed to the left. EX E R C S E S. Write in figures and read the following quantities: 1. Ten millions, seven thousand and eleven. 2. Ninety billions, seven thousand and ten. 3. Eighteen millions, sixty thousand and nine hundred. 4. Forty thousand and nine hundred. ADDITION. 17 5. Eighty-seven millions and one. 6. Ninety thousand, seven hundred and eight. 7. Eleven millions, eight hundred thousand and twenty-four. 8. Six hundred and seven thousand and ninety-seven. 9. Eight hundred and seventy billions, sixty thousand and eighteen. 10. Eleven billions, eleven millions, eleven thousand and eleven. AXIOMS. 4.-AXIOMS used in the sequel: I. Things that are equal to the same thing, or to equals, are equal. II. If equals be added to equals, the wholes are equal. Corollary.-If equals be multiplied by the same, the products are equal. III. If equals be taken from equals, the remainders are equal. Cor.-If equals be divided by the same, the quotients are equal. IV. The whole is greater than its part Cor.-The whole is equal to all its parts taken together. V. Magnitudes which coincide, or occupy the same or equal spaces, are equal. N. B.-This axiom is modified by, but still is the principle of, all business transactions, purchases, sales, barters, exchanges, &c., &c., where the articles traded in are not equals, but equivalents. AD D ITION. 5.-ADDITION is the mode of combining two or more numbers into one. The ppe&ation depends on axiom II. The result is called the sum. Thus: $8+$9+$6=$23. The sign plus (+) indicates addition. To illustrate the operation, let it be required to find the sum of the five numbers of dollars noted in the margin. First, the numbers are placed so that those of the $28,654 same name are in vertical columns, i. e., units under 758287? units, tens under tens, &c. Next, we find that the 612873 494768 sum of the units' column is (Ax. IV., Cor.) 27, i. e., 834195 two tens and seven units. Next, we find that the 2-..-.sum of the tens' column is 35, but, as it is the tens' 2 18 ARITHMETIC. 27 column, we write (Art. 3) 350; in the same man350 ner we find the sum of the hundreds' column to 27000 be 2400; the sums of the others will be seen by 260000 inspection. Having thus obtained the sum of each 2700000 column, each being summed as if units, but placed t....7 in succession towards the left (by Arts. 2 and 5), we now take the sum 6f the partial results, which (Axiom IV. Cor.) is the sum of the whole, viz.: $2,969,777. In practice the operation is much abbreviated in the following manner:-When the units' column $287654 has been added, and we find the sum to be 27, 758287 i. e., 7 units and 2 tens, we write down the 7 units 612873 under the units' column, and add up (Art. 3) the 2 494768 tens with the tens' column, and we find the sum is 836195 3 tens, i. c., tens and 3 hundreds, and we place $298977 the tens under the tens' column, and add up the 3 hundreds with the hundreds' column, and so on. The transferring of the tens, obtained by adding the units' column to the tens' column, and the hundreds obtained by adding the tens' column to the hundreds' column, &c., &c., is called carrying. In all such operations the learner should carefully bear in mind the principle explained in Art. 3., that every figure to the left is ten times the value that it would have if one place farther to the right.* EXERCISES. Find the sums of the following quantities: (1) (2) (3) (4) 895763 99876 49176 987654231 638{79 89765324 283527 123456789 54387 42356798 659845 908760504- 789 56798423 7984 890705063 137568& 28567989 31659 759086391 278652 76842356 968438 670998767 85945 65324897 2896392 4340661745 721096 357655787 * We would strongly recommend every one who wishes to become an expert accountant, to avoid the common practice of drawling up a column of figures in the manner that may be sufficiently illustrated by the adding of the units' column of the above example. Never say 5 and 8 are 13; 13 and,3 are 16: 16 and 7 are 23; 23 and 4 are 27; but run up your column thus: 5, I3, 16, 23, ADDITION. 19 (5) (6) (7) (8) 738 659 47,1 78563 897, 47986 12345 658 5798 67890 918273 856 19843 98765 651928 789 56479 43219 374859 978 28795 87654 263748 654 897 32169 597485 999 1984 78912 986879 888 68195 65439 98765 777 3879 98765 9876 666 698 43288 987 55S 5879 77877 456879 89T 17985 98989 345678 978 336981 80512 4705357 12460 (9) (10) (11) Y12) 189 1298 916 98 47 764 85 89 96 58g7 73 76 83 6495 338 67 59 789 793 281 74 638 49 592 82 546 75 678 97 98 218 58 68 475 36,4 67 75 394113 98 49 89,:: 279 149 76 157 67. 67 54 638: 76 54 78 594 84!' 72 69 789' 1379 298 37 114 5189 2744 1044 19715 27, for that is the mode to secure both rapidity and accuracy. The same remark will apply equally to multiplication, and therefore to every arithmetical operation. To enforce this advice let us add a simple example to caution the student before he approaches multiplication. In multiplying 497 by 6, avoid the tediousness of saying 6 times 7 is 42-2 and carry 4-6 times 9 is 54, and 4 is 58-8 and carry 5-6 times 4 is 24, and 5 is 29; but practice the eye, aided by the memory, to take in at a glance 6 times 7 is 42, &U.The quick operator uses the eye, and not the tongue. 20 ARITHMLETIC. There is no mnethod of proving the correctness of any addition with positive certainty, but a very convenient mode of checking is to add each column both upwards and downwards. Another mode is, to add by parts and take the sum of those. This is a very secure method in the case of long columns, but not so ready as the former. If the same result is found by each method, the sum may be accounted correct. SUBTRACTION. 6. SUBTRACTION is the converse of addition, z. e., it is the mode of finding the difference between two numbers, or, in other words, the excess of one number above another. The number to be subtracted is called the subtrahend, and that from which it is to be taken the minuend, and the result is called the remainder, difference or excess. The sign used for subtraction is a line (-) called minus, or less. Let it be required to find the difference between $578643957 and $235412712. Having placed them in vertical columns, as in addition, it is obvious 578643957 that 2 units taken from 7 units will leave 5 units, 235412712 and that 1 ten taken from 5 tens will leave 4 tens,.-.... — and so on. But if it is required to find the excess.343231245 of $51674208 above $347895319, we find that each figure of the subtrahend, except the last, counting from right to left, is greater than the corresponding one of the minuend, and therefore, to find the correct difference, we have recourse to a simple artifice, which is deduced from the principle of the notation, and may be illustrated in the following manner:- Taking the question in the margin, we are first 333,333 a 177,777 required to subtract 7 units from 3 units. Now, though the algebraic notation furnishes the means of noting the 155.556 difference directly, the ordinary arithmetical form does not, but still it furnishes the means of doing it indirectly. By Art. 3 each figure to the left is ten times the value of the next to its right, therefore we take one of the 3 tens and call it ten units, and add it to the 3 units, and thus we have 13 units, which let 2(12)(12')(12)(12)(13) us enclose in a parenthesis or bracket, 1 7 7 7 7 7 thus: (13), to indicate that the whole 1 5 5 5 5 6 quantity, 13, is to occupy the units' place; when one of the three tens has been thus transferred to the units' SUBTRACTION. 21 place, only two tens remain in the place of tens, and we are now required to take 7 tens from 2 tens; to do this we have recourse to the same artifice, by calling one of the hundreds tens, which gives 10 tens and 2 tens, and so on to the end, the last 3 necessarily becoming 2. We can now subtract 7 from 13, &c., &c. This mode of resolution depends onthe corol200000 lary to Axiom IV. The parts into which the whole 120000 is virtually resolved are shown in the margin. This 12000 1200 artifice is popularly called borrowing. In practice the 120 resolution can be effected mentally as we proceed, and 13 as each figure from which we borrow is diminished --- - by unity, it is usual to count it as it stands, and to 333333i compensate for this to increase the one below it by one, for, as in the example, 7 from 12 is the same as 8 from 13, and 2 from 3 is the same as 1 from 2. We are now prepared to answer the proposed question, as annexed, and we say 9 from 8, we cannot, and $513674208 there are no tens to borrow from, we therefore $347895319 $34789539 take one of the hundreds and call it 10 tens, and $165778889 one of the tens and call it 10 units, which with 8 units makes 18 units, and we take 9 from 18 and 9 remain. We have now only 9 tens left, but we reckon them as ten, and to compensate for the surplus ten, we reckon the 1 below as 2, and say 2 from 10 and 8 remain. We proceed thus to the end, and find the whole remainder to be $165778889. EX ER CISES. REMAINDERS. 1.-From 847639021 take 476584359=371054662. 2. " 1010305061 " 670685093-339619968. 3. " 59638Y43 " 18796854=- 40841889. 4. " 7813i57 " 3745679- 4067578. 5. " 111111111 " 98657293- 12453818. In Subtraction, as in Addition, we have no method of proof that arrives at positive certainty, but either of the two following methods may be generally relied upon. 1.-Add the remainder and subtrahend, and if the sum is equal to the minuend, it is to be presumed that the work is correct. 2.-Subtract the remainder from the minuend, and if this second remainder is the same as the subtrahend, the work may be accounted correct. 22 ARITHMETIC. MULTIPLICATION. 7.-MULTIPLICATION may be simply defined by saying that it is a short method of performing addition, when all the quantities to be added are the same or equal. Thus: 66 +6+6+6+6-+6+6, means that eight sixes are to be added together, or that six is to be repeated as often as there are units in eight, and we say that 8 times 6 is 48, and write it thus: 8X6=48. So also 8+ 8+8+8+8+8 gives 48. So that 6.8=8.6 —48, and thus we can construct a multiplication table. The number to be repeated is called the multiplicand, and the one that shows how often it is to be repeated is called the multiplier, and the result is called the product, or what is produced, and hence the multiplier and multiplicand are also called the factors or makers, or producers, and the operation may be called finding a product when the factors are given. Hence also the mode of carrying is the same in multiplication as in addition. MULTIPLICATION TABLE. Twice 3 times 4 times 5 times 6 times 7 times 1 is 2 1 is 3 1 is 4 1 is 5 1 is 6 1 is 7 2- 4 2 6 2- 8 2 -10 2-12 2 -14 3- 6 3- 9 3-12 3 —15 3-18 3 21 4 - 8 4- 12 4 16 4-20 4- 24 4- 28 5-10 5-15 5-20 5- 25 5- 30 5-35 6-12 6-18 6-24 6- 30 6-36 6 -42 7-14 7 — 21 7 —28 7 - 35 7-42 7- 49 8- 16 8- 24 8- 32 8 40 8- 48 8 -56 9 -18 9 27 9-36 9 -45 9 - 54 9- 63 1.0-20 10 - 30 10 40 10- 50 10 - 60 10- 70 11 — 22 11 - 33 11 44 11- 55 11- 66 11 - 77 12 24 12-36 12 — 48 12-60 12- 72 12- 84 8 times 9 times 10 times 11 times 12 times 1 is 8 1 is 9 1 is10 is 11 is 12 2-16 2 -18 2 — 20 2 - 22 2-24 3-24 3- 27 3- 30 3 — 33 3-36 4-32 4 36 4-40 4-44 4- 48 5-40 5-45 5- 50 5-55 5- 60. 6-48 6 54 6- 60 6 - 66' 6- 72 7- 56.7-63 7- 70 7 - 77 7- 84 8-64 8-72 8-80 8-88 8 —96 9-72 9 -81 9- 90 9-99 9 -108 10 - 80 10 - 90 10 -100 10 -110 10 -120 11 - 88 11 -99 11 -110 11 -121 11 -132 12 -96 12 -108 12-120 12 -132 12-144 MULTIPLICATION. 23 Regarding the following part of this table, see suggestions to Teachers: 13 times 14 times 15 times 16 times 17 times 18 times 19 times 2 is 26 2 is 28 2 is 30 2 is 32 2 is 34 2 is 36 2 is 38 3 — 39 3- 42 3 — 45 3 — 483 - 51 3- 54 3 57 4- 52 4 56 4- 60 4 - 64 4 684- 4 72 4 76 5 - 65 5- 70 5 7 5 80 5 — 85 5 90 5 95. 6 - 78 6- 84 6 - 906 - 96 6 -- 102 6 - 108 6 --- 114 7- 91 7- 98 7 105 7 112 7 -119 7 -126 7 133 8 -104 10 12.8 - 120 8 128 8 136 8 -144 8-152 9 - 117 9 - 1?6 9 - 135 9 - 144 9 - 153 9 - 162 9 - 171 We have in the above table corrected the gross grammatical blunder so common of saying eight times two ARE sixteen. When morQ than two factors are given, the operation is called continued multiplication, as 6X3X2X 5=180. When the factors consist of more figures than one, the most convenient mode of operation is that shown by the annexed example, where the multiplicand is first repeated 8 times, then 60 times, or which is the same thing 6 times when the first figure of the second line is placed under the second figure of the first line, i. e. (art. 2,) in the place of tens, and then the partial products are added, which (Ax. IV. Cor.) gives the full product. 345186 Hence we deduce the 268 2761488 RULE FOR MULTIPLICATION. 2070916 Place the multiplier under the multiplicand, units 690372 under units, tens under tens, &c., &c.,-commencing 92507848 at the right, multiply each figure of the multiplicand by each figure of the multiplier in succession, placing the results in parallel lines, and units, tens, &c., in vertical columns,-add all the lines, and the sum of all the partial products will (Ax. IV. Cor.) be the whole product required. As far as the learner has committed a multiplication table to memory, say.to 12 times 12, the work can be done by a single operation. When any number is multiplied by itself, 5 the product is called the square or second power of that number, and the product of three equal factors is called a cube or third power, the pro~_ - _ 5 duct of four equal factors the fourth power, &c., _~ -| &c. The terms square and cube are derived from superficial and solid measurement. The annexed square has each of its sides divided into 5 equal parts, and it will be found on inspection that the whole figure contains 24 ARITHMETIC. 25 (5 5X5) small squares, all equal in area, and having all their sides equal.-Hence because 5 X 5 represents the whole area, 25 is called the square of 5, or the second power of 5, because it is the product of the two equal factors 5 and 5. A cube is a solid body,' the length, breadth and thickness of which are all equal, and hence, if these dimensions be each represented by 5, the whole solid will be represented by 5X5X5-=125, which is therefore called the cube or third power of 5. The terms square and cube are often used without any reference to superficial and solid measure. For example, in lineal measure an expression'for distance in a straight line is often called the square and cube of a certain number, thus: 81 is called the square, and 729 the cube of 9, although these are only used to show that the distance is not 9 in either case, but in the one 9 X 9, and in the other 9 X 9 X 9. In such cases the terms second and third power are therefore to be preferred, and since no solid can have more than three dimensions, we have no term corresponding to square and cube for the product of four or more equal factors, and therefore we are obliged to use the words fourth power, fifth power, &c., &c. CONTRACTIONS AND PROOF. There are many cases in which multiplication may be performed by contracted methods, but the utility of these, for the purposes of accuracy, is, at least, doubtful. The most secure method in the great majority of cases, is to follow the general rule. Multiplication by 10, 100, &c., is effected at once by adding a cipher for ten, two for 100, &c., &c. The following is, next to the above, the most safe and useful contraction that can be adopted. It ismexhibited in the subjoined examples, but purposely without explanation, as an exer cise for the learner's reflection: ORDINARY METHOD. CONTRACTED METIOD. ORDINARY METHOD. CONTRACTED METHOD 35697X17 35697X17 35697X71 35697X71 17 249879 71 249879 249879 606849 35697 2534487 35697 249879 606849 2534487 The only practically useful proof of the correctness of the product, is the one subjoined, but even it, though it seldom fails, does: not secure positive certainty; MULTIPLICATION. 25 Add together all the figures of each factor separately, rejecting 9 from all sums that contain it, and multiply the remainders together, rejecting every 9 from the result, —add the figures of the product in the same manner, and if the two remainders are equal, the work may be accounted as correct, but if they are not equal, the work must be wrong. The reason of this proof depends on the property of the number 9, that if any number be divided by 9, the remainder will be the same as if the sum of its digits were divided by 9.Thus;:7A12153-9=824683+6, and the sum of the digits is 24, and 24- -9=2+6, i. e. 9 is contained in 24 twice with a remainder 6. Every 9 is rejected because 9 is contained in itself oice evenly, and therefore cannot affect the remainder. Let it now be required to multiply 122 by 24. Now, 122-9X13+5, and 24=9X2+6, and if we multiply together the two factors thus resolved, we get 9X13X9X2+9X2X5+9X13X6+6X5, and since 9 is a factor of all but the last, the last only will give a remainder When divided by 9, and therefore the whole product will give the same remainder when divided by 9, as 6X5- 9, which gives the remiainder 3, for 6X5=-30 and 30)-9 gives 3 with a remainder 3. To test this by trial, we find 122 —9 —13 with a remainder 5, and 24 —9=2 with a remainder 6, and the product of these remainders is 6X5-30, and 30. —9=3 with a remainder 3. Again, 122X24=2928, and 2928-.-9=325 with a remainder 3, as in the case of the factors. EXERCISES. 1. 7896X5=39480. 8. 719864X43=30954152. 2. 581967X8=4655736. 9. 375967X64=24061888. 3. 938746X4 —3754984. 10. 27859X29=807911. 4. 193784X7 —1356488. 11. 679854X83-56427882. 5. 391876X9 —3526884. 12. 759684X187-142060908. 6. 987456X6=5924736. 13. 5372X1634-8777848 7. 496783X52=25832716. 14. Find the second power of 389? Ans. 151321. 15. Find' the third power of 538? Ans. 155.720873. 16. Find the fourth power of 144? Ans. 429991r696. 17. Find the cube of 99? Ans. 970299. 18. 5796 seamen have to be paid 169 dollars each; what is the amount of the treasury order for that purpose? Ans. $979,524. 19. A block of buildings is 87 feet long; 38 feet deep, and 29 feet high; how many cubic yards does it contain? Ans. 35509 cubic yards: 26 ARITHMETIC. 20. If 29 oil wells yield 19 gallons an hour each; how much will they all yield in a year? Ans. 201115 gals. 21. If the rate on each of 1597 4-ouses be $19; what is the whole assessment? Ans. $30343. 22. If 1297 persons have paid up 9 shares each in a railway company, and each share is $15; what is the working capital of the company? Ans. $17095 DIVISION. 8.-DIVISION is the converse operation to multiplication. It is the mode of finding a required factor when a product and another factor are given. It bears the same relation to subtraction that multiplication does to addition, as will be seen below. By Ax. IV. Cor. we may resolve any complex quantity into its component parts; so division is resolving a certain quantity called the dividend into the number of parts indicated by another quantity called the divisor, (divider,) and the result is called the quotient (how often.) Let it be required to*find how often 8 is contained in 279,856. We can resolve 279,8 240000 30000 856 as in the margin; then dividing the 32000 4000 lines separately by 8, we obtain the partial 7200 900 640 980 quotients, the sum of which is the whole 16 2 quotient. But this resolution may be - -:_ _____ done mentally as we proceed. We first 8 279856 34982 see that 8 is not contained in 2, therefore we take 27, and find that 8 is contained in it 3 times, with a remainder 3; next combining this 3 with the next figure 9, we get 39, in which 8 is contained 4 times, with a remainder 7; combining this 7 with the next figure 8, we have 78, in which 8 is contained 9 times, with a remainder 6; combining this with the 5 following, we obtain 65, and 8 is contained in it 8 times, with a remainder 1, which combined with the 6 makes 16, and 8 is contained twice in 16. The correctness of the result may be tested by multiplying the quotient by the divisor. When the divisor consists of more than one figure, the learner must have recourse to a trial quotient, but after some practice he will have little difficulty in finding each figure by inspection. DIVISION. 27 Let it be required to find how often 298 is coi.tained in 431766.The numbers being arranged in the convenient order indicated in the margin, we mark off to the right of the dividend blank spaces for the trial and true quotients. We readily see that 2 is contained twice in 4, but cannot so easily see whether the whole divisor 298 is coftained twice in the same number of figures of the dividend, (viz. 431,) we therefore make trial, and place the 2 in the trial quotient, and multiply the divisor by 2 to find how'much we shall have to subtract from 431. We find 298X2=-596, larger than 431, and therefore we reject 2 and try 1. Now 298X1=298, less than 431, so we subtract and find a remainder of 133, and as this proves correct, we place the 1 obtained in 298)431766(2.1.5.4.5.4.9.8 trials. the true quotient. e find 298 8 te q our next partial divdend by 1448 true quotient. - 1~- treqoin. writing 7, the next figure of 1337 the dividend after the re1192 mainder 133. Our experi1456 ence of the first case sug1192 gests to us that though 2 is contained 6 -times in 13, yet 2646 on multiplying something 2384 will have to be carried from 9262 the 98 which we expect will make the result too large, 298 and therefore we at once try 5, but we find that 298X5=1490, which is larger than 1337, and so we try 4, and find 298X4=1192, which being less than 1337, we subtract and find a remainder of 145; and having placed the 4 in the true quotient, we bring down the next figure of the dividend, giving a partial dividend 1456. By inspection, as before, we see that 6 would be too large, owing to the carrying from 98, we try 5 and find 298X5=1490, which is larger than 1456; we try 4, and find 298X4-1192, which is less than 1456, so we subtract and find a remainder of 264. Having placed this 4 after the other 4 in the true quotient, we bring down 6, the last figure of the dividend, we try 9, and find 298X9=2682, which is greater than our last partial dividend, 2646; we try 8, and find 298X8=2384, and this being less than 2646, we subtract it from 28 ARITHMETIC. that number, and find a final remainder of 262, and close the question by entering 8 in the true quotient. The mode adopted to indicate that the remainder 262 still remains to be divided, which cannot be actually done, as it is less than the divisor, is to write the 298 below the 262, and draw a line between them, thus 29j, as also is seen in 298! 298. 70 ARITHMETIC. 11. Express 40 lbs. as a fraction of 1 cwt. Ans. | cwt. 12. Express 50 lbs. as a fraction of 1 ton. Ans. -- ton. 13. Express 721bs. as a fraction of 1 cwt. Ans., cwt. 14. A day is 23 hours, 56 minutes, 48 seconds, nearly; what fraction of this will 7 hours be? Ans. I5Yg. 15. Express 95 square yards as a fraction of an acre. Ans. -g.9 16. Express 14 yards as a fraction of a mile. Ans. -.8 17. What fraction of a year (3651 days) is one month (30 days)? Ans. 4-0. 18. Express 100 yards as a fraction of a mile. Ans. n5. 19. Express 45 cents as a fraction of a dollar. Ans. 9. 20. Express 60 lbs. as a fraction of a cwt. Ans. 3 21. A man has an income of $3610 a year, and saves 3 of it; how much does he spend? Ans. $2062F. To find the value of a fraction in the denominations which the integer contains, reduce the numerator to the next lower denomination, and divide the result by the denominator; if there be a remainder, reduce to the next denomination, and divide again, and continue the same operation till there is either no remainder, or down to the lowest denomination by which the integer is counted. Thus, since a ton is 20 cwt., 7 of 6 tons is 120 tons divided by 7, which gives 17 cwt., with a remainder of 1, which, reduced to qrs., will give 4, in which 7 is not contained, and the 4 qrs. reduced to lbs., will give 100, and this divided by 7 produces 14-.; so that -7 of a ton is 17 cwt., 0 qrs., 142 lbs. EXERCISES. 1. What is the value of -7 of a ton? Ans. 11 cwt., 2 qrs., 161 lbs. 2. What is the value of lo of a yard? Ans. 2 feet., 8| in. 3. What is the value of 3 of a mile? fL4 4. What is the value of - of a shilling Stg.? Ans. 11d. 5. What is the value of 4 of a ton? Ans. 11 ewt., 1 qr., 17f lbs. 6. What is the value of lb. troy? Ans. 8 oz. 7. What is the value of JL of a shilling? Ans. 57-d. 8. What is the value of $? Ans. 88 cts. 9. What is the value of 4 of $6? Ans. $4.80. 10. What is the value of i of $8? Ans. $6.80. To change a fraction to one of a lower denomination, reduce the numerator to that denomination, and divide by the denominator. Thus: -, of a dollar is 700 cts. divided by 145, which gives 424. DECIMAL FRACTIONS. 71 EXERCISES. 1. Express lo of a foot as a fraction of an inch. Ans. -5. 2. Express -I of a cwt. as a fraction of a lb. Ans. 6. 3.'Express J of a lb. as a fraction of an oz. Ans. 4. 4. Express - of?5 of a yard as a fraction of a foot. Ans. -. 5. Express 24 of a rod as a fraction of a yard. Ans. 22. 6. Express i of -Iof an acre as a fraction of a rood. Ans. i. 7. Reduce 3 cwt. to the fraction of a pound. Ans. 11- lb. 8. Reduce -.l of a day to the fraction of a minute. Ans. 684 min. 9. What part of a second is the one-millionth part of a day? Ans. 5Ad sec. 10. Reduce ~3 to the fraction of a penny. Ans. 62d. 11. Reduce 1 of a pound avoirdupois to the fraction of an oz. Ans. 2 oz. The reducing of a denominate fraction from one of a lower to one of a higher denomination being the converse of the last rule, we must perform the same operation on the denominator as was there performed on the numerator. Thus, ~d. is ~3-, for ~X12Xo 0 1-~ —='F EXERCISES. 1. What part of 1 lb. troy is - of a grain? Ans. G61O 2. What part of 4 days is 3 of a minute? Ans.. 3. What part of 5 bushels is 2 of 4 of a pint? Ans. 4. 4. What part of a rod is 2| of -7 of an inch? Ans. 4. 5. What part of 2 weeks is - of a day? Ans. TI DECIMAL FRACTIONS. 16.-WE have seen already (Art. 3,) that every figure to the right is one-tenth the value it would have if removed one place to the left. Thus, resuming our former example, 8 standing alone m eans 8 units, but if we place another 8 after it, thus 88, it now means 8 tens, so that the last 8 is one-tenth of the first. Now, since the 8 to the right expresses units, another 8 placed to the right will express eight-tenths of the same unit, and another subjoined will express -O of the unit. Thus we see that the decimal notation is directly an extension of the Arabic. Hence arose the convenient mode of writing 8-L in the form 8.7, by which is indicated that all 72 ARITHMETIC. the figures before the decimal point (.) represent integers, and all after it fractions, each being one-tenth of what it would be if one place further to the left. Therefore 888.888 is eight hundreds. eight tens, eight units,-eight-tenths, eight one-hundredths, and. eight one-thousandths; or,; y + To+o' These added will give 800 J_ 80 8 6I T T- O~800O-~ O or ~180, which, for brevity, is written.888, and may be read eight hundred and eighty-eight one-thousandths; or, as is usual, point 888, or decimal 888, but never properly eight hundred and eighty-eight. In the same manner as 80 means 8 tens and no units, so.08 means no tenths, but 8 hundredths, and.008 means no tenths, no hundredths, but eight one-thousandths, &c.Hence we see that for every cipher in the denominator, which is always 10 ofa power of 10, there must be a figure in the numerator when expressed decimally. Thus: ~o must be written decimally.008. From this we see that removing the decimal point one place to the right is the same as multiplying by 10, and removing it one place to the left is the same as dividing by 10; so, also, removing the point two places to the right is the same as multiplying by 100, and removing it two places to the left is the same as dividing by 100. This is the principle already laid down for the reduction of dollars to cents, and cents to dollars. I.-REDUCTION OF COMMON FRACTIONS TO DECIAnLS.-Let it now be required to express the common fraction - as a decimal. We have seen (Art. 14,) that we may multiply the terms of any fraction by the same number without changing the value of the fraction. Let us then multiply the terms of A by 1000, and we get — 000. On the same principle we can divide the terms by the same number without altering the value. Let us then divide by 8, and we get -,o62, where the denominator is a power of 10, and therefore the fraction' is in the decimal form, and may be written.625, the denominator being omitted. But 8)50(0.625 as it is not always apparent by what power of 10 we must multiply, so that when the terms are divi20 ded by the given denominator, that denominator 16 may be transformed into 10 or a power of 10, i. e., -- into 1 followed by a certain number of ciphers, 40 40 we may as well add ciphers, one by one, as we proceed. This is exhibited in the annexed example. From these principles we can deduce a rule for reducing a common fraction to a decimal. DECIMAL FRACTIONS. 73 RULE. Divide the numerator, with a cipher or ciphers annexed, by the denominator. Thus I will give, as in the margin,.6875. In the examples given we find that the addition of three ciphers to the first, and four to the second, 16)110(.6875 16)110s.6875 makes the numerator divisible by the denominator without remainder. Such fractions are cal140 led terminating decimals. From this we see 128 that there are common fractions whose terms --- can be multiplied by such powers of 10 as will 120 112 make the numerator divisible by the denominator without remainder, but it often happens that 80 no power of ten will effect this, and that remain80 ders occur which cannot be made divisible evenly by the denominator, by the addition of any number of ciphers. Such fractions will never terminate, and therefore are called interminate, and the common fraction can never be expressed exactly in the decimal form, and all we can do is to make an approximation more or less close, according to the number of decimal places to which we carry it. Let us take the fraction 9.First, 9 is not contained in 1, and therefore we 7)10(14857 place the decimal point in the quotient, and add a 7 cipher to the numerator, and we find that 9 is con- tained once in 10, with a remainder 1,-annexing 30 another cipher, we again obtain 1 in the quotient, 28 and this will obviously continue ad infinitum.20. This recurrence is marked by a dot or dash over 14 the figure, thus:.i or.1'. If we express a as a decimal, we find that after we have got six figures 60 56 in the quotient, we have a remainder 1, the same - as the original numerator, and therefore we should 40 again obtain the same quotient.142857, and 35 hence this is called a circulating or periodic decie mal, and the first and last of the recurring figures 49 are marked with a point or trait. Thus:.142857 - or.1'42857'. Again, it often happens that some 1 figures do not recur whilst others following them do, as in the annexed example, after we have got 74 ARITHMETIC. five figures the 11500 which gave us.the third figure 3, in the quoti ent recurs, and by pursuing the division we should find 345 recurring without 4111 -— 4111 end. When all the figures recur, the 33300)41110(.12345. fraction is called a pure periodic deci3330 mral; when only some of them recur, it 78100 is called mixed, and the term rapeater is 66600 applied when only one figure recurs, as 115000 =o-.1111, &c..i or 7 —.58333, &c. 99900.583. Since the denominator is always 151000 10, or a power of 10, and since 10 has 133200 no factors but 2 and 5, and therefore powers of 10 no factors but 2 and 5, or 178000 powers of these, it follows that no deci165500 mal will terminate except the denomina11500 tor be expressed by either or both of these, or some power or product of themHence all terminating decimals are derived from common fractions having for denominator some figure of the series 2, 4, 8, 16, 32, &c., or 5, 25, 125, &c., or, 10, 20, 40, 50, 60, 80, 100, &c. EXERCISES. 1. Reduce the common fraction 1 to a decimal. Ans..25. 2. Reduce the common fraction i to a decimal. Ans..5. 3. Reduce the common fraction - to a decimal. Ans..75. 4. Reduce the common fraction i to a decimal. Ans..3. 5. Reduce the common fraction, to a decimal. Ans..i. 6. Reduce the common fraction i to a decimal. Ans..125. 7. Reduce the common fraction i to a decimal. Ans..16. 8. Reduce the common fraction ~ to a decimal. Ans..142857. 9. Reduce the common fraction 5 to a decimal. Ans..2. 10. Reduce the common fraction - to a decimal. Ans..1. 11. Reduce the common fraction ~ to a decimal. Ans..09. 11. Reduce the common fraction - to a decimal. Ans..09. 12. Reduce the common fraction l- to a decimal. Ans..083. 13. Reduce the common fraction ~ to a decimal. Ans..6. 14. Reduce the common fraction 4 to a decimal. Ans..8. DECILMAL FRACTIONS. 75 15. Reduce the common fraction 5 to a decimal. Ans..83. 16. Reduce the common fraction 3 to a decimal. Ans..375. 17. Reduce the common fraction 5 to a decimal. Ans..625. 18. Reduce the common fraction i to a decimal. Ans..875. 19. Reduce the common fraction 4 to a decimal. Ans..4. 9 20. Reduce the common fraction - to a decimal. Ans..714285. 21. Reduce the common fraction l ~to a decimal. Ans..90. 22. Reduce the common fraction Ao to a decimal. Ans..91. 22. Reduce the common fraction 12| to a decimal. Ans..923076. 24. Reduce the common fraction to a decimal. 25. Reduce the common fraction 1-1 to a decimal. Ans..6875. 26. Reduce the common fraction to a decimal. 27. Reduce the common fraction -- to a decimal. Ans..34375. 28. Reduce the common fraction T to a decimal. 26. Reduce the common fraction -5 to a decimal. 27. Reduce the common fraction 4 to a decimal. Ans.34375. 31. Reduce the common fraction -y to a decimal. 29. Reduce the common fraction 37 to a decimal. Ans..4683544303797. 30. Reduce the common fraction 4 to a decimal. Ans..(04J. 31. Reduce the common fraction -419 to a decimal. Ans..020408163265306122448979591836734693877551. 32. Express - decimally. Ans..Oi 33. Express 1 decimally. Ans..01i. 34. Express ~, decimally. 35. Express;-TJ decimally. Ans..00059994. To reduce a denominate number to the form of a decimal fracion, reduce it to the lowest denomination which it contains; reduce he integral unit to the same denomination, and divide the former'y the latter. Thus, to express 18s. 4d. as a decimal of ~1, we must reduce it o pence, the lowest denomination given, and divide it by 240, the lumber of pence in ~1, which gives the fraction 2-0-22 —, 1 and;his reduced to a decimal, gives.916 or ~.916. In like manner k5s. 10d. is reduced to half-pence, viz., 381, and the half-pence in A1 are 480, and a|^4~2-, which expressed decimally is.79375. 4 — y 0, 76 ARITHMETIC. EXERCISES. 1. What decimal of ~1 is l1s. 4~d.? Ans..56875. 2. Express 15s. 9jd. as a decimal of ~1. Ans..790625. 3. What decimal of a square mile is an acre? Ans..0015625. 4. Express 1 pound troy as a decimal of 1 pound, avoirdupois.* Ans..82285714. 5. Reduce 17 cwt. to the decimal of a ton. Ans..85. 6. Express -' of a cwt. as a decimal of a ton. Ans..046875. oz. 11-16=.6875 The operation annexed is often lbs. 22.6875-. 25-.9075 convenient in practice. To reduce r29075-4.726875 11 cwt., 2 qrs., 22 lbs., 11 oz., to 2.9075 —.4 —.726875 cwt. the decimal of a ton. First, we 11.726875 —20= 58634375 divide the 11 oz. by 16,'te number of oz. in, 1 lb., and then annex 16~)11 the 22 lbs., and divide by 25, the lbs in a qr., and so on. The first 25)22.6875 form of the work is best suited for I —-~~- ~illustration, the second is neater in 4)2.9075 practice. The principle is the same 20)11.726875 as that implied in the general rule given above.:58634375 ADDITIONAL E ER CIS ES. 7. Reduce 10 drams to the decimal of i lb. Ans..0390625. 8. Reduce 11 dwt. to the decimal of 1 lb. Ans..04583. 9. Express 1 oz., avoirdupois, as a fraction of 1 oz., troy, (see note.) Ans..9114583. 10. Reduce 5 hours, 48 minutes, 49.7 seconds to the decimal of a day. * A caution seems necessary here, for since the pound (troy,) contains 12 ounces, and the pound (avoirdupois,) 16, the natural conclusion would be that the pound (troy) is 2 or i of the pound avoir dupois. This is not correct, for the ounce troy ex5760-412=480 ceeds the ounce avoirdupois by 42~ grains, though 7000 -16=437j the pound avoirdupois (7000 grs.) exceeds the pound Troy (57+0 grs.) by 1240 grains. This w1ll difference.. 42i- be manifest from the operation on the margin, where the standard weights are given. DECIMAL FRACTIONS. 77 II.-REDUCTION OF DECIMALS TO COMMON FRACTIONS.To find the common fraction corresponding to any given decimal.This involves three cases according as the fraction is a terminating decimal, a pure circulating decimal, or a mixed circulating decimal. The first case scarcely requires proof. We give it, however, in order to assist those unaccustomed to the algebraic notation, to understand more clearly the form of illustration used in the other cases. Let us take the fraction.9375, and use d for decimal. We now write d-=.9375, and multiplying both terms by 10000, we obtain 10000 d=-9375, and therefore d —1937-, which reduced to its low-;est terms is {|, the common fraction required. This is simply putting for denominator 1, followed by a cipher for each figure in the -decimal. To find the value of a pure circulator, suppose.6. Put d=.6, or d=.666, and multiply by 10, which d=.666-+ gives 10 d-6.66, and writing the former expres9 d==6 sion beneath, and subtracting, we get 9 d-6, and:9: d-7-6 consequently d=- or 2, the common fraction sought. Let us now seek the vulgar fraction correspond100a d=72.i iDg to.72 Put d=.72, multiply by 100, and subtract as before, and thcr results a rematin99 d=72 mainder of'99 d=72, or d —=;- 8. Again, to find the vulgar fraction coresd-=.5681 8 ponding to.5681. Multiply first by10000, 10000 d=568.81 and then by 100, and subtract the latter i0 d,-~ 56.8i 10 df 56.81 rom the former, and; you obtain 9900 d= 9900 d=5625 u5625, and hence d -Jjo, whichreduced to its lowest terms is j2. From these investigations the three following rules for the three,cases mentioned are derived: I. If the fraction be a: terminating decimal make it the numerator, and for denominator write 1, followed by as many ciphers as there are figures in the decimal. II. If the decimal be a pure circulator, make the digits of the decimal the numerator, and for denominator write as many nines as there arefigures in the period 6 78 ARITHMETIC. III. If the decimal be a mixed circulator, subtract the'non-circulating partfrom the whole decimal to the end of the first period, both being treated as whole numbers; make the remainder the numerator, and for denominator write as many nines as there are circulatingfigures, and after them as many ciphers as there are non-circulating figures. In all cases reduce to the lowest terms. EXERCISES. 1. Find the vulgar fraction corresponding to.04. Ans. 4,j 2. Find the vulgar fraction corresponding to.54. Ans. -T. 3. Find the vulgar fraction corresponding to.2457. Ans. -83. 4. Find the vulgar fraction corresponding to.1. Ans. ~. 5. Find the vulgar fraction corresponding to.3. Ans. 9. 6. Find the vulgar fraction corresponding to.7. Ans.. 7. Find the vulgar fraction corresponding to.75. Ans. i. 8. Find the vulgar fraction corresponding to.47543. 9. Find the vulgar fraction corresponding to.7683544303797. Ans. 95 9. Find the vulgar fraction corresponding to.4683544303797. Ans. 3. 10. Find the vulgar fraction corresponding to.49. Ans. ~. 11. Find the vulgar fraction corresponding to.162. Ans k, 12. Find the vulgar fraction corresponding to.14. Aiis. 1 13. Find the vulgar fraction corresponding to.0138. Ans.'i. 14. Find the vulgar fraction corresponding to.568i. Ans. i. 15. Find the vulgar fraction corresponding to.592. Ans. ~. The last rule may be deduced from the other two in the following manner:-Let us take the mixed circulator.4i8, and this being multiplied by 10, the four becomes a whole number, and to preserve the same value, 10 is put as a divisor, which gives 4ijL or 4+8, but by rule II. we have.18=-1, and hence the whole may be written 4+ 9 B6* 4,14-23 and this result corresponds to rule III10 ADDITION AND SUBTRACTION OF DECIMALS. 79 IV.-ADDITION & SUBTRACTION OF DECIMALS. From what has been said, it is plain that decimals can be added and subtracted just as whole numbers, care being taken to keep the decimal points in the same vertical line. In all'operations into which repetends enter, it should be observed that in order to have a result true to any given number of places, it is generally desirable to carry out the repetend to one or two places more than the required number. It is often sufficient, however, to allow for what would be carried, which can usually be done by inspection. In all cases, respect should be had to the degree of exactness which the nature of the calculation requires. The figures beyond those required can be estimated and added in. Thus, if only five places are required, and the calculation be carried to six places, and the seventh figure is a large one, it should be added to the sixth figure. This may be stated in the form of a RULE. Add and subtract as in whole numbers, keeping the decimal' points in the same vertical line. EXERCISES. (1.) (2.) (3.) 1.78645 8.58333333+ 51.250000000 3.97863 17.74747474 3.4444444447.84396 112.08080808- 7.6373737 373+ 4.32782? 6.12500000.8855555555 9.54179 15.66666667 11.875000000 11.69857.76969697 7.875875875 — 5.48491 11.00000000 7.111111111+ 44.66213 171.97297979 90.079360724 In exercise 2, the eighth figure of each of the fifth and sixth lines: is made 7 instead of 6, which renders it unnecessary to make any allowance for the repetends that would follow, but this change is not made on any of the last figures of exercise 3, and therefore we add 2 for what would be carried from the tenth decimal place to the ninth. 4. Find in the decimal form the sum of ~, j, i,.. Ans. 2.316. 5. Find in the decimal form the sum of4, i, A 1, J, -4, -3, 12 Ans. 6.0078125. 80 ARITHMETIC. 6. Find in the decimal form the sum of 23, 2, 27, 1. Ans. 2.345. 7. Find in the decimal form the sum of 23, 4j, 53T. Ans. 12.775. 8. What is the sum of.786425,.975324,.176009,.32,.62519375,.4? Ans. 3.28295175. 9. Add to 6 places 18.1276, 11.349, 12.145, 8.648, 15.23. Ans. 65.504414. 10. Find to 6 places the sum of 15.7, 12.4, 18.387,.416,.74687,.9,,.4510.45,.12345. Ans. 59.351152. 11. What is the sum of.76,.416,.45,.648,.23 to five places of decimals? Ans. 2.52087. 12. Reduce to decimals, and find the sum of J, -, 7, 2. Ans. 1.416. 13. Find the sum of.427,.416, 1.328, 3.029, 5.476 to six places of decimals..Ans. 10.678037. 14. Required the sum of 1.25, 1.4, 1.637, 1.885, 1.684, 1.937, 1.148 and 1.764085. Ans. 12.750458. 15. Find the sum of.46321,.81532,.154926,.7532 to true to four places. Ans. 2.1867. 16. From 3.468 substract 1.2591^ and you find the excess 2.2089. 17. What is the excess of 10.008576 above 5.789? Ans. 4.219576. 18. From 11.4 take 1.48, and there remains to six places 9.959596. 19. What is the excess of 7.8 above 1.3754658? Ans. 6.424534i. 20. What is the difference between 9.46574, and 4.18345? Ans. 5.28229. 21. Express, decimally, the difference between qi+-+4+-I+-, and i+4+. Ans. 2.34613+. 22. What is the difference, according to the decimal notation, between p and 3 true to six places of decimals? Ans..636363. MULTIPLICATION OF DECIMALS. 81 23. What is the difference between ~ and 4 expressed decimally true to six decimal places? Ans..071428. 24. What is the difference between the vulgar fractions corresponding to.49 and.5? Ans. 0. 25. Find the value of.786425+.975324+.176009+.32+.62519375 —3.28295175+.4. Ans. 0. 26. What is the difference between 138.6012, and 128.8512? Ans. 9.75 27. What is the excess of 31.6322 above 5.674+1.83+.3125+ 18.62+4.3+.395-.5. Ans. 1.0007. 28. What is the excess, expressed decimally, of 5.83 above 41. Ans. 1.6582. 29. What is the difference between 8.375 and 73 true to six decimal places? Ans..946428. 30. What is the value of 601.050725-441.001-.006253.818475-156.1+.125. Ans..25. V.-MULTIPLICATION OF DECIMALS. If we multiply a decimal by a whole number, the process is precisely the same as if the multiplicand were a whole number, but care must be taken to keep the decimal point in the same relative position. Thus, in the annexed example, as 5.678 there are three decimal places in the multiplicand, we 6 make three also in the product. If we have to multiply a whole number by a decimal, we must mark off a decimal in the product for each decimal in the multiplier.The reason of this will be manifest from the considera5678 tion that if we multiply 8 units by.6, or 6, we get 4,.6 or 4.8, i. e., 4 units and 8-tenths; and again, when we 8 multiply 7 tens by.6 or 6-, we get -4-0-42 units, which 3406.8 T -a I 0 with the 4 units already obtained, make 46 units, and we now have arrived at whole numbers. The same illustration will apply to multiplying by.66, which requires two decimal places to be laid off from the right. Therefore, for every decimal place in the multiplier one must be cut off in the product, and we saw already that for every decimal place in the multiplicand, a dcci 82 ARITHMETIC. mal place must be cut off in the product, and therefore we conclude that for every decimal place in both factors, a decimal place must be marked in the product. It may be well to vary the illustration by observing that as the tenth of a tenth is a one-hundredth, tenths multiplied by tenths give hundredths; so also the product of tenths and hundredths is thousandths, and so on. Thus:.2 or -2, multiplied by.3 or yO, is 00. Now,.6 would not represent this, for that would mean J6L; hence, it is necessary to prefix a cipher, and write.06, and this agrees with what has been already noted (Art. 3) regarding whole numbers, viz., that we are compelled by the nature of the notation to introduce a zero character, and in the present instance the cipher means that there are no tenths, just as it indicated in the case referred to that there were no tens. So, also, lyJ would be written decimally.006, which would mean that there are no tenths, no hundredths, but 6 thousandths. From these explanations we deduce the RULE: Multiply, as in whole numbers, and cut offfrom the right a decimal placefor every one in both multiplier and multiplicand. E XA M P L E S. Multiply.78 by.42. Here we multiply as if the quantities were whole numbers, and in the product point off a decimal figure for each one in both multiplier and multiplicand. In (1.).78 Ex. 1, the number of figures in the product is the.42 same as the number in both factors, and therefore we have no whole number in the result, but four decimal 1562 places. In Ex. 2 there are four decimal places in the factors, and there are six figures in the product,.3276 and consequently two figures represent whole numbers. In Ex. 3, when we multiply 6 by 3, we obtain 18, but if we had (2.).674 (3.) 4.56 carried the repitend out one place far34.6 2.43 ther we should have had 5 to be mul4044 1369 tiplied by 3, and consequently 1 to 2696 1826 carry, so we add 1 to the 18, and in 2022 913 like manner we must allow 2 when multiplying by 4, and 1 when multi23.3204 11.0929 plying by 2. plig y2 rULTIPLICATION OF DECIMALS. 83 EXERCISES. 1. Multiply 7.49 by 63.1. Ans. 472.619. 2. Multiply.156 by.143. Ans..022308. 3. Multiply 1.05 by 1.05, and the product by 1.05. Ans. 1.157625. 4. Find the continual product of.2,.2,.2,.2,.2,.2. Ans..000064. 5. Multiply.0021 by 21. Ans..0441. 6. Multiply 3.18 by 41.7. Ans. 132.606. 7. Multiply.08 by.036. Ans..00288. 8. Multiply.13 by.7. Ans..091. 9. Multiply.31 by.32 Ans..0992. 10. Find the continual product of 1.2, 3.25, 2.125. Ans. 8.2875. 11. Multiply 11.4 by 1.14. Ans. 12.996. 12. Find the continual product of., 1,.1,.1,,.1,.1,.1. Ans..000001. 13. Multiply 1240 by.008. Ans. 9.92. 14. Find the continual product of.101,.0[1,.11, 1.1 and 11. Ans..001478741. 15. Multiply 7.43 by.862 to six places of decimals. Ans..640839. 16. Multiply 3.18 by 11.7, and the product by 1000. Ans. 132606. 17. Multiply.144 by.144. Ans..020736. 18. What is the continual product of 13.825, 5.128 and.001? Ans..0708946. 19. What is the continual product of 4.2, 7.8 and.01? Ans..3276. 20. What is the continual product of.0001, 6.27 and 15.9? Ans..0099693. CONTRACTED METHOD.-In many instances where long lines of figures are to be multiplied together, the operation may be very much shortened, and yet sufficient accuracy attained. We may instance what the student will meet with hereafter, calculations in compound interest and annuities, involving sometimes most tedious operations. By the following method the results in such cases may be obtained with great ease, and correct to a very minute fraction. 1! we are ~computing dollars and cents, and extend our calculation to four 84 ARITHMETIC. places of decimals, we are treating of the one-hundredth part of a cent, or the ten-thousandth part of a dollar, a quantity so minute as to become relatively valueless. Hence we conclude that three or four decimal places are sufficient for all ordinary purposes. Thereare cases, indeed, in which it is necessary to carry out the decimals farther, as, for instance, in the case of Logarithms to be considered hereafter. The principle of the contracted method will be best explained by comparing the two subjoined operations on the same, -quantities. Let it be required to find the product of 6.35642 and 47.6453,. true to four places of decimals: EXTENDED OPERATION. CONTRACTED OPERATION. 6.35642 6,35642. 47.6453 3546.74 19 06926 2542568 317 8210 44494? 2542 568 38138 38138 52 2542 444949 4 317 2542568 19._ —-~~~- --- ~2. carried.. 302.8535 37826 302.8535 RULE FOR THE CONTRACTED METHOD. Place the units' figure of the whole number under the last required' decimal place of the multiplicand, and the other integral figures to the right of that in an inverted order, and the decimal figures, also in an inverted order, to the left of the integral unit; multiply by each figure of the inverted multiplier, beginning with the figure of the multiplicand immediately above it, omitting all figures to the right, but allowing for what would have been carried if the decimal had been carried out oneplacefarther-place thefirst figure of each partial product in the same vertical column, and the others in vertical columns to the left; the sum of these columns will be the required product. Thus, in the above example, we are required to find the product correct to four decimal places, therefore we set the units' figure, 7Y under the fourth decimal figure, and the tens' figure, 4, to the right, and the decimal figures, 6453, to the left in reversed order; then we, MULTIPLICATION OF DECIMALS. 85 multiply the whole line by 4, and then we multiply by,7, omitting the 2 which stands to the right, but allowing 1 for what would have been carried, that is, we say 7 times 4 is 28, and 1 is 29, and we write the nine under the 8, the first figure of the first partial product. By coinparing the contracted method with the figures of the extended form, which are to the left of the vertical line drawn after the fourth decimal figures, it will be seen that the figures of each column are the same but placed in reversed order, which makes no difference in the sum, as 5+3 — 3+5-=8. This is the same principle as the contracted method of multiplying by 17, 71, &c., suggested in the article on simple multiplication * The object of writing the multiplier in a reversed order is simply to make the work come in the usual form, as otherwise we should be crossing and recrossing, so to speak, as will be seen by the operation in the margin.6.35642 Beginning with the left hand figure of the 47.6453 multiplier, and the right hand figure of the'2542568 multiplicand, we find the first partial pro444949 duct; then taking the second figure of the 38138 multiplier from the left, (7) and the second 2542 figure of the multiplicand from the right, 317 we get the second partial product, and so on, 19 2 allowed. moving one place each time towards the right in the multiplier, and one place to302.8535 wards the left in the multiplicand. This is so different from the ordinary mode of operation, as to be excessively awkward and puzzling, and this gave rise to the idea of reversing the order of the digits. We append this remark as most persons cannot at first sight comprehend the reason of the inversion. * Let the learner observe that all the figures of the first column are of the same rank, viz., ten-thousandths, and therefore may be added together, and as the value of each figure is increased or decreased 10 times according to its position to left or right, it follows that all figures at equal distances from the decimal point, whether to right or left, are of the same rank, i. e., units will be under units, tens under tens, tenths under tenths, hundredths under hundredths, &c., &c. The contracted method is not of much use in terminating decimals which extend to only a few places, but it saves iA Vast deal of labour in questions which involve either repetends or terminating decimals expressed by a long line of decimal figures 86 ARITHMETIC. ADDITIONAL EXERCISES: 21. Multiply.26736 by.28758 to four decimal places Ans..0769. 22. Multiply 7.285714 by 36.74405 to five decimal places. Ans. 267.70665. 23. Multiply 2.656419 by 1.723 to six decimal places. Ans. 4.578932. 24. What decimal fraction, true to six places, will express the product of 1 multiplied by -? Ans..113445. 25. What decimal fraction is equivalent to X 3 X? Ans..46748. 26. What is the second power of.841? Ans..707281. 27. What is the product of 1.65 by 1.48, true to five places? Ans. 2.45975. 28. Express decimally 2- X. Ans. 2.393162. 29. What is the product of 73.6371 by 8.143? Ans. 599.6272077. 30..681472X.01286, true to five places, will give.00876. In the last exercise it must be observed that since.681472 68210.0 there is no whole number, and five decimal places are required, we must place a cipher under the fifth decimal 681 figure, and write.01286 in reversed order. That the 136 result is a sufficiently close approximation will be 55 evident from the consideration that the last figure 6 is only six one-hundred-thousandths of the unit, and.00876 consequently the next figure would be only one-millionth nart of the unit. VI.-DIVISION OF DECIMALS. WVe have already seen (1) that we cannot perform any operation except the numbers concerned are of the same denomination, or one of them be abstract; (2) that when a denominate number is used either as a multiplier or a divisor, it ceases to be denominate, and becomes abstract, and (3) that the rules for addition, subtraction, multiplication and division of integers apply equally to decimals, the only additiwnal requirement being the placing and moving of the decimal point. DIVISION OF DECIMALS. 87 lSuppose tnen we are required to divide 1.2321 by 11.1, we must y (1) bring both quantities to the same denomination. Now the dividend is carried down to ten-thousands for 1.2321-1+ I 3W2, and therefore we express 11.1 in the corresponding form, ten-thousandths or 11+-lJ-o —oo-oi or 11.1000, so that we change the form, but not the value of 11.1, the divisor. Again, by (2) the.1, which originally expressed a tenth of some unit, and therefore was in reality denominate, now becomes abstract as one of the figures of the given factor of 1.2321, by means of which we are to find the other factor. Hence by (3) we can now divide 1.2321 by 11.1000, as if both were whole numbers, and this is the reason for omitting the decimal point when we have made the number of decimal places equal. Beginners generally feel a difficulty in conceiving how a fraction divided by a fraction can give a whole number. The difficulty may be easily removed by noticing that i is contained twice in I for — =-, e. g., a half dollar contains, or is equivalent to, two quarter dollars. Thus the fraction i divided by the fraction i, gives the whole number 2. So, also, i is contained 4 times in ~, and therefore I —=-4, a whole number. Hence, when we have reduced the divisor and dividend to the same denomination, we may omit the decimal point, as we have only to find how often the one is contained in the other. Hence the U LE. If the number of decimal places in the divisor and dividend be not equal, make them equal by supplying ciphers or repetends, and then divide as in whole numbers, and the quotient so far will be a whole number, but if there is a remainder, annex ciphers or repetends, and the part of the quotient thus obtained will be a decimal. The decimal places may be supplied as the work proceeds, as it is easy to see how many ciphers or repetends must be supplied; for we have seen in multiplication that the number of decimal places in any product must be equal to all the decimal places in the factors, and, since a dividend must always be viewed as a product, it follows that the difference between the number of decimal places in dividend and divisor will indicate how many ciphers or repetends must be supplied. EXER CISES. 1. Divide 47.58 by 26.175 to six decimal places. Ans. 1.817765. 2. Divide 70.8946 by 13.825 to three places. Ans. 5.128. 88 ARITHMETIC. 3. Divide 468.7 by 3.365 to six places of decimals. Ans. 139.309889. 4. Express decimally 1-~-v. Ans. 233.3. 5~ Express in the decimal form; of $ —4 of - true to six places of decimals. Ans. 1.054687. 6. Divide the whole number 9 by the fraction.008. Ans. 1125. 7. What is the quotient of 5.09 by 6.2? Ans..81 nearly. 8. Divide.54439 by 7777. Ans..00007. 9. What decimal is obtained by dividing 1 by 10.473654? Ans..09547766. 10. What is the difference between -i-. and |- -}l in the decimal form? Ans..24583. CONTRACTED. METHOD. The work may often be much abbreviated in the manner exhibited by the following example:.14736).23748 (1.611 14736)23748(1.611 14736... 14736 9012 0 9012 8841 6 8842 170 40 170 147 36 147 23 040 23 14 736 15 8 304 8 Here it is required to divide.23748 by.14736. Since both divisor and dividend contain the same number of decimal places, no alteration is needed, and so we can at once reject the decimal point, and divide as in whole numbers. The principle of the contraction is simply what has been already explained, viz., that all we look for in such calculations is a sufficiently close approximation, by which we mean an approximation sufficient for all practical purposes. For this reason, when we have obtained the integral part of the quotient, we may omit one figure of the divisor in succession after each operation, as the value of each figure decreases in a tenfold degree as we descend towards the right, and after three decimal figures the error, DIVISION OF DECIMALS. 89 or deficit rather, becomes only thousandths, which are very rarely worth taking into account. For example, if the calculation regards dollars and cents, the error at the fourth decimal place would be only the one-thousandth part of a cent. RULE. Arrange the fractions as in the ordinary mode; find the first figure of the quotient and the first remainder; then, instead of annexing a periodic figure or a cipher, cut off the right hand figure of the divisor, and use the remaining figures to find the next figure of the quotient, and so on. It is usual to mark the figures as they are successively cut off by placing a point below each. In multiplying by each figure of the quotient, allowance must be made for what would have been carried from the figure of the divisor last cut off, had it been used in the division. The vertical line drawn through the ordinary form shows how closely the tio modes correspond. As has already been remarked, it is desirable, in order to secure accuracy, to carry the figures of repetends to one or two places more than are required. EX ERCISE S. (1.) (2.) 43232323)7:640000(170.3355......~ 43232323 30407677 54p37 43682(.7995 30262626.*38246 145051 5436 129697 4917 15354 519 12970 491 2384 28 2162 27 222 1 216 6 Divide 73.64 by.432. and.43682 by.54637 to 4 decimal places each. To show that there will be three integral places in the ad ARITHMETIC. quotient of Ex. 1, we must consider that there are two places of whole numbers in the dividend and none in the divisor, and, therefore, if we divide 73 and 6, the first decimal place of the dividend by.4, the first figure cf the divisor, we get three integral places. Hence, since we are to have four decimal places, we shall have seven figures in all. This contraction is extremely useful when there are many decimal places. 3. Find the quotient of 8.6134-.7.3524 to four decimal places. Ans. 1.1715. 4. Divide.61 by 13.543516 to five decimal places. Ans..04549. 5. Divide.58 by 77.482 to five decimal places. Ans..00756. 6. Divide.812.54567 by 7.34 to three decimal places. Ans. 110.649. 7. Divide 1 by 10.473654 to six decimal places. Ans..09547. 8. Divide 7.126491 by.531 to six decimal places. Ans. 13.420887. 9. Divide 1.77975 by the whole number 25425. Ans..00007. 10. Divide to eight places.879454 by.897. Ans..98043924. VII.-DENOMINATE DECIMAILS. To express one denominate number as a fraction of another of the same kind, reduce both to the lowest denomination contained in either, make the former the numerator and the latter the denominator of a common fraction, and reduce the fraction so found to a decimal in the manner alreadypointed out. EX A P L E S. To express 16 cents as a fraction of a dollar: Iere the lowest denomination mentioned is cents, and we reduce a dollar to cents and write -(;, -4, and, dividing 4 by 25, we get.16. To express 11s. 4~d. as a decimal of ~1, we reduce both to half-pence, and obtain I -- 9,-o which, reduced to a decimal, is.56875. EXERCISES. 1. Reduce 5s. 10d. to the decimal of ~1. Ans. ~.29375. 2. Reduce 10~d. to the decimal of ~1. Ans. ~.04375. 3. Reduce 15s. 9dd. to the decimal of ~1. Ans. ~.790625. 4. Express 3 roods and 11 rods as a decimal of an acre. Ans..81875. ULEDUCTION OF DENOMINATIONS. 91 5. Express: cwt., 1 qr., 7 lbs., as a decimal of a ton. Ans..166. G. Reduce 37 rods to the decimal of a mile, Ans..115625. 7. Reduce 7 ozs. 4 dwts., to the decimal of a pound. Ans..6. 8. Reduce a pound troy to the decimal of a pound avoirdupois; correct to six decimal places'* Ans..822857+-. 9. Reduce 5 hours, 48 minutes, 49.7 seconds, to the decimal of a day, taken as 24 hours. Ans..2422419. 10. Express an ounce avoirdupois as a decimal of a pound troy. Ans..9114583. VIII.-REDUCTION TO DENOMINATIONS. To find the value oJ' a fraction in the lower denominations, expressed as a decimal of any given denomination, multiply in succession by the numbers which express the given and lower denominations, and after each multiplication cut off from the right as many decimal figures as are contained in the given decimal, and the figures to the left of the decimalpoint will give the required value. E X A M P L E S. 1. To find the value of.64379 of a pound (apothecary). We 12 multiply by 12, by 8, by 3 and by 20, which gives 7 ozs., 7.72548 8 5 drs., 2 scrs., and a little over 8 grs. Repetends must 5.80384 be reduced to common frac3 tions, or found approximately. 2.41152 20.77777 8.23040 24 carry 1. 2. To find the value of.7: of a day, which is 18 hours, 39 min. and nearly 311109 59t sees. 155555 18.66659 60 39.99540 60 59.72400 *The standard pounds are meant here, viz.: troy, 5760 grains, and avoirdupois 7000 grains. Taking the ounces would giveo'2=i=.75 92 ARITHMETIC. EXERCISES. 1. What is the value of ~.475? Ans. 9s. 6d. 2. What is the value of.7 of a cwt.? Ans. 3 qrs., 3 lbs., 1 oz., 124 drs. 3. What is the value of.5416 of a shilling sterling? Ans. 6~d. 4. What is the value of.6845 of s cwt.? Ans. 2 qrs., 20 lbs., 10 oz., 91,. 5. What is the value of.4 of 9s. 4~d? We have.4=- and 9s. 4~d., multiplied by 4, and the product, divided by 9, gives 4s. 2d., the exact value. 6. What is the value of.026 of 1~ 15'? Reducing.0,26 to a vulgar fraction, we get -2o 4 _, and multiplying 1~ 15' by 2, and dividing by 75, we find 2'. RATIO AND PROPORTION. 17.-RATIO is the relation which one quantity bears to another of the same kind with respect to magnitude, or the number of times that the less is contained in the greater. Thus, the ratio 7 to 21 is 3 because 7 is contained 3 times in 21, or 21 is 3 times 7. The same result is obtained if we divide 7 by 21, for we then find T-= —, which means that 7 is i of 21, and this expresses the very same relation as before; for, to say that 7 is i of 21 is precisely the same as to say that 21 is 3 times 7. (See note under Inverse Proportion.) And, therefore, 3 is called the measure of the ratio. The numbers thus compared are called the terms of the ratio-the first the antecedent and the second the consequent, and the relation is written 7: 21. The sign ( ) originally indicated division. That the magnitudes must be of the same kind will be obvious from the consideration that 7 bags of flour could have no ratio to 21 dollars, for multiplying 7 bags of flour by 3 would not make them 21 dollars, but 21 bags of flour, and multiplying 7 dollars by 3 would not make them 21 bags of flour, but 21 dollars. Hence, the less could not be increased to make the greater, except they are homogeneous, or of the same kind. Proportion is the equality of ratios. The ratio of 9 to 27 is 3, but we have seen that the ratio of 7 to 21 is also 3, therefore the ratios of 7 to 21 and of 9 to 27 are the RATIO AND PROPORTION. 93 same, or 7- 21-9 —27, and these quantities are, therefore, called proportionals. The sign (::) was formerly used for equality, and is still retained for equality of ratios, and the sign (=) is used for the actual equality of quantities, though occasionally used for equality of ratios. Hence, the usual mode of writing the equality of two ratios is 7.: 21 L.9 f'27. Such a statement is called a proportion, or an analogy, and is read-7 is to 21 as 9 to 27, i. e., 27 exceeds 9 as many times as 21 exceeds 7, and this is expressed by saying 27 is the same multiple of 9 that 21 is of 7, or that 9 is the. same sub-multiple, measure, or aliquot part of 27 that 7 is of 21. The four quantities are called the terms of the proportion;'the first and last are called the extremes, and second and third the means;.also, the first and third are called homologous, or of the same name, i. e., both are antecedents, and so the second and fourth are homologous, for they are both consequents. The last term is called a fourth proportional to the other three, and we shall denote it by F. P. There are two simple ways of testing the correctness of an analogy. The first is to divide the second term by the first, and the fourth by the third, and if the quotients are equal, the analogy is correct. This is manifest from what has been already said. The second principle is, that, if the analogy be correct, the product of the extremes is equal to the product of the means. To prove this, let us resume the analogy, 7: 21:: 9: 27. We have seen that 21-7=27 — 9, or.3=3. Now, if each be multiplied by 63, we have (by Ax. II., Cor.,) 189=189..But 189 is the product ofi27 by 7, the extremes, and also of 21 by 9, the means-these products then are always.equal. From this simple principle we readily deduce a rule for finding a fourth proportional to three given quantities. Let the quan, titles be 48, 96, and 132, written thus: 48: 96:: 132:, the required quantity. Now. 132X96=12672, the product of the means are therefore equal to the product of the extremes. We have, therefore, a product, 12672, and one of its factors, 48, hence, dividing this product by the given factor, we find the other factor to be 264, which is therefore the fourth proportional,'or fourth term of the proportion, and we can now write the whole analogy, thus: 4.: 96:: 132: 264. To prove the correctness of the operation, multiply 264 by 48, and 12672 is obtained, the same as before. Hence, 7 94 ARITHMETIC. THE RULE. Divide the product of the second and third terms by Ihejirst, and the quotient will be the requiredfourth term. To show the order in which the three given quantities are to be arranged, let it be required to find how much 730 yards of linen will cost at the rate of $30 for 50 yards. It is plain that the answer, or fourth term, must be dollars, for it is a price that is required, and in order that the third term may have a ratio to the fourth, the $30 must be the third term. Again, since 730 yds. will cost more than 50 yds., the fourth term will be greater than the third, and therefore the second must be greater than the first, and therefore the statement is yds. yds. $ 50: 730:: 30: 4th proportional, and by the rule 7 3- 0 3 0 _2 lo 9o -438, the fourth term, and we can now write the whole analogy, 50 yds: 730 yds:: $30: $438. This may be called the ascending scale, for the second is greater than the first, and the fourth greater than the third. If the question had been to find what 50 yards of linen willcost at the rate of $438 for 730 yards, we still find that the answer will be dollars, and that therefore, as before, dollars must be in the third place, but we see that the answer will now be less than 438, as 50 yards, of which the price is required, will cost much less than 730 yards, of which the price is given, and that therefore the second term must be less than the first. IHence the statement is 730 yds: 50 yds: $438: 1'. P., and by the rule 4 3 s8X5030, the fourth proportional. We now have the full analogy 730 yds. 50 yds:: $438: $30. As the second is less than the first, and the fourth less than the third, this may be called the descending scale.' If the first should turn out to be equal to the second, and therefore the third equal to the fourth, we should say that the quantities were to each other in the ratio of equality. RULE FOR THE ORDER OF THE TERMS. If the question implies that the consequent of the second ratio must be greater than the antecedent, make the greater term.of the first ratio the consequent, and the less the antecedent, and vice versa. The questions hitherto considered belong to what is called Direct Proportion, to distinguish it from another kind called Inverse Proportion; because, in the former, the greater the npmber given, the less will be the corresponding number required, and vice versa; RATIO AND PROPORTION. 95 whereas, in the latter, the greater the number given, the less will be the number required, and vice versa. To illustrate this, let it be required to find how long a stack of hay will feed 12 horses, if it will feed 9 horses for 20 weeks. Here the answer required is time, and therefore 20 weeks will be the antecedent of the second ratio; but the greater the number of horses, the shorter time will the hay last, and therefore the fourth term will be less than the third, and therefore the statement will not be 9: 12, but the reverse, 12: 9; and hence the name INVERSE, because the term 9, for which the time (20 weeks,) is given, and which therefore we should expect to be in the first place, has to be put in the second; and the term 12, for which the time is required, and which therefore we should expect to be in the second place, has to be put in the first, and thus the whole analogy is 12: 9:: 20: 15.* The principal changes that may be made in the order of the terms, will be more readily and clearly understood by the subjoined scheme, than by any explanation in words: Original Analogy:: 6:: 12: 9 for 8X9=-72=66X12. Alternately: 8: 12:: 6: 9 for 8X9=72=6X12. By Inversion: 6: 8:: 9: 12 for 6X12-72 —X9. By Composition: 8-6: 6::12-9: 9 or 14: 6:: 2: 9 for 14X9=126=6X21. By Division: 8-6: 6::12 —9: 9 or 2: 6::3: 9 for 29= 18-6X3. By Conversion: 8: 8-6::12: 12-9 or 8: 2:: 12: 3 for 8X 3-24-2X12. Simple transposition is often of the greatest use. Let us take an easy practical example. In calculating what power will balance a given A F weight, when the arms of the lever are known, let P be the power, W the weight, A the arm of power, and B ( the arm of weight. The rule is, that < o w the power and weight are inversely as the arms. This solves all the four possible cases by transposition. * Inverse ratio is sometimes spoken of, but in reality there is no sucl thing. It is true that Inverse Proportion requires the terms of one of the ratios to be inverted, but that is a matter of analogy, not of ratio, for we havt seen already that 7- -21 expresses the very same relation as 21-+-7.-(See in 96 ARITHMETIC. A: B::: P, gives the power when the others are known, B: A:: P: W gives the weight when the others are known, W: P:: A: B gives the arm of weight when the others are known, P W:: B: A gives the arm of power when the others are known. The work may often be contracted in the following manner: Resuming our example 48: 96:: 132: fourth proportional, we see that 96 is double of 48, and therefore the ratio of 48 to 96 is the same as that of any two numbers, the second of which is double the first, and 48: 96 is the same as 1: 2, and we reduce the analogy to the simple form of 1: 2:: 132: 4th prop., and we have - 32X2 —264, the term required, as before. In the example 50: 730: 30: 4th term, we have 730X30_73x30X 73Xx573X6f438. This is equivalent to dividing the first and second by. 10, and the first and third by 5. Hence we may divide the first and second, or first and -third by any number that will measure both. The same principle -will also be illustrated by the consideration that the second and third.are multipliers, and the first a divisor; and if we first multiply, and then divide by the same quantity, the one operation will manifestly neutralize the other. Thus: 48: 96:: 132: F. P. may be written 1X48: 2X48:: 132: F. P.; where it is plain that since by first multiplying 132 by 48, and then dividing by the same, the one,operation would neutralize the other, both may be omitted. In proportion, when the means are equal, such as 4: 12:: 12: 36, it is usual to write the analogy thus-4: 12: 36, and 12 is called a mean proportional between 4 and 36. To prove the correctness of this statement, we multiply 36 by 4 and 12 by itself, and as both give 144, the analogy is correct. Now, as 144 is the square or second power of 12, so 12 is called the second root, or square root of 144, or that which produced it, or the root from which it grew; hence, to find a mean proportional between two given quantities, we have the following RULE. Mul:iply them together, and take the square root of the product. Thus, in the above example, 4X36=144, the square root of which is 12. Again, to find a mean proportional between 9 and 49, we multroductory remarks.) The term Reciprocal Ratio is liable to the same objection. for though 3 and i are reciprocals, yet they express the same relation. When the expression Inverse Ratio is legitimately used, it does not refer to a single ratio, but means that two ratios are so related that one of them must be inverted. RATIO AND PROPORTION. 97 tiply 49 by 9, which is 441, the square root of which is 21, which is a mean proportional between 9 and 49, i. e., 9: 21: 49, or, written at full length, 9: 21:: 21: 49. Proof: 49X9-441 and 21X21=441. As the learner is not supposed, at this stage, to know the nxthod of finding the roots of quantities beyond the limits of the multiplication table, we append a table of squares and roots at the end of the book. When each quantity in a series is a mean proportional between two adjacent quantities, the quantities are said to be continued, or continual proportionals. Thus: 2: 4: 8: 16: 32: 64: 128, and 3: 9: 27: 81: 243, are series in which each is a mean proportional between two adjacent ones. Let us take 16 and the two adjacent ones, 8 and 32-the analogy is 8: 16:: 16: 32. Proof: 8X32=256, and 16X16=256. So also, 27 and the adjacent terms, 9 and 81. The analogy is 9: 7:: 27: 81, and the proof, 9X81= 729, and 27 X27=729. This subject will be treated of at length in a subsequent part of the work, but this explanation has been introduced here to fill up the outline and let the learner understand the nature of continued proportionals. EXERCISES. 1. If 6 barrels of flour cost $32, what will 75 barrels cost? Ans. $400. 2. If 18 yards of cloth cost $21, what must be paid for 12 yards? Ans. $14. 3. How much must be paid for 15 tons of coal, if 2 tons can be purchased for $15? Ans. $112.50. 4. If you can walk 84 miles in 28 hours, how many minutes will you require to walk 1 mile? Ans. 20. 5. What will 14 horses cost, if 3 of the average value can be bought for $270? Ans. $1260. 6. What must be paid for a certain piece of cloth, if 3 of it cost $9. Ans. $13.50. 7. If 5 men are required to build a wall in 5 days, how many men will do the same in 2- days? Ans. 10. 8. If 16 sheep are |- of a flock, how many are there in the same? Ans. 24. 9. What must bepaid for 41 cords of wood, if the cost of 3 cords is $10? Ans. $15. 98 ARITHMETIC. 10. What is the height ot a tree which casts a shadow of 125 feet, if a stake 6 feet high produces a shadow of 8 feet? Ans. 93-. 11. How long will it take a train to run from Syracuse to Oswego (a distance of 40 miles), at the rate of 5 miles in 1553 minutes? 12. If 15 men can build a bridge in 10 days, how many men will be required to erect three of the same dimensions in i- the time? Ans. 90. 13. If a man receive $4.50 for 3 days' work, how many days ought he to remain in his place for $25? 14. How much may a person spend in 94 days, if he wishes to save $73.50 out of a salary of $500 per annum? 15. If 3 cwt., 3 qrs., 14 lbs. of sugar cost $36.50, what will 2 qrs., 2 lbs. cost? Ans. $4.879+. 16. 5 men are employed to do a piece of work in 5 days, but after working 4 days they find it impossible to complete the job in less than 3 days more, how many additional men must be employed to do the work il the time agreed upon at first? Ans. 10. 17. A watch is 10 minutes too fast at 12 o'clock (noon) on Monday, and it gains 3 minutes 10 seconds a day, what will be the time by the watch at a quarter past 10 o'clock, A. M.,- on the following Saturday? Ans. 10 h. 40 m. 36- s. -18. A bankrupt owes $972, and his property, amounting to $607.50, is distributed among his creditors; what does one receive whose demand is $11.33? Ans. $7.083+. 19. What is the value of.15 of a hhd. of'lime, at $2.39 per hhd.? Ans. $.3585. 20. A garrison of 1200 men has provisions for 4 of a year, at the rate of, of a pound per day; how long will the provisions last at the same allowance if the garrison be reinforced by 400 men? Ans, 6- months. 21. If a piece of land 40 rods in length and 4 in breadth make an acre, how long must it be when it is 5 rods 5~ feet wide? Ans. 30 rods. 22. A borrowed of B $745, for 90 days, and afterwards would return the favor by lending B $1341; for how long should he lend it? 23. If a man can walk 300 miles in 6 successive days, how many miles has he to walk at the end of 5 days? Ans. 50. RATIO AND PROPORTION. 99 24. If 495 gallons of wine cost $394; how much will $72 pay for? Ans. 90 gal. 25. If 112 head of cattle consume a certain quantity of hay in 9 days; how long will the same quantity last 84 head? Ans. 12 days. 26. If 171 men can build a house in 168 days; in what time will 108 men build a similar house? Ans. 266 days. 27. It has been proved that the diameter of every circle is to the,circumference as 113: 355; what then is the circumference of the moon's orbit, the diameter being, in-round numbers, 480,000 miles? Ans. 1,507,964-~js5 m. 28. A round table is 12 ft. in circumference; what is its.diameter? Ans. 3 ft. 9 — a in. 29. A was sent with a warrant; after he had ridden 65 miles, B'was sent after him to stop the execution, and for every 16 miles that A rode, B rode 21; How far had each ridden when B overtook A? Ans. 273 miles. 30. Find a fourth proportional to 9, 19 and 99. Ans. 209. 31. A detective chased a culprit for 200 miles, travelling at the rate of 8 miles an hour, but the culprit had a start of 75 miles; at vwhat rate did the latter travel? Ans. 5 miles an hour. 32. How much rum may be bought for $119.50, if 111 gallons cost $89.625? Ans. 148 gallons 33. If 110 yards of cloth cost $18; what will $63 pay for? Ans. 385 yards. 34. If a man walk from Rochester to Auburn, a distance (f' (say) 79 miles in 27 hours, 54 minutes; in what time will he vwal at the same rate from Syracuse to Albany, supposing the distance to be 152 miles? 35. A butcher used a false weight'- 14 oz., instead of 16 oz. iUr.a pound, of how many lbs. did he defraud a customer who bought 112 just lbs. from him? Ans. 9| lbs. 36. If 123 yards of muslin cost $205; how much will 51 yards cost? Ans. $85. -37. In a copy of Milton's Paradise Lost, containing 304 p-ecs, tlf-conibat of Michael and Satan commences at the 139th page; at what page may it be expected to commence in a copy containing 328 pages? Ans. The fourth proportional is 1493; and hence the passagc will commence at the foot of page 150 38. Suppose a man, by travelling 10 hours a day, performs a 100 ARITHMETIC. journey in four weeks without desecrating the Sabbath; now manyweeks would it take him to perform the same journey, provided he travels only 8 hours per day, and pays no regard to the Sabbath? Ans. 4 weeks, 2 days.. 39. A cubic foot of pure fresh water weighs 1000 oz., avoirdu-'pois; find the weight of a vessel of water containing 217~ cubic in. Ans. 7 lbs., 13 t| oz.. 40. Suppose a certain pasture, in which are 20 cows, is sufficient to keep them 6 weeks; how many must be turned out, that the same pasture may keep the rest 6 months? Ans. 15. 41. A wedge of gold weighing 14 lbs., 3 oz., 8 dwt., is valued at. ~514 4s.; what is the value of an ounce? Ans. ~3. 42. A mason was engaged in building a wall, when another came up and asked him how many feet he had laid; he replied, that the part he had finished bore the same proportion to one league which. y3 does to 87; how many feet had he laid? 43. A farmer, by his will, divides his farm, consisting of 97 acres, 3 roods, 5 rods, between his two sons so that the share of the younger shall be t the share of the elder; required the shares. Here the ratio of the shares is 4: 3, and we have shown that if four magnitudes are proportionals, the first term increased by thesecond is to the second as the third increased by the fourth is to the fourth. Now, 97 acres, 3 roods, 5 rods, being the sum of the shares, we must take the sum of 4 and 3 for first term, and either 4 or 3 for the second, and therefore 7: 4:: 97 acres, 3 roods, 5 rods: F.P., i. e., the sum of the numbers denoting the ratio of the shares is to one of them as the sum of the shares is to one of them. This gives for the elder brother's share, 55 acres, 3 roods, 20 rods, and the younger's share is found either by repeating the operation, or by subtracting the share thus found from the whole, giving 41 acres, 3 roods, 25 rods. 44. A legacy of $398 is to be divided among three orphans, in, parts which shall be as the numbers 5, 7, 11, the eldest receiving the largest share; required the parts? 23: 5:: 398: 86w-, the share of the youngest. 23: 7:: 398-: 121_, the share of the second. 23: 11:: 398: 1903, the share of the eldest. 45. Three sureties on $5000 are to be given by A, B and C, so that B's share may. be one-half greater than A's, and C's one-half' greater than B's; required the amount of the security of each? COMPOUND PROPORTION. 101 Ans. A's share, $1052.63-13; B's, $1578.94 _; C's, $2368.4 2-. 46. Suppose that A starts from Washington and walks 4 miles an hour, and B at the same time starts from Boston, to meet him, at the rate of 3 miles an hour, how far from Washington will they meet, the whole distance being 432 miles? 47. A certain number of dollars is to be divided between two persons, the less share being |- of the greater, and the difference of the shares $800, what are the shares, and what is the whole sum to be divided? Ans. Less share, $1600; greater, $2400; total, $4000. 48. A certain number of acres of land are to be divided into two parts, such that the one shall be - of the other; required the parts and the whole, the difference of the parts being 716 acres? Ans. the less part 537 acres; the greater, 1253 acres; the whole, 1790. 49. A mixture is made of copper and tin, the tin being, of the copper, the difference of the parts being 75; required the parts and the whole mixture? Ans. tin, 37-}; copper, 1121-; the whole, 150. 50. Pure water consists of two gasses, oxygen and hydrogen; the hydrogen is about - of the oxygen; how many ounces of water will there be when there are 7641- oz. of oxygen more than of hydrogen? Ans. 1000 oz. COMPOUND PROPORTION. Proportion is called simple when the question involves only one condition, and compound when the question involves more conditions than one..s each condition implies a ratio, simple proportion is expressed, when the required term is found, by two ratios, and compound, by more than two. Thus, if the question be, How many men would be required to reap 65 acres in a given time, if 96 men, working equally, can reap 40 acres in the same time? Here there is but one condition, viz., that 96 men can reap 40 acres in the given time, which implies but one ratio, and when the question has been stated 40: 65: 96: F.P., and the required term is found to be 156, and the proportion 40: 65:: 96: 156, we have the proportion, expressed by two ratios. But, suppose the question were, If a man walking 12 hours a day, can accomplish a journey of 250 miles in 9 days, how many days would he require walking at the 102 ARITHMETIC. same rate, 10 hours each day, to travel 400 miles? Here there are two conditions, viz.: first, that, in the one case, he travels 12 hours a day, and in the other 10 hours; and, secondly, that the distances are 250 and 400 miles. The statement, as we shall presently show, would be 10 12.. 177 Here each condition im250: 400 9 * 25 plies one ratio, 10:12 and 250: 400, and when the required term, which is 17-7-, is found, there are four ratios, viz., the two already noted, and 9: 17.7r; gives two more, one in relation to 10:12, and one in relation to 250: 400. This will be evident, when we have shown the method of statement and operation. EXPLANATORY STATEMENT PRACTICAL STATEMENT AND OPERATION. AND OPERATION. 18 3315::12: FP.P. 11: 33": 12': F.P. 1 53 12F 1:3:12: 36 ------—' - 3 5 2:F.P. 18: 5:36:F.P. __3 _ 1:5:: 2:10. 1. 1 1 5:: 2:10. Let the question be, How many men would be required to reap 33 acres in 18 days, if 12 men, working equally, can reap 11 acres in 5 days? We first proceed, as on the left margin, as if there were only one condition in the question; or, in other words, as if the number of days were the same in both cases, and the question were-If 12 men can reap 11 acres in a given time, how many men will be required to reap 33 acres in the same.time. This, then, is a question in simple proportion, and by that rule we have the statement — 11: 33: 12: F. P., which, by contraction, becomes 1: 3: 12: F. P.; and thus, we'find F. P. to be 36, the number of men required, if the time were the same in both cases. The question is now resolved into this: How many men will be required to reap, in 18 days, the same quantity of crop that 36 men- can reap in 5 days? This is obviously a case of inverse proportion, for the longer the time allowed the less will be the number of men required, and hence the statement, 18: 5: 36: F. P., which, by contraction, becomes 1: 5:: 2: F. P., which gives 10 for the number of men. The work may be shortened by making the two statements at once, as on the right margin. We first notice that the last term is to represent a COMPOUND PROPORTION. 103 certain number of men, and, therefore, we place 12 in the third place; next, we see that, other things,eing equal, it will take more men to reap 33 than to reap 11 acres, and that, therefore, as,far as that is concerned, the fourth term will be greater than the third. and so we put 11 in the first place, and 33 in the second. Again we see that, other things being equal, a less number of men will be required when 18 days are allowed for doing the work, than when it is required to be done in 5 days, and that therefore the fourth term, as far as that is concerned, will be less than the third, and therefore we write 18: 5 below the other ratio as on the margin. Then by contraction we get 13: 2: F. P. Now, as 3 in the first term is to be a multiplier, and 3 in the second a divisor, we may omit these also, and we obtain }: 2: 10, the answer as before. The full uncontracted operation would be to multiply 18 by 11, which 1X18: 33X5 gives 198, then to multiply 33 by 5, 198:165::12: P. which gives 165, then multiply 165. 198 ~ 165:: 12 F. P. 165 X12=10 the product of the two second terms, by 12, and divide the result, 1980, 198 by 198, the product of the two first terms, which gives 10 as before. Because in the analogy 198: 165:: 12: 10, the first two terms are products, this kind of proportion has been called compound, and the ratio of 19 to 165 is called a compound ratio. We can show the strict and original meaning of the term compound ratio more easily by an example, than by any explanation in words. Let us take any series of numbers, whole, fractional or mixed, say 5, i, 3, 19, 12, 1, 17, 11, f{, 25, then the ratio of the first to the last is said to be compounded of the ratio of the first to the second, the second to the third, the third to the fourth; &c., &c., &c., to the end. Now the ratio of 5 to 25 is -:-=-5, and the several ratios are in this 7 3 19. 15 25 a 1B~~1 11 or X aer ~X X -" >T<-l 5>< - 7X=Xq X 11 which leaving finally -2-=-5 as before. If we took them in reverse order, viz., o= —, it is obvious that all therein could be cancelled, as each would in succession be a multiplier and a divisor. We would also remark that compound proportion is nothing else than a number of questions in simple proportion solved by one opera 104 ARITHMETIC. tion. This will be evident from our second example by comparing the two operations on the opposite margins. Again, we remarked that every condition implies a ratio, and that therefore the third and fourth terms of our first example really involve two ratios, one in relation to each of the preceding. Hence universally the number of ratios, expressed and implied, must always be double the number of conditions, and therefore always even. As the third ratio is only written once, the number of ratios appears to be odd, but is in reality even. n U L E: Place, as in simple proportion, in the third place the term that is the same as the required term. Then consider each condition separately to see which must be placed first, and which second, other things being equal. EXA 1 PLE. 1. If $35.10 pay 27 men,for 24 days; how much will pay 16 men 1dsdays? Here we first observe that the answer will be money, and therefore $35.10 27: 16: $35.10) "nust be in the third place. Again, it will 24: 18 take less money to pay 16 men than 27 men, 3 2 and therefore, other things being equal, the 3 2 answer, as far as this is concerned, will be less than $35.10, and therefore we put the less 9: 4:: $35.10 quantity, 16, in the second place. So also 4 because it will take less to pay any given num9)14.40 ber of men for 18 days than for 24 days, 9)140.40 /____ therefore we put the less quantity in the second Ans. $15.60 place, which the statement shows in the margin. EXERCISES. 1. If 15 men, working 12 hours a day, can reap-60 acres in 16 days; in what time would 20 boys, working 10 hours a day, reap 98 acres, if 7 men can do as much as 8 boys in the same time? Ans. 26|| days. 2. If 15 men, by working 61 hours a day, can dig a trench 48 feet long, 8 feet broad, and 5 feet deep, in 12 days; how many hours a day must 25 men work in order to dig a trench 36 feet long, 12 feet broad, and 3 feet deep, in 9 days? Ans. 3a. COMPOUND PROPORTION. 105 3. If 48 men can build a wall 864 feet long, 6 feet high, and 3 feet wide, in 36 days; how many men will be required to build a wall 36 feet long, 8 feet high, and 4 feet wide, in 4 days? Ans. 32. 4. In what time would 23 men weed a quantity of potato ground which 40 women would weed in 6 days, if 7 men can do as much as 9 women? Ans. 8-%- days. 5. Suppose that 50 meh can dig in 27 days, working 5 hours a day, 18 cellars which are each 48 feet long, 28 feet wide, and 15 feet deep; how many days will 50 men-require, working 3 hours each day, to dig 24 cellars which are each 36 feet long, 21 feet wide, and 20 feet deep? Ans. 45 days. 6. If 15 bars of iron, each 6 ft. 6 in. long, 4 in. broad, and 3 in. thick weigh 20 cwt., 3 qrs., (28 lbs.) 16 lbs.; how much will 6 bars 4 ft. long, 3 in. broad, and 2 in. thick, weigh? Ans. 2 cwt., 2 qrs., 8 lbs. 7. If 112 men can seed 460 acres, 3 roods, 8 rods, in 6 days; how many men will be required to seed 72 acres in 5 days? Ans. 21. 8. If the freight by railway of 3 cwt. for 65 miles be $11.25; how far should 3525 cwt. be carried for $18.75? 9. If a family of 9 persons can live comfortably in Philadelphia for $2500 a year; what will it cost a family of 8 to live in Chicago, all in the same style, for seven months, prices supposed to be i of what they would be in Philadelphia? 10. If 126 lbs. of tea cost $173.25; whatwill 68 lbs. of a differsnt quality cost, 9 lbs. of the former being equal in value to 10 lbs. of the latter? 11. If 120 yards of carpeting, 5 quarters wide, cost $60; what will be the price of 36 yards of the same quality, but 7 quarters wide? Ans. $25.20. 12. If 48 men, in 5 days of 12} hours each, can dig a canal 139k yards long, 4- yards wide, and 2~ yards deep; how many hours per day must 90 men work for 42 days to dig 491k- yards long, 47 yards wide, and 31 yards deep? Ans. 4. 13. A, standing on the bank of a river, discharges a cannon, and B, on the opposite bank, counts six pulsations at his wrist between the flash and the report; now, if sound travels 1142 feet per second, 106 ARITHMETIC. and the pulse of a person in health beats 75 strokes in a minute, what is the breadth of the river? Ans. 1 mile, 201- feet. 14. If 264 men, working 12 hours a day, can make 240 yards of a canal, 3 yards wide, and 12 yards deep, in 5 days; how long will it take 24 men, working 9 hours a day, to make another portion 420 yards long, 5 yards wide, and 3 yards deep? 15. If the charge per freight train for 10800 lbs. of flour be $16 for 20 miles; how much will it be for 12500 lbs. for 100 miles? Ans. $92-". 16. If $42 keep a family of 8 persons for 16 days; how long, at that rate, will $100 keep a family of 6 persons? Ans. 50'? days. 17. If a mixture of wine and water, measuring 63 gallons, consist of four parts wine, and one of water, and be worth $138.60; what would 85 gallons of the same wine in its purity be worth? Ans. $233.75. 18. If I pay 16 men $62.40 for 18 days work; how much must I pay 27 men at the same rate? Ans. $140.40.. 19. If 60 men can build a wall 300 feet long, 8 feet high, and 6 feet thick, in 120 days, when the days are 8 hours long; in what time would 12 men build a wall 30 feet long, 6 feet high, and 3 feet thick, when the days are 12 hours long? Ans. 15 days. 20. If 24 men, in 132 days, of 9 hours each, dig a trench of four decrees of hardness, 3371 feet long, 5a feet wide, and 31 feet deep; in how many days, of 11 hours each, will 496 men dig a trench of 7 degrees of hardness, 465 feet long, 32 feet wide, and 21 feet deep? Ans. 5~. 21. If 50 men, by working 3 hours each day, can dig, in 45 days, 24 cellars, which are each 36 feet long, 21 feet wide, and 20 feet deep; how many men would be required to dig, in 27 days, working 5 hours each day, 18 cellars, which are each 48 feet long, 28 feet wide, and 15 feet deep? Ans. 50. 22. If 15 men, 12 women, and 9 boys, can complete a certain piece of work in 50 days; what time would 9 men, 15 women, and 18 boys, require to do twice as much, the parts performed by each, in the same time, being as the numbers 3, 2 and 1 Ans. 104 days. 23. If 12 oxen and 35 sheep eat 12 tons, 12 cwt. of hay, in 8 days; how much will it cost per month (of 28 days,) to feed 9 oxen and 12 sheep, the price of hay being $40 per ton, and 3 oxen being iupposed to cat as much as 7 sheep? Ans. $924. MISCELLANEOUS EXERCISES. 107 24. A vessel, whose speed was 9{ miles per hour, left Belleville at 8 o'clock, a. m., for Gananoque, a distance of 74 miles. A second vessel, whose speed was to that of the first as 8 is to 5, starting from the same place, arrived 5 minutes before the first; what time did the second vessel leave Belleville? Ans. 55 min. past 10 o'clock, a. m. 25. If 9 compositors, in 12 days, working 10 hours each day, can compose 36 sheets of 16 pages to a sheet, 50 lines to a page, and 45 letters in a line; in how many days, each 11 hours long, can 5 com. positors compose a volume, consisting of 25 sheets, of 24 pages in a sheet, 44 lines in a page, and 40 letters in a line? Ans. 16 days. MISCELLANEOUS EXERCISES ON TIE PRECEDING RULES. 1. What is the value of.7525 of a mile? Ans. 6 fur., 0 rd, 4 yds, 1 ft., 22 in. 2. What is the value of.25 of a score? Ans. 5. 3. Reduce 1 ft. 6 in. to the decimal of a yard. Ans..5. 4. What is the value of 14 yards of cloth, at $3.375 per yard? Ans. $47.15. 5. What part of 2 weeks is?-f of a day? Ans. 5o. 6. What part of ~1 is 13s. 4d? Ans. |-. 7. Reduce 7 of a day to hours, minutes and seconds. Ans. 2 hours, 52 min., 48 sec. 8. Add - of a furlong to - of a mile. Ans. 7 fur., 31 rds, 0 yd., 1 ft., 10 in. 9. What is the value of.857- of a bushel of rye? Ans. 48 pounds. 10. Reduce 47 pounds of wheat to the decimal of a bushel. Ans..7831. 11. Reduce 9 dozen to the decimal of a gross. Ans..75. 12. Add -7 of a cwt. to 3 of a quarter. Ans. 3 qrs., 10 lbs. 13. Subtract 7 of a day from 4 of a week. Ans. 4 days, 3 hrs. 14. From l of 5 tons take - of 9 cwt. Ans. 2 tons, 17 cwt., 1 qr., gj lbs. 15. Bow many yards of cloth, at $31 a yard, can be bought for $484? Ans. 13-4 yards. 16. A man bought - of a yard of cloth for $2.80; what was the rate per yard? Ans. $3.20. 17. How many tons of hay, at $16~ per ton, can be bought for $1964? Ans. 114 tons. 108 ARITHMETIC. 18. A.t $17 — per week, how many weeks can a family board for $7654? Ans. 43 — weeks. 19. What number must be added to 26-, and the sum multiplied by 71, that the product may be 496? Ans. 374. 20. A man owns i of an oil well. He sells ] of his share for $3500; what part of his share in the well has he still, and what is it worth at the same rate? 21. How long will 119 hhlds. of water last a company of 30 men, allowing each man | of a gallon a day? Ans. 627 days. 22. Reduce 4 of 24, - of 15, and 31 of 24, to equivalent fractions having the least common denominator. Ans., 6 5,5 360. 23. From 4 of 24 of 4, take - of 64 of 1. Ans. 24. 24. What is the sum of -, 4, 1, i, i, i, and? Ans. 1l~-. 25. What is the sum of I4 of 3 4-+ 1 of 85? Ans. 22 4 o. 26. How long will it take a person to travel 442 miles, if he travels 3} miles per hour, and 8- hours a day? Ans. 16 days. 27. Find the sum of 2- of -i, 3i of 4 of c- of 4t and 4. Ans. 62. 28. A has 21 times 84 dollars, and B 64 times 94 dollars; how much more has B than A? Ans. $442. 29. If I sell hay at $1.75 per cwt.; what should I give for 94 tons, that I may make $7 on my bargain. Ans. $329. 30. If 7 horses eat 93- bushels of oats in 60 days; how many bushels will one horse eat in 874 days? Ans. 19-. 31. Bought 14-7 yards of broadcloth for $102.90; what was the value of 874 yards of the same cloth? Ans. $612. 32. How many bushels of wheat, at $2j per bushel, will it require to purchase 168 8 bushels of corn worth 75 cents per bushel? Ans. 471-3-. 33. If in 82- feet there are 5 rods; how many rods in one mile? Ans. 320. 34. Suppose I pay $55 for 4 of an acre of land; what is that per acre? Ans. $88. 35. If 4 of a pound of tea cost $1.664; what will 4 of a pound cost? Ans. $1.55 5. 36. Subtract the sum of 2- and 1 I,., from the sum of J, 74 and 3, and multiply the remainder by 3i.- Ans. 24-1. 37. If 4 lb. cost 23-34 cents; what will 244 cost? Ans. 77Ps cents. MISCELLANEOUS EXERCISES. 109 38. What is the difference between 2 1 X3 and 2 X3? Ans. %. 39. If ]b. cost $j; what will B- lb. cost? Ans. 392 cents. 40. What is the difference between i of i-++4;+ X, and 41. If 4-7 yards cost $1 3, what will 2A yards cost? Ans. 472 cents. 42. Bought -3 of 2000 yards of ribbon, and sold A of it; how much remains? Ans. 2855 yards. 43. Divide the sum of, i, 3, s 5- 3, 12, J by the sum of, i, t-, A.,,?1, Ti-, and divide the quotient by 6, and multiply the result by j of. Ans.. 44. I bought ~ of a lot of wood land, consisting of 47 acres, 3 roods, 20 rods, and have cleared i of it; how much remains to be cleared? Ans. 20 acres, 3 roods, 31J rods. 45. What is the difference between 1 2 5 and 12 3? Ans. 247 46. If $' pay for a 1~ st. of flour; for how much will $. pay? Ans. 1- st. 47. Mount Blanc, the highest mountain in Europe, is 15,872 feet above the level of the sea; how far above the sea level is a climber who is 63 of the whole height from the top, i. e., f of perpendicular hight? Ans. 12896 feet. 48. What will 45.94375 tons cost if 12.796875 tons cost $54.64? Ans- $196.17. 49. If I gain $37.515625 by selling goods worth $324.53125; what shall I gain by selling a similar lot for $520.6635416.? Ans. $60.1884. 50. If 52.815 cwt. cost $22.345; what will 192.664 cwt. cost at the same rate? Ans. $81.512+ 51. Required, the sum of the surfaces of 5 boxes, each of which is 5J feet long, 2J feet high, and 3& feet wide, and also the number of cubic feet contained in each box. The hox supposed to be made from inch lumber? 52. If I pay $9u'for sawing into three pieces wood that is 4 ft. long; how much more should I pay, per cord, for sawing into pieces of the same length, wood that is 8 feet long? Ans. 22} cents. 53. A sets out from Oswego, on a journey, and travels at the rate of 20 miles a day; 4 days after, B sets out from the same place, and travels the same road, at the rate of 25 miles per day; how many days before B will overtake A? 8 Ans. 16. 110 ARITHMETIC. 54. A farmer having 56{ tons of hay, sold 3 of it at $10 per ton, and the remainder at $9.75 per ton; how much did he receive for his hay? Ans. $580 —. 55. If the sum of 871 and 1171 is divided by their difference; what will be the quotient? Ans. 6593. 56. If 81 yards of silk make a dress, and 9 dresses be made from a piece containing 80 yards; what will be the remnant left? Ans. 11 yards. 57. A merchant expended $840 for dry goods, and then had remaining only |X as much money as he had at first; how much money had he at first? Ans. $3430. 58. If a person travel a certain distance in 8 days and 9 hours, by travelling 12 hours a day; how long will it take him to perform the same journey, by traveling 81 hours a day? Ans. 12 days. 59. If 15 horses, in 4 days, consume 87 bushels, 6 qrts; of oats; how many horses will 610 bushels, 1 peck, 2 qrts, keep for the sape time? Ans. 105. 60. Reduce 1 pound troy, to the fraction of one pound avoirdupois. Ans. 4.44 61. Reduce - to a simple fraction. Ans. i. 4k of t 62. What will be the cost of 8 cwt., 3 qrs., 12- lbs. of beef, if 4 cwt. cost $34? Ans. $75-7. 63. If 4 men, working 8 hours a day, can do a certain piece of work in 15 days; how long would it take one man, working 10 hours a day, to do the same piece of work? Ans. 48 days. 64. Divide $1728 among 17 boys and 15 girls, and give each boy -?7 as much as a girl; what sum will each receive? Ans. Each girl, $66~ 6; each boy, $424-. 65. If A can cut 2 cords of wood in 12k hours, and B can cut 3 cords in 17~ hours; how many cords can they both cut in 241 hours? Ans. 832 66. If it requires 30 yards of carpeting, which is t of a yard wide, to cover a floor; how many yards, which is 1- yards wide, will be necessary to cover the same floor? Ans. 18. 67. A person bought 1000 gallons of spirits for $1500; but 140 gallons leaked out; at what rate per gallon must he sell the remainder so as to make $200 by his bargain? 68. What must be the breadth of a piece of land whose length is 40~ yards, in order that it may be twice as great as another piece of ANALYSIS AND SYNTHESIS. Ill land whose length is 145 yards, and whose breadth is 13- yards? Ans. 9i yards. 69. If 7 men can reap a rectangular field whose length is 1,800 feet, and breadth 960 feet, in 9 days of 12 hours each; how long will it take 5 men, working 14 hours a day, to reap a field whose length is 800 feet, and breadth 700 feet? Ans. 3~ days. 70. 124 men dug a trench 110 yards long, 3 feet wide, and 4 feet deep, in 5 days of 11 hours each; another trench was dug by one-half the number of men in 7 days of 9 hours each; how many feet of water was it capable of holding? Ans. 2268 cubic feet. 71. If 100 men, by working 6 hours each day, can, in 27 days, dig 18 cellars, each 40 feet long, 36 feet wide, and 12 feet deep; how many cellars, that are each 24 feet long, 27 feet wide, and 18 feet deep, can 240 men dig in 81 days, by working 8 hours a day? Ans. 256. 72. A gentleman left his son a fortune, 5 of which he spent in 2 months, i of the remainder lasted him 3 months longer, and i of what then remained lasted him 5 months longer, when he had only$895.50 left; how much did his father leave him? Ans. $4477.50.. 73. A farmer having sheep in two different fields, sold i of the number from each field, and had only 102 sheep remaining. Now12 sheep jumped from the first field into the second; then the number remaining in the first field, was to the number in the second field as 8 to 9; how many sheep were there in each field at first? Ans. 80 in first field; 56 in second. 74. A and B paid $120 for 12 acres of pasture for 8 weeks, with an understanding that A should have the grass that was then on the field, and B what grew during the time they were grazing; how many oxen, in equity, can each turn into the pasture, and how much should each pay, providing 4 acres of pasture, together with what grew during the time they were grazing, will keep 12 oxen 6 weeks. and in similar manner, 5 acres will keep 35 oxen 2 weeks? A A should turn into the field 18 oxen, and pay $72. Ans.B should turn into the field 12 oxen, and pay $48, ANALYSIS AND SYNTHESIS. Analysis is the act of separating and comparing all the different parts of any compound, and showing their connection with each other, and thereby exhibiting all its elementary principles. 112 ARITHMETIC. The converse of Analysis is Synthesis. The meaning and use of these terms will probably be most readily comprehended by reference to their derivation. They are both pure Greek words. Analysis means loosing up. The general reader would here probably expect loosing down, as employed in most popular definitions; but we may illustrate the Greek term, loosing up, by our own everyday phrase, tearing up, which means rending into shreds, the English up conveying the same idea here as the Greek ana in analysis. The Greek synthesis means literally placing together; that is, the component parts being known, the word synthesis indicates the act of combining them into one. We might give many illustrations, but one will suffice, and we choose the one which will be most generally understood. When we analyse a sentence, we loose it up, or tear it up, into its component parts, and by synthesis we write or compose, i. e., put together the parts, which, by analysis, we have found it to consist of. When we commence to analyse a problem we reason from a given quantity to its unit, and then from this unit to the required quantity; hence, all our deductions are self-evident, and we therefore require no rule to solve a problem by analysis. Although this part of arithmetic is usually called analysis, yet, as it is really both analysis and synthesis, we have given it a title in accordance with the principles now laid down. EXAMPLE. 1. If 12 pounds of sugar cost $1.80, what will 7 pounds cost? SOLUTION. 12)1.80 If 12 lbs. cost $1.80, one pound will cost the -- of $1.80=15 cents. Now, if 1 lb. cost 15.15 7 cents, 7 lbs. will cost 7 times 15 cents=to $1.05. Therefore, 7 lbs. of sugar will cost $1.05, if 12 $1.05 lbs. cost $1.80. NOTE.-The work may be somewhat shortened, especially in long questions, by arranging it in the following manner, so as to admit of cancelling, if possible:15 l X'1059=_=$1.05. Ans. 2. If 5 bushels of pease cost $5.50, for what can you purchase 19 bushels? Ans. $20.90. ANALYSIS AND SYNTHESIS. 113 3. If 9 men can perform a certain piece of labor in 17 days, how long will it take 3 men to do it? Ans. 51 days. 4. How many pigs, at $2 each, must be given for 7 sheep, worth $4 a head? Ans. 14. 5. If $100 gain $6 in 12 months, how much would it gain in 40 months? Ans. $20. 6. If 4j bushels of apples cost $31, what will be the cost of 7| bushels v SOLUTION. In the first place, 4j bushels —14 bushels, and $3 -$298. Now, since 1-A bushels cost $2g8, one bushel will cost $2 — -4-s a-x__$, and 71 or J5- bushels will cost - times $=-2 X — = $5, the value of 7j bushels of apples, if 42 bushels are worth $39. OPERATION S. 5 5 7. If A of 31 lbs. of tea cost $1; what will be the cost of 5~ pounds? Ans. $4.12,. 8. 100 is 2 of what number? Ans. 150. 9. If 4 of a mine cost $2800; what is the value of | of it? 9 vJ. u~lllllrc~ ~v~u Wuuvv ) 3 Ans. $4200. 10. 2 of 24 is 1 3 times what number? Ans. 10. 11. i of 40 is -5 of how many times i of 4 of 20? Ans. 9. 12. A is 16 years old, and his age is 2 times 2 of his father's age; how old is his father? Ans. 36. 13. A and B were playing cards; A lost $10, which was - times 3 as much as B then had; and when they commenced | of A's money was equal to j of B's; how much had each when they began to play? Ans. A $50; B $40. 14. A man willed to his daughter $560, which was - of j of what he bequeathed to his son; and 4 times the son's portion was - the value of the father's estate; what was the value of the estate? Ans. $13,440. 15. A gentleman spent i of his life in St. Louis, A of it in Boston, and the remainder of it, which was 25 years, in Washington; that age was he when he died? Ens. 60 years. 114 ARITHMETIC. 16. A owns i-, and B -l of a ship; A's part is worth $650 more than B's; what is the value of the ship? Ans. $15,600. 17. A post stands i in the mud, i in the water, and 15 feet above the water; what is the length of the post? Ans. 36 feet. 18. A grocer bought a firkin of butter containing 56 pounds, for $11.20, and sold i of it for $8|; how much did he get a pound? Ans. 20 cents. 19. The head of a fish is 4 feet long, the tail as long as the head and i the length of the body, and the body is as long as the head and tail; what is the length of the fish? Ans. 32 feet. 20. A and B have the same income; A saves - of his; B, by spending $65 a year more than A, finds himself $25 in debt at the end of 5 years; what did B spend each year? Ans. $425. 21. A can do a certain piece of work in 8 days, and B can do the same in.6 days; A commenced and worked alone for 3 days, when B assisted him to complete the job; how long did it take them to finish the work? SOLUTION. If A can do the work in 8 days, in one day he can do the * of it, and if B can do the work in 6 days, in one day he can do the - of it, and if they work together, they would do 7+=- -74 of the work in one day. But A works alone for 3 days, apd in one day he can do ~ of the work, in 3 days he would do 3 times -=7 of the work, and as the whole work is equal to | of itself, there would be ~-1=4= of the work yet to be completed by A and B, who, according to the conditions of the question, labour together to finish the work. Now A and B working together for one day can do -7 of the entire job, and it will take them as many days to do the balance i as 7 is contained in 1, which is equal XX-24 —=2 days. 22. A and B can build a boat in 18 days, but if d assists them, they can do it in 8 days; how long would it take C to do it alone? Ans. 14| days. 23. A certain pole was 25- feet high, and during a storm it was broken, when i of what was broken off, equalled J of what remained; how much was broken off, and how much remained? Ans. 12 feet broken off, and 131 remained. 24. There are 3 pipes leading into a certain cistern; the first will fill it in 15 minutes, the second in 30 minutes, and the third in one hour; in what time will they all fill it together? Ans. 8 min., 34~ sec. ANALYSIS AND SYNTHESIS. 115 25. A. and B. start together by railway train from Buffalo tc Erie a distance of (say) 100 miles. A goes by freight train, at the rate of 12 miles per hour, and B by mixed train, at the rate of 1S miles per hour, C leaves Erie for Buffalo at the same time by exs press train, which runs at the rate of 22 miles per hour, how far from Buffalo will A and B each be when C meets them. 26. A cistern has two pipes, one will fill it in 48 minutes, and the other will empty it in 72 minutes; what time will it require to fill the cistern when both are running? Ans. 2 hours, 24 min. 27. If a man spends -5y of his time in working, i in sleeping, I in eating, and 1~ hours each day in reading; how much time will be left? Ans. 3 hours. 28. A wall, which was to be built 32 feet high, was raised 8 feet by 6 men in 12 days; how many men must be employed to finish the wall in 6 days? Ans. 30 men. 29. A and B can perform a piece of work in 5-A days; B and C in 6{days;. and A and C in 6 days; in what time would each of them perform the work alone, and how long would it take them to do the work together? Ans. A, 10 days; B, 12 days; C, 15 days; and A. B, and C, together, in 4 days. 30. My tailor informs me that it will take 10{ square yards of cloth to make me a full suit of clothes. The' cloth I am about to purchase is 1- yards wide, and on sponging it will shrink Lo in width and length; how many yards of this cloth must I purchase for my "new suit?" Ans. 62 -3 yards. 31. If A can do i of a certain piece of work in 4 hours, and B,can do j of the remainder in 1 hour, and C can finish it in 20 min.; in what time will they do it all working together? Ans. 1 hour, 30 min. 32. A certain tailor in the City of Brooklyn bought 40 yards of broadcloth, 21 yds wide; but on sponging, it shrunk in length upon every 2 yards, -i) of a yard, and in width, 11 sixteenths upon every 11 yards. To line this cloth, he bought flannel l yards wide which, when wet, shrunk 1 the width on every 10 yards in length, and in width it shrunk ~ of a sixteenth of a yard; how many yards of flannel had the tailor to buy to, line his broadcloth? Ans. 71-1T yards. 33. If 6 bushels of wheat are equal in value to 9 bushels of barley, and 5 bushels of barley to 7 bushels of oats, and 12 bushels of 116 ARITHMETIC. oats to 10 bushels of pease, and 13 bushels of pease to - ton of hay, and 1 ton of hay to 2 tons of coal, how many tons of coal are equal in value to 80 bushsls of wheat? SOLUTION. If 6 bushels of wheat are equal in value to 9 bushels of barley, or 9 bushels of barley to 6 bushels of wheat, one bushel of barley would be equal to X of 6 bushels of wheat, equal to 6, or i of a bushel of wheat, and 5 bushels of barley would be equal to 5 times of a bushel of wheat, equal to X5 =-o0- =3 bushels of wheat.. But 5 bushels of barley are equal to seven bushels of oats; hence, 7 bushels of oats are equal to 3~ bushels of wheat, and one bushel of oats would be equal to 3.-7-= ~ bushels of wheat, and 12 bushels of oats would be equal to 12 times 0=_-120-=5$ bushels of wheat. But 12 bushels.of oats are equal in value to 10 bushels of pease, hence, 10 bushels of pease are equal to 54 bushels of wheat, and one bushel of pease would equal 5 — 10= — of a bushel of wheat, and 13 bushels of pease would equal 4-X13 —5 —-7 bushels of wheat. But 13 bushels of pease equal in value i ton of hay, hence, ~ ton of hay equals 74- bushels of wheat, and one ton would equal 7- X2= 146 bushels of wheat. But one ton of hay equals 2 tons of coalr hence, 2 tons of coal are equal in value to 146 bushels of wheat, and one ton would equal 14 6 -2 -7 bushels of wheat. Lastly, if 77 bushels of wheat be equal in value to one ton of coal, it would take as many tons of coal to equal 80 bushels of wheat, as 74 is contained in 80, which gives 10-~ tons of coal. NOTE.-This question belongs to that part of arithmetic usually called Conjoined Proportion, or, by some, the " Chain Rule," which has each antecedent of a compound ratio equal in value to its consequent. We have thought it best not to introduce such questions under a head by themselves, on account of their theory being more easily understood when exhibited by Analysis than by Proportion. Questions that do occur like this will most probably relate to Arbitration of Exchange. Although they may all be worked by Compound Proportion as well asrby Analysis, yet the most expeditious plan, and the one generaly adopted, is by the following RULE. Place the antecedents in one column and the consequents in another, on the right, with the sign of equality between them. Divide the continued product of the terms in the column containing the odd term by the, continued product of the other column, and the quotient will be the answer. ANALYSIS AND SYNTHESIS. 117 Let us now take our last example (No. 33), and solve it by this wle: 6 bushels of wheat=9 bushels of barley. 5 bushels of barley 7 bushels of oats. 12 bushels of oats=10 bushels of pease. 13 bushels of pease=- ton of hay. 1 ton of hay=-2 tons of coal. - tons of coal-80 bushels of wheat. 20 N, 7,, 7,, s1, a = 14:0=jo10. Ans., i, 1,, 34. If 12 bushels of wheat in Boston are equal in value to 12j bushels in Albany, and 14 bushels in Albany are worth 14J bushels in Syracuse; and 12 bushels in Syracuse are worth 124 bushels in Oswego; and 25 bushels in Oswego are worth 28 bushels in Cleveland; how many bushels in Cleveland are worth 60 bushels in Boston? Ans. 754j. 35. If 12 shillings in Massachusetts are worth 16 shillings in New York, and 24 shillings in New York are worth 221 shillings in Pennsylvania, and 71 shillings in Pennsylvania are worth 5 shillings in Canada; how many shillings in Canada are worth 50 shillings in Massachusetts? Ans. 412. 36. If 6 men can build 125 rods of fencing in 4 days, how many days would seven men require to build 210 rods? SOLUTION. If 6 men can build 120 rods of fencing in 4 days, one man could do 1 of 120 rods in the same time; and I of 120 rods is 20 rods. Now, if one man can build 20 rods in 4 days, in one day he would build i of 20 rods, and 1 of 20 rods is 5 rods. Now, if one man can build 5 rods in one day, 7 men would build 7 times 5 rods in one day, and 7 times 5 rods==35 rods. Lastly, if 7 men can build 35 rods in one day, it would take them as many days to build 210 rods as 35 is contained in 210, which is 6; therefore, if 6 men can build 120 rods of fencing in 4 days, 7 men would require 6 days to build 210 rods. 37. If 12 men, in 36 days, of 10 hours each, build a wall 24 feet long, 16 feet high, and 3 feet thick; in how many days, of 8 118 AIITHMETIC. hours each, would the same lot of men build a wall 20 feet long, 12 feet high, and 24- feet thick? Ans. 237. 38. If 5 men can perform a piece of work in 12 days of 10 hours each; how many men will perform a piece of work four times as large, in a fifth part of the time, if they work the same number of hours in a day, supposing that 2 of the second set can do as much work in an hour as 3 of the first set? Ans. 66- men. NOTE.-Such questions as this, where the answer involves a fraction, may frequently occur, and it may be asked how 2 of a man can do any work. The answer is simply this, that it requires 66 men to do the work, and one man to continue on working i of a day more. 39. Suppose that a wolf was observed to devour a sheep in i of an hour, and a bear in i of an hour; how long would it take them together to eat what remained of a sheep after the wolf had been eating 4 an hour? Ans. 10 5 min. 40. Find the fortunes of A, B, C, D, E, and F, by knowing that A is worth $20, which is i as much as B and C are worth, and that C is worth i as much as A and B, and also that if 19 times the sum of A, B and C's fortune was divided in the proportion of i, i and 4, it would respectively give i of D's, ~ of E's, and - of F's fortune. Ans. A, 20 i B, 55; C, 25; and D, E and F, 1200 each. 41. A and B set out from the same place, and in the same direction. A travels uniformly 18 miles per day, and after 9 days turns and goes back as far as B has travelled during those 9 days; he then turns again, and pursuing his journey, overtakes B 221 days after the time they first set out. It is required to find the rate at which B uniformly travelled. Ans. 10 miles per day. 42. A hare starts 40 yards before a greyhound, and is not perceived by him until she has been running 40 seconds, she scuds away at the rate of 10 miles an hour, and the dog pursues her at the rate of 18 miles an hour; how long will the chase last, and what distance will the hare have run? Ans. 60. sec.; 490 yards. 43. A can do a certain piece of work in 9 days, and B can do the same in 12 days; they work together for 3 days, when A is taken sick and leaves, B continues on working alone, and after 2 days he is joined by C, and they finish it together in 14 days; how long would C be doing it alone? Ans. 12 days. 44. A, in a scuffle, seized on - of a parcel of sugar plums; B caught i of it out of his hands, and C laid hold on -3 more; D ran off with all A had left, except I which E afterwards secured slyly for himself; then A and C jointly set upon B, who, in the conflict, leC' PRACTICE. 119 fall -1 he had, which were equally picked up by D and E, who lay perdu. B then kicked down C's hat, and to work they all went anew for what it contained; of which A got {, B ~, D 2, and C and E equal shares of what was left of that stock. D then struck | of what A and B last acquired, out of their hands; they, with some difficulty, recovered - of it in equal shares again, but the other three carried off 8 a piece of the same. Upon this, they called a truce, and agreed that the i of the whole left by A at first, should be equally divided among them; how many plums, after this distribution, had each of the competitors? Ans. A had 2863; B, 6335; C, 2438; D, 10294 and E, 4950. PRACTICE The rule which is called Practice is nothing else tnan a particular ease of simple proportion, viz., when the first term is unity. Thus: if it is required to find the price of 28 tons of coal, at $7 a ton-as a question in proportion, it would be, if 1 ton of coal costs $7, what will 28 tons cost? and the statement would be 1: 28:: 7: F. P. Here the first term being 1, the question becomes one of simple multiplication, but the answer, $196, is really the fourth term of an analogy. $7.62- Again, to find the price of 46 barrels of flour, at 46 $7.62} per barrel, we have only to multiply $7.621 23 by 46. In many cases, however, it is more conveni4572 ent to multiply the 46 by 7, which will give the price 3048 of 46 barrels at $7 each. Now, 50 cents being half -- -- a dollar, the price of 46, at 50 cents, will be $23, $350.75 and 12~ cents being j of 50 cents, the price at 12cents will be the fourth of that at 50 cents, or $5.75, and the whole comes to $350.75. 70 2 4 To find the price of 36 cwt., 2 qrs., 15 lbs., - ___ at $4.871. Iere the question stated at 121 i 322 length would be, if 1 cwt. cost $4.87~, what 23 5 will 36 cwt., 2 qrs., 15 lbs. cost? The 5.75 statement would be 1: 36., 2., 15:: $4.871: $350.75 F. P. This becomes a question of multi 120 ARIITHMETIC. plication because the first term is unity, and divided by 1 would not alter the product of the other two terms. Thus 2 qrs. - of 1 cwt. 4.871 36 18 2922.1461 175.50 = price of 3 cwt., @ $4.87i per cwt. 10 lbs. of 2 qrs. 2.437 2 qrs. " " " 5 " -of 10 lbs..487- " 10 lbs. " ".243= — 5 " "': $178.667= " 36 cwt., 2 qrs., 15 lbs. We would call the learner's special attention to the following direction, as the neglect of it is a fertile source of error. Whenever you take any quantity as an aliquot part of a higher to find the price of the former, be sure you divide the line which is the price at the rate of that higher denomination. To find the rent of 189 acres, 2 roods, 32 rods, at $4.20 per acre. 2 roods=- of 1 acre, 4.20 Since the rent of 1 acre is 189 $4.20, the half of it, $2.10, rod= of 2 rood 210 will be the rent of 2 roods, 20 rods —I of 2 roods, 210 525 the rent of 20 rods will be 10 rods — of 20 rods, 2625.525, the i of the rent of 2 525 roods, the half of that,.2625, 2 rods=3 of 10 rods, 3780 will be the rent of 10 rods, 3360 420 and, lastly,.0525 will be the rent of 2 rods, which is $696.74 the 1 of 10 rods. We then multiply by 189, and set the figures of the product in the usual order, so that the first figure of the product by 9 shall be under the units of cents, &c., and then adding all the partial results, we find the final answer, $796.74, the rent of 189 acres, 2 roods and 32 perches. EXERCISES. 1. What is the price of 187 cwt. at $5.371 per cwt.? Ans. $1005.121. PRACTICE. 121 2. What is the value of 1857 lbs., at $3.871 per lb.? Ans. $7195.87j. 3. What will 4796 tons amount to at $14.50 per ton? Ans. $21582. 1. What is the price of 29 score of sheep, at $7.621 each? Ans. $4422.50. 5. Sold to a cattle dealer 196 head of cattle at $18.75 each, find the amount. Ans. $3675 6. Sold to a dealer 97 head of cattle, at $16.12! each, on the average; find the price of all. Ans. $ 1564.12-. 7. What is the price of 16 tons, 17 cwt., 2 qrs. of coal, at $8.62! per ton? 8. What is the yearly rent of 97 acres, 3 roods, 20 rods, at $4.37i per acre? 9. If a man has $12.50 per week; how much has he per year? Ans. $650. 10. If a clerk has $2.12! salary for every working day in the year; what is his yearly income? Ans. $665.12!. 11. If a tradesman earn $1.64 per day; how much does he earn in the year, the Sabbaths not being reckoned? Ans. $513.32. 12. If an officer's pay is a guinea and a half per day; how much has he a year? Ans. ~574 17s. 6d. 13. What is the price of 479 cwt. of sugar, at $17.90 per cwt. Ans. $8574.10. 14. Find the price of 879 articles, at $1.19 each. 15. Find the cost of 1793 tons of coal, at $7.871 per ton. 16. What is the value of 2781 tons of hay, at $8.62! per ton? 17. What is the rent of 189 acres, 2 roods, 32 rods, at $4.20 per acre? Ans. $795.74. 18. What is the price of 879 hogs, at $4.25 each? 19. What will 366 tons of coal come to at $8.12! per ton? 20. What is the price of 118 acres, 3 roods and 20 rods of cleared land, at $36.75 per acre? Ans. $4368.66. 21. What is the price of 286 acres, 1 rood, 24 rods of uncleared land, at $7.25 per acre? Ans. $2076.40. 22. A has 84 acres, 2 roods, 36 rods of cleared land, worth $24.60 an acre; B has 298 acres, 3 roods, 24 rodsof uncleared land, worth $4.40 an acre-they-exchange, the difference of value to be paid in cash; which has to pay, and how much? Ans. B $989.08. 122 ARITHMETIC. ACCOUNTS AND INVOICES. ACCOUNTS are statements from merchants to customers that have pur. chased goods on credit, and are generally made out periodically, unles, specially called for. An invoice is simply a statement rendered by the seller to the buyer, a' time of purchase, showing the articles bought, and the prices of each. 1. NEW YORK, July 1st, 1866. MR. JAMES ANDERSON, To FRENCH, WHITE & CO., Dr. 1866. 2.25 1.35 2.00 Jany. 4, To 2 lbs. tea, 1.12; 3 lbs. coffee, 45c.; 20 lbs. rice, 10c.... 4.871 2.5 29, " 2 yds.. Amer. tweed, 1.95; 1 vest.................... I.75 2.00 Feb. 10, " 14 lbs. Mus. sugar, 12.c.; 10 lbs. crus. white sugar, 20c. 60c. 28c. 1.8 22, " 1 lb. bk. soda,; 1 lb. car. soda,; 4 lbs. coffee, 45c.. 3.00 87'c. Mar. 11, " 10 yds. print, 30c.; trimming, &c., per bill....... 1.80 85c. 2.00 19, " 2 lbs. tobacco, 90c.; 1 gal coal oil,; 2 gals. syrup, 1.00. ~~1.75Z 1.C5. Aprl 12 " ~ yd. blk. silk, 3.50; yd. blk. velvet, 6.62.......... 3.25 40c. 60c. May 6," 2 lbs. tea, 1.6 2; 1 bottle pickles,; 1 lb. pepper, 35c. 1.0oo.50 " 20, " 1 bag salt,; 10 lbs. sugar, 10c.; 3 lbs. raisins, 50c.... 75c. 2.50 " 31, " 3 lbs. currants, 25c.; 10 lbs. white sugar, 25c......... 1.50 12-c. 2.oo June 10, " 2 lbs. tobacco, 75c.; i lb. B. soda, 25c.; 20 lbs. rice, 10c.. 40c. 10c. 30c. 1.75 17, "'lb.cloves,; - lb.nutmegs,; cinnamon,; 1 lb.tea, $47.61 2. BALTIMORE, Oct. 1st, 1866. JMR. WILLIAM PATTERSON, To MOFFAT & MURRAY, Dr. 1866. July 3, To 14 yds. fancy print, 20c.; 12 yds. col'd silk, 2.75....... " 14, " 2 ladies' felt hats, 2.oo; 2 prs. kid gloves, 1.8........ " 22, " 4 prs. cotton hose, 40c.; 3 yds. red flannel, 80c....... Aug. 19, " 21 yds. blk. cassimere, 2.95; 21 yds. cotton, 20c...... " 27, " 1I yds. white flannel, 75c.; buttons, 10c.; twist, 15c.... Sept. 1, " 2 suits boys' clothes, 9.00; 2 felt hats, 1.8 0.......... 8, " 2 prs gloves 80c.; 2 neckties, 62e................... " 22, " Y doz. prs. cotton hose, 7.50; ~ doz. shirts, 26.00..... Contra. Cr. 20.00 15.00 Aug.18, By Cash,; 27, Cash,.......................$35.00 " 25. " firkin butter, 95 lbs., at 22c.................. 20.90 55.90 Balance due.......................... $41.714 Received payment in full, MOFFAT & IURRAY. ACCOUNTS AND INVOICES. 123 ROCHESTER, Jan. 2nd, 1866. MR. JOHN DEANS. To WOOD & FUDGER, Dr. 1866. July 4, To 12 lbs. sugar, lOc.; 3 lbs. tea, 1.25; 2 lbs. tobacco, 87-c. 11, " 1 bbl. salt, 225; 2 lbs. indigo, 25c.; 1 lbs. pepper, 30c. 18, " 2 prs. socks, 45c.; 1 neck-tie, 75c.; 2 scarfs, 25c...... 25, " 10 lbs. sugar, lie.; 201bs. dr'd apples, 10c.; 2 lbs. coffee 28c I" " " 18 Ibs. dried peaches, 12~c.; 1 bush. onions, 112....... Aug. 4, ( 12 lbs. rice, 7c.; 2 gals. syrup, 75c.; 14 lbs. sugar, 12c... " "'; 13 lbs. mackerel, 12c.; 2 lbs. ginger, 20c.; 2 lbs. tea, 1.25 " 21.' 2 prs. kid gloves, 125; 2 boxes collars, 37c........ Sept. 12, " 10 lbs. sugar, 15c.; 2 lbs. coffee, 35c.; 1 lb. chocolate, 40c. Oct. 4, " 2 felt hats, 125 shoe blacking, 25c....................' 21. " 2 lbs. pepper,'15c.; soda, 40c.; salpetre, 30c.; salt, 75c. Contra. Cr. 10.00 5.00 Sept. 14. By Cash,; Oct. 4, Cash..................... Oct. 17.' 2 bbls. winter apples, 225..................... $ 18.42 BOSTON, Nov. 1st, 1866. MR. WM. REID. To CAMPBELL, LINN & Co., Dr. Aug- 4, To 2 prs. kip boots, 325; 2 prs. cobourgs, 225............. " 17. " 7 yds. fancy tweed, 225; trimmings, 100; buttons 25c.. Sept. 4, " 2 prs. gloves, 75c.; 3 prs. socks, 35c.; 2 straw hats, 49c, 26. "10 yds. print, 35c.; trimmings, 125; ribbons, 75c....... Oct. 11, " 3 neck-ties, 62c.; 2 prs. boys'gaiters, 275; shoeties,121c. " 22, " 1 business coat, 1400; 2 felt hats, 175; 1 umbrella, 250 " 27, " 2 flannel shirts, 425; 1 pr. pants, 85s; over-coat, 1600. " 30.: 2 lace scarfs, 225; 3 prs. woollen mits 75c.; pins, 25c. Contra. Cr. 10o 800 Sept. 12. By Cash; Oct. 4, Cash...................... Oct. 24. " 300 lbs. cheese, 10c.; 75 lbs. butter, 25c........ Balance due........ $37.60 Received payment. CAMPBELL, LINN & CO. 124 ARIITHMETIC. AUBURN, Sept. 1st, 1866. MR. S. SMITH To WILsoN, RAY & Co., Dr. 1866. Jan. 15, To 6 yds B. cloth, 4.50; 2 doz. buttons, 30c.; 9 ozs. thread,15c.;' 20, " 40 yds. fac. cot., 16c.; 7 spools cot., 4c.; 12 yds. rib., 35c. " 30. " 15 yds. B. silk, 2.3 o; 16 yds. lining, 15c.; 3 silk spools, l1c. Feb. 20. " 3 yds. drill, 31c.; 5 yds. cob'rg, 34c.; 2 papers need. 18c. Mar. 18, " 9 yds. coating, 5.10; 1I yds. vesting, 1.90; 5 pr. hose, 40c. " 31. " 21 yds. print, 20c.; 19~ yds. mlslin, 30c.; 2 prs. gloves, 1.40 Apr.15, " 4 prs. gloves, 1.10; 16 yds. ribbon, 18c.; 6 hand'k. 36c. " 25. " 3 prs. blankets 6.3 0; 4 counterpanes, 3.20; 15 yds. cot.,25c. May29, " 2 summer hats, 1.05; 6 yds. ribbon, 40c.; 2 feathers, 230 June 5, " 4 prs. slippers, 1.40; 4 prs. hose, 60c.; 3 prs. hose, 40c. " 15. " 3 wool shawls, 5.30; 1 B. suit, 30.50; 9 ozs. thread, 18c. July 6. " 40 yds. cotton, 30c.; 3 spools, 12c.; 2 spools, lOc........ Aug. 10. " 13 yds. flannel, 75c.; 4 hand'ks., 35c.; 12 yds. tape, 13c. Contra. Cr. 15.00 10.oo Jan. 15. By Cash,; 22. Cash,....................... Feb. 20. " 50 lbs. butter, 40c.; 6 cwt. pork, 1040............ May 15. " 6 geese, 80c.; 14 fowls, 40c............... June 5. " 60 lbs. wool, 50c.; 16 lbs. wool, 60c............. 30.00 10.oo July 6. "Cash,; Aug. 10, Cash,................... Balance due................... $82.73 BROOKLIN, July 15th, 1866. MRR. R. R HLLS. To J. WrLLMUs, Dr. 1866. Jan. 10, To 10lbs.M.sugar,15c.; 161bs.W.sugar,20c.; 121bs.C.sugar,18c. " 30. " 15 lbs. raisins, 16c.; 13 lbs. raisins,15c.; 10 lbs. raisins,18c. Feb. 12. " 9 lbs. cur'nts, 13c.; 12 lbs. cur'nts,14c.; 6 lbs. cur'nts,20c. Mar. 30. " 60 lbs. salt, 2c.; 2 lbs. wash. soda, 23c.; 1 lb. bak. soda,25c. Apr. 5, " 6 lbs. D. apples,12c.; 10 lbs. bisc'ts, 17c.; 5 lbs. bisc'ts,21c. " 25. " 3 cwt. flour. 4.2o0;2 cwt. C. meal, 2.3 o; 3 lbs. butter, 25c. May 1. " 16 lbs. pork, 20c.; 19 lbs. cheese, 10c.; 14 lbs. sugar,15c. Junel5. " 5 lbs. tea, 1.3 8; 9 gals. molasses, 40c.; 6 doz. eggs, 12c. July 12, " 5 lbs.sugar,16c.; 9~ lbs.raisins, 16c.; 10 lbs.cur'nts,12ic. " 29, " 14 lbs. bacon, 12c.; 5 lbs. cheese, 16c.; 4 lbs. butter,25c. "31, " 4 lbs. tea, 1.60; 2 lbs. tea, 1.3; 6 lbs. coffee, 35c....... " " " 40 lbs. salt, lic.; 3 lbs. indigo, 90c.; 1~ lbs. blue, 30c. "" " 3 lbs. salt petre, 35c.; 4 doz. eggs,12ic.; 6 lbs. butter,15c. $83.1( leceived payment. J. WILLIAMS. ACCOUNTS AND INVOICES. 125 ALBANY, Dec. 1, 1866 MR. GEO. SIMPSON, To TAYLOR & GRANT, Dr. 1866. July 7, To 12 lbs. sugar, 15c.; 2 lbs. tea, 1.25 3 lbs. coffee, 35c... 12, " 2 lbs.tobacco,87~c.; 3 lbs.raisins,30c.; 12 lbs.currants, 15c. "24, " 3 lbs. gunpowder, 62kc.; 6 lbs. shot, 18c.; 2 lbs. glue, 25c. Aug. 4, " 12 lbs.'washing soda, 15c.; 4 lbs. baking soda, 25c...... 12, " 1 box mustard, 1.50; 2 lbs. filberts, 30c.; 2 lbs. alm'ds, 35c. Sept.21, " 8 lbs. sugar, 14c.; 1 lb. tea, 1.12; 3 lbs. chocolate, 40c. Oct. 12,." 4lbs. figs, 15c.; 2 lbs. orange peel, 30c.; spices 40c..... "20, " 2 lbs. but. blue, 18c.; 2 lbs. sulphur, 20c.; 3 lbs. soda, 35c. 18.00 Nov. 4, " 2 lbs. smok. tobacco, 90c.; 2 lbs. snuff, 20c.; 1 business suit, Contra. Cr. 8.00 5.00 Aug. 12, By Cash,; Sept 21, Cash,;.............. -Oct. 20, " 100 lbs. dried apples, 15c.; 60 lbs. peaches, 200.. Balance due.................... $7.91 DETROIT, bept. 30th, 1866. MR. S. SMITE, To RAY, HILL.& Co., Dr., 1866. Jan. 1, To 5 lbs. tea, 1.70; 15 lbs. sugar, 15c.; 1l Ibs. cinnamon, 2.50. "10, " 18 lbs. rice, lOc.; 16 lbs. salt, 4c.; 34 lbs. oat meal, 6c... " 13, " 12 lbs. raisins, 18c.; 3 lbs. tobacco, 58c.; lb. snuff, 34c.. Feb. 2, " 10 lbs. cur'nts, 17c.; 10 lbs. ginger, 41c.; 5 lbs. mustard, 42c. 8, " 6 lbs. sugar, 18c.; 13 lbs. rice, 8c.; 21 lbs. dr'd apples, 16c. 13, " 25 lbs. raisins, 18c.; I lb. B. soda, 30c.; I lb. nutmegs, 22c..Mar. 4, " 12 lbs. coffee, 36c.; 6 lbs.M. sugar, 15c.; 4 lbs.W. sugar, 20c. " 15,' 4 lbs. mustard,30c.; 3 lbs. tobacco,30c.; 12 lbs. ginger,27c. Aprl. 6, " 2 lbs. currants, 20c.; 14 lbs. rice, 8c.; 9 lbs. tur. seed, 45c.' 14, " l lbs. cin'mon, 70c.; 12 lbs. sago, 31c.; 14 lbs. sugar, 21c. May 10, " 16 lbs. salt, 3c.; 2 lbs. indigo, 90c.; 61 lbs. corn starch. 14c. -June 12, " 40 lbs. flour, 4c.; 30 lbs. corn meal, 3c.; 25 lbs. coffee, 38c. $88.46 9 ~1W~26 ~ ARITHMETIC. 31. CHICAGO, Jan. 4th, 1866. MR. ELIAS G. CONKLIN, Bought of J. BUNTIN & Co., 12 reams of foolscap paper......................... $3.25 15 dozen school books.............................@ 4.50 23 slates................................... 1.30 7 " photograph albums.........................( 15.00 3 " Bullion's grammar..................... 7.00 8 " fifth reader.................................. 3.50 5 gallons of black ink.............................. 1.10 4 dozen American Commercial Arithmetic....... 18.00 $367.90 Received payment, J. BUNTIN & Co. 32 TORONTO, Jan. 12tn, 1866. MR. JAMES H. BURRITT, Bought of MORRISON, TAYLOR & Co., 15 cwt. of cheese................................ $9.00 4 cwt. of flour....................................... 4.25 120 pounds of bacon.................................. 0.14 7 bushels of corn meal............................. 0.75 12 firkins of butter................................ @ 13.50 20 bushels of dried apples.........................@ 2.25 13 " " peaches................... 4.00 11 cwt. of buck-wheat flour....................... 5.50 15 cwt. maple sugar.................................. 8.00 25 bags of common salt............................. 1.15 57 barrels of mess pork............................. 13.00 68 " beef................................... 9.75 13 bushels of clover seed........................... 7.50 $2143.80 Received payment by note at 30 days. FoR MORRISON, TAYLOR & Co., A. C. HENRY. BILLS OF PARCELS. 127 33 HAMILTON, January 2nd, 1866. MR. M. MCCULLOCH, To JOSEPH LIGHT, Stationer, Dr. For 500 French envelopes...................... $3.00 per thousand 12 doz. British American copy books...@ 1.15 6 " B. B. lead pencils.............50 " 5 gross mourning envelopes........... @ 1.05 " 2 reams mourning note paper.......... 3.15 4 " tinted note paper................@ 3.15 " 2j " Foreign note paper............ 1.40 " 3 i" letter paper............ 3.00 " 1 doz. First Books.........................15 " 5 boxes Gillott's No. 303 pens...........90 " 5 doz. Third Books................. 1.62k " 10 quires blank books, half bound......@.35 " 2 packs visiting cards.....................37 $71.98. NoTE.-Bills should not be signed until settled. 34. BROOKVILLE, Jan. 5th, 1866. N. D. GALBREAITH, To R. FITZSIMMONS & CO., Dr. For 24 lbs. Mackerel........................ 05c. " 3 gallons Molasses..............................@ 45 13 lbs. Young Hyson Tea....................... 87" 13 lbs. brown Sugar............................... 11 " 15 bushels of Potatoes........................... 45 $22.2a CR. For 10 lbs. Butter..................................... 17c. " 5 doz. Eggs.................................... @ 12i 3 gallons Maple Molasses..............@ 95 " Note at 20 days, to balance................ 17.05 $22.23 R. FITZSIMMONS & Co. NOTE.-Such a Bill as this would be termed a Barter Bill. 128 ARITHMETIC. 35 KINGSTON, Jan. 2nd, 1866. JAMES THOMPSON, ESQ., To A. JARDINE & Co., Dr. For 3 doz. Buttons................ $0.12 " 5~ yards of black Broadcloth.......... 5.50 " 20 yards Sheeting............15 " 1. chest Y. II. Tea, 83 lbs........ @.95 "18 yards French Print'.................20 2 skeins of Silk Thread.....09 5 yards black Silk Velvet..................... 3.50 " 20 lbs. Loaf Sugar................................18 " 2 gallons Molasses................................40 " 1 bag of common Salt........................ 1.15 25 lbs. Rice..................................... 09 " 3 sacks.Coffee, 70 lbs.-each......................12 CR. $166.74 By Cash............................................................. 50.00 Balance due.................................... $116.74 36 ALGONQUIN, Jan. 15th, 1865, W. FLEMING & CO., Bought of J. & A. WRIGHT, 1500 lbs. Canadian Cheese........................... $.09 300 bushels Fall Wheat.......................... 1.25 9 brls Pot Ash, net 7056 lbs................... 5.75 per cwt. 150 bushels Spring Wheat................... 1.15 200' Potatoes.................................45 600' Oats.....................................(.37~ 150 " Pease...............................65 50 " Indian Corn............................50 60 " Apples...................................60 3 kegs Butter, 110 lbs. each.....................18 50 bushels Rye..........70 50 bushels Eye.....................................70 40 " Barley.................................80 $1688.12 Received payment, J. & A. WRIGHT. PERCENTAGE. 129 PERCENTAGE. 18.-PERCENTAOE is an allowance, or reduction, or estimate of a certain portion of each 100 of the units that enter into any given calculation. The term is a contraction of the Latin expression for one hundred, and means literally by the hundred. In calculating dollars and cents, 6 per cent. means 6 dollars for every 100 dollars, or 6 cents for every $1, or 100 cents. If we are estimating the rate of yearly increase of the population of a rising village, and find that at the end of a certain year it was 100, and at the end of the next it was 106, we say it has increased. 6 per cent. i. c., 6 persons have been added to the 100. So, also, if a large city has a population of 100,000 at the end of a certain year, and it is found that it has 106,000 at the end of the following year, we say it has increased 6 per cent., which means that if we count the population by hundreds we shall find that for every 100 at the end of the one year, there are 106 at the end of the next; because one hundred thousands is the same as one thousand hundreds, and we have supposed the increase in every 100 to be 6, the total increase will be one thousand sixes or 6,000, giving a total population of 106,000 as above, or an increase at the rate of 6 per cent. A decrease would be estimated in the same manner. Thus, a falling off in the population of 6 persons in the hundred would be denoted by 100-6=94, as an increase of 6 in the hundred would be denoted by 100+6=106. So, also, in our first example, a deduction of $6 in $100 would be $100-6=$94, and a gain would be $100+$6=$106. The portion of 100 so allowed or estimated, is called the rate per cent., as in the examples given, 6 denotes the rate per cent., or the allowance or estimate on every 100. Should the sum on which the estimate is made not reach 100, we can, nevertheless, estimate what is to be allowed on it at the same rate. Thus, if 6 is to be allowed for 100, then 3 must be allowed for 50, and 1~ for 25, &c. The number on which the percentage is estimated is called the basis. Thus, in the example given regarding the population of a city, 100,000 is the basis. When the basis and percentage are combined into one, the result is called the amount. If the rate per cent. be an increase or gain, it is to be added to the basis to get the amount, and if it is a decrease, or loss, it is to be subtracted from the basis to get the amount. This latter result is sometimes called the remainder. 130 ARITHMETIC. From what has been said, it is plain that percentage is nothing else than taking 100 as a standard unit of measure-(See Art. 1)and making the rate a fraction of that unit, so that 6 per cent. is f-==(Art. 15, V.).06. We may obtain the same result by the rule of proportion. Thus, in our illustrative example of an increase of 6 persons for every 100 on a population of 100,000, the analogy will be 100 persons: 100,000 persons: 6 (the increase on 100): 6,000, the increase on 100,000. It is manifest that the same result will be obtained whether we multiply the third by the second, and divide by the first, or whether we divide the third by the first, and multiply the result by the second; or, which is the same thing, multiply the second by the result. Now, we already found that 6 —100= —=.06, the same as before. So also, 7 per cent. of any loss is seven one-hundredths of it, i. e., T-==.07. It should be carefully observed that such decimals represent, not the rateper cent., but the rate per unit. Though this is easily comprehended, yet we know by experience that learners are constantly liable to commit errors by neglecting to place the decimal point correctly. We would therefore direct particular attention to the above caution, whiac', with the rule already laid down, under the head of decimal fractions, should be sufficient to guide any one who takes even moderate pains. EXERCISES ON FINDING THE RATE PER UNIT. At i per cent., what is the rate per unit? Ans..00|. At i per cent., what is the rate per unit? Ans..00. At 1 per cent., what is the rate per unit? Ans..01. At 2 per cent., what is the rate per unit? Ans..02. At 4 per cent., what is the rate per unit? Ans..04. At 7j per cent., what is the rate per unit? Ans..07J. At 10 per cent., what is the rate per unit? Ans..10. At 121 per cent., what is the rate per unit? Ans..12~. At 17 per cent., what is the rate per unit? Ans..17. At 25 per cent., what is the rate per unit? Ans..25. At 33~ per cent., what is the rate per unit? Ans..33k. At 66- per cent., what is the rate per unit? Ans..66%. At 75 per cent., what is the rate per unit? Ans..75. At 100 per cent., what is the rate per unit? Ans. 1.00. At 112- per cent., what is the rate per unit? Ans. 1.121. At 150 per cent., what is the rate per unit? Ans. 1.50. At 200 per cent., what is the rate per unit? Ans. 2.00. PERCENTAGE. 131 I. To find the percentage on any given quantity at a given rate: On the principles of proportion, we have as 100: given quantity:: rate: percentage, and as the third term, divided by the first, gives the rate per unit, we have the simple RULE: Multiply the given quantity by the rate per unit, and the product will be the percentage. EXAMPLES. To find how much 6 per cent. is on 720 bushels of wheat, we,have 6' —100-.06; the rate per unit, and 720X.06=435 bushels, the percentage. To find 8 per cent. of $7963-75, in like manner, we have.08, the rate per unit, and $7963.75X.08 gives $637.10, the percentage. Instead of per cent the mark (~/o) is now commonly used. EXERCISES ON THE RULE. 1. What does 6 per cent. of 450 tons of hay amount to? Ans. 27. 2. What is 10 per cent. of $879.62j? Ans. $87.96. 3. If 12 per cent. of an army of 47,800 men be lost in killed and wounded; how many remain? Ans. 42,064. 4. What is 5 per cent. of 187 bushels of potatoes? Ans. 9.35. 5. What is 2j per cent. of a note for $870? Ans. 21.75. 6. Find 121 per cent. of 97 hogsheads? Ans. 12.12k. II. To find what rate per cent. one number is of another given number:-Let us take as an example, to find what per cent. 24 is of 96. Here the basis is 96, and we take 100 as a standard basis, and these are magnitudes of the same kind, and 24 is a certain rate on 96, and we wish to, find what rate it is on 100, and by the rule of proportion, we have the statement 96: 100:: 24 F. P.2==-a ~=25. Therefore 24 is 25 per cent. of 96. From this we can deduce the simple RULE. Annex two ciphers to the given percentage, and divide that by the basis, the quotient will be the rate per cent. 7. What per cent. of 150 is 15? Ans. 10. 8. What per cent. of 240 is 36? Ans. 15. 132 ARITHMETIO. 9. What per cent. of 18 is 2? Ans. 1110. What per cent. of 72 is 48? Ans. 66g. 11. What per cent. of 576 is 18? Ans. 38. 12. What per cent. is 12 of 480?, Ans. 2~. 13. Bought a block of buildings in King street for $1719, and sold it at a gain of 18 per cent.; what was the gain? Ans. $309.42. 14. Vested $325 in an oil well speculation, and lost 8 per cent.; what was the loss? Ans. $26.00. 15. In 1841 the population of Cleveland was about 15,000, it is now about 50,000; what is the rate of increase? Ans. 233.16. An estate worth $4,500 was sold; A bought 30 per ebnt. of it; B, 25 per cent.; 0, 20 per cent.; and D purchased the remainder; what per cent. of the whole was D's share? Ans. 25. 17. If a man walk at the rate of 4 miles an hour; what per cent. is that of a journey of 32 miles? Ans. 12~. 18. What is the percentage on $1370 at 21 per cent.? Ans. 37.67~. III. Given, a number, and the rate per cent. which it is of another number, to find that other number,.400 is 40 per cent. of a certain number, to find that number. As 40: 100:: 400: F. P.40 —X)9I —11,000. Hence we derive the RULE. Annex two ciphers to the given number; and divide by the rate per cent. EXERCISES. 1. A bankrupt can pay $2600, Gich is 80 per cent of his debts; how much does he owe? Ans. $3250. 2. A clerk pays $8 a month for rent, which is 16 per cent. of his salary; what is his yearly salary? Ans. $600. 3. In a manufacturing district in England, 40,000 persons died of cholera in 1832, this was 25 per cent. of the population; what was the population? Ans. 160,000.. 4. Bought a certain number of bags of flour, and sold 124 of them, which is 12~ per cent. of the whole. Required, the number' of bags purchased. Ans. 992. 5. In a shipwreck 480 tons are lost, and' this amount is 15 per cent. of the whole cargo. Find the cargo. Ans. 3200 tons. PERCENTAGE. 133 6.' A firm lost $1770 by the failure of another firm; the loss was 30 per cent. of their capital; what was their capital? Ans. $5900. IV. To find the basis when the amount and rate are given:Suppose a man buys a piece of land for a certain sum, and by selling it for $300, gains 25 per cent.; what did he pay for it at first?Here it is plain that for every dollar of the cost, 25 cents are gained by the sale, i. e., 125 cents for every 100, which gives us the analogy, 125: 100:: 300: F. P.; or, dividing the two terms by 100, 1.25: 1.00:: 300: F. P., which by the rules for the multiplication and division of decimals, gives _sl-=OO$240, the original cost. Again, suppose the farm had been sold at a loss of 25 per cent. This being a loss, we subtract 25 from 100, and say, as 75: 100:: 300: P. P.=-00oaQ-Q$400, the prime cost in this case. Hence we derive the R ULE. Divide the given amount by one increased or diminished by the given rate per unit, according as the question implies increase or decrease, gain or loss. EXERCISES. 1. Given the amount $198, and the rate of increase 20 per cent. to find the number yielding that percentage. Ans. $165. 2. A field yields 840 bushels of wheat, which is 250 per cent. on the seed; how many bushels of seed were sown? Ans. 240 bushels. 3. At 5 per cent. gain; what is the basis if the amount be $126? Ans. $120. 4. At 10 per cent. loss; what is the basis, the amount being $328.5? Ans. $365. 5. A ship is sold for $12045, which is a gain of i- per cent. on the sum originally paid for it; for how much was it bought at fErst? Ans. $12000. 6. A gambler lost 10 per cent. of his money by a venture, and had $279 left; how much had he at first, and how much did he lose? Ans. He lost $31, and had $310 at first. 7. A grocer bought a lot of flour, and having lost 20 per cent. of the whole, had 160 bags remaining; how many bags did he buy? Ans. 200. 8. A merchant lost 12 per cent. of his capital by a bankruptcy, and had still $2200 left; what was his whole capital? Ans. $250Q. 134 ARITHMETIC. 9. Sold a sheep for $5, and gained 25 per cent.; what did I pay for it? Ans. $4. 10. Lost $12000 on an investment, which was 30 per cent. of the whole; what was the investment? Ans. $40000. INTEREST. From a transition common in language, the word interest has been inappropriately applied to the sum paid for the use of money, but its original and true meaning is simply the use of money. To illustrate this, we will suppose that A borrows of B $100 for one year, and at the end of the year, when A wishes to settle the account, he gives B $107. Were we to ask the question of almost any person except an accountant, whether A or B received the interest, we should undoubtedly receive for an answer that B received it. But such is not the case. A having had the use of that money for one year, paid B $7 for that use or interest; hence A received the interterest or use of that money, and B received $7 in cash for the same. It is only by considering this subject in its true light that accountants are able to determine upon the proper debits and credits that arise from a transaction where interest is involved. If an individual borrows money, he receives the use of that money, and when he pays for that use or interest, he places the sum so paid to that side of his "interest account" which represents interest received, and if he lends money, he has parted with the use of that money, and when he receives value for that use or interest, he places tl.e sum so received to that side of his " interest account" which represents interest delivered. We think that this explanation is sufficiently clear to illustrate the difference between interest and the value received or paid for it. It will also be noticed that we have given many of the exercises in the usual form, e. g., we say what is the interest on $100 for one year, instead.of saying what must be paidfor the interest of $100 for one year, but we have done this more in accordance with custom than from any intention to deviate from the true meaning of the word interest. Interest is reckoned on a scale of so many units on every $100 for one year, and hence it is called so much per cent. per annum, from the Latin per centum, by the hundred, and per annum, by the year. Thus, $6 a year for every $100, is called six per cent. per INTEREST. 13i annum. The term is also extended to designate the return accruing from any investment, such as shares in a joint stock company. To show the object and use of such transactions, we may suppose a case or two. A person'feels himself cramped or embarrassed in his circumr stances and operations, and he applies to some friendly party thai lends him $100 for a year, on the condition that at the expiration ol the year he is to receive $106, that is, the $100 lent, and $6 more as a return for the use of the $100; or, if the borrower gets $600; he pays at the and of the stipulated time not only the $600, but alsc $36 ($6 for each $100) in return for the use of the $600. By this means the borrower gets clear of his difficulty, and maintains his credit at a small sacrifice. -The sum on which interest is paid is called the principal. The sum paid for the use of money is called the interest. The sum paid on each $100 is called the rate. The sum of the principal and interest is called the amount. When interest is charged on the principal only, it is called simple interest. When interest is charged on the amount, it is called compound interest. When a certain rate per cent. is established by law, it is called legal interest. When a higher rate per cent. is charged than is allowed by law. it is called usury. The legal rate per cent. differs in different States and in different countries, so also does the mode of calculation d:ffer. In some States it is considered legal, to reckon the month as consisting of 30 days, in the calculating of interest on any sum for a short period, in others it is considered illegal. We have given the different modes of calculation in order to make the work applicable to all the States. For the legal rate per cent. of each State, see " Laws of the States," at the end of this work. SIMPLE INTEREST. As simple interest, when calculated for one year, differs in no way from a percentage on a given sum, we have only four things to consider, viz., the principal, the rate (100 being the basis), the inter 136 ARITHMETIC. est, and the time, any three of which being known, the fourth can be found. The finding of the interest includes by far the greatest number of cases. We shall first show the general principle, and from it deduce an easy practical rule. Let it be required to find the interest on $468 for one year, at 6 per cent. As 100 is taken as the basis principal in relation to which all calculations are made, it is plain that 100 will have the same ratio to any given principal that the rate, which is the interest on 100, has to the interest on the given principal. Hence, in the question proposed, we have as $100: $468:: $6: interest=$468XTl —= $468X.06=$28.08. Now.06 is the rate per unit, and from this we can deduce rules for all cases. CASE I. To find the interest of any sum of money for one year, at any given rate per cent. RULE. Multiply the principal by the rate per unit. EXERCISES. 1. What is the interest on $15, for 1 year, at 3 per cent.? Ans. $0.45. 2. What is the interest on $35, for 1 year, at 5 per cent.? Ans. $1.75. 3. What is the interest on $100, for 1 year, at 7 per cent.? Ans. $7.00. 4. What is the interest on $2.25, fer 1 year, at 8 per cent.? Ans. $0.18. 5. What is the interest on $6.40, for 1 year, at 8i per cent.? Ans. $0.54. 6. What is the interest on $250, for 1 year, at 9i per cent.? Ans. $23.75. 7. What is the interest on $760.40, for 1 year, at 71 per cent.? Ans. $57.03. 8. What is the interest on $964.50, for 1 year, at 6} per cent.? Ans. $62.69. 9. What is the interest on $568.75, for 1 year, at 71 per cent.? Ans. $41.23. INTEREST. 137 CASE II. To find the interest of any sum of money, for any number of years, at a given rate per cent. RULE. Find the interest for one year, and multiply by the number of years. EXERCISES. 10. What is the interest of $4.60, for 3 years, at 6 per cent.? Ans. $0.83. 11. What is the interest of $570, for 5 years, at 7. per cent.? Ans. $213.75. 12. What is the interest of $460.50, for 3 years, at 61 per cent.? Ans. $86.34. 13. What is the interest of $17.40, for 3 years, at 8j per cent.? Ans. $4.35. 14. What is the interest of $321.05, for 8 years, at 5 per cent.? Ans. $147.68. 15. What is the interest of $1650.45, for 2 years, at 9 per cent.? Ans. $297.08. 16. What is the interest of $964.75, for 4 years, at 10 per cent.? Ans. $385.90. 17. What is the interest of $1674.50, for 3 years, at 10J per cent.? Ans. $527.47. 18. What is the interest of $640.80, for 5 years, at 41 per cent.? Ans. $152.19. 19. What is the interest of $965.50, for 7 years, at 5j per cent.? Ans. $371.72. 20. What is the interest of $2460.20, for 4 years, at 7 per cent.? Ans. $688.86. CASE III. To find. the interest on any sum of money for any number of months, at a given rate per cent. RULE. Find the interest for one year, and take aliguot parts for the months; or, Fintd the interest for one year, cdivide by 12, and multiply by the number of months. 138 ARITHMETIC. EXERCISES. 21. What is the interest on $684.20, for 4 months, at 6 per cent.? Ans. $13.68. 22. What is the interest on $760.50, for 5 months, at 7 per cent.? Ans. $22.18. 23. What is the interest on $899.99, for 2 months, at 8 per cent.? Ans. $12.00. 24. What is the interest on $964.50, for 4 months, at 9 per cent.? Ans. $28.94. 25. What is the interest on $1500, for 7 months, at 10 per cent.? Ans. $87.50. 26. What is the interest on $1560, for 11 months, at 7- per cent.? Ans. $107.25. 27. What is the interest on $1575.54, for 8 months, at 6] per cent.? Ans. $65.65. 28. What is the interest on $1728.28, for 9 months, at 8~ per cent.? Ans. $110.18. 29. What is the interest on $268.25, for 13 months, at 7 per cent.? Ans. $20.34. 30. What is the interest on $1569.45, for 1 year, 3 months, at 8 per cent.? Ans. $156.95. 31. What is the interest on $642.99, for 1 year, 5 months, at 10 per cent.? Ans. $91.09. 32. What is the interest on $560.45, for 1 year, 6 months, at 9~ per cent.? Ans. $79.86. 33. What is the interest on $48.50, for 3 years, 9 months, at 109 per cent.? Ans. $19.10: 34. What is the interest on $560.80, for 2 years, 8 months, at 11 per cent.? I Ans. $175.72. 35. What is the interest on $2360.40, for $onths, at 12 per cent.? Ans. $448.48. CASE IV. To find the interest on any sum of money, for any number of months and days, at a given rate per cent. RULE. Find the interestfor the months, and take aliquot parts for the days, reckoning the month as consisting of 30 days. EXAMPLE. 36. What is the interest on $875.50, for 8 months, 18 days, at 11 per cent.? SIMPLE INTERET. 139 SOLUTION. Principal.......................................................... $875.50 Rate per unit......................................................11 Interest for 1 year............................................... 96.3050 Interest for 6 months; or, i of interest for 1 year......... 48.1525 Interest for 2 months; or, ~ of interest for 6 months...... 16.0508 Interest for 15 days; or, i of interest for 2 months...... 4.0127 Interest for 3 days; or, - of interest for 15 days.........8025 Interest for 8 months, 18 days............................... $69.0185 We find the interest for 1 year to be $96.305, and as 6 months are the i of 1 year, the interest for 6 months will be the i of the interest for 1 year; likewise the interest for 2 months will be the i of the interest for 6 months, and as 15 days are the i of 2 months or 60 days, the interest-for 15 days will be the i of the interest for 2 months, and likewise the interest for 3 days, will be the - of the interest for 15 days. Adding the interest for the months and days together, we obtain $69.02, the sum to be paid for the use of $875.50, for 8 months, 18 days, at 11 per cent. EXERCISES. 37. What is the interest on $468.75, for 4 months, 15 days, at 7 per cent.? Ans. $12.30. 38. What is the interest on $1654.40, for 3 months, 8 days, at 5 per cent.? Ans. $22.52. 39. What is the interest on $345.65, for 11 months, 25 days, at 6 per cent.? Ans. $20.45. 40. What is the interest on $74.85, for 5 months, 22 days, at 9 per cent.? Ans. $3.22. 41. What is the interest on $673.75, for 8 months, 19 days, at 7~ per cent.? Ans. $36.35. 42. What is the interest on $57.45, for 1 year, 2 months, 12 days, at 6 per cent.? Ans. $4.14. 43. What is the interest on $2647, for 1 year, 5 months, 18 days, at 6i per cent.? Ans. $242.64. 44. What is the interest on $268.40, for 2 years, 1 month, 1 day, at 8 per cent.? Ans. $44.79. 45. What is the interest on $2345.50, for 3 years, 7 months, 20 days, at 10 per cent.? Ans. $853.50. 140 ARTHMETIC. 46. What is the interest on $4268.45, for 4 years, 11 months, 11 days, at 11j per cent.? Ans. $2481.24. 47. What is the interest of $642.20, for 2 years, 7 months, 24 days, at 12 per cent.? 48. What is the interest of $64.50, for 2 years, 11 months, 2 days, at 7 per cent.? Ans. $13.19. 49. What is the amount of $746.25, for 1 year, 10 months, 12 days, at 5 per cent.? 50. What is the interest of $680, for 4 years, 1 month, 15 days, at 6 per cent.? Ans. $168.30. CASE V. To find the interest on any sum of money, for any number of days, at a given rate per cent.* RULE. Find the interest for one year, and say, as one year (365 days,) is to the given number of days, so is the interest for one year to the interest required; or, Having found the interest for one year, multiply it by the given number of days, and divide by 365. E X-E R I S E S. 51. What is the interest on $464, for 15 days, at 6 per cent.? Ans. $1.14. 52. What is the interest on $364, for 12 days, at 7 per cent.? Ans. 84 cents. 53. What is the interest on $56.82, for 14 days, at 8 per cent.? Ans. 17 cents. * To find how many years elapse between any two dates, we have only to subtract the earlier from the later date. Thus, the number of years from 1814 to 1865 is 51 years. To find months, we must reckon from the given date in the first named month, to the same date in each successive month. Thus, five months from the 10th of March brings us on to the 10th of August. To find days, we require to count how many days each month contains, for to consider every month as consisting of 30 days, in the calculation of interest, is not strictly correct, although for portions of a single month it causes no serious error. Thus, the correct time from March 2nd to June 14th, would be 104 days, viz., 29 for March, 30 for April, 31 for May, and 14 for June. A very convenient plan for reckoning time between two given dates is to count the number of months and odd days that intervene. Thus, from June 14th to November 20th, would be 5 months and 6 days. SIMPLE INTEREST. 141 54. What is the interest on $75.50, for 18 days, at 8i per cent.? Ans. 32 cents. 55. What is the interest on $125.25, for 20 days, at 5 per cent.? Ans. 34 cents. 56. What is the interest on $150.40, for 33 days, at 6 per cent.? Ans. 82 cents. 57. What is the interest on $56.48, for 45 days, at 6~ per cent.? Ans. 45 cents. 58. What is the interest on $75.75, for 65 days, at 7 per cent.? Ans. 94 cents. 59. What is the interest on $268.40, for 70 days, at 7i per cent.? Ans. $3.86. 60, What is the interest on $464.45, for 80 days, at 8 per cent.? Ans. $8.14. 61. What is the interest on $15.84, for 120 days, at 9 per cent.? Ans. 47 cents. 62. What is the interest on $240, for 135 days, at 9i per cent.? Ans. $8.43. 63. What is the interest on $2460, for 145 days, at 10 per cent.? Ans. $97.73. 64. What is the interest on $1568, for 170 days, at 11 per cent.? Ans. $80.33. 65. What is the interest on $2688, for 235 days, at 114 per'Pant.? Ans. $203.35. 66. What is the amount of $364.80, for 320 days, at 11 per 3ent.? Ans. $401.58. CASE VI. To find the interest on any sum of money, for any time, at 6 per cent. Since.06 would be the rate per unit, or the interest of $1 for 1 year, it follows that the interest for one month would be the -y. of.06, or 6 of a cent, equal to - cent or.005, and for 2 months it would equal i cent, or.005X2 —.01. Therefore, when interest is at the rate of 6 per cent., the interest of $1, for every 2 months, is one cent. Again, if the interest of $1, for one month, or 30 days, is i cent or.005, it follows that the interest for 6 days will be the 5 of.005 or.001. Therefore, when interest is at the rate of 6 per cent., the interest of $1 for every 6 days is one mill. Hence the 10 142 ARITHMETIC. RULE. Find the interest of $1 for the given time by reckoning 6 cents for every year, 1 cent for every 2 months, and 1 millfor every 6 days; then multiply the given principal by the number denoting that interest, and the product will be the interest required. NoTE.-This method can be adopted for any rate per cent. by first finding the interest at 6 per cent., then adding to, or subtracting from the interest so found, such a part or parts of it, as the given rate exceeds, or is less than 6 per cent. This method, although adopted by some, is not exactly correct as the year is considered as consisting of 360 days, instead of 365; so that the interest, obtained in this manner, is too large by ag or i, which for every $73 interest, is $1 too much, and must therefore be subtracted if the exact amount be required. EXAMPLE. 67. What is the interest of $24, for 4 mouths, 8 days, at 6 per cent.? SOLUTION. The interest of $1, for 4 months, is..............................02 The interest of $1, for 8 days, is.................................001Hence the interest of $1, for 4 months, 8 days, is...........021~ Now, if the interest of $1, for the given time, is.021k, the interest of $24 will be 24 times.021i, which is $.512. EXERCISES. 68. What is the interest on $171, for 24 days, at 6 per cent.? Ans. 68 cents. 69. What is the interest on $112, for 118 days, at 6 per cent.? Ans. $2.20. 70. What is the'interest on $11, for 112 days, at 6 per cent.? Ans. 21 cents. 71. What is the interest on 50 cents, for 360 days, at 6 per cent.? Ans. 3 cents. 72. What is the interest on $75.00, for 236 days, at 6 per cent.? Ans. $2.95. 73. What is the interest on $111.50, for 54 days, at 6 per cent.? Ans. $1.00. 74. What is the interest on $15.50, for 314 days, at 6 per cent.? Ans. 81 cents. SIMPLE INTEREST. 143 75. What is the interest on $174.25, for 42 days, at 6 per cent.? Ans. $1.22. 76. What is the interest on $10, for 1 month, 18 days, at 6 per cent. Ans.g cents. 77. What is the interest on $154, for 3 months, at 6 per cent.? Ans. $2.31. 78. What is the interest on $172, for 2 months, 15 days, at 6 per cent.? Ans. $2.15. 79. What is the interest on $25, for 4 months, at 6 per cent.? Ans. 50 cents. 80. What is the interest on $36, for 1 year, 3 months, 11 days, at 7 per cent.? Ans. $3.23. 81. What is the interest on $500, for 160 days, at 6 per cent.? Ans. $13.33. 82. What is the interest on $92.30, for 78 days, at 5 per cent.? Ans. $1.00. 83. What is the interest on $125, for 3 years, 5 months, 15 days,. at 10 per cent. Ans. $43.23.. 84. What is the amount of $200, for 9 months, 27 days, at 6 per cent.? Ans. $209.90.. 85. What is the interest on $125.75, for 5 months, 17 days, at: 7 per cent.? Ans. $4.08. 86. What is the interest on $84.50, for 1 month, 20 days, at 5 per cent.? Ans. 59 cents. 87. What is the amount of $45, for 1 year, 1 month, 1 day, at 8 per cent.? Ans. $48.91. 88. What is the interest on $175, for 7 months, 6 days, at 5per cent.? Ans. $5.78. 89. What is the interest on $225, for 3 months, 3 days, at 9 per cent.? Ans. $5.23. 90. What is the interest on $212.60, for 9 months, 8 days, at 8S per cent.? Ans. $13.95. CASE VII. To find the interest on any sum of money, in pounds, shillings, and pence, for any time, at a given rate per cent. RULE. Multiply theprincipal by the rate per cent., and divide by 100. 144 ARITHMETIC. EXAMPLE. 91. What is the interest of ~47 15s. 9d., for 1 year, 9 months, 15 days, at 6 per cent.? SOLUTION. ~ s. ~. D. Interest for 1 year............................ 2 17 4 47 15 9 Interest for 6 mos., or i of int. for 1 year, 1 88 6 Interest for 3 mos., or ~ of int. for 6 mos., 0 14 4 Interest for 15 days, or.ofint. for 3 mos., 0 2 41 2)86 14 6 20 Interest for 1 year, 9 months, 15 days....~5 2 8 --- 17)34 12 4)14 92. What is the interest of ~25, for 1 year, 9 months, at 5 per cent.? Ans. ~2 3s. 9d. 93. What is the interest of ~75 12s. 6d., for 7 months, 12 days, at 8 per cent.? Ans. ~3 14s. 7jd. 94. What is the amount of ~64 10s. 3d., for 3 months, 3 days, at 7 per cent.? Ans. ~65 13s. 7d. 95. What is the interest of ~35 4s. 8d., for 6 months, at 10 per sent.? Ans. ~1 15s. 2jd. 96. What is the amount of ~18 12s., for 10 months and 3 days, at 6 per cent.? Ans. ~19 10s. 9jd. CASE VIII. To find the PRINCIPAL, the interest, the time, and the rate per cent. being given. EXAMPLE. 97. What principal will produce $4.50 interest in 1 year, 3 months, at 6 per cent.? SOLUTION. If a principal of $1 is put on interest for 1 year, 3 months, at 6 per cent., it will produce.075 interest. Now, if in this example,.075 be the interest on $1, the number of dollars required to produce $4.50, will be represented by the number of times that.075 is contained in $4.50, which is 60 times. Therefore, $60 will produce $4.50 interest in 1 year, 3 months, at 6 per cent. Hence the SIMPLE INTEREST. 145 RULE. Divide the given interest by the interest of $1 for the given time, at the given rate per cent. EXERCISES. 98. What principal will produce 77 cents interest in 3 months, 9 days, at 7 per cent.? Ans. $40. 99. What principal will produce $10.71 interest in 8 months, 12 days. at 71 per cent.? Ans. $204. 100. What principal will produce $31.50 interest in 4 years, at 31- per cent.? Ans. $225. 101. What sum of money will produce $79.30 interest in 2 years, 6 months, 15 days, at 6j per cent.? Ans. $480. 102. What sum of money is sufficient to produce $290 interest in 2 years and 6 months, at 7j per cent.? Ans. $1600. CASE IX. To find the RATE PER CENT., the principa the interest, and the time being given. EXAMPLE. 103: If $3 be the interest of $60 for 1 year, what is the rate per cent.? SOLUTION. If the interest of $60 for 1 year, at 1 per cent, is.60, the required rate per cent. will be represented by the number of times that.60 is contained in 3.00, which is 5 times. Therefore, if $3 is the interest of $60 for 1 year, the rate per cent. is 5. Hence the RULE. Divide the given interest by the interest of the given principal at 1 per cent. for the given time. EXERCISES. 104. If the interest of $40, for 2 years, 9 months, 12 days, is $13.36; what is the rate per cent.? Ans. 12. 105. If I Arrow $75 for 2 months, axnd pay $1 interest; what is the rate per cent.? Ans. 8. 146 ARITHMETIC. 106. If I give $2.25 for the use of $30 for 9 months; what rate per cent. am I paying? Ans. 10. 107. At what rate per cent. will $150 amount to $165.75, in 1 year, 4 months, 24 days? Ans. 71. 108. At what rate per cent. must $1, or any sum of money, be on interest to double itself in 12 years? Ans. Ans. 8-. 109. At what rate per cent. must $425 be lent'to gain $11.73 in 3 months, 18 days? Ans. 91. 110. At what rate per cent. will any sum of money amount to three times itself in 25 years? Anis. 8. 111. If I give $14 for the interest of $125 for 1 year, 7 months, 6 days; what rate per cent am I paying? Ans. 7. CASE X. To find the TIME, the principal, the interest, and the rate per cent. being given. EXAMPLE. 112. How long must $75 be at interest, at 8 per cent., to gain $12? SOLUT ION. The interest for $75, for 1 year, at 8 per cent., is $6. Now, if $75 require to be on interest for 1 year to produce $6, it is evident that the number of years required to produce $12 interest, will be represente.l by the number of times that 6 is contained in 12, which is 2. Therefore, $75 will have to be at interest for 2 years to gain $12. Hence the RU LE. Divide the given interest by the interest of the principal for one year, at the given rate per cent. EXERCISES. 113. In what time will $12 produce $2.88 interest, at 8 per cent? Ans. 3 years. 114. In what time will $25 produce 50 cents interest, at 6 per cent.? Ans. 4 months. 115. In what time will $40 produce 75-cents interest, at 61 per cent.? Ans. 3 months, 18 days. SIMPLE INTEREST. 147 116. In what time will any eum of money double itself, at 6 per cent.? Ans. 16 years, 8 months. 117. In what time will any sum of money quadruple itself, at 9 per cent.? Ans. 33 yars; 4 months. 118. In what time will $125 amount to $I.75, at 8 per cent.? Ans. 1 year, 4 months, 15 days. 119. Borrowed, January 1, 1865, $60, at 6 per cent, to be paid as soon as the interest amounted to one-half the principal. When is it due? Ans. May 1, 1873. 120. A merchant borrowed a certain sum of nmoney on January 2, 1856, at 9 per cent., agreeing to settle the account when the interest equalled the principal. When should he pay the same? Ans. Feb. 1I 1867. MERCHANTS' TABLE For showing. in what time any sum of money will double itself, at any rate per cent., from one to twenty, simple interest. Per cent. Years. Per cent. Years. Per cent. Years. Per cent. Years. 1 100 1 6 164 11 9 1 16 6f 2 50 7 14~ 12 71 17 5 - 3 33j 8 12~ 13 7 9 18 58 4 25 9 11- 14 7 19 55 5 20 10 10 15 6 20 5 MIXED EXERCISES. 121. What is the interest on $64.25 for 3 years, at 7 per cent.? Ans. $13.49. 122. What is the interest on $125.40 for 6 months, at 6 per oent.J. Ans. 3.76. 123. What is the amount of $369.29 for 2 years, 3 months, 1 day, at 9 per cent.? Ans. $444.16. 124. What must be paid for the use of 75 cents for 6 years, 9 aonths, 3 days, at 10 per cent.?. Ans. 51 cents. 125. What will $54 amount to in 254 days, at 10 per cent.?* Ans. $57.81. * This and the following exercises (marked with a *) are to be worked by Case VI. 148 ARITHMETIC. 126. What must be paid for the interest of $45 for 72 days, at 9 per cent.?* Ans. 81 cents. 127. What is the interest of $240 from January 1, 1866, to June 4, 1866, at 7 per cent.? Ans. $7.14. 128. What will $140.40 amount to from August 29, 1865, to November 29, 1866, at 61 per cent.? Ans. $151.83. 129. What principal will give $4.40 interest in 1 year, 4 months, 15 days, at 8 per cent.? Ans. $40. 130. In what time will $40 amount to $44.40, at 8 per cent.? Ans. 1 yr., 4 mos., 15 days. 131. At what rate per cent. will $40 produce in 1 yr., 4 mos., 15 days, $4.40 interest? Ans. S. 132. What must be paid for the interest of $145.50 for 240 Ifays, at 9~ per cent.?* Ans. $9.22. 133. What will $160 amount to in 175 days, at 6 per cent.?* Ans. $164.67T 134. At what rate per cent. must asy sum of money be ox interest to quadruple itself in 33 years and 4 months? Ans. 9. 135. In what time will ans-sum of money -drbte itself, at 10 per cent.? Ans. 10 years. CASE XI. To find the interest on bonds, notes, or other documents drawing 7,3 per cent. interest. Since.07 3 or,.073 would be the rate per unit, or the interest of $1 for L year or 365 days, it follows that the interest for 1 day would be the 3l' part of.073 which is.0092, equal to two tenths of a mill, hence the RULE. Multiply the principal by the number of days, and the product by two tenths of a mill the result will be the answer in mills. EXAMPLE. What must be paid for the use of $75 for 36 days at 7-T3 per sent.? SOLUTION. The interest on $75 for 36 days would be the same as the inter — -st on $75X36=$2700 for 1 day, and at - of a mill per day would be $2700X.0002=54 cents. 2. What would be the interest oQ^$118.30 for 42 days at 7 6 per cent. Ans. 99cts.. COMMERCAL PAPER. 149 COMMERCIAL PAPER COMMERCIAL paper is divided into two classes-NEGOTIABLE and NON-NEGOTIABLE. NEGOTIABLE C3OMM ERCIAL PAPER. 2Negotiable commercial paper is that which may be freely transferred from one owner to another, so as to pass the right of action to the holder, without being subject to any set-offs, or legal or equitable delences existing between the original parties, if transferred for a valuable consideration before maturity, and received without any detect therein. Negotiable paper is made payable to the payee therein named, or to his order, or to the payee or bearer, or to bearer; or some similar term is used; showing that the maker intends to give the payee authority to transfer it to a third party, free from all set-offs, or equitable or legal defences existing between himself and the payee. NON-NEGOTIABLE COMMERCIAL PAPER. Nron-negotiable commercial paper is that which is made payable to the payee therein named, without authority to transfer it to a third party. It may be passed from one owner to another by assignment, or by indorsement, but it passes subject to all set-offs, and legal or equitable defences existing between the original parties. 1HOW TIIE TITLE PASSES. The title to negotiable paper passes from one owner to another by delivery, if made payable to payee or bearer, or to bearer. It passes by indorsement and delivery, if made payable to payee or order. The title to non-negotiable paper passes by a mere verbal assignment and delivery, or by indorsement and delivery. PRIMARY DEBTOR. in a promissory note there are two original parties —the maker and the payee. The obligation of the maker is absolute, and continues until the note is presumed to have been paid under the Statute of Limitations. The maker is the primary debtor. In a bill of exchange there are three parties. When the drawer accepts the bill, he becomes the primary debtor upon the bill of exchange. PROMISSORY NOTE NOT PAYABLE IN MONEY. When a promissory note is payable in anything but money, it does not come within the Statute. There is no presumption that it is founded upon a valuable consideration. A consideration must be 150 ARITHMETIC. alleged in the complaint, and proved on the trial. The acknowledgment of a consideration in such promissory note, by inserting the words' value received," is sufficient to cast upon the defendant the burden of proof that there was no consideration. The acknowledgment of "value received," raises the presumption that the note was given for value; but this presumption may be rebutted by the defendant. A negotiable instrument is a written promise or request for the payment of a certain sum of money to order or bearer. A negotiable instrument must be made payable in money only, and without any condition not certain of fulfillment. The person, to whose order a negotiable instrument is made payable, must be ascertainable at the time the instrument is made. A negotiable instrument may give to the payee an option between the payment of the sum specified therein, and the performance of another act. A negotiable instrument may be with or without date; with or without seal; and with or without designation of the time or place of payImen t. A negotiable instrument may contain a pledge ot collateral security, with authority to dispose thereof: A negotiable instrument must not contain any other contract than such as is specified.'Two different contracts cannot be admitted. Any date may be inserted by the maker ef a negotiable instrument, whether past, present, or future, and the instrument is not invalidated by his death or incapacity at the time of the nominal date. There are several classes of negotiable instruments, namely:1. Bills of Exchange; 2. Promissory Notes; 3. Bank Notes; 4. Cheques on Banks and Bankers; 5. Coupon Bonds; 6. Certificates of Deposit; 7. Letters of Credit. A negotiable instrument that doss nit specify the time of payment, is payable immediately. A negotiable instrument which does not specify a place of payment, is payable wherever it is held at its maturity. An instrument, otherwise negotiable in form, payable to a person named, but adding the words, " or to his order," or " to bearer," or equivalent thereto, is in the former case fryable to the written order of such person, and in the latter caae, payable to the bearer. A negotiable instrument, made payable to the order of the maker, or of a fictitious person, if issued by the maker for a valid consideration, without indorsement, has the same effect against him and all other persons having notice of the facts, as if payable to the bearer. A negotiable instrument, made payable to the order of a person obviously fictitious, is payable to the bearer. The signature of every drawer, acceptor and indorser of a nego COMMERCIAL PAPER. 151 tiable instrument, is presumed to have been mace for a valuable consideration, before the maturity of the instrument, and in the ordinary course of business, and the words "value received," acknowledge a consideration. One who writes his name upon a negotiable instrument, otherwise than as a maker or acceptor, and delivers it, with his name thereon, to another person, is called an indorser, and his act is called an indorsement. One who agrees to indorse a negotiable instrument is bound to write his signature upon the back of the instrument, if there is sufficient space thereon for that purpose. When there is not room for a signature upon the back of a negotiable instrument, a signature equivalent to an indorsement thereof may be made upon a paper annexed thereto. An iidorsement may be general or special. A general indorsement is one by which no indorser is named. A special indorsement specifies the indorsee. A negotiable instrument bearing a general indorsement cannot be afterwards specially indorsed; but any lawful holder may turn a general indorsement into a special one, by writing above it a direction for payment to a particular person. A special indorsement may, by express words for that purpose, but not otherwise, be so made as to render the instrument not negotiable. Every indorser of a negotiable instrument warrants to every subsequent holder thereof, who is not liable thereon to him: 1. That it is in all respects what it purports to be; 2. That he has a good title to it; 3. That the signatures of all prior parties are binding upon them; 4. That if the instrument is dishonored, the indorser will, upon notice thereof duly given unto him, or without notice, where it is excused 6y law, pay so much of the same as the holder paid therefor, with interest. One who indorses a negotiable instrument before it is delivered to the payee, is liable to the payee thereon, as an indorser. An indorser may qualify his indorsement with the words, " without recourse," or equivalent words; and upon such indorsement, he is responsible only to the same extent as in the case of a transfer without indorsement. Except as otherwise prescribed by the last section, an indorsement " without recourse" has the same effect as any other indorsement. An indorsee of a negotiable instrument has the same right against every prior party thereto, that he would have had if the contract had been made directly between them in the first instance. An indorser has all the rights of a guarantor, and is exonerated from liability in like manner. 152 a ARITHMETIC. One who indorses a negotiable instrument, at the request, and for the " accommodation" of another party to the instrument, has all the rights of a surety, and is exonerated in like manner, in respect to every one having notice of the facts, except that he is not entitled to contribution from subsequent indorsers. The want of consideration for the undertaking of a maker, acceptor, or indorser of a negotiable instrument, does not exonerate him from liability thereon, to an indorsee in good faith for a consideration. An indorsee in due course is one who in good faith, in the ordinary course of business, and for value, before its apparent maturity or presumptive dishonor, acquires a negotiable instrument duly indorsed to him, or indorsed generally, or payable to the bearer. An indorser of a negotiable instrument, in due course, acquires an absolute title thereto, so that it is valid in his hands, notwithstanding any provision of law making it generally void or voidable, and notwithstanding any defect in the title of the person from whom he acquired it. One who makes himself a party to an instrument intended to be negotiable, but which is left wholly or partly in blank, for the purpose of filling afterwards,-is liable upon the instrument to an indorsee thereof in due course, in whatever manner, and at whatever time it may be filled, so long as it remains negotiable'in form. It is not necessary to make a demand of payment upon the principal debtor in a negotiable instrument in order to charge him; but if the instrument is by its terms payable at a specified place, and he is able and willing to pay it there at maturity, such ability and willingness are equivalent to an offer of payment upon his part Presentment of a negotiable instrument for payment, when necessary, must be made as follows, as nearly as by reasonable diligence it is practicable: 1. The instrument must be presented by the holder, or his authorized agent. 2. The instrument must be presented to the principal debtor, if he can be found at the place where presentment should be made, and if not, then it must be presented to some other person of discretion, if one can be found there, and if not, then it must be presented to some other person of discretion, if one can be found there, and if not, then it must be presented to a notary public within the State; 3. An instrument which specifies a place for its payment, must be presented there, and if the place specified includes more than one house, then at the place of residence or business of the principal debtor, if it can be found therein; 4. An instrument which does not specify a place for its payment, must be presented at the place of residence or business of the principal debtor, or wherever he mav be found, at the option of the presentor; and, COMMEAIAL PAPER. 153 a 5. The instrument must be presented upon the day of its apparent maturity, or, if it is payable on demand, at any time before its apparent maturity, within reasonable hours, and, if it is payable at a banking house, within the usual banking hours of the vicinity; but, by the consent of the person to whom it should be presented, it may be presented at any hour of the day. The apparent maturity of a negotiable instrument, payable at a particular time, is the day on which by its terms it becomes due; or, when that is a holiday, it should be paid the previous day. A bill,of exchange, payable at a specified time after sight, which is not accepted within ten days after its date, in addition to the time which would suffice, with ordinary diligence, to forward it for acceptance, is presumed to have been dishonored. The apparent maturity of a bill of exchange, payable at sight or on demand. is: 1. If it bears interest, one year after its date; or, 2. If it does not bear interest, ten days after its date, in addition to the time which would suffice, with ordinary diligence, to forward it for acceptance. The apparent maturity of a promissory note, payable at sight or on demand, is: 1. If it bears interest one year after its date; or, 2. If it does not bear interest, six months after its date. When a promissory note is payable at a certain time after sight or demand, such time is to be added to the periods mentioned in the last paragraph. A party to a negotiable instrument may require, as a condition concurred to its payment by him: 1. That the instrument be surrendered to him, unless it is lost or destroyed, or the holder has other claims upon it; or, 2. If the holder has a right to retain the instrument, and does not retain it, then that a receipt for the amount paid, or an exoneration of the party paying, be written thereon; or, 3. If the instrument is lost, then that the holder give to him a bond, executed by himself and two sufficient sureties, to indemnify hin against any lawful claim thereon; or 4. If the instrument is destroyed, then that proof of its destruction be given to him. A negotiable instrument is dishonored when it is either not paid, or not accepted, according to its tenor, or presentment for the purpose, or without presentment, where that is excused. Notice of the dishonor or protest of a negotiable instrument may be given: 1. By a holder thereof; or, 2. By a party to the instrument who might be compelled to pay it to the holder, and who would, upon taking it up, have a right to reimbursement from the party to whom the notice is given. 154 a ARITHMETIC. A notice of dishonor mray be given in any form which describes the instrument with reasonable certainty, and substantially informs the party receiving it that the instrument has been dishonored. A notice of dishonor may be given: 1. By delivering it to the party to be charged, personally, at any place; or, 2. By delivering it to some person of discretion at the place of residence or business of'such party, apparently acting for him; or, 3. By properly folding the notice, directing it to the party to be charged, at his place of residence, according to the best information that the person giving the notice can obtain, depositing it in the post-office most conveniently accessible from the place where the presentment was made, and paying the postage thereon. In case of the death of a party to whom notice of dishonor should otherwise be given, the notice must be given to one of his personal representatives; or, if there are none, then to any member of his family who resided with him at his death, or, if there is none, then it must be mailed to his last place of residence, as prescribed by subdivision 3 of the last praragraph. A notice of dishonor sent to a party after his death, but in ignorance thereof, and in good faith, is valid. Notice of dishonor, when given by the holder of an instrument, or his agent, otherwise than by mail, must be given on the day of dishonor, or on the next business day thereafter. When notice of dishonor is given by mail, it must be deposited in the post-office in time for the first mail which closes after noon of the first business day succeeding this dishonor, and which leaves the place where the instrument was dishonored, for the place to which the notice should be sent. When the holder of a negotiable instrument, at the time of its dishonor, is a mere agent for the owner, it is sufficient for him to give notice to his principal in the same manner as toan indorser, and his principal may give notice to any other party to be charged, as if he were himself an indorser. And if an agent of the owner employes a sub-agent, it is sufficient for each successive agent or subagent to give notice in like manner to his own principals. Every party to a negotiable instrument, receiving notice of its dishonor, has the like time thereafter to give similar notice to prior parties, as the original holder had after its dishonor. But this additional time is available only to the particular party entitled thereto. A notice of the dishonor of a negotiable instrument, if valid in favor of the party giving it, inures to the benefit of all other parties thereto, whose right to give the like notice has not then been lost. COMMERCIAL PAPER 155 a FORMS OF FOREIGN BILLS OF EXCHANGE. FRENCH. Lille, le 28 Septembre, 1848. Bou pour ~158 9 Sterlings. Au vingt-cinq Decembre prochain, II vous plaira payer par ce mandat a l'ordre de nousme'mes la somme de cent cinquante-huit livres sterlings 9 shellings valeur en nousmemes et que passerez suivant l'avis de 4 Messieurs a Londres. GERMAN. Niurnberg, den 28 October, 1838. Pro ~100 Sterling. Zwei monate nach dato zahlen Sie gegen diesen Prima Wechsel an die Ordre des fHerrn Ein Hundert Vfund Sterling den Wferth erhalten. Sie bringen solche auf Rechnung laut Bericht von Iierrn__ London. ITALIAN. Livorno, le 25 Seltembre, 1848. Per ~500 Sterline. A Tre mesi data pagate per questa prima de Cambio (una sol volta) all' ordine -, la somma di Lire cinque cento sterline valuta cambiata,oe ponete in conto M. S. secondo l'avviso Addio. Al__ Londra. SPANISH. Malaga, a 20 de Setbre de 1848, Son ~300. A noventa diasfecha se serviran Vs mandar pagar por esta primera de cambio a la orden de los Sres____ Tres cientas libras Esterlinas en oro o plata valor recibido de dhos Sre que anotaran valor en cuenta segun aviso de A los Sres Londres. PORTUGUESE. ~600 Esterlinas. Lisbon, aos 8 de Dezembro de 1848. A Sessenta dias de vista precizos pagardi V. por esta nossa unica via de Letra Segura, a nossa Ordem a quantia acima de Seis Gentas Livras Esterlinas valor de nos recibido em Fazendas, que passera em Comnta segundo o aviso de Ao Sen Londres, BILL OF EXCHANGE ON LONDON. ~347 19s. 10d. Philadelphia, Oct. 25th, 1866. Sixty days after sight of this, my first Bill of Exchange (second and third oj the same date, and tenor unpaid), pay to the order of Williams & Mann, Three Hundred and Forty-seven Pounds, Nineteen Shillings and Ten peuce, Sterling, value received, with or withoutfurther advice. KERR, BROWN & Co. To R. II. GLADSTONE, Banker. London. INLA.ND DRAFT. $97 1-2d-5 Chicago, Sept. 10th, 1866. Ninety days after sight, pay to the order of Manning and Munson, Nine hIundred and Seventy-one and -?o Dollars, value received, and charge the sanme to our account. SMITH & EVANS. To SAMUEL SMALL & Co., Baltimore, Md. 156 a ARITHMETIC. Bills of Exchange are the highest class of commercial paper known to the law, and it has never been the cherished object of the law merchant,-which has been permitted by the English courts to insinuate itself into the common law, till it now forms a part of that code,-to uphold them inviolate, as far as possible. While the lexmercatoria (or mercantile law) is deeply impregnated with the principles of equity, those principles have been chiefly marked to enable courts of law to enforce equitable rights, and upon this principle was the negotiability of bills of exchange insisted upon and finally maintained at the common law; but when equitable principles have been invoked for the purpose of destroying the validity and security of bills of exchange, they have been listened to with great disfavor and only admitted as exceptional cases CHECKS. 1. A check is substantially the same as an inland bill of exchange; it passes by delivery, when payable to bearer, and the rules as to presentment, diligence of the holder, &c., which are applicable to the one, are generally applicable to the other. 2. A check is an appropriation of the drawer's funds, in the hands of the banker, to the amount thereof, and, consequently, the drawee has no right to withdraw them before the check is paid. 3. The characteristics which distinguish checks from bills of exchange are, that checks are always drawn on a bank or banker; that they are payable immediately upon presentment, and without days of grace; and that they are not presentable for acceptance, but only for payment. The want of due presentment of a check, and notice of the non-payment thereof only exonerates the drawer in so far as actual damages have thereby resulted to him. LETTERS OF CREDIT. In addition to the commercial paper before mentioned, there is an extensive business done by the issue of " Letters of Credit." These are issued by prominent bankers in London, Paris, New York and other cities, to travellers who are about to visit foreign countries, and who are thus saved the risk and expense of carrying any large amount of cash about them. These LETTERS OF CREDIT are addressed by the banker to his correspondents abroad, authorizing any one or more of them to pay to the person named, any portion of the sum mentioned in the letter. Thus a person leaving New York for the Pacific Ports, South America or Arctic Ports, or any city or place in Europe or other portions of the world, need carry very little cash. At the first port of arrival he is able to realize such funds as may be necessary to pay COMMERCIAL PAPER. 157 a lhis expenses to a further port by using his Letters of Credit. A traveller may go round the world, wtth the aid of such a CrEDIT, and never have more than one hundred dollars in his pocket. No loss from exchange need occur, in such cases: bills on London being in demand throughout the civilized world. The usual charge by the bankers for such " Letters of Credit," is one per cent. where the trader does not pay the amount of the Letter in advance. Where he pays in advance, no charge is made; the use of the money in the banker's hands being an equivalent for the cost of the credit. Letters of Credit are also extensively used by importers when travelling abroad for the purchase of goods; also by supercargoes and captains of vessels for the purchase of cargoes in foreign ports; also as remittances to distant ports in Asia, Australia, &c., for the purchase of cargoes of foreign goods. Before Letters of Credit were adopted or in circulation, it was the practice among American and other merchants to remit specie to remote parts for investment in foreign merchandize DAYS OF GRACE. 1. In most countries, when a bill or note is payablet at a certain time after date, or after sight, or after demand, it is not payable the precise time mentioned in the bill or note, but days of grace are allowed. 2. The days of grace are so called, because they were formerly gratuitous, and not to be claimed as a right by the person on whom it was incumbent to pay the bill, and were dependant on the inclination of the holder; they still retain the name of days of grace, though the custom of merchants, recognized by law, has long reduced them to a certainty, and established them as a right. 3. In England, Scotland, Wales, and Ireland, three days grace are allowed; in other countries they vary from three to twelve days. 4. The days of grace as allowed in England, are generally allowed in the United States, at least no traces can be found of a contrary decision, except in the State of Massachusetts, where it has been held that no days of grace are allowable, unless stipulated in the contract itself. It is probable that a bill of exchange was, in its original, nothing more than a letter of credit from a merchant in one country, to his debtor, a merchant in another, requesting him to pay the debt to a third person, who carried the letter, and happened to be travelling to the place where the debtor resided. It was discovered, by experience, that this mode of making payments was extremely convenient to all parties:-to the creditor, for he could thus receive his debt without trouble, risk or expense-to the debtor, for the facility of 11 152 ARITHMETIC. payment was an equal accommodation to him, and perhaps drew after it facility of credit to the bearer of the letter, who found himself in funds in a foreign country, without the danger and incumbrance of carrying specie. At first, perhaps, the letter contained many other things besides the order to give credit. But it was found that the original bearer might often, with advantage, transfer it to another. The letter was then disencumbered of all other matter; it was opened and not sealed, and the page on which it was written, gradually shrunk to the slip now in use. The assignee was, perhaps, desirous to know beforehand whether the party to whom it was addressed would pay, and sometimes showed it to him for that purpose; his promise to pay was the origin of acceptances. These letters or bills, the representatives of debts due in a foreign country, were sometimes more, sometimes less, in demand; they became, by degrees, articles of traffic; and the present complicated and abstruse practice and theory of exchange was gradually formed. PARTIAL PAYMENTS Partial payments, as the term indicates, are the part payments of promissory notes, bonds, or other obligations. When these payments are made the creditor specifies in writing, on the back of the note, or other instrument, the sum paid, and the time when it is paid, and acknowledges it by signing his name. The method approved of by the Supreme Court of the United States, for casting interest upon bonds, notes, or other obligations, upon which partial payments have been made, is to apply the'payment, in the first place, to the discharge of the interest then due. If the payment exceeds the interest, the surplus goes towards discharging the principal, and the subsequent interest is to be computed on the balance of the principal remaining due. If the payment he less than the interest, the surplus of interest must not be taken to augment the principal, but interest continues on the former principal until the time when the payments, taken together, exceed the interest due, and then the surplus is to be applied towards discharging the principal. RULE. Find the amount of the principal to the time of the first payment; subtract the payment from the amount,, and then find the amount of the remainder to thetime of the second payment; deduct the payment as before; and so on to the tilhe of settlement. But if any payment is less than the interest then due, find the amount of the sum due to the time when the payments, added together, shdll be equal, at least, to the interest already due; then find the balance, and proceed as before. PARTIAL PAYMENTS. 153 EXA M PLE. 1. On the 4th of January, 1865, a note was given for $800, payable on demand, with interest at 6 per cent. The following payments were receipted on the back of the note: February 7th, 1865, received......... $150 April 16th, "......... 100 Sept., 30th, " "....... 180 January 4th, 1866. "....... 170 March 24th, " "........ 100 June 12th, " "......... 50 Settled July 1st, 1867. How much was due? SOLUTION: Face of the note, or principal.................................. 800.00 Interest on the same to February 7th, 1865 (1 month, 3 days)......................................................... 4.40 Amount due at time of 1st payment......................... 804.40 First payment to be taken from this amount............... 150.00, Balance remaining due February 7th, 1865.......... 654.40; Interest on the same from February 7th, 1865, to April 16th, 1865................................. 7.525 Amount due at time of 2nd payment........................ 661.925 Second payment to be taken from this amount............. 100.000 Balance remaining due April 16th, 1865............561.925 Interest on the same from April 16th, 1865, to September 30th, 1865.................................. 15.359 Amount due at time of 3rd payment......................... 577.284 Third payment.to be taken from this amount.............. 180.000 Balance remaining due Sept. 30th, 1865............. 397.284 Interest on the same from Sept. 30th, 1865, to January' 4th, 1866........................................ 6.290 Amount due at time of 4th payment......................... 403.574 Fourth payment to be taken from this amount............ 170.000 Balance remaining due January 4th, 1866........... 233.574 154 ARITHMETIC. Interest on the same from Jan. 4th, 1866, to March 24th, 1866................................................ 3.114 Amount due at time of 5th payment........................ 236.688 Fifth payment to be taken from this amount............... 100.000 Balance remaining due, March 24th, 1866............ 136.688 Interest on the same from March 24th, 1866, to June 12th, 1866.................................................. 1.799 Amount due at time of 6th payment......................... 138.487 Sixth payment to be taken from this amount............... 50.000 Balance remaining due June 12th, 1866............. 88.487 Interest on the same from June 12th, 1866, to July 1st, 1867...................................... 5.589 Amount due on settlement.................................... 94.076 2. $1600. CHARLESTON, February 16th, 1865. On demand, Ipromise to pay Jacob Anderson, or order, one thousand six hundred dollars, with interest, at 7 per cent. JOHN FORTUNE JR. There was paid on this note, April 19th, 1865........................$460 July 22nd "........................ 150 August 25th, 1866....................... 50 Sept. 12th, "....................... 100 Dec. 24th....................... 700 How much was due December 31st, 1866? SOLUTION. Face of the note or principal................................... $1600.00 Interest on the same from Feb. 16th, 1865, to April 19th, 1865............................. 19.60 Amount due at time of 1st payment.................... 1619.60 First payment to be taken from this amount............... 460.00 Balance remaining due, April 19th. 1865............. 1159.60 PARTIAL PAYMENTS. 155 Interest on the same from April 19th, 1865, to July 22nd, 1865................... 20.969 Amount due at time of 2nd payment......................... 1180.569 Second payment to be taken from this amount............. 150.000 Balance remaining due, July 22nd, 1865.............. 1030.569 Interest on the same from July 22nd, 1865, to Aug. 25th, 1866, greater than 3rd payment,* Interest on the same fro n July 22nd, 1865, to Sept. 12th, 1866.......................................................... 82.359 Amount due at time of 4th payment........................ 1112.928 Third and fourth payments to be taken from this amount, 150.000 Balance remaining due Sept. 12th, 1866.............. 962.928 Interest on the same from Sept. 12th, 1866, to Dec. 24th, 1866..................................................... 19.098 Amount due at time of last payment......................... 982.026 Last payment to be taken from this amount................ 700.000 Balance remaining due Dec. 24th, 1866............... 282.026 Interest on the same from Dec. 24th, 1866, to Dec. 31st, 1866..........................................382 Amount due at time of settlement, Dec. 31st, 1866....... $282.408 3. $350. BOSTON, May 1st, 1864. On demand I promise to pay William Brown, or order, three hundred and fifty dollars, with interest, at 6 per cent. JAMES WESTON. There was paid on this note, December 25th, 1864................... $50 June 30th, 1865................... 5 * The interest on $1030.569, from July 22nd, 1865, to August 25th, 1866. is $78.752, and the payment made at this date, is only $50, not enough to pay the interest, so if we proceeded, as in the former case, to add the interest to the principal, and subtract the payment from the amount obtained, we would be taking aiterest, until the next payment, on the excess of the interest, $78.752, over the payment, $50, which would be in effect interest upon interest, or compound interest which the law does not allow. 156 ARITHMETIC. 4ugust 22nd, 1866................... 15 June 4th, 1867................... 100 How much was due April 5th, 1868? Ans. $251.67. 4. $609.65. BRANTFORD, June 8th, 1861. Six months after date, we jointly and severally promise to pay John Anderson, or order, six hundred and nine 165 dollars, at the Royal Canadian Bank in Toronto, with interest at 6 per cent. after maturity. SAMUEL GRAHAM. T. B. BEARMAN. There was paid on this note, October 4th, 1862.................... $25.00 March 15thI63............... 16.25 August 24th, 1864................... 36.5,6 What was due December 19th, 1865? Ans. 679.27. 5. $874.95. KINGSTON, May 9th, 1863. Three months after date, I promise to pay Harmon Cummings, or order, eight hundred and seventy-four -95 dollars, with interest after maturity at 6 per cent. THOMAS GOODPAY. There was paid on this note, April 12th, 1864...................... $56.30 July 14th, 1865...................... 24.80 Sept. 18th, 1866...................... 240.60 What was due February 9th, 1868? Ans. $773.07. When the interest accruing on a note is to be paid annually adopt the following RULE. Compute the interest on the principal to the time of settlement, and on each year's interest after it is due, then add the sum of the' When notes, bonds, or other obligations, are given, " with interest payable annually," the interest is due at the end of each year, and may be collected, but if not collected at that time, the interest due draws only simple interest, and the original principal must not be increased by any addition of yearly interest. If nothing has been paid until maturity on a note drawing annual interest, the amount due consists of the principal, the total annual interest, or the simple interest, and the simple interest on each item of annual intemest from the time it became due until paid. PARTIAL PAYMENTS. 157 interests on the annual interests to the amount of the principal, and from this amount take the payments, and the interest on each, from the time they were paid to the time of settlement, the remainder will be the amount due. 6. $500. PRESCOTT, May 1st, 1864. One yeur after date, for value received, Ipromise to pay Musgrove & Wright, or order, Five Hundred Dollars, at their ofice, in the city of Toronto, with interest at 6 per cent.,payable annually. JAMES MANNING. There was. paid on this note: May 4th, 1865..................... $150 Dec. 18th, ".......................... 300 How much was due June 1st, 1866? SOLUITIO N. Face of note, or principal................................ $500.00 Interest on the same from May 1st 1864, to June 1st, 1866......................................................... 62.50 Amount of the principal at time of settlement............ 562.50 First year's interest on principal............... $ Interest of the same from May lIt, 1865, to June 1st, 1866....................................... $1.95 Second year's interest on principal............ $30 Interest on the same from May 1st, 1866, to June 1st, 1866.................................15 Amount of interest upon annual interest.................. 2.10 Total amount of principal....................................... $5640 First payment, May 4th, 1865..................... $150.00 Interest on the same from May 4th, 1865, to June 1st, 1866................................. 9.70 Second payment, December 18th, 1865.......... 300.00 Interest on the same from December 18th, 1865, to June 1st, 1866.............................. 8.20 Payments and interest on the same......................... 467.90 Amount due June 1st, 1866................................ $96.70 11 158 ARITHMETIC. 7. $700. CINCINNATI, January 2nd, 1863. Eighteen months after date, I promise to pay'to the order of J. H. Wilson, Seven Hundred Dollars, for value received, wzith interest at 6 per cent., payable annually. THos. A. BRYCE. There was paid on this note: January 15th, 1864..................... $350 July 2nd, 1864.......................... 300 What amount was due January 2nd, 1865? Ans. $107.22. 8. $950. INDIANAPOLIS, Jan.3rd, 1863. Two years after date, Ipromise to pay A. R. Tennison, or order, Nine hundred and Fifty Dollars, with interest at 9 per cent., payable annually, value received. JAMIES S. PARMENTER. The following payments were receipted on the back of this note: February 1st, 1864, received................ $500 May 14th, " "............... 100 January 12, 1865,............... 300 What was due May 6th, 1865? Ans. $188.94. 9. $250. MOBILE, January 2nd, 1863. Three years from date. for value received, I promise to pay Michael Wright, or order, Two Hundred and Fifty Dollars with interest, payable annually, at 6 per cent. CALVIN W. PEARSONS. At First National Bank here. What was the amount of this note'at maturity? Ans. $297.70.. CONNECTICUT RULE. The Supreme Court of the State of Connecticut has adopted the following RULE. Compute the interest on theprincipal to the time of the firstpayment; if that be one year or more from the time the interest commenced, add it to the principal, and deduct thepayment from the sum total. If there be after payments made, compute the interest on the balance due to the next payment, and then deduct the payment as above, and; like manlner from one payzment t,? a.ot.er, t;71 a1? the PARTIAL PAYMENTS, 159 payments are absorbed, provided the time between one payment and another. be one year or more. If any pacyments be made before one year's interest has accrued, then compute the interest on the principal sum due on the obligation for one year, add it to the principal, and compute the interest on the sum paid, from the time it was paid, up to the end of the year; add it to the sum paid, and deduct that sum from the principal and interest, added as above. If any payments be made, of a less sum than the interest arising at the time of such payment, no interest is to be computed, but only on the principal sum for any period. NOTE.-If a year extends beyond the time of settlement, find the amount of the remaining principal to the time of setilement; find also the amount of the payment or payments, if any, from the time they were paid to the time of settlement, and subtract their sum from the amount of the principal. V:) A M P L ES. 10. $900, KINGSTON, June 1st, 1862. On demand wzje promise to pay J. R. Smith & Co., or order, nine hundred dollars, for value received, with interest from date, at 6 per cent. JONES & WRtIGCHT. On the back of this note were receipted the following payments: June 16th, 1863, received............... $200 August 1st, 1864, "............... 160 Nov. 16th, 1864, ".............. 75 Feby. 1st, 1866,............... 220 What amount was due August 1st, 1866? SOL TION. Face of note or principal...................................... $900.00 Interest on the same from June 1st, 1862, to June 16th, 1863.......................................................... 56.25 Amount of principal and interest, June 16th, 163......... 956.25 First payment to be taken from this amount.................. 200.00 Balance due....................................................... 756.25 Interest on the same from June 16th, 1863, to August 1st, 1864................................... 51.046 Amount due uust st,186............................... 807.29046 Amount due August 1st, 1864................................. 807.296 160 ARITHMETIC. Second payment to be taken from this amount............. 160.000 Balance due....................................................... 647.296 Interest on the same for one year............................ 38.837 Amount due August 1st, 1865............................. 686.133 Amount of 3rd payment from Nov. 16th, 1864, to August 1st, 1865.................................................... 78.187 Balance due....................................................... 607.946 Interest on the same from August 1st, 1865, to August 1st, 1866................................................ 36.476 Amount due August 1st, 1866.................... 644.422 Amount of 4th payment from February 1st, 1866, to August 1st, 1866........................................ 226.600 Balance due August 1st, 1866................................. $417.822 MEROCHANTS' RULE. It is customary among merchants and others, when partial payments of notes or other debts are made, when the note or debt is settled within a year after becoming due, to adopt the following R ULE. Find the amount of the principal from the time it became due until the time of settlement. Then find the amount of each payment from the time it was paid until settlement, and subtract their sum from the amount of the principal. EXAMPLE. 11. $400. MAITLAND, January 1st, 1865. For value received, Ipromise to pay J. B. Smith & Co., or order, on demand, four hundred dollars, with interest at 6 per cent. A. R. CASSELS. The following payments were receipted on the back of this note: February 4th, 1865, received........ $100 May 16th, " "......... 75 August 28th "......... 100 November 25th,......... 80 What was due at time of settlement, which was Deccmber 28th, 1865? PARTIAL PAYMENTS. 11 SOLUTION. Principal or face of note...................................... $400.00 Interest on the same from Jan. 1st, 1865, to Dec. 28th, 1865........................................ 3.80 Amount of principal at settlement........................... 423.80 First payment........................................$100.00 Interest on the same from Feb. 4th, 1865, to Iec. 28th, 1865............................... 5.40 Second payment.................................... 75.00 Interest on the same from May 16th, 1865, to Dec. 28th, 1865............................... 2.77{Third payment....................................... 0.00 Interest on the same from August 28th, 1865, to Dec. 25th, 1865............................ 2.00 Fourth payment................................... 80.00 Interest on the same from Nov. 25th, 1865, to Dec. 28th, 1865................................44 Amount of payments to be taken from amount of principal................................... 365.61, Balance due, December 28th, 1865.............................. $58.18~ 12. $500. CLEVELAND, January 1st, 1865. Three months after date, I promise to pay James JManning, or order, five hundred dollars, for value received, at the First National Bank of Buffalo. CYRUS KING. Mr. King paid on this note, July 1st 1865, $200. What was due April 1st, 1866, the rate of interest being 7 per cent? Ans. $324.50. 13. $240. PHILADELPHIA, May 4th, 1865. On demand, Ipromise to pay A. K. Frost &( Co., or order, two hundred and forty dollars, for value received, with interest at 6 per cent. DAVID FLOOK. The following payments were receipted on the back of this note: September 10th, 1865, received............. $60 January 16th, 1866,............ 90 What wGs dfti at the time of settlement, which was May 4th, 1866? Ans. $100.44. 162 APRTHMETIC. 14. $340. LOWELL, June 16th, 1864. Three months after date, 1 promise to pay Thomas Culverwell, or order, three hundred and forty dollars, with interest, t.6 per cent. WILLIAIM MANNING. On this note were receipted the following payments: October 14th, 1864, received...............$86 February 12th, 1865,............... 40 What was due at time of settlement, Aug. 10, 1865? Ans. $232.0.6 COMPOUND INTEREST. When interest is unpaid at the end of the year, it may, by special agreement, be added to the principal, and in its turn bear interest, and so on from year to year. Whlen added to the principal in this way, it is said to be compound. A person may take compound interest and not be liable to the charge of usury, provided the person to whom he lends money chooses to pay compound interest, but he cannot legally collect it unless there has been a previous agreement to that effect. EX A M P LE. 1. What is the compound interest of $60, for 4 years, at 7 per cent.? SOLUTION. Principal.......................................................$6.00 Interest on the same for one year............................... 4.20 New principal for 2nd year........................................ 64.20 Interest on the same for one year............................... 4.494 New principal for 3rd year..................................... 68.694 Interest on the same for one year............................. 4.808 New principal for 4th year................................. 73.502 Interest on the same for one year............................... 5.145 Amount for 4 years....................................... 78.647 Principal to be taken from same.................................. 60.000 Compound interest for 4 years................................... $18647 The method of finding compound interest is usually much shortened by the following table, which shows the amount of $1 or ~1 for any number of years not exceeding 50, at 3, 3), 4, 5, 6 and 7 per cent. The amount of $1 or ~1 thus obtained, being multiplied by the given principal, will give the required amount, from lwhich, if the principal be taken, the remainder will be compound interest: COMPOUND INTEREST. 163 TABLE, SHOWING THE AMOUNT OF ONE DOLLAR AT COMPOUND INTERMET FOR A.NY NUMBER OF YEARS NOT EXCEEDING FIFTY. No. 3 per cent. 3}4 per cent. 4 per cent. 5 per cent. 6 per cent. 7 per cent. | 1 1.0 0 1.0035 000 1.040 00U 1.050 000 1.060 000 1.070 000 2 1.060 900 1.071 225 1.081 600 1.102 500 1.123 600 1.144 900 3 1.092 727 1.108 718 1.124 864 1.157 625 1.191 016 1.225'043 4 1.125 509 1.147 523 1.169 859 1.215 506 1.262 477 1.310 796 5 1.159 274 1.187 686 1.216 653 1.276 282 1.338 226 1.402 552 6 1.194 052 1.229 255 1265 319 1.340 096 1.418 519 1.500 730 7 1.229 874 1.272 279 1.315 932 1.407 100 1.503 630 1.605 781 8 1.266 770 1.316 809 1.368 569 1.477 455 1.593 848 1.718 186 9 1.304 773 1.362 897 1.423 312 1.551 328 1.689 479 1.838 459 10 1.343 916 1.410 599 1.480 244 1.628 895 1.790 848 1.967 151 11 1.384 234 1.459 970 1.539 454 1.710 339 1.898 299 2.104 852 12 1.425 761 1.511 069 1.601 032 1.795 856 2.012 196 2.252 192 13 1.468 534 1.563 956 1.665 )74 1.885 649 2.132 928 2.409 845 14 1.512 590 1.618 694 1.731 676 1.979 932 2.260 904 2.578 534 15 1.557 967 1.675 349 1.800 944 2.078 928 2.396 558 2.759 032 16 1.604 706 1.733 986 1.872 981 2.182 875 2.540 352 2.952 164 17 1.652 848 1.794 675 1.947 901 2.292 018 2.692 773 3.158 815 18 1.702 433 1.857 489 2.025 817 2.406 619 2.854 339 3.379 932 19 1.753 506 1.922 501 2.106 849 2.526 950 3.025 600 3.616 526 20 1.806 111 1.989 789 2.191 123 2.653 298 3.207 135 3.869 684 21 1.860 295 2.059 431 2.278 768 2.785 963 3.399 564 4.140 562 22 1.916 103 2.131 512 2.369 919 2.925 261 3.603 537 4.430 402 23 1.973 587 2.206 114 2.464 716 3.071 524 3.819 750 4.740 530 24 2.032 794 2.283 328 2.563 304 3.225 100 4.048 935 5.072 367 25 2.093 778 2.363 245 2.665 836 3.386 355 4.291 871 5.427 433 26 2.156 591 2.445 959 2.772 470 3.555 673 4.549 383 5.807 353 27 2.221 289 2.531 567 2.883 369 3.733 456 4.822 346 6.213 868 28 2.287 928 2.620 177 2.998 703 3.920 129 5.111 687 6.648 838 29 2.356 566 2.711 878 3.118 651 4.116 136 5.418 388 7.114 257 30 2.427 262 2.806 794 3.243 398 4.321 942 5.743 491 7.612 255 31 2.500 080 2.905 031 3.373 133 4.538 039 6.088 101 8.145 113 32 2.575 083 3.006 708 3.508 059 4.764 941 6.453 387 8.715 271. 33 2.652 335 3.111 942 3.648 381 5.003 189 6.840 590 9.325 340 34 2.731 905 3.220 860 3.794 316 5.253 348 7.251 025 9.978 114 35 2.813 862 3.333 590 3.946 089 5.516 015 7.686 087 10.676 581 36 2.890 278 3.450 266 4.103 933 5.791 816 8.147 252 11.423 942 37 2.985 227 3.571 025 4.268 090 6.081 407 8.636 087 12.223 618 38 3.074 783 3.696 011 4.438 813 6.385 477 9,154 252 13.079 271 39 3.167 027 3.825 372 4.616 366 6.704 751 9.703 507 13.994 820 40 3.262 038 3.959 260 4.801 021 7.039 989 10.285 718 14.974 458. 41 3.359 899 4.097 834 4.993 061 7.391 988 10.902 861 16.022 670 42 3.460 696 4.241 258 5.192 784 7.761 588 11.557 033 17.144 257 43 3.564 517 4.389 702 5.400 495 8.149 667 12.250 455 18.344 355 44 3.671 452 4.543 342 5.616 515 8.557 150 12.985 482 19.628 460 45 3.781 596 4.702 358 5.841 176 8.985 003 13.764 611 21.002 452 46 3.895 044 4.866 941 6.074 823 9.434 258 14.590 487 22.472 623 47 4.011 895 5.037 284 6.317 816 9.905 971 15.465 917 24.045 707 48 4.132 252' 5.213 589 6.570 528 10.401 270 16.393 872 25.728 907 49 4.256 219 5.396 065 6.833 349 10.921 333 17.377 504 27.529 930 50 4.383 906 5.584 927 7.106 683' 11.467 400 18.420 154 29.457 025 NoTE.-lf each of the numbers in the table ba diminished by 1, the remainder will denote the interest of $1, instead of its amount. 164 ARITHMETIC. EXERCISES. 2. What is the compound interest on $75, for 2 years, at 7 per cent? Ans. $10.87. 3. What will $50 amount to in 3 years, at 6 per cent., compound interest? Ans. $59.55. 4. What is the compound interest on $600, for 2 years, at 6 per cent., payable half-yearly? Ans. $75.31. 5. What will $320 amount to in 21 years, at 7 per cent., compound interest? Ans. $379.19. 6. What is the compound interest of $150,'for 3 years, at 9 per cent.? Ans. $44.25. 7. What is the compound interest on $1,000, for 2 years, at 3per cent, payable quarterly? Ans. $72.18. 8. What will $460 amount to in 3 years, 4 months, 10 days, at 6 per cent., compound interest? Ans. $559.74. 9. What is the compound interest on $1860, for 8 years, at 7 per cent.? Ans. $1335.83. 10. What will be the compound interest on $75.20, for 20 years, at 31 per cent.? Ans. $74.43. 11. How much more will $500 amount to at compound than simple interest, for 20 years, 3 months, 15 days, at 7 per cent.? Ans. $764.14. 12. What sum will $50, deposited in a savings bank, amount to at compound interest, for 21 years, at 3 per cent, payable half-yearly? Ans. $173.03. 13. If a note of $60.60, dated October 25th, 1856, with the interest payable yearly, at 6 per cent., be paid October 25th, 1860; what will it amount to-at compound interest? Ans. $76.51. 14. What remains due on the following note, April 1st, 1863, at 7 per cent. compound interest? $1,000. CLEVELAND, January 1, 1858. For value received, Ipromise to pay A. B. Smith & Co., or order, one thousand dollars on demand, with interest at 7 per cent. J, D. DFOSTER. On the back of this note were receipted the following payments: June 10, 1858, received............... $70 Sept. 25, 1859, "................ 80 July 4, 1860, "............... 100 DISCOUNT AND PRESENT WORTH. 165 Nov. 11 1861, "............ 30 June 5, 1862, "............ 50 Ans. $1022.34. DISCOUNT AND PRESENT WORTH. Discount being of the same nature as interest, is, strictly speaking, the use of money before it is due. The term is applied, however, to a deduction of so much per cent. from the face of a bill, or the deducting of interest from a note before any interest has accrued. This is the practice followed in our Banks, and is therefore called Bank discount, in order to distinguish. it from true discount. The method of computing bank discount differs in no way from that of computing simple interest, but the method of finding true discount is quite different, e. g., a debt of $107, due one year hence, is considered to be worth $100 now, for the reason that $100 let out at interest now, at 7 per cent., would amount to $107 at the end of a year. In calculating interest, the sum on which interest is to be paid is known, but in computing discount we have to find what sum must be placed at interest so that that sum, together with its interest, will amount to the given principal.' The sum thus found is called the "Present Worth." We have already seen that $1.00 is the present worth of $1.07 due one year hence, at 7 per cent., therefore, to get the present worth of any sum due one year hence, at 7 per cent., it is only necessary to find how many times $1.07 is contained in the given sum, and we have the present worth; hence To find the present worth of any sum, and the discount for any time, at any rate per cent., we have the following RULE. Divide the given sum by the amount of $1 for the given time and rate, and the quotient will be the present worth. From the given sum subtract the present worth, and the remainder will be the discount. EXERCISES. 1. What is the present worth of $224, due 2 years hence, at 6 per cent.? Ans. $200. 166 ARITHMETIC. 2. What is the discount on $670, due 1 year and 8 months hence, at 7 per cent.? Ans. $70. 3. What is the discount on $501, due 1 year and 5 months hence, at 8 per cent.? Ans. $51. 4. What is the present value of a debt of $678.75, due 3 years and 7 months hence, at 7j per cent.? Ans. $534.97~. 5. What is the discount on $88.16, due 1 year, 8 months, and 12 days hence, at 6 per cent.? Ans. $8.16. 6. If the discount on $1060, for 1 year, at 6 per cent., is $60; what is the discount on the same sum for one-half the time? Ans. $30.87. 7. How much cash will discharge a debt of $145.50, due 2 years, 6 months and 12 days hence, at 6 per cent.? Ans. $126.30. 8. If I am offered a certain quantity of goods for $2500 cash, or for $2821.50, on 9 months credit; which is the best offer, and by how much? Ans. Cash by $200, 9. What is the difference between the interest and discount of $46.16, due at the end of 2 years, 6 months, and 24 days, at 6 per cent.? Ans. 95 cents. 10. A merchant sold goods to the amount of $1500, one-half to be paid in 6 months, and the balance in 9 months; how much cash ought he to receive for them after deducting 1I per cent. a month? Ans. $1331.25. 11. Suppose a merchant contracts a debt of $24000, to be paid in four instalments, as follows: one-fifth in 4 months; one-fourth in 9 months; one-sixth in 1 year and 2 months, and the rest in 1 year and 7 months; how much cash must he give at once to discharge the debt, money being worth 6 per cent.? Ans. 22587.66. 12. Bought goods to the amount of $840, on 9 months credit; how much money would discharge the debt at the time of purchasing the goods, interest being 8 per cent.? Ans. $792.45. 13. A bookseller marks two prices in a book, one for ready money, and the other for one year's credit, allowing discount at 5 per cent. If the credit price be marked $9.80; what ought to be the price marked for cash? Ans. $9.33. 14. A man having a horse for sale, offered it for $225, cash; or, $230 at 9 months credit; the buyer chose the latter; did the seller lose or make by his bargain, and how much, supposing money to be worth 7 per cent.? Ans. He lost $6.47. 15. A. B. Smith owes John Manning as follows:-$365.87, to BANKS AND RANKING. 167 be paid December 19th, 1863; $161.15, to be paid July 16th, 1864; $112.50, to be paid June 23rd, 1862; $96.81, to be paid April 19th, 1866, allowing discount at 6 per cent.; how much cash should Manning receive as an equivalent, January 1st, 18632? Ans. $653.40. 16. I buy a bill of goods amounting to $2500 on six months' credit, and can get 5 per cent. off by paying cash; how much would 1 gain by paying the bill now, provided I have to borrow the money, and pay 6 per cent. a year for it? Ans. $53.75. BANKS AND BANKING. General Principles of Banking.-Banks are commonly divided into the two great classes of banks of deposit and banks of issue. This, however, appears at first sight to be rather an imperfect classification, inasmuch as almost all banks of deposit are at the same time banks of issue, and almost all banks of issue also banks of deposit. But there is in reality no ambiguity; for by banks of deposit are meant banks for the custody and employment of the money deposited with them or entrusted to their care by their customers, or by the public; while by banks of issue are meant banks which, besides employing or issuing the money entrusted to them by others, issue money of their own, or notes payable on demand. The Bank of England is principally a bank of issue; but it, as well as the other banks in the different parts of the empire that issue notes, is also a great bank of deposit. The private banking companies of London, and the various provincial banks, that do not issue notes of their own, are strictly banks of deposit. Banking business may be conducted indifferently by individuals, by private companies, or by joint stock companies or associations. Utility and Functions of Banks of Deposit.-Banks of this class execute all that is properly understood by banking business; and their establishment has contributed in no. ordinary degree to give security and facility to commercial transactions. They afford, when properly conducted, safe and convenient places of deposit for the money that would otherwise have to be kept, at a considerable risk, in private houses. They also prevent, in a great measure, the necessity of carrying money from place to place to make payments, and enable them to be made in the most convenient and least expensive manner. The objects of banking. —Correct sentiments beget correct conduct. A banker ought, therefore, to apprehend correctly, the objects of banking. They consist in making pecuniary gains for the stockholders ty legal operations. The bnsiness is eminently 12 168 ARITHMETIC. beneficial to society; but some bankers have deemed the good of society so much more worthy of regard than the private good of stockholders, that they have supposed all loans should be dispensed with direct reference to the beneficial effect of the loans on society, irrespective, in some degree, of the pecuniary interests of the dispensing bank. Such a banker will lend to builders, that houses or ships may be multiplied; to manufacturers, that useful fabrics may be increased; and to merchants, that goods may be seasonably replenished. He deems himself, ex-officio, the patron of all interests that concern his neighbourhood, and regulates his loans to these interests by the urgency of their necessities, rather than by the pecuniary profits of the operations to the bank, or the ability of the bank to sustain such demands. The late Bank of the United States is a remarkable illustration of these errors. Its manager seemed to believe that his dutes comprehended the equalization of foreign and domestic exchanges, the regulation of the price of cotton, the upholding of State credit, and the control, in some particulars, of Congress and the President-all vicious perversions of banking to an imagined paramount end. When we perform well the direct duties of our station, we need not curiously trouble ourselves to effect, indirectly, some remote duty. Results belong to Providence, and by the natural catenation of events (a system admirably adapted to our restricted foresight), a man can usually in no way so efficiently promote the general welfare, as by vigilantly guarding the peculiar interests committed to his care. If, for instance, his bank is situated in a region dependent for its prosperity in the business of lumbering, the dealers in lumber will naturally constitute his most profitable customers; hence, in promoting his own interest out of their wants, he will, legitimately, benefit them as well as himself, and benefit them more permanently than by a vicious subordination of his interests to theirs. Men will not engage permanently in any business that is not pecuniarily beneficial to them personally; hence, a banker becomes recreant to even the manufacturing and other interests that he would protect, if he so manage his bank as to make its stockholders unwilling to continue the employment of their capital in banking. This principle, also, is illustrated by the late United States Bank, for the stupendous temporary injuries which its mismanagement inflicted on society, are a smaller evil than the permanent barrier its mismanagement has probably produced against the creation of any similar institution. Bank of England Notes Legal Tender.-According to the law as it stood previously to 1834, all descriptions of notes were legally payable at the pleasure of the holder in coin of the standard weight and purity. But the policy of such a regulation was very questionable; and we regard the enactment of the Stats. 3 & 4, Will. 4, c. 99, which makes Bank of England notes legal tender, everywhere BANKS AND BANKING. 169 except at the Bank and its branches, for all sums above ~5, as a great improvement. Savings Banks have been in use in Europe over fifty years, and in Canada and the United States, almost as long. They are established for the purpose of receiving from people in moderate circumstances, small sums of money on interest. In England the deposits are held by the Government, and invested in the three per cent. funds. In New England, New York and other States, the deposits are generally loaned on bond and mortgage at six or seven per cent. interest. Friendly Societies.-Friendly Societies are associations, mostly in England, of persons chiefly in the humblest classes for the purpose of making provision by mutual contribution against those contingencies in human life, the occurrence of which can be calculated by way of average. The principal objects contemplated by such societies are the following: The insurance of a sum of money to be paid on the birth of a member's child, or on the death of a member or any of his family; the maintenance of members in old age and widowhood; the administration of relief to members incapacitated for labor by sickness or accident; and the endowment of members or their nominees. Friendly Societies are, therefore, associations for mutual assurance, but are distingushed from assurance societies, properly so called, by the circumstance that the sums of money which they insure are comparatively small. BANK DISCOUNT. The Bank Discount of a note-is the simple interest on the sum. for which it is given from the time it is discounted to the time it becomes due, including three days of grace. Suppose, for example, in getting a note of $200 discounted at a bank I am charged $12 for discount, which being deducted, I receive but $188, so that I pay interest on $12 which I did not receive. From this it is clear that I am paying a higher rate of interest in discounting a note at a bank, than I would pay were I to borrow money at the same rate. As bank discount is the same as interest., we derive the following RULE. Find the interest on the sum, specified in the note at the given rare, and for the given time, including three days of grace, and thiwill be the BANK DISCOUNT. Subtract the discount from the face of the note, and the remain der will be the PROCEEDS OR PRESENT WORTH. 170 ARITHMETIC. EXERCISES 1. What is the bank discount on a note, given for 60 days, for $350, at 6 per cent.?* Ans. $3,67. 2. What is.the bank discount on a note of $495, for 2 months, at 5 per cent.? Ans. 4.33. 3. What is the present value of a note of $7840 discounted at a bank for 4 months and 15 days, at 6 per cent.? Ans. $7659.68. 4. How much money should be received on a note for $125, payable at the end of 1 year, 3 months, and 15 days, if discounted at a bank at.8 per cent.? Ans. $112. 5. A note, dated December 3rd, 1860, for $160.40, and having 6 months to run, was discounted at a bank, April 3rd, 1861, at 6 per cent.; how long had it to run, and what were the proceeds? Ans. 64 days; proceeds $158.71. 6. On the first day of January, 1866, I received a note for $2405 at 60'days, and on the 12th of-the same month had it discounted at a bank at 7 per cent.; how much did I realize upon it. Ans. $237.61. 7. A merchant sold 240 bales of cotton, each weighing 280 pounds, for 12- cents per pound, which cost him, the same day, 10 cents per pound; he received in payment a good note, for 4 months' time, which he discounted immediately at a bank. at 7 per cent.; what will be his profits? Ans. $1479.10: 8. I hold a note against Clemes, Rice & Co., to the amount of $327.40 dated April 11th, 1866, having six months to run after date, and drawing interest at the rate of 6 per cent. per annum. What are the proceeds if discounted at the Girard Bank on the 10th of August, at 71- per cent.? Ans. $332.99. NOTE. When a note drawing interest, is discounted at a bank, the interest is calculated on the face of the note from its date to the time of maturity, and added to the face of the note, and this amount discounted for the length of time the note has still to run. 9. What will be the discount on the following note if discounted at the City Bank on the 17th of Nouember, at 6 per cent. (360 days to a year), * Throughout all the exercises, unless otherwise specified, the year is to be considered as consisting of 365 days. Since it is customary in business when a fraction of a cent occurs in and result to reject it. if less than half a cent, and if not less, to call it a cent, we have adopted this principal through. out the bool rANK DISCOUNT. 171a $527.. OBERLIN, Oct. 4, -S66. VNinety aays after date for value received, twe promis( to pay to the order of Smith, Warren & Co., five hundred twentyseven and _' dollars at'the City Bank, Oberlin, with interest ai eight per cent. THOMPSON & BURNS. 10. What will be the discount at 7-,% per cent. on a note for $227.41, drawing interest at 8 per cent., dated May 1st, 1865, at 1 year after date, if discounted on March 7th, 1866? 11. What amount of money will I receive on the following note, if discounted at the First National Bank of Detroit on June 21st, at 9 per cent.? $473.80. DETROIT, May 17, 1866. Three months after daie Ipromise to pay to the order of J. R. Sing & Co., four hundred and seventy-three and 8o Dollars, at the First National Bank, Detroit, for value received with interest at 7 y3 per cent. RICHARD DUNN. 12. What must I pay for the following note on August, 15th, 1866, so as to make at the-rate of 30 per cent. interest per annum on the money I pay for it? Ans. $708.54. $746.75. ADRIAN, January 19, 1866. Ont yearfron date, for value received, we promise to pay James Ames, or order, seven hundred andforty-six -,7 dollars, at the Commercial Bank, Adrian, with interest at 7-a3 per cent. per annum. WILSON & CUMMINGS. 13. A holds a note against B to the amount of $478.92, dated May 10th, 1865 at 1 year after date drawing 7,3 per cent. interest. I purchase this note from A. on August 18th, paying for it such a sum that will allow me 20 per cent. interest on my money. What shall I pay for it? 14. I got my note for $2000 discounted at a bank, May 20, 1862, for 2 months, and immediately invested the sum received in flour. June 7, 1862, I sold half the flour at 10 per cent. less than cost, and put the money on interest at 9 per cent. August 13, 1862, I sold the remainder of the flour at 18 per cent. advance, and expended the money for cloth at $1 per yard; 12 days after I sold the cloth at $1.16'] per yard, receiving half the pay in cash, which I lent on interest at 7- per cent. and a note for the other half, to be on inter 172a ARITHMETIC. est from October 4, 1862, at 6 — per cent. When my note at the bank became due I renewed it for 5 months, and when this note became due I renewed it for 2 months, and when this note became due I renewed it for such a time that it became due July 20, 1863, at which time I collected the amount due me, and paid my note at the bank. Required the loss or gain by the transaction. It is.sometimes necessary to know the amount for which a note must be given, in order that it shall produce a given sum when discounted at a bank. EXAMPLLE. 1. Suppose we require to obtain $236.22 from a bank, and that we are to give our note, due in two months; for what amount must we draw the note, supposing that money is worth 9 per cent.? SOLUTION. From the nature of this example we can readily perceive that such a sum must be put on the face of the note, that when discounted the proceeds will be exactly $236.22. If we were to take a one dollar note and discount it at a bank for the given time, and at the given rate, the proceeds would be.98425. Hence, for every dollar we put upon the face of the note we receive.98425, and to receive $236.22 we would have to put as many dollars on the face of the note as are represented by the number of times that.98425 is contained in $236.22, which is 240. Therefore, we must put $240 on the face of a note due at the end of two months to produce $236.22 when discounted at a bank at 9 per cent. From this we deduce the following RULE. Deduct the bank discount on $1, for the given time and rate, from $1, and divide the desired amount by the remainder. The quotient will be the face of the note required. 2..For what sum must a note be given, having 4 months to run, that shall produce $1950, if discounted at a bank at 7 per cent.? Ans. $1997.78. 3. What must be the face of a note, so that when discounted for 5 months and 21 days, at 7 per cent., it will produce $57.97, cash? Ans. $60. BANK I DISCOUNT. 171 4. Suppose your note for 6 months is discounted at a bank at 6 per cent., and $484.75 placed to your credit, what must have been the face of the note? Ans. $500. 5. A merchant bought a quantity of goods for $600. For what sum must he write his note, to be discounted at a bank for 6 months, at 6 per cent.? Ans. $618.88. 6. A farmer bought a farm for $5000 cash, and having only onehalf of the sum on hand. he wishes to obtain the balance from the bank. For what sum must lhe give his note, to be discounted for 9 months, at 6 per cent.? Ans. $2619.17. 7. If a merchant wishes to obtain $550 of a bank, for what sum must he give his note, payable in 60 days, allowing it to be discounted at - per cent. per month? Ans. $555.75. 8. I sold A. Mills, merchandize valued at $918.16, for which he was to pay me cash, but being disappointed in receiving money expected, he gave me his endorsed note at 90 days, for such an amount that when discounted at the bank at 7 per cent. it would produce the price of the merchandize. What was the face of the note? 9. I am owing R. Harrington onaccount, now due, $168.45; he also holds a note against me for $210, due in 34 days, including days of grace; he allows a discount of 8 per cent. on the note, and if I give him my note at 60 days for an amount that will be sufficient if discounted at 6 per cent., to produce the amount of account and note. What will be the face of new note? 10. Samuel Johnson has been owing me $274.48 for 84 days. I,charge him interest at 6 per cent. per annum for this time, and he gives me his note at 90 days for such an amount that when discounted at the Girard Bank, at 8 per cent., the proceeds will equal the amount now due. What is the face of the note? From the many dealings business men have, in regard to discount and interest, it is frequently required to know what rate of interest corresponds to-a given rate of bank discount. EXAMPLE. 1. What rate of interest is paid when a note, payable in 362 days, is discounted at 10 per cept.? 172 ARITHMETIC. SOLUTION. If we discount $1 for the given time, and at the given rate, the proceeds will be.90, or 90 cents. Hence, the discount being 10 cents, we are paying 10 cents for the use of 90 cents. Now, if we pay 10 cents for the use of 90, for the use of 1 cent we must pay J- of 10 cents, or - of a cent, and for $1, or 100 cents, we must pay 100 times of a cent, or - 0 o~-=.11, and for $100, $111, or 11 per cent. Therefore, to find the rate of interest corresponding to,a given rate of bank discount, we deduce the following RULE. Divide the given rate per cent., expressed decimally, or the rate per unit, by the number denoting the proceeds of $1 for the given time and rate. The quotient will be the rate of interest required. EXE RCISES. 2. What rate of interest is paid when a note, payable in 60 days,. is discounted at 7 per cent.? Ans. 7,21f 3. What rate of interest is paid when a note, payable in 3 months, is discounted at 6 per cent.? Ans. 6 8 11 4. A note, payable in 6 months, is discounted at 1 per cent. a month; what rate of interest is paid? Ans. 12 4 5. What rate of interest is paid, when a note of $200, payable in 70 days, is discounted at - per cent. a month? Ans. 94s~1-. 6. When a note of $45, payable in 65 days, is discounted at 7 per cent., to what rate of interest does the bank discount correspond? Ans. 7o3 3 7. A bank, by discounting a note at 6 per cent., receives for its money a discount equivalent to 6~ per cent. interest; how long must. the note have been discounted before it was due? Ans. 1 yr., 3 mos, 12d. C OMM ISS ION. COMMISSION is the term applied to money paid to an agent to remunerate him for his trouble in buying, selling, valuing, or for forwarding merchandise or other property. The goods sent to a commission merchant or agent, to be sold on account and risk of another, are termed a consignment. COMMISSION. 173 The person to whom these goods are consigned is called the conshoree or correspondent. The term shipment is sometimes used instead of consignment. EX A 1 PLE. A mmmission merchant sells for me goods worth $1200, and chargce 4 per cent.; what have I to pay him? SOLUTION. 4 per cent. of $1200 is equal to $1200X.04=-$48. Hence I would have to pay $48, and from this we deduce the following RULE. Find the percentage on the given sum at the given rate, whieh will be the commission. EXERCISES. 1. Consigned to A.K.Boomer, Esq., Syracuse, by the Troy, N.X., foundry, agricultural implements which are sold for $1875.75, what is the agent's commission at 2~ per cent.? Ans. $46.89. 2. Bought in Boston 12 chests of tea, containing 64 lbs. each, at $1.121 per lb., on a commission of 1 per cent.; what was my commission? Ans. $15.12. 3. My Toledo correspondent has bought for me 2768 lbs. of bacon, at 12} cts. a pound; what is his commission at 31 per cent.? $11.25. 4. Bought a carriage and pair of horses, per the order of S. Williams, Portland; paid for the horses $240, and charged 4- per cent., and paid for the carriage $160, and charged 1~ per cent.; how much did I earn? Ans. $13.20. 5. A commission agent in a Southern State bought cotton worth $2284 for an English manufacturer, and charged 51 per cent.; what is his commission? Ans. $125.62. 6. On another occasion the manufacturer gave the commission merchant $165.78, for purchasing for him cotton worth $3684; what was the rate per cent? Ans. 4-~ 7. An English commission merchant buys for a Portland house, ~576 10s. Od. worth of provisions, and charges 4~ per cent.; what is his commission? Ans. ~25 18s. 10-d. 8. A New York provision merchant instructs a Belfast (Ireland) commission merchant to purchase for him ~534 4s. Od. worth of 174 ArITHMETIC. bacon and hams, and offershim 7t per cent.; what does the agent get? Ans. ~38 14s. 7d. 9. A book agent in Cincinatti, sells $487.50 worth of books for Day & Co., of Montreal, and receives $72.05 for his trouble; at what rate per cent. was he paid? Ans. 15 nearly. 10. An agent sells 84 sewing machines at $25 each, and his commission amounts to $262.60; what is the rate? Ans. 124. When a sum has to be sent to a commission agent, such that it will be equal both to the sum to be invested, and the agent's commission, it is plain, as already noted, that this is merely a case of percentage. It is the same as the first part of case IV., and we will have the corresponding RULE. Divide the given amount by 1, increased by the given rate per unit, and the quotient will be the sum to be invested; subtract this from the given amount, and the remainder will be the commission. EXAMPLE. If I send $1890 to a commission merchant, and instruct him to buy merchandise with what is left after his commission at 5 per cent. Is deducted; what will be the sum invested, and the agent's commission? SOLUTION. It is plain that for every dollar of the proposed investment I must remit 105 cents, 100 towards the investment, and 5 towards the commission, and hence the number of dollars which can be invested from the sum remitted will be the same as the number of times that 1.05 is contained in 1890. Now, $1890 — 1.05 gives $1800, the sum to be invested, and this subtracted from $1890, leaves $90, the commission to which thteagent is entitled. EXERCISES. 1. Remitted to A. B., St. Pauls, $988 to purchase flour for me with the balance that remains after deducting his commission at 4 per cent.; required the purchase money and percentage? Ans. $950 and $38. 2. Received a commission to buy wheat with $779, less by my commission at 24 per cent.; required the price of the wheat and my commission. Ans. $760, and $19. BROKERAGE. 175 3. Remitted to my corr},pndent to Augusta $266.76, to pay for lumber which he purchased for me, and to pay his own commission at 4 per cent.; what was the price of the lumber, and what the commission? Ans. $256.50, and $10.26. 4. John Jones, Newmarket, commissions W. Orr, Portland, to procure for him a quantity of fine flour, and remits $917.61; how much flour can he have, after allowing 4- per cent., and what will the commission amount to? Ans. $876, and $41.61. 5. John Stalker, London, commissions J. Fleming New York, to purchase for him as much butter as he can procure for the balance between $779.52, and his own commission at 12 per cent.; how many pounds butter did he get at 25 cents per lb.; what the whole price, and what was the commission? Ans. 3072 lbs., $768, and $11.52. 6. Dr. Gallipot is about to remove to England, and sends to a London cabinet maker $4005.45 towards getting his house furnished, he is charged 3~ per cent. over and above the price of the furniture, Ibr time and labour, what does the furniture cost? Ans. $3870. 7. Graham Bros., of Newbury, send to R. White, Charleston, bacon and hams worth $1560, they charge 5~ per cent. commission, and the charge for lading is $75.15; how much does R. White owe them? Ans. $1720.95. 8. P. Robson, commission merchant, Albany, buys for T. Black & Co., Baltimore, groceries, the price of which, together with their commission at 4 per cent. comes to $475.02; what was the price of the goods, and what was the amount of the commission? Ans. $456.75, and $18.27. BROKERAG-E. BROKERAGE is a per centage paid to an agent for negociating bills, collecting accounts, exchanging money, buying and selling shares and stocks, and all similar transactions. Such an agent is called a Broker. A smaller percentage is usually allowed to a broker than to a commission merchant, because the work he has to do requires less time and labour. Like commission, brokerage is merely a particular case of percentage, and hence the R U L E. To find the brokerage on any sum, find the percentagye on the given sum at the given rate, which will be the brokerage 176 ARITHMETIC. 1. A broker in Buffalo has bought for me $1275 worth of Erie R. R. stock; what will be the brokerage at 2- per cent. Ans. 27 -9. 2. I pay a collector of accounts 2 per cent. for collecting $118.50; how much does it cost me? Ans. $2.37. 3. I pay a broker 1- per cent. for selling $2716.75 government stock; how much do I give him? Ans. $50 94, nearly. 4. Advised R. P., broker, to collect two bills amounting to $897, he has collected -- of it, and I have given him 1,- per cent. on the amount collected; how much have I paid him? Ans. $8.97. 5. A. B. sent me $756 to purchase flour for him. I have charged 2- per cent. commission on the whole sum, and purchased flour with the remainder; what is my commission, and how much do I vest in flour for A. B.? Ans. $738.99, and $17.01. 6. The school taxes on all the sections of a country amount to $1180, and collectors get 2j per cent.; how much remains available for school purposes? Ans. $1149.03. 7. Instructed a broker in Syracuse to sell for me 200 shares of N. Y. C. R. R. stock, at 1141; what will be my proceeds, broker's commission being i per cent. 8. I am charged 1 per cent. by a broker in Raleigh, for negociating a draft for $750; what are the proceeds coming to me? Ans. $748.121-. 9. Bought G. W. R. shares to the amount of $578, and paid my broker 21 per cent.; how much did I give him? Ans. $13.01. 10. Gave D. F. 8- per cent. for collecting accounts for me to the amount of $639; how much did I give him? Ans. $21.30. To find the sum that can be invested when the given amount includes both the brokerage and the investment. For example, if I wish a broker to invest for me $700, and his charge is 2 per cent., I must obviously remit to him $714, as $14 is 2 per [cent. on. $700; conversely, if I send him $714, and instruct him to invest for me that sum, minus his own perceptage, he will have to calculate how much he will have remaining to invest after deducting his own charge. Now, since his percentage is $2 on every $100, he should get from me $102 for every $100 he is to invest, and therefore the sum he can invest will be the 102nd part of what I remit, i. e., $714 —1.02=-$700. Ience the BROKERAGE. 177 RULE. Divide the given amount by one, increased by the given rate per unit of brokerage, and the quotient will be the sum to be invested; subtract this from the given amount, and the remainder will be the brokerage. EXERCISES. 1. A broker receives $574, with instructions to invest what remains after deducting brokerage at 2j per cent., in R. R. shares; how much has he to invest? Ans. $560. 2. The assessment on a certain district, together with the percentage for collection at 2{ per cent., is $1717.80; what is the amount of the assessment, and what the expense of collection? Ans. $1680, and $37.80. 3. A tax amounting to $3276.52, ineluding collector's fees at 4 per cent., is levied on a certain town; what is the amount of the tax, and how much is the collector entitled to? ~ Ans. $3150.50, and $126.02. 4. A gentleman once invested in U. S. government bonds, a certain sum which, with the broker's fee at 1{ per cent., amounted to $18,315; what was the amount of the investment? Ans. $18,000. 5. A Portland broker negociates a draft for $1218 for a Hamilton merchant, at 1, per cent.; what are the proceeds? Ans. $1199.73. 6. A broker, after deducting his charge at 11 per cent., invests the balance of $2450.25 for his employer in bank stock; how much does he invest? Ans. $2420. 7. My broker invests for me in oil well shares, at $83 each, what remains after deducting his fee at - per cent. from $8341.50; how much does he invest. and how many shares does he purchase? Ans. $8300, and 100 shares. 8. A broker's charge is $285, at 1~ per cent., on a certain sum invested; what is the sum?-(See Percentage, Case II.) Ans. $19000. 9. A broker sells stocks for me, and the sum which is realized, together with the brokerage at 4 per cent., amounts to $910; what is the sum procured, and what the brokerage? Ans. $875 and $35. 178 ARITHMETIC. 1. My agent in Richmond has purchased cotton for me to the amount of $1785.80 and charges me a commission of j per cent.; how much have I to remit him to pay for the cotton and commission. Ans. $1801.421. 2. I have received from a correspondent in Troy $4783.11, with instructions to invest the same in Five-twenties at 105~, first deducting my commission of i per cent. What is the commission, and what amount of Five-twenties can I purchase? Ans. Commission $35.61; invested in Five-twenties, $4500. 3. A collector receives $20 for collecting $900; at what per cent. is he paid? Ans. 2|. 4. A purchased per the order of Andrew Camphell & Co., Nashville, Tenn., 14872 lbs. C. C. bacon, at 131 cts. per lb., charging a commission of 1, per cent. A wishes to draw on them for reimbursement; what must be the face of the draft if it cost ~ per cent. to get it cashed, and what is the commission on purchase? Ans. Face of dft. $2010.15; Commission $29.56. 5. A broker invests for me $1750, and I pay him for his trouble $43.75, at what rate per cent. do I pay him? Ans. 21. 6. An Auctioneer valued the furniture of a deceased gentleman, and charging 4 per cent., he was paid $53.86; what was the value of the furniture? Ans. $1346.50. 7. I sent to Taylor & Morrison, Com. merchants, Neiv York, 250 firkins butter, containing on an average 56 lbs. each, at 15 cts. per lb. They sold at an advance of 10 per cent.; freight, &c., deducted $10.45, commission 2~ per cent. They have remitted me a sight draft for net proceeds, which they purchased at 8- per cent. premium, charging ~ per cent. commission on face of draft. What amount of draft did I receive, and what amount of commission charged? 8. A certain district pays $800 school taxes, the collector gets $38 for collecting; what per centage does he get? Ans. 4. 9. B. instructed a broker to sell for him 100 shares of the N. Y. C. R. R. at 112~, how much would the broker's commission be at I per cent. 10. ApaWountant is entrusted to make schedules of the debts and assets 6f a bankrupt; he charges only 2- per cent. on the debts, on the principle that he will have little trouble in getting the accounts due by the bankrupt sent in; but as he knows very well that he will have trouble in getting correct statements sent in of accounts due to the bankrupt, he stipulates for 51 per cent. on these; how much does he get altogether, the debts being $2786, and the assets $618? Ans. $103.64. INSURANCE. 179 INSURANCE. INSURANCE is an engagement by which one party is bound, in consideration of receiving a certain sum, to indemnify another for something in case it should in any way be lost. The party undertaking the risk is seldom, if ever, an individual, but a joint stock company, represented by an agent or agents, and doing business under the title of an " Insurance Company," or " Assurance Company," such as the "Globe Insurance Company," the "Mutual Insurance Company." Some companies are formed on the principle that each individual shareholder is insured, and shares in the profits, and bears his portion of the losses. - Such a company is usually called a.Mutual Insurance Company. The sum paid to the party taking the risk is called the Premium of Insurance, or simply the Premium. The document binding the parties to the contract, is called the Policy of Insurance, or simply the Policy. The party that undertakes to indemnify is called the Insurer, or underwriter after he has written his name at the foot of the policy. The person or party guaranteed is called the Insured. As there are many different kinds of things that may be at stake or risked, so there are different kinds of insurance which may be classified under three heads. Fire Insurance, including all cases on land where property is exposed to the risk of being destroyed by fire, such as dwelling houses, stores and factories. lMarine Insurance.-This includes all insurances on ships and cargoes. Such an insurance may be made on the ship alone, and in that case it is sometimes called hull insurance, and sometimes bottomry, the ship's bottom representing the whole ship, just as we say fifty sail for fifty ships. The insurance may be made on the cargo alone, and is then usually called Cargo Insurance. It may be made on both ship and cargo, in which case the general term JlMarine Insurance will be applicable. This kind, as the name implies, insures against all accidents by sea. Life Insurance.-This is an agreement between two parties, that in case' the one insured should die within a certain stated time, the other shall, in consideration of having received a stipulated sum annually, pay to the lawful heir of the deceased, or some one men 180 ArrTHETIC. tioned in his will, or some other party entitled thereto, the amount recorded in the policy. For instance, a man may, on the occasion of his marriage, insure his life for a certain sum, so that should he die within a certain time, his widow or children shall be paid that sum by the other party. Again, a father may insure the life of his child, so that in case of the child's death within a specified time, he shall be paid the sum agreed upon, or that the child, if it lives to a certain age, shall be entitled to that sum. One person may insure the life of another. Supposing that A owes B a certain sum, there is the risk that A may die before he is able to pay B; another party engages, for a certain yearly sum, to pay B in case A should fail to do so during his life time. In some instances, insurances are effected to gain a support in case of sickness.' Such a contract is called a Health Insurance. Inaurances are now also effected for compensation in case of railway accidents. These we may call Railway Accident Insurances. A policy is often transferred from one party to another, especially as collateral security for debt or some analogous obligation. If the payments agreed upon are not regularly kept up, the policy lapses, that is, becomes null and void, so that the holder of it forfeits not only his claim to the sum insured, but also the instalments previously paid. In many companies a person can insure in such a.way as to be entitled to have a share of the profits. The date at which the system of insurance began cannot be clearly ascertained; but, whatever its date, its origin seems to have been protection against the perils of the sea. We know that it was practised, in a certain way, by the ancient Greeks and Romans. If a Roman merchant sent a cargo to a distant port, he made a contract with some one engaged in such business, that he would advance a certain sum, to be repaid with interest, if the vessel reached her destination in safety, but should the vessel or cargo, or both be lost, the lender was to bear the loss. This was termed respondentia, (a respondence) a term corresponding pretty nearly to the English word repayment. It was lawful to charge interest in such cases, above the legal interest in ordinary cases, on account of the greatness of the risk. The lender of the money usually sent an agent of his own on board the vessel to look after the cargo, and receive the repayment on the safe delivery of the goods. This agent'corrmos ponded pretty nearly to our more modern supercargo. As the art of navigation advanced, and the securities afforded by law became INSURANCE. 181 more stringent, and also facilities of communication increased, this system gradually gave way, and has eventually been supplanted by communications by post, and telegraphic messages to agents at the ports of destination. With regard to the equitableness of insurances, and their utility in promoting commercial exterprise, we may remark that they make the interest of every merchant, the interest of every other. To show this, we may c mpare an insurance office to. a club. Suppose the merchants of a town to form a club, and establish a fund, out of which every member, if a loser, was to be indemnified, it is plain that no loss would fall on the individual, except his share as a member of the club. Even so the insurance system causes that each speculator, by insuring his own stake, contributes so much to the funds of a company, which is bound to indemnify each loser. On the other hand, the insurer or insuring company, gains in this way, that the profits accruing from cases where no loss is sustained, far exceed the cases where loss is sustained, and the trifling expense of insuring is of no moment to the insured, in comparison with the damage of a disastrous voyage, or consuming conflagration. By the insurance system, loss is virtually distributed over a large community, and therefore falls heavily on no individual, from which we draw our conclusion, that it is equivalent to a mutual mercantile indemnification club. We must now show the rules of the club, and principles on which its calculations are made. The principal thing to be taken into account, in all insurances, is the amount of risk. For example, a store, where nothing but iron is kept, would be considered safe; a factory, where fire is used, would be accounted hazardous, and one where inflammable substances are used would be designated extra hazardous, and the rates would be higher in proportion to the increased risks. As, however, the degrees of risk are so very varied, only a rough scale can be made, and hence the estimate is nothing more than a calculation of probabilities. In life insurances, the rates are regulated chiefly by the age, and general health of the individual, and also by the general health of the family relations. Connected with this is the cal-culation of the average length of human life. Almost all the calculations in insurance come under two heads. FIRST, to find the premium of insurance on a given amount, and at a given rate; and, SECONDLY, to find how much must be insured at a 13 182 ARITHMETIC. given rate, so that in case of loss, both the principal and premium may be recovered. As the premium is reckoned as so much by the hundred, insurance is merely a particular case of percentage. Hence to find the premium of insurance on any given amount, at a given rate per cent., we deduce the following RULE. Multiply the given amount by the rate per unit.~ EX A PLES. 1. To find the cost of insuring a block of buildings valued at $2688, at 6 per cent.? Here we have.06 for the rate per unit, and $2688X.06-=$161.28, the answer. 2. What will be the cost of insuring a cargo worth $3679, at 3 per cent.? The rate per unit is.03, and $3679X.03=$110.37, the answer. 3. A gentleman employed a broker to insure his residence and outhouses, valued at $2760, the rate being 8 per cent., and the broker's charge 1- per cent,; how much had he to pay? The cost of insurance is $2760X.08=$220.80, and the brokerage $41.40, which added to $220.80, will give $262.20, the answer. EXERCISES. What will be the premium of insurance on goods worth $1280, at 5~ per cent.? Ans. $70.40. 2. A ship and cargo, valued at $85,000, is insured at 2{ per cent.; what is the premium? Ans. $1912.50. 3. A ship worth $35,000, is insured at 1i per cent., and her cargo, worth $55,000, at 2~ per cent.; what is the whole cost? Ans. $1900.00. 4. What will be the cost of insuring a building valued at $58,000, at 2~ per cent.? Ans. $1450.00. * It is plain that the rate can be found, if the amount and premium are given, and the amount can be found if the rate and premium are given. In the case of insuring property, a professional surveyor is often employed to value it, and likewise in the case of life insurance, a medical certificate is required, and in each case the fee must be paid by the person insured. As 300, the basis of percentage, is a constant quantity, when any two of the other quantities are given, the third can be found. INSURANCE. 183 5. What must I pay to insure a house valued at $898.50, at j per cent.? 6. A village store was valued at $1180; the proprietor insured it for six years; the rate for the first year was 3j per cent., with a reduction of - each succeeding year; the stock maintained an average value of $1568, and was insured each of the six years, at 2} per cent.; how much did the proprietor pay for insurance during the six years? Ans. $397.53. 7. A store and yard -were valued at $1280, and insured at 1 per cent.; the policy and surveyor's fee came to $2.25; what was the whole cost of insuring? Ans. $16.65. 8. W. Smith, Port Huron, requests R. Tomlinson, Toronto, to insure for him a building valued at $976; R. Tomlinson effects the insurance at 43 per cent., and charges | per cent commission; how much has W. Smith to remit to R. Tomlinson, the latter having paid ~he premium? Ans. $46.36. 9. The cost of insuring a factory, valued at $25,000, is $125; what is the rate per cent.? Ans. ~. 10. A 1 per cent. insuring my dwelling house cost me $50; what is the value of the house? Ans. $4000.00. To find how much must be insured for, so that in case of loss, both principal and premium may be recovered. Here it is obvious that the sum insured for must exceed the value of the property in the same ratio that 100 exceeds the rate. EX A P L E. To find what sum must be insured for on property worth $600, at 4 per cent., to secure both property and premium, we have as $100 —4-$96: $100:: $600: F. P.% —6~Ox o-Q$625, the sum required. Taking the rate per unit we find 100- 94 6. 96. This gives the RULE. Divide the value of the property by 1, diminished by the rate per, unit, and the quotient will be the sum required. EXAM PLES. 1. A foundry is valued at $874: for what sum at 8 per cent. must it be insured to secure both the value of the property and the premium? One minus the rate or 1.00-.08=.92, and $874-.92 =$950, the answer. 684: ARITHME'1'I;. The premises of a gunsmith, who sells gunpowder, are valued at $2618.85: for how. much, at 15 per cent., must they be insured in order to recover the value of the property and also the premium of insurance? Subtract.15, the rate per unit, from 1, and the remainder is.85 and $2618.85 —.85 gives $3081, the sum required. EXERCISES 1. A chemist's laboratory and appurtenances are valuec ao $26,250, for what sum should he insure them at 61 per cent., to secure both property and premium? $28,000. 2. A New York merchant sent goods worth $1,186 by water conveyance to Chicago; he insured them from New York to Buffalo at 1 per cent., and from Buffalo to Chicago at 21 per cent., and in both cases so as to secure the premium as well as the cargo; how much did the insurance cost him? Ans. $45.42. 3. A person owned a flour mi!l, valued at $1846.05, which he insured at 1} per cent. He also owned a flax mill, valued at $846.30, which he insured at 2~ per cent., and in both cases at such a sum as to secure both property and premium. Which cost him.most, and how much more? Ans. The flour mill cost him $1.67 more than the other. 4. Collins & Co., of Philadelphia, ordered a quantity of pork from G. S. Coates & Son, Cincinnati, which amounts to $2423.10. They insure it to Pittsburg at i per cent., and from Pittsburg to Philadelphia at 3 per cent., and in all cases so as to secure both the price and premium. How much does the whole insurance come to? Ans. $87.12. 5. In order to secure both the value of goods shipped and the premium, at 1 per cent., an insurance is effected on $1526.72. What is the value of the goods? Ans. $1500.00. 6. The Mechanics' Institute is valued at $18,000; it is-insured at 1 per cent., so that in case of fire, the property and premium may both be recovered. For how mnch is it insured? Ans. $18,227.85. 7. How much must be insured on a cargo worth $40,000, at i per cent., to secure both the value of the cargo and the cost of insurance? Ans. $40,201.00. LIFE INSURANCE. 185 8. The Rossin House, King-street, Toronto, is valued at, say, $150,000, and is insured at 1 per cent, so that in case of another conflagration, both the value of the property and the premium of insurance may be recovered. For how much must it be insured? Ans. $152,671.76, nearly. 9. A jail and court-house, adjoining chemical works, and therefore deemed hazardous, will not be insured under 21 per cent. How much will secure both property and premium, the valuation being $17,550.00? Ans. $18,000.00. 10. A cotton mill is insured for $12,000, at 4 per cent., to secure both premium and property. What is the value of the property? 11. What sum must be insured on a vessel and cargo valued at $40,000, at 5~ per cent., in order to secure both the premium and property? Ans. $42,328.04. 12. How much must be insured on property worth $70,000, at 4~ per cent., to secure both premium and property, a commission of i per cent. having been charged? Ans. $73,848.17. LIFE INSURANCE. A LIFE INSURANCE may be effected either for a term of years or for the whole period of life. The former is called a Temporary Insurance, and binds the insurer to pay the amount to the legal heir or legatee or creditor, if the insured should die within the specified time. The latter is called a Life Insurance, because it is demandable at death, no matter how long the insured may live. The rate per annum that the insured is to pay is reckoned from tables constructed on a calculation of the average duration of life beyond different ages. This calculation is made from statistical returns called BILLS OF MORTALITY, and the result is called THE EXPECTATION OF LIFE. The annual premium is fixed at such a rate as would, at the end of the expectation of life, amount to the sum insured. From tables of the expectation of life other tables are constructed, showing the premium on $100 for one year, calculated on the supposition that it is to be paid annually in advance. 186 ARITHMETIC. LIFE INSURANCE TABLE. Age next Age next 1 year. 7 years. For Life. 1 year. 7 years. For Life. Birthday. Birthday. 15.83.85 1.44 38 1.19 1.28 2.75 16.84.86 1.47 39 1.22 1.31 2.85 17.85.87 1.51 40 1.24 1.36 2.95 18.86.88 1.54 41 1.27 1.41 3.07 19.87.90 1.58 42 1.31 1.47 3.19 20.88.91 1.62 43 1.35 1.54 3.32 21.89.92 1.66 44 1.40 1.62 3.45 22.90.93 1.70 45 1.47 1.71 3.60 23.91.95 1.74 46 1.54 1.80 3.75 24.92.96 1.79 47 1.62 1.90 3.92 25.93.98 1.84 48 1.71 2.02 4.09 26.95.99 1.89 49 1.81 2.14 4.27 27.96 1.01 1.94 50 1.91 2.28 4.46 28.98 1.03 2.00 51 2.03 2.42 4.67 29.99 1.05 2.06 52 2.15 2.59 4.89 30 1.01 1.07 2.12 53 2.29 2.76 5.12 31 1.03 1.09 2.18 54 2.44 2.95 5.36 32 1.05 1.11 2.25 55 2.60 3.15 5.62 33 1.07 1.14 2.32 56 2.78 3.38 5.89 34 1.09 1.16 2.40 57 2.96 3.62 6.19 35 1.11 1.19 2.48 58 3.17 3.87 6.50 36 1.14 1.21 2.56 59 3.39 4.17 6.83 37 1.16 1.24 2.65 60 3.64 4.50 7.18 E X A M P L ES. Supposing a young man, on coming of age, wishes to effect an insurance for $3000 for the whole period of his life. To find the annual premium which he must pay, we look for 21 in the left hand column, and opposite that, in the column headed FOR LIFE, we find the number 1.66, which is the premium for one year on $100, and 6 --.0166 is the premium on $1 for 1 year, and hence $3000X.0166=$49.80, is the whole annual premium. If the insurance is to lastfor seven years only, we find under that heading.92, and -9 —.092, and $3000X.092-$27.60, the annual premium. If the insurance is to be for one year only, we find.89 under hat head, and $3000X.089-$26.70, the premium. LIFE INSURANCE. 187 From these explanations we can now derive a rule for finding the annual premium, when the age of the individual and the sum to be insured for are known. RULE..Find the age in the left hand column of the table, and opposite this in the vertical column for the given period will be found the premium on $100 for one year, and this divided by 100 will give the premium on $1 for one year, and the given sum multiplied by this will be the whole annual premium. EXERCISES. 1. What will be the annual premium for insuring a person's life, who is 18 years old, for $1000 for 7 years? Ans. $8.80. 2. What amount of annual premium must be paid by A. B. Smith, who wishes to insure his life for 7 years for $2000, his age being 25 years? Ans. $19.60. 3. John Jones, 35 years of age, wishes to effect an insurance for life for $1500. What amount of annual premium must he pay? Ans. $37.20. 4. A gentleman in Chicago, 32 years of age, being about to start for Australia, and wishing to provide for his family in case of his death, obtains an insurance for seven years for $3000. What amount of annual premium must he pay? Ans. $33.30. 5. Amos Fairplay, 48 years of age, being bound on a dangerous voyage, and wishing to provide for the support of his widowed mother, in case of accident to himself, insures his life for 1 year for $2500. What amount of premium must he pay? Ans. $42.75. 6. A gentleman, 50 years of age, gets his life insured for $3000, by paying an annual premium of $4.46 on each $100 insured; if he should die at the age of 75 years, how much less will be the amount of insurance than the payments, allowing the latter to be without interest? Ans. $345, 7. A gentleman, 45 years of age, gets his life insured for $5000, for which he pays an annual premium of $180, and dies at the age of 50years. Suppose we reckon simple interest at 7 per cent. on his payments, what is gained by the insurance? Ans. $3911. 188 ARITHMETIC. PROFIT AND LOSS. IN the language of arithmetic, the expression Profit and Loss is usually applied to something gained or something lost in mercantile transactions, and the most important rule relating to it directs how to find at what increased rate above the cost price goods must be sold to produce a fair remuneration for time, labour and expenditure; or, in case of loss by unforeseen cfrcumstanees, to estimate the amount of that loss as a guide in future transactions. There are other cases, however, which we shall consider in detail. CASE I. When the prime cost and selling price are known, to find the gain or loss. RULE. Find, by the rule of practice, the price at the difference between the prime cost and selling price, which will be the gain or loss according as the selling price is greater or less than the prime cost; or, Find the price at each rate, and take the difference. EXAMPPLES. To find what is gained by selling 4 cwt. of sugar, which cost 12~ cents per lb., at 15 cents per lb. Here the difference between the two prices is 21 cents per lb., and 400 lbs., at 21 cents per lb., will give $10. Also, 400 lbs. at 15 cents per lb.-$60, and at 12' cents=-$50, and $60-$50=$10. Again, if 120 lbs. of tobacco be bought at 92 cts. per lb., and, being damaged, is sold at 75 cents per lb., the loss will be a loss of 17 cents in the pound, and 120 lbs., at 17 cents per lb., is $20.40; or, 120 lbs., at 92 cents, will come to $110.40, and at 75 cents, to $90, and $110.40-$90-$20.40. EXERCISES. 1. If 224 lbs. of tea be bought at 60 cents per lb., and sold at 95 cents per lb.; how much is gained? Ans. $78.40. 2. A grocer bought 24 barrels of flour, at $5.80 per barrelj and sold 12 barrels of it at $6.10 per barrel, 9 barrels at $6.20 per barrel, and the rest at $6.25; how much did he gain? Ans. $8.55.'3. If a person is obliged to sell 216 yards of flannel, which cost him $86.40, at 37i- cents per yard; how much does he lose? Ans. $5.40. PROIT AND LOSS. 189 4. If a dealer buys 78 bushels of potatoes, at 62~ cents per bushel, and retails them at 87- cents per bushel; how much does he gain? Ans. $19.50. 5. A wine merchant bought 374 gallons of wine, at $3.20 per gallon, and sold it at $3.35 per gallon; how much did he gain? Ans. $56.10. CASE II. To find at what price any article must be sold, to gain a certain rate per cent., the cost price, and the gain or loss per cent. being known. RULE. Multiply the cost price by 1 plus the gain, or 1 minus the loss. EXAMPLE. If a quantity of linen be bought for 75 cents a yard; at what price must it be sold to gain 16 per cent.? Since 16 per cent. is 16 cents for every dollar, each dollar in the cost price would bring $1.16 in the selling price, so that we have $1.16X.75-.8 7, or 8 7 cents. EXERCISES. 1. Railroad shares being purchased for $2500, and sold at a gain of 20 per cent.; for what amount were they sold? Ans. $3000. 2. A property having been bought for $2000 was sold at a gain of 10 per cent. For what was it sold? Ans. $2200. 3. A horse was bought for $50, but, proving lame, was sold at a loss of 15 per cent. At what price was he sold? Ans. $42.50. 4. Bought a horse for $897 and sold it at a loss of 11 per cent; for what sum was it sold? Ans. $798.33. 5. A merchant buys dry gopds for $1562. and sells them at a profit of 22 per cent. For what were they sold? Ans. $1905.64. CASE III. To find the cost when the selling price and the gain per cent. are known. RULE. Divide the selling price by 1 plus the gain, or 1 minus the loss. To find what was the first cost of a quantity of flour which produced 8 per cent. profit by being sold for $127.44. 13 190 ARITHMETIC. Since the gain is 8 per cent. of the cost, it follows that each dollar laid out has brought in a return of $1.08, and therefore the cost must have been as many dollars as the number of times that 1.08 is contained in 127.44, which is 118, and therefore the first cost must have been $118. EXERCISES. 1. If flaxseed is sold at $17.40 per bushel, and 13 per cent. lost, what was the first cost? Ans. $20.00. 2. A dealer bought 116 hogs for $580, and sold them at a gain of 25 per cent.; at what price did he sell each on an average? 3. If 13 sheep be sold for $52.65, and 25 per cent. gained on the first cost, how much was paid for each at first? Ans. $3.24. 4. If.161 per cent. be lost on the sale of linen at $1.25, what was the first cost? Ans. $1.50. 5. If a quantity of glass be sold for $4, and 10 per cent. gained, for what sum was it bought? Ans. $3.64, nearly. CASE IV. To find the gain or loss per cent. when the first cost and selling price are known. RULE. Divide the gain or loss by the first cost. E X A M P L E. If a web of linen be bought for $20 and sold for $25, what is the gain per cent? Here $5 are gained on $20, and $20 is 5 of $100, therefore $25 will be gained on $100, i. e., 25 per cent. EXERCISES. 1. If a quantity of goods be bought for $318.50, and sold for $299.39, how much per cent. is lost? Ans. 6 per cent. 2. If two houses are bought, the one for $150 and the other for $250; and the first sold again for $100 and the latter for $350, what per cent. is gained on the whole? Ans. 12g. A grocer buys butter at 24 cents per lb. and sells it at 30 cents per lb., what does he gain per cent? Aus. 25. PROFIT AND LOSS. 191 4. If a cattle dealer buys 20 cows, at an average price of $20, and pays 50 cents for the freight of each per railroad, what per cent. does he gain by selling them at $25.621 each? Ans. 25. 5. A tobacconist bought a quantity of tobacco for $75, but a part of it being lost, he sold the remainder for $60: what per cent. did he lose? Ans. 20. CASE v. Given the gain or loss per cent. resulting from the sale of goods at one price, to find the gain or loss per cent. by selling the same at another price. RULE. Find by case II. the first cost, and then by case iv. the gain or loss per cent. on that cost at the second selling price. EXAMPLE. If a farmer sells his hogs at $5 each, and realizes 25 per cent.; what per cent. would he realize by selling them at $7 each. We find by case III., that the cost was $4, and then by case IV. what the gain per cent. would be on the seeond supposition, that is $3 —4=.75, or 75 per cent. EXERCISES. 1. If a grocer sells rum at 90 cents per bottle, and gains 20 per cent.; what per cent. would he gain by selling it at $1.00 per bottle? Ans. 33g. 2. If a hatter sells hats at $1.25 each, and loses 25 per cent.; what per cent. would he lose by selling them at $1.60 each? Ans. 4. 3. If a storekeeper sells cloth at $1.25, and loses 15 per cent.; would he gain or lose, and how much, by selling at $1.65? Ans. He would gain 12 per cent. nearly. 4. A milliner sold bonnets at $1.25, and thereby lost 25 per cent.; would she have gained or lost by selling them at $1.40? Ans. She would have lost 16 per cent. 5. A merchant sold a lot of goods for $480, and lost 20 per cent.; would he have gained or lost by selling them for $720, and how much? Ans. He would have gained 20 per cent. 6. A quantity of grain was sold for $90, which was 10'per cent. less than the cost; what would have been the gain per cent. if it had been sold for $150? Ans. 50. 192 ARITHMETIC. 7. A grocer sold tea at 45 cents per pound, and thereby gained 121 per cent.; what would he have gained per cent. if he had sold the tea at 54 cents per pound? Ans. 35. 8. A farmer sold corn at 65 cents per bushel, and gained 5 per cent.; what per cent. would he have gained if he had sold the corn at 70 cents per bushel? Ans. 13-1. MIISCELLANEOUS EXERCISES. 1. If I buy goods amounting to $465, and sell them at a gain of 15 per cent.; what are my profits? 2. Suppose I buy 4001 barrels of flour, at $16.75 a barrel, and sell it at an advance of 3 per cent.; how much do I gain? Ans. $25.14. 3. If I buy 220 bushels of wheat, at $1.15 per bushel, and wish to gain 15 per cent. in selling it; what must I ask a bushel? 4. A grocer bought molasses for 24 cents a gallon, which he sold for 30 cents; what was his gain per cent.? Ans. 25. 5. A man bought a horse for $150, and a chaise for $250, and sold the chaise for $350, and the horse for 100; what was his gain per cent..? Ans. 12k. 6. A gentleman sold a horse for $180, and thereby gained 20 per cent.; how much did the horse cost him? Ans. $150. 7. In one year the principal and interest of a certain note amounted to $810, at 8 per cent.; what was the face of the note? Ans. $750. 8. A carpenter built a house for $990, which was 10 per cent. less than what it was worth; how much should he have received for it so as to have made 40 per cent.? Ans. $1540. 9. A broker bought stocks at $96 per share, and sold them at $1.02 per share; what was his gain per cent.? Ans. 6{. 10. A merchant sold sugar at 6~ cents a pound, which was 10 per cent. less than it cost him; what was the cost price? Ans. 7| cents per pound. 11. A merchant sold broadcloth at $4.75 per yard, and gained 12. per cent.; what would he have gained per cent. if he had sold it at $5.25 per yard? Ans. 24-13. 12. I sold a horse for $75, and by so doing, I lost 25 per cent.; whereas, I ought to have gained 30 per cent.; how much was he sold for under his real value? Ans. $55. PROFIT AND LOSS. 193 13. A watch which cost me $30 I have sold for $35, on a credit of 8 months; what did I gain by my bargain, allowing money to be worth 6 per cent.? Ans. $3.65. 14. Bought 84 yards of broadcloth, at $5.00 per yard; what must be my asking price in order to fall 10 per cent., and still make 10 per cent. on the cost? Ans. $6.111. 15. A farmer sold land at 5 cents per foot, and gained 25 per cent. more than it cost him; what would have been his gain or loss per cent. if he had sold it at 3~ cents per foot? Ans. 12~ per cent. loss. 16. What must I ask per yard for cloth that cost $3.52, so that I may.fall 8 per cent., and still make 15 per cent., allowing 12 per cent. of sales to be in bad debts? Ans. $5. 17. A merchant sold two bales of cotton at $240 each; for one he received 60 per cent. more than its cost, and for the other 60 per cent. less than its cost. Did he gain or lose by the operation, and how much? Ans. loss $270. 18. Bought 2688 yards of cloth at $2.16 per yard, and sold one-fourth of it at $2.54 per yard; one-third of it at $2.75 per yard, and the remainder at $2,90 per yard. Find the whole gain, and the gain per cent. Ans. $1612.80 and 27-8-8 per cent. 19. A flour merchant bought the following lots: 118 barrels at.................................$9.25 per barrel 212 "............................... 9.50 " 315 "................................ 9.12~ " 400 "........................... 10.00 " The expenses amounted to $29.50, besides insurance at i per cent. At what price must he sell it per barrel to gain 15 per cent? Ans., $11.05. 20. Bought 100 sheep at $5 each; having resold them at once and received a note at six months for the amount; having got the note discounted at the Fourth National Bank, at six per cent., I found I Had gained 20 per cent. by the transaction. What was the selling price of each sheep? Ans., $6.19. 194 ARITHMETIC. ST ORAGE. When a charge is made for the accommodation of having goods kept in store, it is called storage. Accounts of storage contain the entries showing when the goods were received and when delivered, with the number, the description of the articles, the sum charged on each for a certain time, and the total amount charged for storage, which is generally determined by an average reckoned for some specified time, usually one month (30 days). EX A P LES. 1. What will be the cost of storing wheat at 3 cents per bushel per month, which was received and delivered as follows:-Received, August 3rd, 1865, 800 bushels; August 12th, 600 bushels. Delivered, August 9th, 250 bushels; September 12th, 360 bushels; September 15th, 400 bushels, and October 1st, the balance. SOLUTION. 1865. Bush. Days. Bush. August 3. Received...... 800 X 6 4800 in store for one day. " 9. Delivered...... 250 Balance....... 550 X 3 1650 in store for one day. " 12. Received...... 600 Balance.......1150 X 31 -35650 in store for one day. Sept. 12. Delivered...... 350 Balance........ 800 X 3 - 2400 in store for one day. " 15. Delivered....... 400 Balance........ 400 X 16 = 6400 in store for one day. Oct. 1. Delivered....... 400 Total.................................. 50900 in store for one day. 50,900 bushels in store for one day would be the same as 50900-30=-16964 bushels in store for one month of 30 days, and the storage of 1697 bushels for one month, at 3 cents per month, would equal 1697X.03-$50.91. It is customary, in business, when the number of articles upon which storage is to be charged, as found, contains a fraction less STORPAGE. 195 than a half, to reject the fraction; but if it is more than a half, to regard it as an entire article. From the solution.of the foregoing example, we deduce the following RULE. Multiply the number of bushels, barrels, or other articles, by the number of days they are in store, and divide the sum of the products by 30, or the number of days in any term agreed upon. The quotient will give the number of bushels, barrels, or other articles on which storage is to be charged for that term. 2. What will be the cost of storing salt at 3 cents a barrel per month, which was put in store and taken out as follows:-Put in, January 2, 1866, 450 barrels; January 3, 75 barrels; January 18, 300 barrels; January 27, 200 barrels; February 2, 75 barrels. Taken out, January 10, 60 barrels; January 30, 150 barrels; February 10, 190 barrels; February 20, 300 barrels; March 1, 250 barrels; and on March 12, the balance, 150 barrels? Ans. $39.44. 3. Received and.delivered, on account of T. C. Musgrove, sundry bales of cotton, as follows:-Received January 1, 1866, 2310 bales; January 16, 120 bales; February 1, 300 bales. Delivered February 12, 1000 bales; 5March 1, 600 bales; April 3, 400 bales; April 10, 312 bales; May 10, 200 bales. Required the number of bales remaining in store on June 1, and the cost of storage up to that date, at the rate of 5 cents a bale per month. Ans. 218 bales in store; $321.18 cost of storage. 4. W. T. Leeming & Co., Commission Merchants, Albany, in account with A. B. Smith & Co., Oswego, for storage of salt and gunpowder, received and delivered as follows: Received, January 18, 1866, 400 kegs of gunpowder and 50 barrels of salt; January 25, 250 barrels of salt; February 4, 150 barrels of salt, and 50 kegs of gunpowder; February 15, 100 kegs of gunpowder; March 5, 64 kegs of gunpowder; April 15, 50 kegs of gunpowder, and 75 barrels.of salt. Delivered, February 25, 15 kegs of gunpowder, and 40 barrels of salt; March 10, 150 kegs of gunpowder, and 285 barrels of salt; April 20, 200 kegs of gunpowder.; April 125, 50 barrels of salt, and 200 kegs of gunpowder. R.equired the number of barrels of salt and kegs of gunpowder in store May 1, and the bill of storage up to that date. The rate of 196 ARITHMETIC. storage for salt being 3 cents a barrel per month, and for gunpowder 12 cents at keg per month. Ans. In store, 50 barrels of salt and 99 kegs of gunpowder; bill of storage, $200.01. GENERAL AVERAGE, THIS is the term used to denote the contribution of all persons interested in a ship, frdight, or cargo, towards the loss or damage incurred by any particular part of a ship, or cargo, for the preservation of the rest. This sacrifice of property is called jettison, from the goods being cast into the sea to save the vessel; although not only property destroyed in that way is the subject of general average, but also any damages or expenses voluntarily incurred for the good of all. For example, the expense of unloading the cargo that the ship may be repaired; masts or sails cut away and abandoned to save the ship. The only articles exempt from contribution are provisions, wearing apparel of passengers, and wages of the seamen. The owners contribute according to the chear value of the ship and freight at the end of the voyage, after deducting the wages of the crew and other expenses. In New York ~, and in other States -g of gross freight is sometimes deducted for seamen's wages; but as a general custom the exact amount is ascertained and deducted. Goods that have been subject to jettison, and are lost, are valued, when the average is calculated at the place of the ship's destination, at the price they could have sold for there; but when the average is to be ascertained at the port of lading, the invoice price is the standard of value. In making an account of the articles which are to contribute, the property lost or sacrificed must be included, and its owners must suffer the same proportionate loss as the rest. The losses to the different parties interested in the vessel, freight and cargo, are paid by their insurers. When repairs have to be made to a ship-new sails, masts, or rigging, for example-one-third of the expense is deducted on account of 7mlioration, or the improved condition of the ship by these repairs. When the ship is new, and on her first voyage, the full amount of the cxpcnse of repairs is allowed in computation of the loss. GENERAL AVERAGE. 197 EXAMPLE. On the 26th June, 1865, the steamer Cuba left New York for Liverpool with a cargo, as follows:-Shipped by T. A. Collins, $7480; R. Evans & Co., $5365; H. C. Wright, $9218; W. Manning & Co., $11428; E. Carpenter, $7559. When off Sandy Hook.a heavy gale was experienced, during which cargo to the value of $3498 was thrown overboard; of this $1123.40 belonged to R. Evans & Co., and the balance to E. Carpenter. The necessary,repairs of the steamer cost $876, and the expenses in port, while getting repaired, were $253. The steamer was valued at $100,000; gross freight, $4310. The seamen's wages were $860. What was the loss per cent., and what was the loss of each contributory in*terest? SOLUTION. Loss for general benefit. Contributory interests. Cargo thrown overboard,$3498 Value of steamer..........$100,000 Repairs to steamer less i 584 Invoice price of cargo....'41,050 Expenses in port.......... 253 Fr'ght, less seamen's wages 3,460 Total loss............$4335 Total contrib. int....$144,500 $48;35-144,500-=.03 loss per unit, or 3 per cent. $100,000 X.03=$3000.00, steamer's share of loss. 7,480 X.03= 224.40, T. A. Collins' share of loss. 5,365X.03- 160.95, R. Evan & Co.'s share of loss. 9,218X.03= 276.54, H. C. Wright's share of loss. 11,428X.03= 342.84, W. Manning & Co.'s share of loss. 7,559 X.03= 226.77, E. Carpenter's share of loss. 3,450 X.03= 103,50, Freight's share of loss. $4335.00, Total loss. $3000.00-837.00=$2163.00, balance payable by steamer. 1123.40-160.95=_ $962.45,balance receivable by R. Evans & Co. 2374.60-226.77= 2147.83, balance receivable by E. Carpenter. NOTE.-It is evident that since the steamer lost $837 ($584 by repairs, and $253 by expenses),-that the net amount required from the steamer will be $3000-837=$2163. R. Evans & Co. having lost by merchandize being thrown overboard $1123.46, a sum greater than their share of the general loss, so that there must be due them $1123.40 -160.95=$962.45; so also the amount of E. Carpenter's share of the general loss must be deducted from his individual loss in order to find the balance due him. 14 198 ARITHMETIC. RULE. Find the rate per unit of loss, by which mtultiply the value oj at h contributory interest, and the product will be the share of loss to be sustained by each. EXERCISES. 1. The steamship Ocean Queen on her trip from Philadelphia to Liverpool, was crippled in a storm, in consequence of which the captain had to throw overboard a portion of the cargo, amounting in value to $4465.50, and the necessary repairs of the vessel cost $423. The contributory interests were as follows:-Vessel, $30,000; gross freight, $6225; cargo shipped by J. Jones & Co., $3650; by Henry Anderson, $6500; by George Millan, $2000; by J. Foster & Son, $550; by Brown Brothers, $5450; and by Wilson & Carter, $8500. Of the cargo thrown overboard, there belonged to Henry Anderson the value of $3000, and to Brown Brothers the remainder, $1465.50. The cutt of detention in port in consequence of repairs, was $116.50; seaman's wages, $2075. How ought the loss to be shared among the contributory interests? Ans. 8 per cent. 2. The steamer Persia left Boston for Halifax, June 30th, loaded with 7210 bushels of spring wheat, shipped by'J. M. Musgrove, and invoiced at 95 cents per bushel; 4815 bushels of corn, shipped by Thomas A Bryce & Co., and invoiced at 60 cents per bushel; 2180 barrels of flour, shipped by A. B. Smith & Co., and invoiced at $5.50 per barrel. When near Halifax, the steamer collided with the Bay State, and the captain found it necessary to throw overboard 1600 bushels of wheat, 1280 bushels of corn, and 720 barrels of flour. On estimating the proportionate loss, it was allowed that the wheat would have sold in Montreal at an advance of 10 per cent., the corn at an advance of 15 per cent., and the flour for $5 per barrel. The contributory interests were:-Steamer, $95,000; cargo, $; gross freight, $2361.20. The cost of repairs to steamer was $2198.15; cost arising from detention during repairs, $318; seamen's wages, $1252.50. How much of the loss had each contributory interest to bear? 3. The steamer Edith left Baltimore for New Orleans with 7600 bushels of wheat, valued at $1.25 per bushel, shipped by Dunn, Lloyd & Co., and insured in the Hartford Insurance Company at 1j per cent., 9200 bushels of corn, valued at 75 cents per bushel, TAXES AND CUSTOM DUTIES. 199 shipped by J. W. Roe and insured in the ZE tna Insurance Company at 1! per cent.; 14,800 bushels of oats, valued at 37~ cents per bushel, shipped by Morris, Wright & Co., and insured in the Mutual Insurance Company at 1~ per cent.; 1,800 barrels of flour, valued at $5.25 per barrel, shipped by Smith & Worth, and insured in the Beaver Insurance Company at 1; per cent. In consequence of a violent gale in the Gulf of Mexico, it was found necessary to throw overboard the flour, 4,600 bushels of oats, and 3,150 bushels of wheat. The propeller was valued at $45,000, and insured in the Beaver Insurance Company for $12,000, at 2 per cent., and in the Western for $25,000, at 21 per cenit. The gross freight was $4950; seamen's wages, $340, and repairs to the boat, $3953.75; what was the loss sustained by each of the contributory interests, the propeller being on her first trip? TAXES AND CUSTOMS DUTIES. A tax is a money payment levied upon the subjects of a State: or the members of any community, for the support of the government. A tax is either levied upon the property or the persons of inrdividuals. When levied upon the person, it is called apoll tax. It may be either direct or indirect.'When direct, it is levied from the individuals, or the property in the hands of the ultimate owners. When indirect, it is in the nature c-f a customs' or excise duty, which is levied upon imports, or manuilctures, before they reach the consumer, although in the end they are paid by the latter. Customs' duties are paid by the importer cf goods at the port of entry, where a custom-house is stationed, with government employees. called custom-house officers, to collect these dues. Excise duties are those levied upon articles manufactured in the country. An invoice is a complete list of the particulars and prices of goods sent from one place to another. A Specific duty is a certain sum paid on a ton, hundred weight, yard, gallon, &c., without regard to the cost of the article. An ad valorem duty is a percentage levied on the actual cost, or fair market value of the goods in the country from which they are imported. 200 ARITHMETIC. Gross weight is the weight of goods, upon which a specific duty is to be levied, before any allowances are deducted. Net weight is the weight of the goods after all allowances are deducted. Among the allowances made are the following: Breakage-an allowance on fluids contained in bottles or break. able vessels. Draft-the allowance for waste. Leakage-an allowance for waste l;y leaking. Tare and tret are the deductions made for the weight of the case -or barrel which contains the goods. When goods, invoiced at gold value, upon which duty is payable, are imported into this country from any foreign country, the custom house duties are payable in gold, for else manifest injustice might be done. If the duty were payable in greenbacks, it would be necessary, in order to obtain uniformity, either to increase or decrease the rate per cent. of duty, as greenbacks fluctuated in value, compared with gold (the invoice price of the goods), or else the goods imported -would require to be reduced to their value in greenbacks at time of delivery. To avoid all this trouble and confusion, goods that are invoiced at their gold value, the duties are made payable in the same currency. When goods are imported from any country which has a depre-,iated currency, a note is attached to the invoice, certifying the amount of depreciation. This is the duty of the Consul representing the country to which the goods are exported, and residing at the portfrom which they are exported. E XAMPLES. To find the specific duty on any quantity of goods. Suppose an Albany Provision Merchant imports from Ireland 59 casks of butter, each weighing 68 lbs., and that 12 lbs. tare is allowed on each cask, and 2 cents per lb. duty on the net weight. We find the gross is......................59X68 —4012 lbs. " tare is............... 59X 12= 708 lbs. Hence the net weight is................... 3304 lbs The duty is 2 cents per lb........................... 2 The duty, therefore, is........................ 66.08 TAXES AND CUSTOM DUTIES. 201 To find the ad valorem duty on any quantity of goods. Suppose a Troy dry goods merchant to import from Montreal 436 yards.of silk, at $1.75 per yard, and that 35 per cent. dnty is charged on them. Here we find the whole price by the rule of Practice to be $763, then the rest of the operation is a direct case of percentage, aud therefore we multiply $763 by.35, which gives $267.05, the amount of duty on the whole. Hence we have the following RULE FOR SPECIFIC DUTY. Subtract the tare, or other allowance, and multiply tne remainder by the rate of duty per box, gallon, &c. RULE FOR AD VALOREM DUTY. Multiply the amount of the invoice by the rate per unit. EXERCISES. 1. Find the specific duty on 5120 lbs. of sugar, the tare being 14 per cent., and the duty 2j cents per lb. Ans. $121.09. 2. What is the ad valorem duty on a quantity of silks, the amount of the invoice being $95,800, and the duty 62- per cent? Ans. $59,875. 3. At 30 per cent., what is the ad valorem duty on an importation of china worth $1260.? Ans. $378. 3.'What is the specific duty, at 10 cents per lb., on 45 chests of tea, each weighing 120 lbs., the tare being 10 per cent.? Ans. $486. 5. What is the ad valorem duty on a shipment of fruit invoiced at $4560, the duty being 40 per cent.? Ans. $1824. 6. What is the specific duty on 950 bags of coffee, each weighing 200 lbs., the duty being 2 cents per lb., and the tare 2 per cent? Ans. $3724. 7. What is the ad valorem duty on 20 casks of wine, each containing 75 gallons, at 18 cents a gallon? Ans. $270. 8. A. B. shipped from Oswego 24 pipes of molasses, each containing 96 gallons; 2 per cent. was deducted for leakage, and 12 cents duty per gallon charged on the remainder; how much was the duty? Ans. $270.95. 202 AR1ITHIMETIC. 9. Peter Smith & Co., Brooklin, import from Cadiz, 80 baskets of port wine, at 70 francs per basket; 42 baskets of sherry wine, at 35 francs per basket; 60 casks of champagne, containing 31 gallons each, at 4 francs per gallon. The waste of the wine in the casks was reckoned at a gallon each cask, and the allowance for breakage in the baskets was- 5 per cent.; what was the duty at 30 per cent., 183 cents being taken as equal to 1 franc? Ans. $776.54. 10. J. Johnson & Co., of Boston, import from Liverpool 10 pieces of Brussels carpeting, 40 yards each, purchased at 5s. per yard, duty 24 per cent.; 200 yards of hair cloth, at 4s. per yard, duty 19 cwt.; 100 woollen blankets, at 2s. 6d., duty 16 per cent.;. and shoe-lasting to the cost of ~60, duty 4 per cent. Required the whole amount of duty, allowing the value of the pound sterling to be $4.84. Ans. $173.64. 11. John McMaster & Co., of Collingwood, Canada West., bought of A. M. Smith, of Buffalo, N. Y., goods invoiced at $5440.50, which should have passed through the custom-house during the first week in May, when the discount on American invoices was 43{ per cent., but they were not passed until the fourth week in May, when the discount was 36; per cent. The duty in both cases being 20 per cent.; what was the loss sustained by McMaster & Co. on account of their goods being delayed? Ans. $70.60. STOCKS AND BONDS. CAPITAL is a term generally applied to the property accumulated by individuals, and invested in trade, manufactures, railroads, buildings, government securities, banking, &c. The capital of incorporated companies is generally termed its "capital stock," and is divided into shares; the persons owning one or more of these shares, being called stockholders. The shares in England, are usually ~100, ~50, or ~10 each. In the United States they are generally $100, $50, or $10 each. The management of incorporated companies is generally vested in officers and directors, as provided in the law or laws, who are elected by the stockholders or shareholders; each stockholder, in most cases, being entitled to as many votes as the number of shares he holds; but sometimes the holder of a few shares votes in a larger proportion than the holder of many. The accumulating profits which are distributed among the stockholders, once or twice a year, are called "dividends," and when' declared," are a certain percentage of the par value of the shares. In aininm, ail somelo other companies, where the shares are only a STOCKS AND BONDS. 2)3 few dollars each, the dividend is usually a fixed sum "per share." Certificates of stock are issued by every company, signed by the proper officers, indicating the number of shares each stockholder is entitled to, and as an evidence of ownership; these are transferable, and may be bought and sold like any other property. When the market value equals their nominal value they are said to be "at par." When they sell for more than their nominal value, or face, they are said to be above par, or at a " premium"; when for less, they are below par, or at a " discount." Quotations of the market value are generally made by a percentage of their par value. Thus, a share which is $25 at par, and sells at $28, is quoted at twelve per cent. premium, or 112 per cent. When states, cities, counties, railroad companies, and other corporations, borrow large amounts of money, for the prosecution of their objects, instead of giving common promissory notes, as with the mercantile community, they issue bonds, in denominations of convenient size, payable at a specified number of years, the interest usually payable semi-annually at some well known place. These are usually payable to " bearer," and sometimes to the " order" of the owner or holder. When issued by Governments or States, these bonds are frequently called Government stocks or State stocks, under authority of law. To these bonds are attached, what are called " coupons," or certificates of interest, each of which is a due bill for the annual or semi-annual interest on the bond to which it is attached, representing the amount of the periodical dividend or interest; which coupons were usually cut off, and presented for payment as they become due.. These bonds and coupons are signed by the proper officers, and like certificates of capital stock, are negotiable by delivery. The loan is obtained by the sale of the bonds, with coupons attached, but they are sometimes negotiated at par. Their market value depends upon the degree of confidence felt by capitalists of their being paid at maturity, and the rate of interest compared with the rate in the market. Treasury notes are issued by the United States Government, for the purpose of effecting temporary loans, and for the payment of contracts and salaries, which resemble bank notes, and are made payable without interest generally. Recently such notes have been issued bearing one year or three years' interest. "Consols" is a term abbreviated from the expression "consolidated," the British Government having at various times borrowed money at different rates of interest and payable at different times, " consolidated" the debt or bonds thus issued, by issuing new stock, drawing interest at three per cent, per annum, payable semi-annually, and redeemable only at the option of the Government, becoming practically perpetual annuities. With the proceeds of this, the old stock was redeemed. The quotations of these three per cent. perpetual annuities, or " consols," indicate ordinarily the state of the 204 A"ITHMETIC. money market, as they form a large portion of the British public debt. " Mortgage Bonds" are frequently issued by owners of real property, with coupons attached, which render the bonds more saleable as well as more convenient for the collection of interest. "Coupon Bonds," being negotiable by delivery, are payable to the holder; and in case of loss or theft, the amount cannot be. recovered from the government or corporation issuing them, unless ample notice is given of the loss. "Registered Bonds" are those payable only to the "order" of the holder or owner, and are more safe for investment. By law, stockholders are liable for the whole debts of the corporation, in case of failure. In some States the law provides that they are liable only to an amount equal to their stock. In England the statute provides for "Limited" liability, by an Act passed in 1862 termed the " Limited Act." CASE I. The premium or discount being known, to find the market valueof any amount of stock. EXAM PLES. If G. W. R. shares are at 7 per cent. premium, to find the value' of 30 shares of $100. Here it is plain that each $100 will bring $107, and that each $1 will bring $1.07, and as the par value is $3000, the advanced value will be 3000 times 1.07, which gives $3210, the market value, and $3210-$3000=$210, the gain. Again, if the same are sold at a discount of 7 per cent., it is plain that each $100 would bring only $93, and therefore each $1 would bring only $0.93, and therefore as the par value is $3000, the depreciated value will be 3000 times.93, which gives $2790, and therefore the loss would be $3000-2790-210. From this we derive the It ULE. Multiply the par value by 1 plus or minus tne rate per unit, according as the shares are at a remium or a discount. STOCKS AND BONDS 205 EX ERCISE S. 1. What is the market value of $450 stock, at 8- per cent. discount? Ans. $411.75. 2. What is the value of 29 shares of $50 each, when the shares are 11 per cent. below par? Ans. $1290.50. 3. A man purchased 60 shares of $5 each, from an oil well company, when the shares were at a discount of 8 per cent., and sold them when they were at a premium of 10 per cent; how much did he gain? Ans. $54. 4. A man purchased $10,000 stock when it was at an advance of 8 per cent., and sold when it was at a discount of 8 per cent.; how much did he lose? Ans. $1600. 5. If a man buys 15 shares of $100 each, when the shares are at a premium of 5 per cent., and sells when they have advanced to 12 per cent., how much does he gain? Ans. $105. CASE II. To find how much stock a given sum will purchase at a given premium or discount. Let it be required to find how much stock can be purchased for $21,600 when at a premium of 8 per cent. In this case it will require $108 to purchase $100 stock, and therefore $1.08 to purchase $1 stock, and hence the amount that can be purchased for $21600 will be represented by the number of times that $1.08 is contained in 21600, which gives $20000. Again: Let it be required to find how much stock can be purchased for $5520, when at a discount of 8 per cent. When stocks are 8 per cent. below par, $92 will purchase $100 stock, and therefore $0.92 will purchase $1, and hence the amount that can be purchased for $5520 will be represented by the number of times that.92 is contained in 5520, which gives $6000 stock. Hence we derive the U L E. Divide the given sum by 1 plus or minus the rate per unit, according as the shares are at a premium or a discount. EXERCISES. 6. When stocks arc at a premium of 12 per cent., how much can be purchased for $8064? Ans. $7200. 14 206 ARITHMETIC. 7. When stocks are at a discount of 9 per cent., how much can be bought for $3640? Ans. $4000. 8. When G. T. R. stock is at 18 per cent. below par, how much can be bought for $42,640. Ans. $52000. 9. When G. W. R. stock is at a premium of 9 per cent., how much will $4578 purchase? Ans. $4200. 10. When government stock is selling at 92i, what amount of stock will $28,675 purchase, and to what will it amount with brokerage at i per cent.? Ans. $31077.50. CASE III. The premium or discount being known, to find the par value. To find the par value of $1,296, when stock is at a premium of 8 per cent. At 8 per cent. premium, each $1 brings $1.08, hence the par value will be represented by the number of times 1.08 is contained in 1296, which gives $1200 for the par value. To find the par value of $1104, when stock is at a discount of 8 per cent. Each $1 will bring $0.92, and therefore the par value will be represented by the number of times that.92 is contained in 1104, which gives $1200, the par value. Hence the RULE. Divide the market value by 1 plus or minus the rate per unit, according as the stocks are selling above or below par. EXERCISES. 11. What is the par value of $24420, when stock is 11 per cent. above par? Ans. $22000. 12. What is the par value of $10800, when stocks are at a discount of 4 per cent.? Ans. $11250. 13. When government stocks are at 6 per cent. premium; how much will $20246 purchase at par value? Ans. $19100. 14. The shares in a canal company are at 15 per cent. discount; how many shares of $100 will $11390 purchase? Ans. 134. 15. The shares of a British gas company -were selling in 1848, at a discount of 12 per cent.; a speculator purchased a certain number of shares for ~792; the value of the shares suddenly rose to par; how many shares did he purchase, and how much did he gain? Ans. 9 shares; ~108 gain. STOCKS AND BONDS. 207 CASE IV. To find to what rate of interest a given dividend corresponds. If a person receives a dividend of 12 per cent. on an investment made at 20 per cent. above par, the corresponding interest may be calculated thus: As the stock was bought at 20 per cent., or.20 above par, $1.20 of market value corresponds to $1 of par value, and as every $1 of par value correspondT to 12 per cent. interest, or.12, it follows that the per cent. which was invested will be represented by the number of times that 1.20 is contained in.12, which is.10 or 10 per cent. Hence the RULE. Divide the rate per unit of dividend by 1 plus or minus the rate per cent. premium or discount, according as the stocks are above or belovw p.ar.* EXERCISES. 16. If a dividend of 10 per cent. be declared on stock vested at 25 per cent. advance; what is the corresponding interest? Ans. 8 per cent. 17. If a dividend of 4 per cent. be declared on stock invested at 12 per cent. below par, what is the corresponding interest? Ans. 4iT. 18. If money invested at 24 per cent. yields a dividend of 15 per cent., what is the rate of interest? Ans. 12-3. 19. If railroad stock is invested at 18 per cent. above par, and a dividend of 6 per cent. be declared, what is the rate of interest*? Ans. 55g. 20. If bank stock be invested at 15 per cent. below par, and a dividend of 10 per cent. declared, what is the rate of interest? Ans. 11-}. MISCELLANEOUS EXERCISES. 1. What must be paid for 20 shares of railway stock, at 5 per cent. premium, the shares being $100 each? Ans. $2100. * To find at what price stock paying a given rate per cent. dividend can be purchased, so that the money invested shall produce a given rate of interest, divide the rate per unit of dividend by the rate ver unit of interest. 208 ARITHMETIC. 2. What is the par value of bank stock worth $8740, at a premium of 15 per cent.? Ans. $7600. 3. Railway stock was bought at 15 below par, for $1895.624; how many shares were there, each share being $150? Ans. 15 shares. 4. If 6 per cent. stock yields 8 per cent. on an investment, at what per cent. discount was it bought? Ans. 25. 5. If bank stock which pays 11 per cent. dividend, is 10 per cent. above par, what is the corresponding rate of interest on any investment? Ans. 10. 6. When 1 per cent. stocks were at 17J discount, A bought $1000; how much did he pay, and how much did he gain by selling when stock had risen to 861? Ans. $821.25, and $41.25. 7. What will $850 bank stock cost at a discount of 9g per cent., I per cent. being charged for brokerage? Ans. $771.38. 8. On the data of the last example, how much would be lost by selling out at 10~ per cent.? Ans. $10.03. 9. What incomre should I get by laying out $1620 in the purchase of 3 per cent. stock at 81? Ans. $60. 10. What sum must be invested in the 4 per cent. stocks at 84, to yield an income of $280? Ans. $5880. 11. What rate of interest' will a person receive by investing in the 4~ per cent. stocks at 90? Ans. 5 per cent. 12. A person transfers his capital from the 3~ per cent. stocks at 77, to the 4 per cent. at 89; what is the increase or decrease per cent. in his income? Ans. Decrease 25. 13. A person sells out of the 3 per cent. stock at 96, and invests his money in railway 5 per cent. stock at par; how much per cent. is his income increased? Ans. 60. 14. What must be the market value of 5j per cent. stock, so that after deducting an income tax of 2 cents on the dollar, it may produce 5 per cent. interest? Ans, 1074. 15. A gentleman invested $7560 in the 34 per cent. stocks at 94~, and on their rising to 95 sold out, and purchased G. T. R. 4 per cent. stock at par; what increase did he make in his annual income? Ans. $24. 16. How much more may a person increase his annual income by lending $3800, at 6 per cent., than by purchasing railway 5 per cent. stock at 95? Ans. $28. PARTNERSHIP. 209 17. A person sells $4200 railway stock which pays 6 per cent. at 115, and invests one-third of the proceeds in the 3 per cent. consols at 80k, and the balance in savings' bank stock, which pays 9 per,ent. at par; what is the decrease or increase of his annual income? Ans. Increase $97.80. 18. A person having $10,000 consols, sells $5000 at 947, and on their rising to 98J he sells $5000 more; on their again rising he buys back the whole at 96; how much does he gain? Ans. $75. 19. The sum of $4004 was laid out in purchasing 3 per cent. stocks at 89j, and a whole year's dividend having'been received upon it, it was sold out, the whole increase of capital being $302.40; at what price was it sold out? Ans. 93k. 20. Suppose a person to have been an original subscriber for 500 shares of $50 each, in the First National Bank, payable by instalments, as follows: — in three months, which he sold for 51 per cent. advance; ~ in 6 months, which brought him 63 per cent. advance, and the balance in nine months, which he was compelled to sell at 8j per cent. discount; what did he gain br the whole transaction? Ans. $808.33. PARTNERS HIP. Partnership has been defined to be the result of a contract, under which two or more persons agree to combine property, or labour, for the purpose of a common undertaking, and the acquisition of a common profit. A dormant, or sleeping partner, is one who shares in the concern, but does not appear to the world as such. A nominal partner is one who lends his name and credit to a firm, without having any real interest in the profits. All the partners may contribute equally to the business; or the capital may be contributed by some or one, and the skill and labour by the other. Or, unequal proportions may be furnished by each. The contract need not be in writing, but all parties to be bound must assent to it, and it is usually contained in an instrument called "Articles of Partnership." A dissolution can take place at any time by mutual consent. A partnership at willis one in which there is no limited time affixed for its continuance, and the whole firm may be dissolved 210 ARITHMETIC. by any of its members at a moment's notice. A document is, however, generally drawn up and signed upon a dissolution, called a settlement, which, contains a statement of the mode of adjustment of the accounts, and the apportionment of profits or losses. E X A M P L E. Two persons, A. and B., enter into partnership. A. invests $300 and B. $400. They gain during one year $210; what is each man's share of the profit? SOLUTION BY PROPORTION. A.'s stock, $300 B.'s " 400 Entire stock $700: 300:: $210: $90 A.'s gain. "t " 700: 400::$210: 120 B.'s " SOLUTION BY PERCENTAGE. Since the entire amount invested is $700, and the gain $210, the gain on every $1 of investment will be represented by the number of times that 700 is contained in $210, which is.30 or 30 cents on the dollar. Now if each man's stock be multiplied by.30 it will represent his share of the gain thus: $300X.30=$ 90 A.'s gain. 400X.30 — 120 B.'s " Entire stock....... 700 210 Entire gain. Hence,-To find each partner's share of the profit or loss, when there is no reference to time, we have the following RULE. As the whole stock is to eachpartner's stock, so is the whole gain or loss to each partner's gain or loss; or, divide the whole gain or loss by the number denoting the entire stock, and the quotient will be the gain or loss on each dollar of stock; which multiplied by the number denoting each partner's share of the entire stock, will give his share of the entire gain or loss. E XERCIS ES. 1. Three, persons, A., B., and C., enter into partnership. A. advances $500, B. $550, and 0. $600; they gain by trade $412.50. What is each partner's share of the profit? Ans. A.'s $125; B.'s $137.50; C.'s $150. PARTNERSHIP. 211 2, A, B, C and D purchase an oil well. A pays for 6 shares, B for 5, 0 for 7, and D for 8. Their net profits at the end of three months have amounted to $7800; what sum ought each to receive? Ans. A, $1800; B, $1500; C, $2100; D, $2400. 3. A and B purchased a lot of land for $4500. A paid i of the price, and B the remainder; they gained by the sale of it 20 per cent.; what was each man's share of the profit? Ans. A, $300; B, $600. 4. A captain, mate, and 12 sailors, won a prize of $2240, of which the captain took 14 shares, the mate 6, and the remainder was equally divided among the sailors; how much did each receive? Ans. The captain, $980; the mate, $420; each sailor, $70. 5. A and B invest equal sums in trade, and clear $220, of which A is to have 8 shares on account of transacting the business, and B only 3 shares; what is each man's gain, and what allowance is made A for his time? Ans. Each man's gain $60; A $100 for his time. 6. A, B, C and D enter into partnership with a joint capital of $4000, of which A furnishes $1000; B $800; C $1300, and D the balance; at the end of nine months their net profits amount to $1700; what is each partner's share of the gain, supposing B to' receive $100 for extra services? Ans. A, $400; B, $320; C, $520; D, $360. 7. Six persons, A, B, C, D, E and F, enter into partnership, and gain $7000, which is to be divided among them in the following manner:-A to have ^; B, - C, A as much as A and B, and the remainder to be divided between D, E and F, in the proportion of 2, 2- and 31; how much does each partner receive? Ans. A, $1400; B, $1000; C, $800; D, $950; E, $1187.50; F, $1662.50. 8. A, B and C enter into partnership with a joint stock of $30,000, of which A furnished an unknown sum; B furnished 1, and C 11 times as much. At the end of six months their profits were 25 per cent. of the investment; what was each man's share of the gain? Ans. A's, $2000; B's, $3000; and C's, $2500. 9. A, B, C and D trade in company with a joint capital of $3000; on dividing the profits, it is found that A's share is $120; B's, $255; C's, $225; and D's, $300; what was each partner's stock? Ans. A's, $400; B's, $850; C's, $750; and D's, $1000. 10. Three labouring men, A, B and C, join together to reap a certain field of wheat, for which they agree to take the sum of 212 ARTHMETIC. $19.84; A and B calculate that they can do 4 of the work; A and C *; B and C 3 of it; how much should each receive according to these estimates? Ans. A, $8.32; B, $7.04; and C, $4.48. To find each partner's share of the gain or loss, when the capital is invested for different periods. EXAMPLE. Two merchants, A and B, enter into partnership. A invests $700 for 15 months, and B $800 for 12 months; they gain $603; what is each man's share of the profits? SOLUTION. $700X15=-$10500 $800X12- 9600 20100: 10500:: $603: $315 A's gain. 20100: 9600:: $603: $288 B's gain. The reason for multiplying each partner's stock by the time it was in trade, is evident from the consideration that $700 invested for 15 months would be equitalent to $700X15 equal to $10500 for one month, that is $10500 would yield, in one month, the same interest that $700 would in fifteen months. Likewise $800 invested for 12 months would be the same as $9600 for one month; hence the question becomes one of the previous case, that is, their investments are the same as if they had invested respectively $10500 and $9600 for equal times; hence the RULE. Multiply each man's stock by the time he continues it in trade; then say, as the sum of the products is to each particular product, so is the whole gain or loss to each man's share of the gain or loss. EXERCISES. 11. A, B and C are associated in trade. A furnished $300 for 6 months; B, $350 for 7 months, and C, $400 for 8 months. Their profits amounted to $1490 at the time of dissolution; what was the profit belonging to each partner? Ans. A, $360; B, $490; C, $640. PARTNERSHIP. 213 12. A, B and C contract to perform a certain piece of work; A employs 40 men for 4i months; B 45 men for 3- months, and C 50 men for 21 months. Their profits, after paying all expenses, are $850; how much of this belongs to each? Ans. A, $340; B, $297.50; 0, $212.50. 13. Four men, A, B, C and D, hired a pasture for $27.80; A puts in 18 sheep for 4 months; B, 24 for 3 months; C, 22 for 2 months; and D, 30 for 3 months; how much ought each to pay? Ans. A and B each, $7.20; C, $4.40; D, $9. 14. On the first day of January A began business with a capital of $760, and on the first of February following he took in B, who invested $540; and on the first of June following they took in C, who put into the business $800. At the end of the year they found they had gained $872; how much of this was each man entitled to? Ans. A, $384.93; B, $250.71; C, $236.36. 15. Three merchants, A, B and 0, entered into partnership with a joint capital of $5875, A investing his stock for 6 months, B his for 8 months, and C his for 10 months; of the profits each partner took an equal share; how much of the capital did each invest? Ans. A, $2500; B, $1875; C, $1500. 16. Two merchants, A and B, entered into partnership for two years; A at first furnished $800, and at the end of one year, $500 more; B furnished at first $1000, at the end of 6 months, $500 more, and after they had been in business one year, he was compelled to withdraw $600 At the expiration of the partnership their net profits were $2550; how much must A pay B who wishes to retire from the business? Ans. $2190. 17. Three persons, A, B and C, form a partnership for one year, commencing January 1st, 1865; A puts in $4000; B, $3000; and C, $2500; April 1st, A withdraws $500, and B withdraws $600; June 1st, C puts in $800 more; September 1st, A furnishes $700 more, and B $400 more. At the end of the year they find they have gained $1500; what is each partner's share of it? Ans. A, $608.68; B, $423.31; C, $468.01. 18. John Adams commenced business January first, 1865, with a capital of $10000, and after some time formed a partnership with William Hickman, who contributed to the joint stock the sum of $2800 cash. In course of time they admitted into the firm Joseph Williams, with a stock worth $3600. On making a settlement January first, 1866, it was found that Adams had gained $2250; 15 214 AFITTIMETIC. Hickman, $420; and Williams, $405; how long had Hickman's and Williams' money been employed in the business, and what rate of interest per annum had each of the partners gained on their stock? Ans. Hickman's 8 months; Williams' 6 months. Gain, 22~ per cent. interest. BANKRUPTCY. When any person is unaDle to meet his liabilities, he makes an assignment of his property to some other person or persons, called official assignee or assignees, whose office it is to distribute the available property, after paying expenses, rateably among the creditors. An allowance for maintenance is generally made to the insolvent, but sometimes lie is compelled to surrender all his estate, but only in case of manifestfraud, which the word bankrupt originally implied, though now it is used as nearly synonimous with insolvent. The property to be divided is called the assets. The shares of the property which are divided among creditors, are called dividends. EXAMPLE. A bankrupt owes A $400; B. $350, and C, $600; his net assets amount to $810 cash; how much is hle able to pay on the $1, and how much will each creditor receive? SOLUTION. $400 —$350+$600-=$1350, total liabilities. Now, if he nas $1350 to pay, and only $810 to pay it with, he will only be able to pay $810- 1350=.60 or 60 cents on the $1. Therefore, A will receive $400X.60-$240; B, $350 X.60=$210, and C, $600 X.60 =$360. Hence the RULE. Divide the net assets by the number denoting the total amount of the debts, and the quotient will be the sum to be paid on each dollar, then multiply each man's claim by the sunmpaid on the dollar, and the product will be the amounet he is to receive. BANKRRUPTCY. 215 EXERCISES. 1. A becomes bankrupt. He owes B, $800; C, $500; D, j1100, and E, $600. The assets amount to $1110; how much can he pay on the dollar, and how much does each creditor receive? Ans. He can pay 37 cents on the dollar, and B receives $296; C, $185; D, $407; and E, $222. 2. A house becomes bankrupt; its liabilities are $17940; its assets are $8970; what is the dividend, and what is the shase of the chief creditor to whom $1282 are due? Ans. The dividend is 50 cents on the dollar, and the principal creditor gets $641. 3. A shipbuilder becomes bankrupt, and his liabilities are;303000; the premises, building and stock are worth $220000, and he has in cash and notes $12875; the creditors allow him $3000 for maintenance of his family; the costs are 31 per cent. of the amount available for the creditors; what is the dividend, and how. much does a creditor get to whom $1360.60 are due? Ans. The dividend is 75 cents on the dollar, and the creditor specified gets $1020. 4. Foster & Co. fail. They owe in Alhany, $22000; in Baltimore, $18000; in Philadelphia, $17100; in Charleston, $16000; in Boston, $4400, and in Newark, $4200. Their assets are: house property, $14000; farms, $2200; cash in bank, $4400; railway stock, $4200; sundry sums due to them, $20135; what is the dividend, and how much goes to each city? Ans. Dividend, 55 cents on the dollar; to be paid in Albany, $12100; in Baltimore, $9900; in Philadelphia, $9405; in Charleston, $8800, in Boston, $2420; in Newark, $2310. 5. The firm of Reuben Ring & Nephews becomes bankrupt. It owes to Buchanan & Ramsay, $1080; to Kinneburgh & McNabb,. $850; to Collier Bros., $1720; to David Bryce & Son, $1580; to: Sinclair & Boyd, $970. The assets are: house and store, valued at $848; merchandise in stock, $420; sundry debts, $220. What can the estate pay, and what is the share of each creditor? Ans. The estate pays 24 cents on the dollar, and the payments are: to Buchanan & Ramsay, $259.20; to Kinneburgh & McNabb, $204; to Collier Bros., $412.80;.to David Bryce & Son, $379.20; to Sinclair & Boyd, $231.80. 216 ARTHMETIC. EQUATION OF PAYMENTS. Equation of Payments is the process of finding the average or mean time at which the payment of several sums, due at different times, may all be made at one time, so that neither the debtor nor creditor shall be at any loss. The date to be found is called the equated time. The mode of finding equated time almost universally adopted is very simple, though, as we shall show in the sequel, not altogether correct. It is known as the mercantile rule. Let us observe, in the first place, that the standard by which men of business reckon the advantage that accrues to them from receiving money before the time fixed for its payment, and the loss they sustain by the -payment being deferred beyond the appointed time, is the interest of money for each such period. Thus, if $50 be a year overdue, the loss is $,; at 6 per cent.; and, if $50 be paid a year in advance of the time agreed upon, the gain to the payee is $3, at the same rate. In the former case, the person receiving the money charges the payer $3 interest for the inconvenience of lying out of his money, but, in the latter case, he deducts $3 from the debt, for the advantage of having the money in hand. If, on the 1st May, A gives B two notes, one for $50, at a term of three months, and the other for $80, at a term of seven months, the first will be legally.due on the 1st August, and the 2nd on the 1st December; but A is not able to meet the first at August, and it is held over till the 1st November, when A finds himself in a position to pay both at once. The first is then three months over-due, and accordingly B claims interest for that time, which, at 6 per cent., is 75 cents, but as A tenders payment of the whole debt at once, and the second note will not be due f6ioanother month, A claims a deduction of one month's interest, which, at the same rate, is 40 cents, and accordingly A, in addition to the debt, pays B 35 cents. Let us now suppose another case. A owes B $130, as before, and he gives B two notes-one for $50, on 1st May, at 3 months, and another, on the 6th May, for $80, at 8 months. The first falls due on 1st August, and the other on the 6th January, but A and B agree to settle at such a time that neither shall have interest to pay, but that A shall simply have to pay the principal. Supposing that a settlement is made on 6th November, we find that the 1st note is EQUATION OF PAYMENTS. 217 3 months and 6 days over due, and the interest on it for that period is 80 cents, while the second will not be due for 2 months, and the interest on it for that period is also 80 cents; consequently, the interest that A should pay, and that which B should allow being equal, they balance each other, and the principal only has to be paid. There are, then, three methods for the payment of several debts, or a debt to be paid by instalments. The first is to pay each instalment as it becomes due. This needs no elucidation, nor is it often practised, except in the case of small debts, due by persons of contracted means. The second is what has been illustrated above by the first example, viz., that interest is added for overdue money, and deducted for sums paid in advance of the stipulated time. The third has been illustrated by the second example, viz., to fix on such a time that the interests on the overdue and underdue sums shall be equal, so that the debtor has only to give the principal to the creditor. If, in this last case, the time should come out as ji mixed number, the fraction must be taken as another day, or thrown off, making the payment fall due a day earlier. The principle on which all such settlements are made is, that the interest of any sum paid in advance of a stipulated time is equivalent to the interest of the same sum overdue for a like time. With thlse explanations we are now ready to investigate a rule for the Equation of Payments. For this purpose let us suppose a case. PR. Evans owes J. Jones $200, which he undertakes to pay by two'instalments of $100 each, (basis of interest 6 per cent.,) thle first payment to be made at once, and the second at the expiration of two years. But the first payment is not made till the end of the first year, at which time R. E. tenders payment of the whole amount. For the accommodation of having the first payment deferred for one year he is to pay $6, i. e., $106 in all, and in return for making the second payment a year before it is due, he claims a discount at the same rate, which gives $6. He has therefore, by the mercantile rule, to pay $106+94 —$200, so that the $6 in the latter case balances the $6 in the former. This takes one year as the equated time, and is the mode usually adopted on account of its simplicity, though not strictly accurate. To find the equated time when there are several payments to be made at different dates. 218 ARITHMETIC. If A owes B $300, payable at the end of 4 months; $500, payable at the end of 6 months, and $400, payable at the end of 10b months, to find at what time the whole may be paid, so that interest shall be chargeable to neither party. The interest of $300 for 4 months is the same as the interest of $1 for 1200 months; the interest of $500 for 6 months is the same as the interest of,$1 for 3000 months, and the interest of $400 for 101 months is the same as the interest of $1 for 4200 months. The sum of all these is 8400 months, and the interest of the whole is the same as the interest of $1 for 8400 months, and if $1 requires 8400 months to produce a certain interest, the sum of all the principals will require only the -J-Ao part of 8400 months to produce the same interest, and 8400-1200=-7, and hence the equated time is 7 months. RULE. lMultiply each payment by the time that must elapse before it becomes due, and divide the sum of these products by the sum of the payments. E X A 3I PL E. To find the equated time for the payment of three debts, the first for $45, due at the end of 6 months; the second for $70, due at the end of 11 months, and the third for $75, due at the end of 13 months. $45X 6-$270 70X11= 770 75X13- 975 190 2015 and 2015 —19-10|-, so that the equated time will be 10 months and 18 days, the small remaining fraction being rejected. Let us suppose that nothing is paid until the end of the 13 months, and all paid at once, then the amount to be paid will be, at 6 per cent., For first debt overdue 7 months, $45+1.57-, interest for 7 months.$...457...................................... $46.57 For second debt overdue 2 months, $70-+.70, interest for 2 months............................................... 70.70 For third debt just due, $75, no interest..................... 75.00 $192.271 EQUATION OF PAYMENTS. 219 The work may often be somewhat shortened by counting the differences of time from the date at which the first payment becomes due, the mean time between the dates when the first and last become due being alone required. If a person owes $1200 to be paid in four instalments, $100 in 3 months; $200 in 10 months; $300 in 15 months, and $600 in 18 months, then the excesses of time of the last three above the first are 7, 12 and 15 months, and the work will stand as below. $100 (no time.) 200X 7-1400 300 X 12-3600 600 X 15=9000 1200) 14000(11~ and 11jX3=14' mnonths. This gives the RULE. Multiply each debt, except the one first due, by the difference be-.tween its term and the term of the first; divide the sum of the products by the sum of the debts, the quotient with the term of the first added to it will be the equated time. Another method, which is often convenient, may be illustrated by the example already given, as the two operations will give the same result. Interest on $300 for 4 months=$ 6.00 Interest on 500 for 6 " - 15.00 Interest on 400 for 101 - 21.00 Interest on 1200 for 1 month=6)42.00(7 months as before. R U L E. Find the interest on each instalment for the given time, and divide the sum of these by the interest of the whole debt for one month, and the quotient will be the equated time. As the sum of the instalments is equal to the debt, the result will be the same for any rate of interest. For the first instalment, $300, overdue 3 months, A has to pay............................................................. $4 50.For the second instalment, $500, overdue 1 month, A has to pay.......................................................... 2 50 $7 00 220 ARITHMETIC. For the third instaiment, $400, not due for 3} months, A has to get...................................................... $7 00 so that the amounts of interest exactly balance, and the paying of the whole, at the end of 7 months, is precisely equivalent to the paying of each instalment as it falls due. The only difference that could arise is, that it might be inconvenient for the creditor to lie out of the first instalment for the three months. In all other respects the settlement is strictly equitable, according to the understanding that exists among business men. In the first place, the difference between this and what is called " the accurate rule," is insignificantly small; and, in the second place, the C mercantile rule" saves much time, and time is equivalent to so much capital in mercantile transactions. Independently, however, of any other consideration, we may remark that when the mode of reckoning is conventionally understood, it becomes perfectly equitable, because every merchant knows the terms on which he can do business with any other, just as bank discount becomes -perfectly equitable, because every man, before going to a bank for the discounting of a note, knows perfectly well on what terms he can have it. Much warm discussion has been indulged in on this subject; but, as we consider the discussion more subtle than profitable, we shall dismiss the subject in a few words. We shall adopt the usual case, thatA owes B $200, one-half to be paid at the present time, and the remainder at the end of two years. It is perfectly obvious that, at the end of the first year, A should pay $106, that is, the principal, plus the interest agreed upon. Regarding the settlement of the second instalment, if A proffers payment of the whole at once, he is clearly entitled to claim a reduction for the unexpired term. Now, the question is, what ought the reduction to be. By the mercantile rule he should pay $94, but the true present worth of $100, due at the end of the year, would be 94.33w-, so that he would have to pay $106 on the instalment over due, and $94.33| on the one not due, making $200.335, whereas the object is to find at what time interest should be chargeable to neither party. As a further illustration of the general rule, let us suppose that J. Smith owes R. Evans $1300, of which $700 are to be paid at the end of 3 months, $100 at the end of 4 months, and the balance at the end of 8 months, to find the equated time. We shall suppose that J. Smith agrees to pay R. Evans the whole amount at the time the debt was contracted; thcn J. Smith would EQUATION OF PAYMENTS. 221 owe R. Evans $1300, minus the discount for the length of time the amount was paid before it became due, viz., three months, equalling the discount on $210 for 1 month; $100, less the discount for 4 months, equalling the discount on $400 for 1 month; $500, less the discount for 8 months, equalling the discount on $4000 for 1 month. This gives a total of $2100+$400+$4000=$6500, for 1 month. Now, it is evident that if J. Smith wished to pay the whole amount at such a time that there should be no loss to either party, he must retain this amount for such a length of time as it will take this amount to equal the discount on $6500 for 1 month, which will be -lQ' of $6500, that is, for 5 months. To prove that 5 months must be the equated time, we have recourse to the principles laid down under the head of Interest. If a settlement is not made until the expiration of 5 months from the time the debt was contracted, then J. Smith would owe R. Evans $700, plus the interest of that principal during the.time it remained unpaid after becoming due, viz., two months, which would give an amount of $707. So also, $100, plus the interest for 1 month, would be $100.50, and $500, minus its discount for 3 months (the length of time paid before due), would give $7.50, leaving $492.50, and $707+$100.50+$492.50z=$1300. EXERCISES. 1. T. C. Musgrove owes H. W. Field $900, of which $300 are due in 4 months; $400 in 6 months, $200 in 9 months; what is the equated time for the payment of the whole amount? Ans. 6 months. 2. E. P. Hall & Co. have in their possession 5 notes drawn by G. W. Armstrong, all dated 1st January, 1865; the first is drawn at 4 months, for $45; the second at 8 months, for $120; the third at 10 months, for $75; the fourth at 11 months, for $60; and the fifth at 15 months, for $90; for what length of time must a single note be drawn, dated 1st May, 1865, so that it may fall due at the properly equated time? Ans. 6 months. 3. A merchant sold goods as follows, on a credit of 6 months:May 10, a bill of $600; June 12, a bill of $450; September 20, a bill of $900; at what time will the whole become due? Ans. January 16. 4. A merchant proposed to sell goods amounting to $4000 on 8 months' credit, but the purchaser preferred to pay i in cash and { in 3 months; what time should be allowed him for the payment of the remainder? Ans. 2 years, 5 months. 222 ARITHMETIC. 5. A gentlemon left his son $1500, to be paid as follows: i in 3 months, ~ in 4 months, - in 6 months, and the remainder in 8 months; at what time ought the whole to be paid at once? Ans. 4 mos., 15 days. 6. A merchant bought goods amounting to $6000. He agrees to pay $500 in cash, $600 in six months, $1500 in 9 months, and the remainder in 10 months; at what time ought he to pay the whole in one payment? Ans. 8 3 months. 7. There is due to a merchant $800, one-sixth of which is to be paid in 2 months, one-third in 3 months, and the remainder in 6 months; but the debtor agrees to pay one-half in cash; how long may he retain the other half, so that neither party may sustain loss? Ans. 8 months. 8. A merchant sold to W. L. Brown, Esq., goods to the amount of $3051, on a credit of 6 months, from September 25th, 1864. October 4th Brown paid $476; November 12th, $375; December 5th, $800; January 1st, 1865, $200. When, in equity, ought the merchant to receive the balance? 9. A having sold B goods to the amount of $1200, left it optional with him either to take them on 8 month's credit, or to pay one-half in cash, one-fifth in two months, one-sixth in four months, and the remainder at an equated time, to correspond with the terms first named; what was the time? Ans. 4 years, 4 mos. 10. A grocer sold 484 barrels of rosin, as follows: February 6th, 35 barrels @ $3.12., on 4 months' time. March 12th, 38 barrels @ 3.00, on 4 months' time. March 12th, 411 barrels @ 2.62~, on 4 months' time. What is the equated time for the payment of the whole? Ans. July 9th. 11. Bought of A. B. Smith & Co. 1650 barrels of flour, at different times, and on various terms of credit, as by the following statement; what is the equated time for the payment of the whole? May 6th, 150 barrels, at $4.50, on 3 months' credit. May 20th, 400 barrels, at 4.75, on 4 months' credit. July 10th, 500 barrels, at 5.00, on 5 months' credit. August 4th, 600 barrels, at 4.25, on 4 months' credit. Ans. November 7th. 12. J. B. Smith & Co. bought of A. Iamilton & Son 576 barrels of rosin, as follows: May 3rd, 62 barrels ~ $2.50, on 6 months' credit. AVERAGING ACCOUNTS. 223 May 10th, 100 barrels @ 2.50, on 6 months' credit. May 18th, 10 barrels @ 2.50, as cash. May 26th, 50 barrels @ 2.75, on 30 days' credit. May 26th, 345 barrels @ 2.50, on six months' credit. May 26th, 9 barrels @ 2.00, on six months' credit. What is the equated time for the payment of the whole? Ans. November 3rd. 13. Purchased goods of J. R. Worthington & Co., at different times, and on various terms of credit, as by the following statement: March 1st, 1863, a bill of $675.25, on 3 months' credit. July 4th, 1863, a bill of 376.18, on 4 months' credit. September 25th, 1863, a bill of 821.75, on 2 months' credit. October 1st, 1863, a bihl of 961.25, on 8 months' credit. January 1st, 1864, a bill of 144.50, on 3 months' credit. February 10th, 1864, a bill of 811.30, on 6 months' credit. March 12th, 1864, a bill of 567.70, on 5 months' credit. April 15th, 1864, a bill of 369.80, on 4 months' credit. What is the equated time for the payment of the whole? Ans. March 16th, 1864. AVERAGING ACCOUNTS. WHEN one merchant trades with another, exchanging merchandise, or giving and receiving cash, the memorandum of the transactions is called an Account Current. If the goods be purchased at different dates, or for different terms of credit, and some are not due while others are overdue, the fixing on a time when all may be settled, so that no interest shall be chargeable to either party, is called Averaging the Account. Since interest is the standard to which is referred the benefit of receiving money before it is due, so that in the meantime it can be used in trade, and also the damage of not getting it when due, it is fair and proper that interest should be charged on all sums overdue, and deducted from all not due. In illustration, let us suppose that A sells goods to B, March 2, on 4 months' credit, and again an equal amount on March 20, on 6 months' credit; the first will be due on July 2, and the second on September 20. Should B tender payment of the whole on June 2, he would be entitled to claim interest for 224 ARITHMETIC. one month on the first purchase, and for three months and eighteen days on the second. But if payment be delayed till August 2, A would be entitled to one month's interest on the first purchase, and B to the interest on the second for one month and eighteen days, so that there would be in favour of B, on the whole, a balance of interest for eighteen days. Again, supposing the settlement is not made till September 20, when all is due, no interest can be either charged or claimed on the second purchase, the term of credit having just then expired; but as the first debt is two months and eighteen days overdue, A is entitled to interest on it for that period. If neither is paid till after September 20, A has a right to claim interest on each for the period it has been overdue. But this regulates only one side of the account. In order to settle the other, let us suppose that B has, in the meantime, sold goods to A, it is obvious that B's claims on A must be settled on the very same principle, and that therefore the final result must be simply the finding of the balance. It is more usual, however, in accounts current, to fix on a time such that the interest due by A shall exactly balance that due by B. To illustrate this, let us suppose a case corresponding to a ledger account: R. EVANS. 1865. DR. July 21, To Merchandise on 2 months' credit...$200 July 25, To Cash.................................... 150 Aug. 24, To Merchandise on 4 months' credit... 100 Sept. 21, To Merchandise on 3 months' credit... 250 $700 1865. CR. August 1, By Cash................................$100 August 20, By Merchandise at 22 days......... 110 Sept'r 30, By Cash................................. 180 B alance................................................. 310 $700 To find in this case at what time the account may be settled so that interest shall be chargeable to neither party. Equating the time, as in equation of payments, we have the following operation: AVERAGING ACCOUNTS. 225 DR. CR. 1865 1865. July 25....;.150X 0 August 1.......100X 0 Sept. 21.......200X 58=11600 Sept. 12........110X22= 2420 Deer. 24......250X152=38000 Sept. 30........180X71=12780 Deer. 21.......100X149-14900 390 15200 700 64500 15200- 390=39 days. 64500 —700=92 days.. Due 39 days from August 1, Due 92 days from July 25, viz., viz., on September 9. on October 25. Time from September 9, to October 25=46 days. Excess of debit above credit 700-390=310. 390X46=17940, and 17940- -310=58 days, nearly. Counting 58 days forward, from October 25, will bring us to December 22, the time required for a settlement, with interest chargeable to neither party. Here the time is counted forward from the average date of the larger side which becomes due last, but had it become due first, we should have counted backward. The first transaction on the debit side being two months' credit from July 21,-is not to be taken into consideration till September 21. The second transaction, being a cash one, and therefore considered as so much due, will therefore mark the date from which all others shall be reckoned; and, since there is no interval of time, we write it without a multiplier. The next transaction has a term of credit extending to 152 days, and therefore we write 250X152= 11600. The term of the next extends from September 21 to December 21, a period of 149 days, and we write 100X149=14900. The sum of the debits is $700, and the sum of the results obtained by multiplying each item by the number of days it has to run from July 25 is $64500. Then 64500- 700=92, the equated time in days for the debit side. Now, as already explained, the interest for $700 for 92 days will be the same as the interest of $64500 for 1 day. Hence, the debits are due 92 days from July 25, viz., on October 25. In like manner, on the credit side, the first transaction being a cash one, we start from its date, August 1, and, as there is no interval, we have no multiplier. The second being merchandise, on 22 226 ABITHMETIC. days' credit, we write 110X22-2420. The third is cash paid 71 days after August 1, and we write 180X71=12780. Had tihe account been settled on September 9, the debits would have been paid 46 days before coming due, and the credit side would have gained and the debit side lost the interest for that time. Again, we must consider how long it would take the balance, $310, to produce the same interest that $390 would produce in 46 days. It is obvious that whatever interest $390 gives in 46 days will require 46 times $390 for $1 to produce the same interest, that is, 390X46==17940 days, and it will require 17940 —310=58 days, for $310 to produce the same interest. If the settlement is made on October 25, the latest date, then the credit has been due 46 days, and therefore bearing interest; and in order that the debit side may be increased by an equal amount, the time must k'e extended beyond October 25, that is, it must be countedforward. For the same reason, if the greater side had become due first, then the balance must be considered as due at a previous date, and'therefore'e must count backward. An account may be averaged from any date, but either the first or the last will be found the most convenient. The first due is generally used. On the principles now explained may be founded the following RU LE. Find the equated time when each side becomes due. Multiply the amount of the smaller side by the number of days between the two average dates, and divide the product by the balance of the account. The quotient thus obtained will be the time that the balance becomes due, counted from the average date of the larger side, FORWARD when the amount of that side becomes due LAST, but BACKWARD when it becomes due FIRST. The cash value of a balance depends on the time of settlement. If the settlement be made before the balance is due, the interest for the unexpired time is to be deducted; but if the settlement is not made till after the balance is due, interest is to be added for the time it is overdue. EXERCISES. In J. H. Marsden's Ledger, we find the following accounts, which, AVERAGING ACCOUNTS. 227 on being equated, stand as follows; at what time should the respective balances commence to draw interest: 1. Dr. J. S. PECKHAM. Cr. May 16th, 1865.........$724.45. I July 29th, 1865.........$486.80. Ans. December 15th, 1864. 2. Dr. NELSON BOSTFORD. Cr. November 19th, 1865......635. 1 December 12th, 1865......$950. Ans. January 27th, 1866. 3. Dr. JAMES CROW & CO. Cr. February 24th, 1866....$512.25. | June 10th, 1865.........$309,70. Ans. March 27th, 1867. 4. Dr. J. H. BURRITT & Co. Cr. March 17th, 1866..........$145; i January 15th, 1866.....$695.60. Ans. December 30th, 1865. 5. Dr. MC. DONALD. (r. August 27th, 1865...$......$341. 1 November 7th, 1865.......$247. 6. Dr. JAMEs I. MUSGROVE. Cr. July 20th, 1866.............$711. I April 14th, 1866..........1260. Ans. December 9th, 1865. 7. Dr. THos. A. BRYCE & CO. Cr. June 24th, 1864...........$1418. 1 September 7th, 1865......$2346. 8. Dr. E. R. CARPENTER. Cr. December 2nd, 1865...$1040.80. 1 August 13th, 1865....$1112.40. 9. Required the time when the balance of the following account becomes subject to interest, allowing the merchandise to have been on 8 months' credit? Dr. A. B. SMIITH & CO. Cr. 1864. 1T 1s865.. May 1, To Mdse............ $300.00 Jan. 1, By Cash.... $500.00 July 7, "........ 759.96 Feb. 18, " Mdse.. 481.75 Sep. 11, "............ 417.20 Mar. 19, " Cash... 750.25 Nov. 25,............ 287.70 April 1, " Draft... 210.00 Dec. 20, "......... 571.10May 25, Cash... 100.00 Ans. August 5, 1865. 228 ARITHMETC. 10. When will the balance of the following account fall due, the merchandise items being on 6 months' credit? Dr. J. K. WHITE. Cr. 1865. 1865. May ],To Mdse........... $312.40 June 14, By Cash.... $200.00 May 23,........... 85.70 July 30, " Mdse.... 185.90 June 12, " Cash paid dft.. 105.00 Aug. 10, "Cash.... 100.00 July 29, " Mdse........... 243.80 Aug. 21, " Mdse.... 58.00 Aug. 4, " "........... 92.10 Sept. 28,.... 45.10 Sept. 18, " Cash........... 50.00 Ans. January 12, 1866. 11. When does the balance of the following account become subject to interest? Dr. W. H. MUSGROVE. Cr. 1864. 1864. Aug. 10, To Mdse 4 mos. $285.30 Oct 13, By Cash...... $400.00 Aug. 17, " 60 days 192.60 Oct. 26 " "..... 150.00 Sept. 21, " " 30 " 256.80 Dec. 15, " Mdse2mos 345.80 Oct. 13, " Cash p'ddft. 190.00 Dec. 30, " " 4 " 230.40 Nov. 25, Mdse 6 mos. 432.20 1865. Nov. 30," " 90 days 215.25 Jan. 4, " Cash....... 340.30 Dec. 18 " " 2 mos. 68.90 Jan. 21, ".....180.00 1865. Jan. 31, C Cash.......... 100.00 12. In the following account, when did the balance become due, the merchandise articles being on 6 months' credit? Dr. R. J. BRYCE in account with D. IICKS & CO. Cr. 1864. 11 1864. Jan. 4, To Mdse........... $ 96.57 Jan. 30, By Cash... $240.00 Jan. 18,........ 57.67 April 3,... 48.88 Feb. 4, Cashpaid draft. 80.00 May 22, "... 50.00 Feb. 4, " Mdse............ 38.96 Feb. 9, " Cash paid draft. 50.26 Mar. 3, " Mdse............ 154.46 Mar. 24."............ 42.30 April 9, " "............ 23.60 May 15, " "............ 28.46 May 21," ",........177.19 Ans. December 22nd, 1864. AVERAGING ACCOUNTS. 229 13. When, in equity, should the balance of the following account be payable? Dr. J. McDoNALD & Co. Cr. 1865. 1864. Jan. 3, To Cash.... $200 Sept. 20, By Mdse, 6 mos.. $583.17 Jan. 31, ".... 300 Oct. 27," " 4 ".. 321.00 Feb. 8, ".... 75 Dec. 5, 6 ".. 137.00 Feb. 21, "a.... 100 1865. Mar. 10,' ".... 350 Jan. 18, " 60 days. 98.75 Mar. 24... 25 Feb. 26," " 6 mos.. 53.98 Apr. 12," ".... 40 Apr. 15.: " 4 i.. 634.00 June 1, " 80 June 12, 2.. 97.23 June20, " 125 Sept. 21," " 6.. 84.00 July 4, 268 Dec. 29, " 6 ".. 132.14 Sept. 27,.. 250 Dec. 9,.... 100 Ans. October 10, 1866. CASH BALANCE. To find the true cash balance of an account, when each item draws interest. EXAMPLE. What is the balance of the following account on January 19th, 1866, a credit of three months being allowed on the merchandise, money being worth 6 per cent.? Dr. MUSGROVE & WRIGHT. Cr. 1865. 1865.! Mar. 12, To Merchandise.... $340.00 Apr. 20, By Mdse... $200.00 Apr. 21,1 ".... 150.00 May 4, " Cash.... 110.00 May 6,1 " Cash paid draft 165.00 June 15, " 2... 230.00 May 27, " Mdse............ 215.00 Aug. 10,'" Mdse... 180.00 July 16, " Cash........... 100.00 Sept. 23, " Cash.... 50.00 Sept. 10,' Mdse............310.00 Nov. 12, ".. 50.00 Oct. 19,1 "....... 120.00 Dec. 15, c.. 100.00 16 230 ARITHMETIC. SOLUTION. Debits. Credits. Due. Due. June 12, $340X221= 75140 July 20, $200X183= 36600 July 21, 150X182= 27300 May 4, 110X260= 28600 May 6, 165X258= 42570 June 15, 230X218= 50140 Aug. 27, 215X145= 31175 Nov. 10, 180X 70= 12600 July 16, 100X187=- 18700 Sept. 23, 50X118= 5900 Dec. 10, 310X 40- 12400 Nov. 12, 50X 68= 3400 Jan. 19, 120X 0= 0 Dec. 15, 100X 35- 3500 $1400 6)207285 $920 6)140740 $34.547 $23.456 The different items on the debit and credit sides of the account being on interest from the date on which it becomes due until the time of settlement, the total interest of all the debit items will be the same as the interest of $207285 for one day, or the interest of $1 for 207285 days, which is $34.547. So also, the total interest of all the credit items will be the same as the interest of $140740 for one day, or the interest of $1 for 140740 days, which is $23.456. Now, since each side of the account is to be increased by its interest, the cash balance will be represented by the number denoting the difference between the two sides of the account, after the interest is added; thus, $1400+$34.547-=$1434.547, amount of debit side, and $920 +$23.456=$943.456, amount of credit side, then $1434.547$943.456=$491.09, cash balance. SECOND METHOD. Debits. Credits. Days. Int. Days. Int. Int. on $340 for 221==$12.523 Int. on $200 for 183= $6.100 " 150 " 182- 4.550 " 110 " 260- 4.766'" 165 " 258= 7.095 " 230 " 218= 8.356 " 215 " 145- 5.195 " 180 " 70= 2.100 C" 100 " 187- 3.116 " 50' 118-.983 310 " 40- 2.066 " 50 " 68=.566'" 120 " 0 " 100 " 35=.583 $1400 $34.545 $920 $23.454 Now, $34.545 debit interest-$23.454 credit interest=$11.09, CASH BALANCE. 231 the balance of interest, and $1400, amount of debit items+$11.09 -$1411.09, and $1411.09-$920 amount of credit items=$491.09 the cash balance, which is the same as obtained by the first solution. Hence from the foregoing we deduce the following RULE. Multiply each item of debit and credit by the number of days intervening between its becoming due and the time of settlement. Then consider the sums of the products of the debit and credit items as so many dollars, and find the interest on each for one day, which will be the interest, respectively, of the debit and credit items. Place the balance of interest on its own side of the account, and the difference then between the two sides will be the true balance; or, Find the interest on each item from the date on which it becomes due to the time of settlement. The difference of the sums of interests, on the debit and credit sides of the account will represent the balance of interest, which is placed on its own side of the account, and the difference then between the two sides will be the true balance. NOTE.-If any item should not come due until after the time of settlement, the side upon which it is, should be diminished, or the opposite side increased by the interest of such item from the time of settlement until due.. EXERCISES. 1. What will be the cash balance of the following account if settled on January 1, 1865, allowing interest at 8 per cent. on each item after it is due? Dr. R. EVANS in account with JOHN JONES. Cr. 1864. 1864. June 11, To Mdse, 4 mos. $315.00 Apr. 15, By Mdse, 3 mos. $350.00 June 29, L 6 "i 180.00 May 10, c. 4'c 120.00 July 18, " Cashp'd dft. 200.00 June 12, " 6' 240.00 Aug. 25, " Cash......... 75.00 June30, " Cash......... 100.00 Aug. 31, " Mdse,2 mos. 50.00 July 15,......... 90.00 Sept. 3, "' 1 i" 00.00 July 27, "..... 80.00 Sept. 20, " Cash......... 80.00 Aug. 6, " Mdse,as cash 100.00 Oct. 14, "..... 150.00 Aug. 20, " Cash......... 175.00 Oct. 19. M" dseas cash 300.001 Aug.30, Mdse,3mos. 75.00 Ans. $110.86. 2. A. B. Smith is in account and interest with J. K. Amos & Co., as follows:-Debtor, January 1, 1865, to merchandise, on 6 months, 232 ARITHMETIC. $156.10; February 3, to cash paid draft, $100; March 20, to merchandise, oa 4 months, $316.90; March 3Q, to merchandise, on 4 months, $162; May 15, to cash paid draft, $100; August 20, to merchandise, on 6 months, $213. Creditor, February 1, by cash, $120; March 20, by merchandise, on 4 months, $420.16; May 1, by merchandise, on 6 months, $300: July 1, by merchandise, on 4 months, $50; September 10, by merchandise, on 4 months, $99.84. Required, the true balance, if settled on December 1, 1865, interest being at 6 per cent.? Ans. $61.36. 3. Required the true balance, March 25, 1865, on the following account, each item drawing 7 per cent. interest from its date. A. B. Lyman in account and interest with John Russell & Co.:Debtor, July 4, 1864, to merchandise, $200; September 8) to merchandise, $300; September 25, to merchandise, $250; October 1, to merchandise, $600; November 20, to merchandise, $400; December 12, to merchandise, $500; January 15, 1865, to merchandise, $100; March 11, to merchandise, $120. Creditor, July 20, 1864, by cash, $300; August 15, by cash, $350; September 1, by cash, $400; November 1, by cash, $320; December 6, by merchandise, $600; December 20, by cash, $100; February 1, 1865, by cash, $200; February 28, by merchandise, $150. Ans. $50.64. ALLIGATION. Alligation is the method of making calculations regarding the compounding of articles of different kinds or different values. It is a Latin word, which means binding to, or binding together. It is usual to distinguish alligation as being of two kinds, medial and alternate. ALLIGATION MEDIAL. Alligation medial relates to the average value of articles compounded, when the actual quantities and rates are given. E X A P L E. A miller mixes three kinds of grain: 10 bushels, at 40 cents a bushel; 15 bushels, at 50 cents a bushel; and 25 bushels, at 70 cents a bushel; it is required to find the value of the mixture. ALLIGATION. 233 10 bushels, at 40 cents a bushel, will be worth 400 cents., 15 bushels, at 50 cents a bushel, will be worth 750 cents., 25 bushels, at 70 cents a bushel, will be worth 1750 cents., giving a total of 50 bushels and 2900 cents, and hence the mixture is 2900 —50=58 cents, the price of the mixture per bushel. Hence the RULE. Find the value of each of the articles, and divide the sum of their values by the number denoting the sum of the articles, and the quotient will be the price of the mixture. EXERCISES. 1. A farmer mixes 20 bushels of wheat, worth $2.00 per bushel, with 40 bushels of oats, worth 50 cents per bushel; what is the price of one bushel of the mixture? Ans. $1. 2. A grocer mixes 10 pounds of tea, at 40 cents per pound; 20 pounds, at 45 cents per pound, and 30 pounds, at 50 cents per pound; what is a pound of this mixture worth? Ans. 46- cents. 3. A liquor merchant mixed together 40 gallons of wine, worth 80 cents a gallon; 25 gallons of brandy, worth 70 cents a gallon; and 15 gallons of wine, worth $1.50 a gallon; what was a gallon of this mixture worth? Ans. 90 cents. 4. A farmer mixed together 30 bushels of wheat, worth $1 per bushel; 72 bushels of rye, worth 60 cents per bushel; and 60 bushels of barley, worth 40 cents per bushel; what was the value of 21 bushels of the mixture? Ans. $1.50. 5. A goldsmith mixes together 4 pounds of gold, of 18 carats fine; 2 pounds, of 20 carats fine; 5 pounds, of 16 carats fine; and 3 pounds, of 22 carats fine; how many carats fine is one pound of the mixture? Ans. 18-. ALLIGATION ALTERNATE. Alligation alternate is the method of finding how much of several ingredients, the quantity or value of which is known, must be combined to make a compound of a given value. CASE I. Given, the value of several ingredients, to make a compound of a given value. 234 ARITHMETIC. E X A P L E How much sugar that is worth 6 cents, 10 cents, and 13 cents per pound, must be mixed together, so that the mixture may be worth 12 cents per pound? SOLUTION. {1 ib., at 6 cents, is a gain of 6 cents. Gain. 12 cents. 1 lb., at 10 cents, is a gain of 2 cents. 8 1 lb., at 13 cents, is a loss of 1 cent. Loss1 7 lbs. more, at 13 cents, is a loss of...... 7 Gain 8 Loss 8 It is evident, in forming a mixture of sugar worth 6, 10 and 13 cents per pound so as to be worth 12 cents, that the gains obtained in putting in sugar of less value than the average price must exactly balance the losses sustained in putting in sugar of greater value than the average price. Hence in our example, sugar that is worth 6 cents per pound when put in the mixture will sell for 12, thereby giving a gain of 6 cents on every pound of this sugar put in the mixture. So also sugar that is worth 10 cents per pound, when in the mixture will bring 12, so that a gain of 2 cents is obtained on every pound of this sugar used in the compound. Again, sugar that is worth 13 cents per pound, on being put into the mixture will sell for only 12 cents, consequently a loss of 1 cent is sustained on every pound of this sugar used in forming the mixture. In this manner we find that in taking one pound of each of the different qualities of sugar there is a gain of 8 cents, and a loss of only 1 cent. Now, our losses must equal our gains, and therefore we have yet to lose 7 cents, and as there is only one quality of sugar in the mixture by which we can lose, it is plain that we must take as much more sugar at 13 cents as will make up the loss, and that will require 7 pounds. Therefore, to form a mixture of sugar worth 6, 10 and 13 cents per pound, so as to be worth 12 cents per pound, we will require 1 pound at 6 cents, 1 pound at 10 cents, and 1 pound at the 13 cents+7 pounds of the same, which must be taken to make the loss equal to the gain. By making a mixture of any number of times these answers, it will be observed, that the compound will be correctly formed. Hence we can readily perceive that any number of answers mav be obtained ALLIGATION ALTERNATE. 235 to all exercises of this kind. From what has been said we deduce the following RULE. Find how much is gained or lost by taking one of each kind of the proposed ingredients. Then take one or more of the ingredients, or such parts of them as will make the gains and losses equal. EXERC ISES. 1. A grocer wishes to mix together tea worth 80 cents, $1.20, $1.80 and $2.40 per pound, so as to make a mixture worth $1.60 per pound; how many pounds of each sort must he take? Ans. 1 lb. at 80 cents; 1 lb. at $1.20; 2 lbs. at $1.80, and 1 lb. at $2.40. 2. How much corn, at 42 cents, 60 cents, 67 cents, and 78 cents per bushel, must be mixed together that the compound may be worth 64 cents per bushel? Ans. 1 bush. at 42 cts.; 1 bush. at 60 cts.; 4 bush. at 67 cts.; and 1 bush. at 78 ets. 3. It is required to mix wine, worth 60 cents, 80 cents, and $1.20 per gallon, with water, that the mixture may be worth 75 cts. per gallon; how much of each sort must be taken? Ans. 1 gal. of water; 1 gal. of wine at 60 cts.; 9 gal. at 80 cts.; and 1 gal at $1.20. 4. In what proportion must grain, valued at 50 cents, 56 cents, 62 cents, and 75 cents per bushel, be mixed together, that the compound may be 62 cents per bushel? Give, at least, three answers, and prove the work to be correct. 5. A produce dealer mixed together corn, worth 75 cents per bushel; oats, worth 40 cents per bushel; rye, worth 65 cents per bushel, and wheat, worth $1 per bushel, so that the mixture was worth 80 cents per bushel; what quantity of each did he take? Give four answers, and prove the work to be correctly done in each case. CASE II. When one or more of the ingredients are limited in quantity, to find the other ingredients. E X A M PLE. How much barley, at 40 cents; oats, at 30 cents, and corn, at 60 236 ARITHMETIC. cents per bushel, must be mixed with 20 bushels of rye. at 85 cents per bushel, so that the mixture may be worth 60 cents per bushel? SOLUTION. Bush. Cents. Gain. Loss. 1 at 40, gives.................20 1 at 30, gives...................30 1 at 60, gives.................00.00 20 at 85, gives..................... 5.00.50 5.00 9 at 40, gives............. 1.80 9 at 30, gives.................. 2.70... $5.00 $5.00 By taking 1 bushel of barley, at 40 cents, 1 bushel of oats at 30 cents, and 1 bushel of corn at 60, in connection with 20 bushels of rye at 85 cents per bushel, we observe that our gains amount to 50 cents and our losses to $5.00. Now, to make the gains equal the losses, we have to take 9 bushels more at 40 cents, and 9 bushels more at 30 cents. This gives us for the answer 1 bushel4-9-10 bushels of barley, 1 bushel+9=10 bushels of oats, and 1 bushel of corn. From this we deduce the RU LE. Find how much is gained or lost, by taking one of each of the proposed ingredients, in connection with the ingredient which is limited, and if the gain and loss be not equal, take such of the proposed ingredients, or such parts of them, as will make the gain and loss equal. EXERC ISES. 6. How much gold, of 16 and 18 carats fine, must be mixed with 90 ounces, of 22 carats fine, that the compound may be 20 carats fine? Ans. 41 ounces of 16 carats fine, and 8 of 18 carats fine. 7. A grocer mixes teas worth $1.20, $1, and 60 cents pet pound, with 20 pounds, at 40 cents per pound; how much of each sort must he take to make the composition worth 80 cents per pound? 8. How much barley, at 50 cents per bushel, and at 60 cents per bushel, must be mixed with ten bushels of pease, worth 80 cents' AITGATION ALTERNATE. 237 per bushel, and 6 bushels of rye, worth 85 cents per bushel, to make a mixture worth 75 cents per bushel? Ans. 3 bushels, at 50 cents; 2~ bushels, at 60 cents. 9. How many pounds of sugar, at 8, 14, and 13 cents per pound, must be mixed with 3 pounds, worth 9{ cents per pound; 4 pounds, worth 10~ cents per pound; and 6 pounds, worth 13~ cents per pound, so that the mixture may be worth 12~ cents per pound? Ans. 1 lb., at 8 cts.; 9 lbs., at 14 cts.; and 51 lbs., at 13 cts CASE III. To find the quantity of each ingredient, when the sum of the ingredients and the average price are given. EXA P L E. A grocer has sugar worth 8, 10, 12 and 14 cents per pound, and he wishes to make a mixture of 240 pounds, worth 11 cents peo pound; how much of each sort must he take? SOLUTION. Gain. Loss. 1 lb., at 8 cents, gives..................... 3 1 lb., at 10 cents, gives.................... 1 1 lb., at 12 cents, gives..................... 1 1 lb., at 14 cents, gives.................... 3 4 lbs. 4 4 240 lbs. - 4-60 lbs. of each sort. By taking 60 lbs. of each sort we have the required quantity, and it will be observed that the gains will exactly balance the losses, consequently the work is correct. Hence the RULE. Find the least quantity of each ingredient by CASE I., Then divide the given amount by the sum of the ingredients alreadyfound, and multiply the quotient by the quantities found for the proportional quantities. 10. What quantity of three different kinds of raisins, worth 15 cents, 18 cents, and 25 cents per pound, must be mixed together to fii a box containing 680 lbs., and to be worth 20 cents per pound? Ans. 200 lbs., at 15 cents; 200 lbs., at 18 cents; and 280 lbs., at 25 cents. 16 238.-JR3.'' IT'I —TI!H METIC. iI. HTow much sugar, at 6 cents, 8 cents, 10 cents, and 12 cents p'r pound, must be mixed together, so as to form a compound of 200 pounds, worth 9 cents per pound? Ans. 50 lbs. of each. 12. Iow much water must be mixed with wine, worth 80 cents per gallon, so as to fill a vessel of 90 gallons, which may be offered.t 50 cents per gallon? Ans. 56| gals. wine, and 33- gals. water. 13. A wine merchant has wines worth $1, $1.25, $1.50, 81.75,and $2. per gallon, and he wishes to form a compound to fill a 150 gallon cask that will sell at $1.40 per gallon; how many gallons of each sort must he take? Ans. 54 of $1, and 24 of each of the others. 14. A grocer has sugars worth 8 cents, 10 cents 12 cents, and 20 cents per pound; with these he wishes to fill a hogshead that would contain 200 pounds; how much of each kind must he take, so that the mixture may be worth 15 cents per pound? Ans. 33} lbs. of 8, 10, and 12 cents, and 100 lbs. of 20 cents. 15. A grocer requires to mix 240 pounds of different kinds of raisins, worth 8 cents, 12 cents, 18 cents, and 24 cents per lb., so that the mixture shall be worth 10 cents per pound; how much must be taken of each kind? Ans. 192 lbs. of 8 cents, and 16 lbs. of each of the other kinds. MONEY; ITS NATURE AND VALUE. [MONEY is the medium through which the incomes of the different members of the community are distributed to them, *and the measure by which they estimate their possessions. The precious metals have, amongst almost all nations, been the standard of value from the earliest time. Except in the very rudest state of society, men have felt the necessity of having somo article, of more or less intrinsic value, that can at any, time be exchanged for different commodities. No other substances were so suitable for this purpose as gold and silver. They are easily divisible, portable, and among the least imperishable of all substances. The work of dividing the precious metals, and marking or coining them, is generaHy undertaken by the Government of the country. Money is a commodity, and its value is determined, like that of other commodities, by demand and supply, and cost of production. When there is a large supply of money it becomes cheap; in other words, more of it is required to purchase other articles. I' all the MONEY: ITS NATUPRE AND VALUE. 239 money in circulation were doubled, prices would be doubled. The usefulness of money depends a great deal upon the rapidity of its circulation. A ten-dollar bill that changes hands ten times in a month, purchases, during that time, a hundred dollars' worth of goods. A small amount of money, kept in rapid circulation, does the same work as a far larger sum used more gradually: Therefore, whatever may be the quantity of money in a country, only that part of it will effect prices which goes into circulation, and is actually exchanged for goods. Money hoarded, or kept in reserve by individuals, does not act upon prices. An increase in the circulating medium, conformable in duration and extent to a temporary activity in business, does not raise prices, it merely prevents the fall that would otherwise ensue from its temporary scarcity. PAPER CURRENCY. PAPER CURRENCY may be of two kinds-convertible and inconvertible. When it is issued to represent gold, and can at any time be exchanged for gold, it is called convertible. When it is issued by the sovereign power in a State, and is made to pass for money, by merely calling it money, and from the fact that it is received in payment of taxes, and made a legal tender, it is known as an inconvertible currency. Nothing more is needful to mace a person accept anything as money, than the persuasion that it will be taken from him on the same terms by others. That alone would ensure its currency, but would not regulate its value. This evidently cannot depend, as in the case of gold and silver, upon the cost of production, for that is very trifling. It depends, then, upon the supply or the quantity in circulation. While the issue of inconvertible currency is limited to something under the amount of bullion in circulation, it will on the whole maintain a par value. But as soon as gold and silver are driven out of circulation by the flood of inconvertible currency, prices begin to rise, and get higher with every additional issue. Among other commodities the price of gold and silver articles will rise, and the coinage will rise in value as mere bullion. The paper currency will then become proportionably depreciated, as compared with the metallic currency of other countries. It would be 240 ARITHMETIC. quite impossible for these results to follow the issue of convertible paper for which gold could at any time be obtained. All variations in the value of the circulating medium are mischievous; they disturb existing contracts and expectations, and the liability to such disturbing influences renders every pecuniary engagement of long date entirely precarious. A convertible paper currency is, in many respects, beneficial. It is a more convenient medium of circulation. It is clearly a gain to the issuers, who, until the notes are returned for payment, obtain the use of them as if they wire a real capital, and that, without any loss to the community. THE CURRENCY OF CANADA. IN Canada there are two kinds of currency; the one is called the old or Halifax currency, reckoned in pounds, shillings, pence and fractions of a penny; the other is reckoned by dollars and cents as already explained under the head of Decimal Coinage. The equivalent in gold of the pound currency is 101.321 grains Troy weight of the standard of fineness prescribed by law for the gold coins of the united kingdom of Great Britain and Ireland. The only gold coins now in circulation in Britain are the sovereign, value one pound, or twenty shillings sterling; and the half sovereign, ten shillings. The dollar is one-fourth of the pound currency, and the pound sterling is equal to $4.86-. In the year 1786, the congress of the United States adopted the decimal currency, the dollar being the unit, and the system was introduced into Canada in 1858. By the term legal tender is meant the proffer of payment of an account in the currency of any country as established by law. Copper is a legal tender in Canada to the amount of one shilling or twenty cents, and silver to the amount of ten dollars. The British sovereign of lawful weight passes current, and is a legal tender to any amount paid in that coin. There is a silver currency proper to Canada, though United States' coins are most in circulation. The gold eagle of the United States, coined before July 1, 1834, is a legal tender for $10.661 of the coin current in this province. The same coin issued after that is a legal tender for $10. EXCHANGE. 241 EXCHANGE. IT often becomes necessary to send money from one town or country to another for various purposes, generally in payment for goods. The usual mode of making and receiving payments between distant places is by bills of exchange. A merchant in Liverpool. whom we shall call A. B., has received a consignment of flour from C. D., of Chicago; and another man, E. F., in Liverpool, has shipped a quantity of cloth, in value equal to the flour, to G. IT. in Chicago. There arises, in this transaction, an indebtedness to C}licago for the flour, as well as an indebtedness from Chicago for the cloth. It is evidently unnecessary that A. B., in Liverpool, should send money to C. D. in Chicago, and that G. -I., in Chicago, should send an equal sum to E. F. in Liverpool. The one debt may be applied in payment of the other, and by this plan the expense and risk attending the double transmission of the money may be saved. C. D. draws on A. B. for the amount which he owes to him,.^ and G. IH. having an equal amount to pay in Liverpool, buys this bill from C. D., and sends it to E. F., who, at the maturity of the bill, presents it to A. B. for payment. In this way the debt due from Chicago to Liverpool, and the debt due from Liverpool to Chicago are both paid without any coin passing from one place to the other. An arrangement of this kind can always be made when the debts due between the different places are equal in amount. But if there is a greater sum due from one place than from the other, the debts cannot be simply written off against one another. Indeed, when a person desires to make a remittance to a foreign country, he does not make a personal search for some one who has money to receive from that country, and ask him for a bill of exchange. There are exchange brokers and bankers whose business this is. They buy bills from those who have money to receive, and sell bills to those who have money to pay. A person going to a broker to buy a bill Ilay very likely receive one that has been bought the same day from a merchant. If the broker has not on hand any exchange that he ohus bought, he will often give a bill on his own foreign correspondent; and to place his correspondent in funds to meet it, he will remit to him all the exchange which he has bought and not re-sold. 242 ARITIIMETIO. When brokers find that they are asked for more bills than are offlred to them, they do not absolutely refuse to give them. To enable their correspondents to meet the bills at maturity, as they have no exchange to send, they have to remit funds in gold and silver. There are the expenses of freight and insurance upon the specie, besides the occupation of a certain amount of capital involved in this; and an increased price, or premium, is charged upon the exchange to cover all. The reverse of this happens when brokers find that more bills are offered to them than they can sell or find use for. Exchange on the foreign country then falls to a discount, and can be purchased at a lower rate by those who require to nmke payments. There are other influences that disturb the exchange between different countries. Expectations of receiving large payments from a foreign country will have one effect, and the fear of having to make larger payments will have the opposite effect. AME RIC ANT EXCIIANGEE Exchange between Canada and the United States, especially the northern, is a matter of every day occurrence on account of the proximity of the two countries, and the incessant intercourse between them, both of a social and commercial character. The exigencies of the Northern States arising from the late war, compelled them to issue, to an enormous extent, an inconvertible paper currency, known by the name of " Greenbacks." As the value of these depended mainly on the stability of the government and the issue of the war, public confidence wavered, and in consequence, the value of this issue sunk materially. This caused a gradual rise in the value of gold until it reached the enormous premium of nearly two hundred per cent., or a quotation of nearly three hundred per cent., that is, it took nearly three hundred dollars in Greenbacks to purchase one hundred dollars in gold. It is to be hoped and expected, however, that as peace is now restored, matters will soon find their former level. It has been deemed essential that this should be distinctly explained, as it has brought about a necessity for a constant calculation AMERIICAN EXCHANGE. 243 of the relative values of gold and greenbacks, and has generated an extensive business in that species of exchange. When the term " American currency" is used in the following exercises it is understood to be Greenbacks. CASE I. To find the value of $1, American currency, when gold is at a premium. EXA MPLE. When gold is quoted at 140, or 40 per cent. premium, what is the value of $1, American currency? SOLUTION. Since gold is at a premium of 40 per cent., it requires 140 cents of American funds to equal in value $1, or 100 cents in gold. Hence the value of $1, American money, will be represented by the number of times 140 is contained in 100, which is.71- or 71 -3 cents. Hence to find the value of $1 of any depreciated currency reckoned in dollars and cents, we deduce the following RULE. Divide 100 cents by 100 plus the rate of premium on gold, and the quotient will be the value of $1. Subtract this from $1, and the remainder will be the rate of discount on the given currency. CASE II. To find the value of any given sum of American currency when gold is at a premium. E XA M P L ES. What is the value of $280, American money, when gold is quoted at 140, or 40 per cent. premium? SOLUTION. We find by Case I. the value of $1 to be 71- cents. Now, it is evident that if 71- cents be the value of $1, the value of $280 will be 280 times 71 cents, which is $200, or $280 [.40_-28000 — 140=$200. Hence we have the following 244 ARITHMETIC. RULE. Multiply the value of $1 by the number denoting the given amount of American money, and theproduct will be the gold value; or, Divide the given sum of American money by 100 (the number of cents in $1,) plus the premium, and the quotient will be the value in gold. CASE III. To find the premium on gold when American money is quoted at a certain rate per cent. discount. EXA MPLE.'When the discount on American money is 40 per cent., what is the premium on gold? SOLUTION. If American money is at a discount of 40 per cent., the discount on $1 would be 40 cents, and consequently the value of $1 would be equal to $1.00-40 cents, equal to 60 cents. Now, if 60 cents in gold be worth $1 in American currency, $1 or 100 cents in gold would be worth 100 times I of $1, which is $1.66I, from which if we subtract $1, the remainder will be the premium. Therefore, if American currency be at a discount of 40 per cent., the premium on gold would be 661 per cent. Hence we deduce the following RU L E. Divide 100 cents by the number denoting the gold value of $1, American currency, and the quotient will be the value, in American currency, of $1 in gold, from which subtract $1, and the remainder will be the premium. CASE IV. To find the value in American currency of any given amount of gold. EXAMPLE. What is the value of $200 of gold, in American currency, gold being quoted at 150? SOLUTION. When gold is quoted at 150, it requires 150 cents, in American currency, to equal in value $1 in gold. Now, if $1 in gold be worth $1.50 in American currency, $200 will be worth 200 times $1.50, which is $300. Hence the AMERICAN EXCHANGE. 245 RULE. Multiply the value of $1 by the number denoting the amount of gold to be changed, and the product will be the value in American currency; or To the given sum add the premium on itself at the given rate, and the result will be the value in American currency. EXERC IS ES. 1 If American currency is at a discount of 50 per cent., what is the value of $450? Ans. $225. 2. The quotation of gold is 140, what is the discount on Ameri6an currency? Ans. 284 per cent. 3. A person exchanged $750, American money, at a discount of 35 per cent. for gold; how much did he receive? Ans. $427.50. 4. Purchased a draft on Montreal, Canada East, for $1500 at a premium of 64" per cent.; what did it cost me? Ans. 5. If American currency is quoted at 334- per cent. discount; what is the premium on gold? Ans. 50 per cent. 6. Purchased a suit of clothes in Toronto, Canada West, for $35, but on paying for the same in American funds, the tailor charged me 32 per cent. discount; how much had I to pay him? Ans. $51.47. 7. What would be the difference between the quotations of gold, if greenbacks were selling at 40 and 60 per cent. discount? Ans. 83J per cent. 8. P. Y. Smith borrowed from C. R. King, $27 in gold, and wished to repay him in American currency, at a discount of 38 per cent.; how much did it require? Ans. $43.55. 9. J. E. Pekham bought of Sidney Leonard a horse and cutter for $315.50, American currency, but only having $200 of this sum, he paid the balance in gold, at a premium of 65 per cent.; how much did it require? Ans. $70. 10. A cattle drover purchased of a farmer a yoke of oxen valued at $135 in gold, but paid him $112 in American currency, at a discount of 27~ per eent.; how much gold did it require to pay the balance? Ans. $53.80. 11. W. H. Hounsfield & Co., of Toronto, Canada West, purchased in New York City, merchandise amounting in value to $4798.40, on 3 monthbs credit premium on gold being 793 per cent. At the 17 rmimo 246 ARITHMETIC. expiration of the three months they purchased a draft on Adams, Kimball and Moore, of New York, for the amount due, at a discount of 57i per cent.; what was the gain by exchange? Ans. $647.75. 12. A makes an exchange of a horse for a carriage with B; the horse being valued at $127.50, in gold, and the carriage at $210, American currency. Gold being at a premium of 65 per cent.; what was the difference, and by whom payable? Ans. B pays A 23 cents in gold, or 37 cents in greenbacks. 13. A merchant takes $63 in American silver to a broker, and wishes to obtain for the same greenbacks which are selling at a discount of 30 per cent. The broker takes the silver at 31 per cent. discount; what amount of American currency does the merchant receive? Ans. $86.85. 14. I bought the following goods, as per invoice, from John McDonald & Co., of Montreal, Canada East, on a credit of 3 months: 11201 yards Canadian Tweed at...............95 cents per yard. 2190 " long-wool red flannel at...........60' " " 3400 " " white flannel at.........55 " " Paid custom house duties, 30 per cent.; also paid for freight, $37.40. Gold at time of purchase was at a premium of 63j per cent.; what shall I mark each piece at per yard to make a net gain of 20 per cent. on full cost? Ans. C. tweed, $2.44; red flannel, $1.54; white flannel, $1.41. 15. A merchant left Toronto, Canada West, for New York City to purchase his stock of spring goods, taking with him to defray expenses $95 in gold. After purchasing his ticket to the Suspension Bridge for $2.40, he expended the balance in greenbacks, which were at a discount of 41~ per cent. When in New York he drew from this amount $23.85 to "square" an old account then past due. On arriving home he found that he still had in greenbacks $16.40, which he disposed of at a discount of 43; per cent., receiving in payment American silver at a discount of 3~ per cent.,which he passed off at 2- per cent. discount for gold. What were his expenses in gold; the actual amount in greenbacks paid for expenses, and the amount of silver received? Ans. Total expenses in gold, $71.76; expenses in greenbacks, $118.04; silver received, $9.53. EXCHANGE WITH GREAT BRITAIN. 247 EXCHANGE WITH GREAT BRITAIN. In Britain money is reckoned by pounds, shillings and pence, and fractions of. a penny, and is called Sterling money, the gold sovereign or the pound sterling, consisting of- 22 parts gold and 2 alloy, being the standard, and the slilling, one-twentieth part of the pound, a silver coin of 37 parts silver and 3 copper, and the penhy, one-twelfth part of the shilling, a copper coin, the ingredients and size of which have frequently been altered. The comparative value of the gold sovereign in the United States previous to the year 1834 was $4.444, but by Act of Congress passed in that year it was made a legal tender at the rate of 94y8 cents per pennyweight, because the old standard was less than theintriLsic value and also because the commercial value, though fluctuating, was always considerably higher. Hence, the full weight of the sovereign being 5 dwts. 3.274 grs., it was made equivalent to 4 dollars and 861- cents..The increase in the standard value was, therefore, equal to 9- per cent. of its nominal value. The real par of exchange between two countries is that by which an ounce of gold in one country can be replaced by an ounce of gold of equal fineness in the other country. If the course of exchange at New York on London were 108k per cent.; and the par of exchange between England and America 109k per cent., it follows that the exchange is 100 per cent. against England; but the quoted exchange at New York being for bills at, 60 days sight, the interest must be deducted from the above difference. The general form for the quotation of exchange with England is: 108, 108k, 109, 109~ &c., which indicates that it is at 8, 8j, 9, or 9D per cent. premium on its nominal value. EXA M PLE. What amount of decimal money will be required to purchase a draft on London for ~648 17s. 6d.?-exchange 108. The old par value or nominal value is $4.44 =4-04 = of $40 248 RITHMETIC. by reducing to an improper fraction. Now, the quotation is.108, or 8 per cent. above the nominal value, we find the premium on $40 at 8 per cent., which is $3.20, which added to $40 will give $43.20, and $43.20. —9=$4.80 to be remitted for every pound sterling, and therefore ~648 17s. 6d. multiplied by 4.80 or 4.8 will be the value in our money. 17s. 6d.-.875 of a pound, and the operation is as follows: ~648,875 4.8 5191000 2595500 $3114.6000 RULE. Po $40 add the premium on itsel at the quoted rate, multiply the sum by the number representing the amount of sterling money, and divide the result by 9, the quotient will be the equivalent of the sterling money in dollars and cents. NOTE.-If there be shillings, pence, &c., in the sterling money, they are to be reduced to the decimal of ~1. To find the value of decimal money in sterling money, at any given rate above par. Let it be required to find the value of $465 in sterling money, at 8 per cent above its nominal value. Here we have exactly the converse of the last problem, and therefore, having found the value of ~1 sterling, we divide the given sum instead of multiplying; thus the premium on $40, at 8 per cent., is $3.20, which added to $40 makes $43.20, and 43.20-9=4.80, and $465 —4.80 —~96.17.6. RULE. Divide the given sum by the number denoting the value of one pound sterling at the given rate abovepar, and if there be a decimal remaining reduce it to shillings and pence. EXERCISES. 1. When sterling exchange is auoted at 108, what is the value of ~1? Ans. $4.80; EXCHANTGE WITH GREAT ERITAIN. 249 2. If ~1 sterling be worth $4.844, what is the premium of exchange between London and America. Ans. 9 per cent. 3. At 10 per cent. above its nominal value, what is the worth of ~50 sterling, in decimal currency? Ans. $244.44. 4. When sterling exchange is quoted at 9j per cent. premium, what is the value of $1000? Ans. ~205 18s. 1d. 5. At 12 per cent. above its nominal value, what will a bill for ~1800 cost in dollars and cents? Ans. $8960. 6. A merchant sold a bill of exchange on London for ~7000, at an advance of 11 per cent; what did he receive for it more than its real value? Ans. $466.662. 7. Bought a bill on London for ~1266 15s. at 9- per cent. premium; what shall I have to pay for it? Ans. $6164.85. 8. A merchant sells a bill on London for ~4000, at 8 per cent. above its nominal value, instead of importing specie at an expense of 2 per cent.; what does he save? Ans. $122.662. 9. A merchant in Kingston paid $7300 for a draft of ~1500 on Liverpool; at what per cent. of premium was it purchased? Ans. 91. 10. Exchange on London can be purchased in Detroit at 108~; in New York at 1081. At which place would it be the most advantageous to purchase a bill for ~358 14s. 9d., supposing the N.Y. broker charges - per cent. commission for investing and gold drafts mn New York are at a premium of I per cent. Ans., Detroit by $6.82. 11. A broker sold a bill of exchange for ~2000, on commission, at 10 per cent. above its nominal value receiving a commission of -Q per cent. on the real value, and 5 per cent. on what he obtained for the bill above its real value; what was his commission? Ans. $11.95tj. 12. I owe A. N. McDonald & Co., of Liverpool, $7218, net proaeeds of sales of merchandise effected for them, which I am to remit them in a bill of exchange on London for such amount as will close the transaction, less i per cent. on the face of the bill for my commission for investing. Bills on London are at 8 per cent. premium. Required the amount of the bill, in sterling money, to be remitted. Ans. ~11500. 250 ARITHMETIC. TABLE OF FOREIGN MONEYS. CITIES AND COUNTRIES. DENOMLNATIONS OP MONEY. VALUE. London, Liverpool, &c. 12 pence-=l shilling; 20 shillings =1 pound..................... $4.86 Paris, Havre, &c........ 100 centimes=l franc.............183 Amsterdam, Hague, &c. 100 cents-1 guilder or florin....40 Bremen.................. 5 swares-1 grote; 72 grotes=l rix dollar.......................=.781 Hamburg, Lubec, &c... 12 pfennings=-1 schilling; 16s.1 mark banco....................35 Berlin, Dantzic.......... 12 pfennings=l groschen; 30 gro. = 1 thaler.......................69 Belgium.................. 100 centimes=l1 franc...........18 St. Petersburg........ 100 kopecks 1 ruble.............75 Stockholm............ 12 rundstycks 16 skillings; 48s. =1 rix dollar specie.......... 1.06 Copenhagen......... 16 skillings==1 mark; 6 m.= — rix dollar..........................=. 1.05 Vienna, Trieste, &c.... 60 kreutzers=1 florin.............481 Naples................. 10 grani —1 carlino; 10 car.=l ducat..............................80 Venice, Milan, &c....... 100 centesimi-l lira.........16 Florence, Leghorn, &c. 100 centesimi=1 lira............ —.16 Genoa, Turin, &c....... 100 centesimi-1 lira......... 18 Sicily..................... 20 grani=1 taro; 30 tari==1 oz.= 2.40 Portugal............... 000 reas- 1 millrea............- 1.12 34 maravedis==1 real vellon=.05 Spami.................... 68 maravedis=1 real plate..10 Constantinople........100 aspers=l piaster..........-.05 British India............ 12 pice-1 anna; 16 annas —1 rzpeee...........................=.44 Canton............... 100 candarines=1 mace; 10 m.1 tae........................... 1.48 Mexico......8 ials= — dollar................= 1.00 Monte Video............. 00 centesimas=-1 rial; 8 rials-1 dollar..........................83-f3 Brazil................... 1000 reas=1 milrea.............=.824 Cuba.................... 8 reals plate or 20 reals vellon 1 dollar......................... 1.00 Turkey.................. 100 aspers=1 piaster............05 United Stafes............ 10 mills= cent 10 cents dime; 10 dimes —l' dollar.....= variable. New Brunswick......... 4 farthings penny; 12 pence Nova Scotia......... 1 shilling; 20 shillings=l Newfoundland........ pound.*.................. 4.00 * The Government of New Brunswick now issues postage stamps in the decimal currency, but so for as we have been able to ascertain, the currency of ARBITRATION OF EXCHANGE. 251 ARBITRATION OF EXCHANGE. Arbitration of Exchange is the methol of finding the rate of exchange between two countries through the intervention of one or more other countries. The object of this is to ascertain what is the most advantageous channel through which to remit money to a foreign country. Three things have here to be considered. First, what is the most secure channel; secondly, what is the least expensive, and thirdly, the comparative value of the currencies of the different countries. Regarding the two first considerations no general rule can be given, as there must necessarily be a continual fluctuation arising from political and other causes. We are therefore compelled to confine our calculation to the third, viz., the comparative value of the coin current of different countries. For this purpose we shall investigate a rule, and append tables. Let us suppose an English merchant in London wishes to remit money to Paris, and finds that owing to certain international relations, he can best do it through Hamburg and Amsterdam, and that the exchange of London on Hamburg is 13~ marcs per pound sterling; that of Hamburg on Amsterdam, 40 marcs for 361 florins, and that of Amsterdam on Paris, 56j florins for 120 francs, and thus the question is to find the rate of exchange between London and Paris. SOLUTION: We write down the equivalents in ranks, the equivalent of the first term being placed to the right of it, and the other pairs below them in a similar order. Hence the first term of any pair will be of the same kind as the second term of the preceding pair. As the answer is to be the equivalent of the first term, the first term in the last rank corresponds to the third term of an analogy, and is therefore a multiplier, it must be placed below the second rank. The these three Provinces is, as usual, in pounds, shillings and pence. It is to be hoped that when the Confederation of the British Provinces takes place, the decimal currency will be speedily adopted in the Lower Provinces,, and that the efforts now being made in Britain to adapt the same currency will prove successful. 252 ARITHMETIC. terms being thus arranged, we divide the product of the second rank by that of the first, and the quotient will be the equivalent, as exhibited below: ~1 sterling= 131 marcs. 40 marcs = 36 florins. 56J florins =120 francs ~1 stg. As it is most convenient to express the fractions decimally, we have 1 3.5 36.2X5X120X1 25.87 francs. 1X40X5 6.75 The foregoing explanations may be condensed into the form of a RULE. Write down the first term, and its equivalent to the right of it, ana the other pairs in the same order, the odd term being placed under the second rank, and then divide the product of the second rank by the product of the first, the quotient will be the required equivalent. NOTE.-The true principle on which this operation is founded, is that each pair consists of the antecedent and consequent which are to each other in the ratio of equality IN POINT OF INTRINSIC VALUE, though not in regard to THE NUMBERS BY WHICH THEY ARE EXPRESSED, and therefore the required term and its equivalent must have the same relation to each other, that is, they will be an antecedent and a consequent in the ratio of equality as regards their value, but not as regards the numbers by which they are expressed. EXE RCISES. 1. If the exchange of London on'Paris is 28 irancs per pound sterling, and that of America on Paris 18 cents per franc; what is the rate of exchange of America on London, through Paris? Ans. $5.04 per ~ sterling. 2. If exchange between New York and London is at 8 per cent. premium, and between London and Paris 251 francs per pound sterling; what sum in New York is equal to 7000 francs in Paris? 3. When exchange between Portland and Hamburg is at 34 cents per mark banco, and between Hamburg and St. Petersburg is 2 marks, 8 schillings per ruble; how much must be paid in St. Petersburg for a draft on Portland for $650? Ans. 764 rubles, 7010 kopecks. I I EXCHANGE. 253 4. If a merchant buys a bil in London, drawn on Paris, at the rate of 25.87 francs per pound sterling, and if this bill be sold in Amsterdam at 120 francs for 561 florins, and the proceeds be invested in a bill on Hamburg, at the rate of 36, florins for 40 marcs; what is the rate of cxahange between London and Hamburg, or what is ~1 sterling worth in Hamburg? Ans. 13.449+marcs. 5. A merchant of St. Louis wishes to pay a debt of $5000 in New York; the direct exchange is 1j per cent. in favour of New York, but on New Orleans it is ~ per cent. discount, and between New Orleans and New York at a i per cent. premium; how much would be saved by the circular exchange compared with the direct? Ans. $87.566. A merchant in Detroit wishes to remit to J. B. Gladstone & Co., of London, ~3600 sterling. Exchange on London, in Detroit, is at a premium of 10 per cent. Exchange on London can be obtained at New York for 9 per cent. premium. If Detroit bills on New York are at a discount of J per cent., and the merchant remits a draft to New York, and pays his agent i per cent. for investing it in bills on London; what will he gain over the direct exchange? Ans. $123.80.'7. A merchant in London remits to Amsterdam ~1000, at the rate of 18 pence per guilder, directing his correspondent at Amsterdam to remit the same to Paris at 2 francs, 10 centimes per guilder, less 1 per cent. for his commission; but the exchange between Amsterdam and Paris happened to be, at the time the order was received, at 2 francs, 20 centimes per guilder. The merchant at London, not apprised of this, drew upon Paris at 25, francs per pound sterling. Did he gain or lose, and how much per cent.?, Ans. 16 6, per cent. gain. MIXED EXERCISES IN EXCHANGE. 1. When gold is quoted at 150 per cent. premium; what is the reason greenbacks are not at a discount of 50 per cent.? 2. Bar gold in London is 77s. 9d. per ounce standard; required, the arbitrated rate of exchange produced by its import to this country for coinage, at the rate of 232-1 grains of fine gold for the eagle of 10 dollars. 3. What sum in decimal money must I pay for a bill on London of ~76 14s. Id., exchange being 91- per cent. premium, and the broker's commission for negotiating the bill being ~ per cent.? 254 ARITHMETIC. 4. A merchant shipped 2560 barrels of flour to his agent in Liverpool, who sold it at ~1 8s. 6d. per barrel, and charged 2 per cent. commission; what was the net amount of the flour in decimal money, allowing exchange to be at a premium of 8 per cent.? Ans. $17160.19. 5. What is the cost of a 30 days' bill on Montreal, at ~ per cent. premium, the face of the bill being $1500? Ans. $1507.50. 6. What must be the face of a 60 days' draft on New Orleans to yield $1641.75, when sold at a discount of ~ per cent.? Ans. $1650. 7. What is the cost of a 30 days' bill on Chicago, at - per cent. premium, and interest off at 6 per cent.; the face of the bill being $9256.40?* Ans. $9240.20. 8. A merchant paid $14400.12 for a bill on Havre for $79000 francs; how much was exchange below par? Ans. 2 per cent. 9. I have in possession the net proceeds of a sale of cotton amnounting to $3765, which my correspondent desires me to'remit to him in New Orleans; exchange on New Orleans is at a discount of 2z per cent., and I invest the whole in a draft at that rate, which I remit to him; what is the face of the draft? Ans. $3861.54. 10. The proceeds of a sale of goods, consigned to me from Bremen, is $2764.67, on which I am to charge a commission of 10 per cent., and remit the balance to my consignor in such a way as shall be most advantageous to him. Exchange on Paris can be had at 92 cents per 5 francs, and in Paris exchange on Bremen is 17 francs to 4 thalers. Exchange on Liverpool can be had a 9 per cent. premium, and in Liverpool exchange on Bremen is 6 thalers to the pound sterling. Direct exchange is 804 cents per thaler. Which course will be the best, allowing -per cent. brokerage to correspondents both in Liverpool and Paris? Ans. By way of Paris. 11. A, of Buffalo, sent articles to the World's Fair in London, which were afterwards sold by B, of London, on A's account, net proceeds ~1266 15s. sterling. B was instructed to invest this amount in bills on New York, and remit to A, which was accordingly done. B charged i per cent. brokerage on the face of the bills for investing, and purchased the bills at 7 per cent. discount. Required *When there is interest to be computed, it must be reckoned on the face of the bill or draft. When other than the value or cost of the bill is to be found, proceed as in percentage. EXCHANGE. 255 the amount of the bill A must receive in dollars and cents to close the transaction. Ans. $6037.53 nearly. 12. A merchant in Boston having to remit ~434 15s. to Liverpool, wishes to know which is the most profitable, to buy a set of exchange on Liverpool at 10- per cent. premium, or send it by way of France; exchange on the latter place being 19| cents per franc, and exchange on Liverpool can be bought in France at the rate of 241 francs per pound sterling, and he has to pay his correspondent in France 4 of 1 per cent. for purchasing the bill on Liverpool. Ans. By way of France, $15.69. 13. John DcDonald & Co., of Toronto, Canada West, wish to remit to a creditor in London ~1241 15s. 9d. Exchange on London can be bought in Toronto at 109j, but Exchange on London can be purchased in New York for gold at 108~. In New York it takes $1.85 greenbacks to equal $1 in gold. The broker in New York charges j per cent. on the greenback value for investing. If Exchange on New York is at 47 per cent. discount, at which place would it be the most advantageous to purchase, and how much gain, and if the remittance be made by way of New York, what would be the face of the draft? Ans. New York by $141.72; face of draft, $11161.21. [4. Find the arbitrated rate of exchange between London and Amsterdam when the exchange of London on Madrid is 37 pence for one dollar of plate, and that of Amsterdam on Madrid is 100 florins, 75 cents, for 40 ducats of plate. 15. Hughes Bros. & Co., purchase of E. Chaffey & Co., a sterling bill at 60 days on Gladstone & Hart, of London, for ~3956 10s. They remit this bill to James Alder, in London, where it is accepted by Gladstone & Hart, and falls due on the 20th November, at which time it is protested causing an expense of ~2 19s. Gladstone & Hart having failed, E. Chaffey & Co.'s agent in London pays James Alder on the 20th November, ~2000 on account. How much must E. Chaffey & Co., pay to Hughes, Brothers & Co., on the 24th December, to cover the amount still due in London, allowing interest at the rate of 10 per cent. from November 20th, to the maturity of a 60 days' bill at date of 24th December, and - of 1 per cent. commission for their trouble in negociating a new bill? Ans. $9815.91. 250 ARITHMETIC.!NVOLUTION. Involution is the process of finding a given power of a given number. We have noted already, under the head of multiplication, that the product of any number of equal factors is called the second, third, fourth, &c., power of the number, according as the factor is taken two, three, four, &c., times. Thus: 9=3 X3 is the second power of 3; 27=3X3X3 is the third power of threes 81-=33X3X 3 is the fourth power of 3. These are often written thus: 32, 33, 34, &c. The small figures, 2, 3, 4, indicate the number of factors, and therefore each is called the index or exponent of the power. Hence to find any required power of a given quantity, we have the RULE. Multiply the quantity continually by itself until it has been used as a factor as often as there are units in the index. Since the first multiplication exhausts two factors, the number of operations will be one less than the number of factors. Involution, then, is nothing more than multiplication, and for any power above the second, it is a case of continual multiplication. For the sake of uniformity the original quantity is called the first power, and also the root in relation to higher powers. Again, if we multiply 3X3 by 3X3X3, we have five factors, or 3X3X3X3X3, but this being an inconvenient form, it is written briefly 35, the 5 indicating the number of times that 3 is to be repeated as a factor. Hence, since 3X3 is written 32, and 3X33 is written 33, it follows that 32 X33=35, and therefore we may multiply quantities so expressed by adding their indices, and so also we may divide such quantities by subtracting the index of the divisor from that of the dividend. For example 33.-32=3 or 31. If we divide 31 by 31 by subtracting the index of the divisor from that of the dividend, we obtain 30, but 3 or 31 divided by 3 or 31 is equal to 1, and therefore any quantity with an index zero is equal to unity. When high powers are to be found, the operation may be shortened in the following manner:-Let it be required to find the sixteenth power of 2. We first find the second power of 2, which is 4, INVOLUTION. 257 then 4X4=16, which is the fourth power, and 1GX16=256, the eighth power, and 256X256= 65536, the sixteenth power. If we wished to find the nineteenth power, we should only have to multiply the last result by 8, which is the third power of 2, for 216 X 2 3 21 9. EXE R CISES. 1. Find the second power of 697. Ans. 485809. 2. What is the third power of 854? Ans. 622835864. 3. What is the second power of 4.367? Ans. 19.070689. 4. Find the fourth power of 75. Ans. 31640625. 5. What is the sixth power of 1.12? Ans. 1.9738+. 6. What is the second power.7, correct to six places? Ans..060893+. 7. What is the fifth power of 4? Ans. 1024. 8. Find the third power of.3 to three places? Ans..036963. 9. What is the third power of 7? Ans. 4 10. What is the fifteenth power of 1.04?* Ans. 1.800943. 11. Raise 1.05 to the thirty-first power. Ans. 4.538039. 12. What is the eighth power of 3? Ans. 3s561 13. What is the second power of 4g? Ans. 23 4. 14. Expand the expression 65. Ans..7776. 15. What is the second power of 5j? Ans. 1-i — 301. 16. What part of 83 is 26? Ans. a. 17. What is the difference between 56 and 46? And. 11529. 18. Expand 35 X24. Ans. 3888. 19. Express, with a single index, 473X475 X476? Ans. 4714. 20. How many acres are in a square lot, each side of which is 135 rods? Ans. 113 acres, 3 roods, 25 rods. 21. What is the sixth power of.1? Ans..000001. 22. What is the fourth power of.03? Ans..00000081. 23. What is the fifth power of 1.05? Ans..1.2762815625. 24. What is the third power of.001? Ans..000000001. 25. What is the second power of.0044? Ans..00001836. The second power of any number ending with the digit 5 may be readily found by taking all the figures except the 5, and multio This exercise will be most readily worked by finding the sixteenth power, and dividing by 1.04. So in the next exercise, find the thirty-second power, and divide by 1.05. A still more easy mode of working such questions will be found under the head of logarithms. 258 ARITHMETIC. plying that by itself, increased by a unit, and annexing 25 to the result. Thus, to find the second power of 15, cut off the 5, and 1 remains, and this increased by 1 gives 2, and 2X1-=2, and 25 annexed will give 225, the second power of 15. SD also, 2,5 3,5 6,5 10,5 21,5 57,5 3 4 7 11 22 58 625 1225 4225 11025 46225 330625 EXERCISES ON THIS METHOD. 26. What is the second power of 135? Ans. 18225. 27. What is the second power of 205? Ans. 42025. 28. What is the second power of 335? Ans. 112225. 29. What is the second power of 455? Ans. 207025. 30. What is the second power of 585? Ans. 342225. 31. What is the second power of 795? Ans. 632025. NOTE.-The square root of any quantity ending in 9, must end in either 3 or 7. No second power can end in 8, 7, 3 or 2. The second root of any quantity ending in 6, must end in 4 or 6. The second root of any quantity ending in 5, must end also in 5. The second root of any quantity ending in 4, must end either in 8 or 2. The second root of any quantity ending in 1, must end either in 1 or 9. The second root of any quantity ending in 0, must also end in 0. EVOLUTION. The root of any quantity is a number such that when repeated, as a factor, the specified number of times, will produce that quantity. Thus, 3 repeated twice as a factor gives 9, and therefore 3 is called the second root of 9, while 3 taken three times as a factor will give 27, and therefore 3 is called the third root of 27, and so also it is called the fourth root of 81. There are two ways of indicating this. First, by the mark I/ which is merely a modified form of the letter r, the initial letter of the English word root, and the Latin word radix (root). When no mark is attached, the simple quantity or first root is indicated. When the second root is meant, the mark i/ alone is placed before the quantity, bu) if the third, fourth, &c., roots are to be indicated, SECOND OR SQUARE ROOT. 259 the figures 3, 4, &c., are written in the angular space. Thus: 4 5 3=-/9 —27 —1/81= 1/243, &c., &c. The other method is to write the index as a fraction. Thus, 92 means the second root of the first power of 9, i. e. 3. So also, 273 is the third root of the 2 first power of 27. In the same manner 64T means the third root of the second power of 64, or the second power of the third root of 64. Now the third root of 64 is 4, and the second power of 4 is 16, or the second power of 64 is 4096, and the third root of 4096 is 16, so that both views give the same result. Evolution is the process of finding any required root of a given quantity. SECOND OR SQUARE ROOT. Extracting the square or second root of any number, is the finding of a number which, when multiplied by itself, will produce that number. To find the second root, or square root of any quantity. By inspecting the table of second powers, it will be found that the second power of any whole number less than 10, consists of either one or two digits; the second power of any number greater than 9, and less than 100, will in like manner be found to consist of three or four digits; and, universally, the second power of any number will consist of either twice the number of digits, or one less than twice the number of digits that the root itself consists of. Hence, if we begin at the units' figure, and mark off the given number in periods of two figures each, we shall find that the number of digits contained in the root will be the same as the number of periods. If the number of digits is even, each period will consist of two figures, but if the number of digits be odd, the last period to the left will consist of only one figure. Let it now be required to find the second root of 144. We know by the rule of involution that 144 is the second power of 12. Now 12 may be resolved into one ten and two units, or 10+2, and 10+2 multiplied by itself, as in the margin, gives 100+40+4, and since 100 is the second power of 10, and 4 the second power of 2, and 40 is twice the product of 10 and 2, we conclude that the second 260 ARITHMETIC. power of any number thus resolved is equal to the sum of the seeond powers of the parts, plus twice the product of the parts. Hence to find the second root of 144, let us 10+2 resolve it into the three parts 100+40+4, and we 10+2 find that the second root of the first part is 10, and since 40 is twice the product of the parts, 40 100+20 20+4 divided by twice 10 or 20 will give the other part 2, and 10+2=12, the second root of 144. We 100+40+4 should find the same result by resolving 12 into 11+1, or 9+3, or 8+4, or 7+5, or 6+6, but the most convenient mode is to resolve into the tens and the units. In the same manner, if it be required to find the second root of 1369, we have by resolution 900+420+49, of which 900 is the second power of 30, and 30X2=60, and 420 —60=7, the second part of the root, and 30+7=37, the whole root. Again, let it be required to find the second root of 15129. This may be resolved as below: 10000 is the second power of 100. 400 is the second power of 20. 9 is the second power of 3. 4000 is twice the product of 20 and 100. 600 is twice the product of 100 and 3. 120 is twice the product of 20 and 3. 15129 is the sum of all, and hence 1 is the root of the hundreds, 2 the root of the tens, and 3 the root of the units. Generalizing these investigations, we find that the second power of a number consisting of units alone is the product of that number by itself; that the second power of a number consisting of tens and units is the second power of the tens, plus the second power of the units, plus twice the product of the tens and units; that the second power of a number, consisting of hundreds, tens and units, is the sum of the squares of the hundreds, the tens, and the units, plus twice the product of each pair. Now since the complement. of the full second power, to the sum of the second powers of the parts, is twice the product of the parts, it follows that, when the first figure of the root has been found, it must be doubled before used as a divisor to find the second term, and for the same reason each figure, when found, must be doubled to give correctly the- next divisor. Hence the SECOND OR SQUARE ROOT. 261 RULE. Beginning at the units' figure, mark off the whole line in periods of two figures each; find the greatest power contained in the left hand period, and subtract it from that period; to the remainder annex the next period; for a new dividend, place thefigure thus obtained as a quotient, and its double as a divisor, and find how often that quantity is contained in the second partial dividend, omitting the lastfigure; annex the figure thus found to both divisor and quotient, multiply and subtract as in common division, and to the remainder annex the next period; double the last obtained figure of the divisor, and proceed as before till all the periods are exhausted,-if there be a remainder, annex to it two ciphers, and the figure thence obtained will be a decimal, as will everyfigure thereafter obtained. EXAMPLES. 1. To find the second root of 797449. First, commencing with the units' figure, we divide the line into periods, viz., 49, 74 and 79,-we then note that the greatest square contained in 79 is 64,-this we subtract from 79, and find 15 remaining, to which 8 797449 893 we annex the next period 74, and place 64 8, the second root of 64, in the quotient, 169 1574 and its double 16 as a divisor, and try 1783 1521 how often 16 is contained in 157, which we find to be 9 times; placing the 9 in 5349 both divisor and quotient, we multiply 5349 and subtract as in common division, and find a remainder of 53, to which we annex the last period 49, and proceeding as before, we find 3, the last figure of the root, without remainder, and now we have the complete root 893. 2. This operation may be illustrated as follows: To find the second root of 273529. 500 273529 500+20+3=523 500X2=1000+20, or 250000 1020 23529 1000+2X20+3=-1043 20400 3129 1s8 1 3129 262 ARITHMETIC. 3. To find the second root of 153687. Here we obtain, by the same process as in the last example, the whole number 392, with a remainder of 23, which can produce only a fraction. 3 153687 392.029+ 9___~_ ~We now annex two ciphers, 69 636 placing the decimal point after 782 621 the root already found, but as ---- the divisor is not contained in 1587 78402 1564 this new dividend, we place a cipher in both quotient and di784049 230000 visor, and annex two ciphers 156804 more to the dividend, and by — 9- - continuing this process we find 7319600 the decimal part of the root, and the whole root is 392.029+. 263159 EXERCISES. 1. What is the second root of 279841? Ans. 529. 2. What is the second root of 74684164? Ans. 8642. 3. What' is the second root of 459684? Ans. 678. 4. What is the second root of 785? Ans. 28.01785+. 5. What is the second root of 1728? Ans. 41.569219+. 6. What is the second root of 666? Ans. 25.8069+. 7. What is the second root of 123456789? Ans. 11111.11106+. 8. What is the second root of 5 to three places? Ans. 2.236. 9. What is the side of a square whose area is 19044 square feet? Ans. 138 feet. 10. What is the length of each side of a square field containing 893025 square rods? Ans. 945 linear rods. The second root of a fraction is found by extracting the roots of its terms, for 16_4 X 4 and therefore V/-=V/ X4=-. So also, /49==. Again, since V /8 — 9 — 09, and.3 X.3=.09, the second root of.09 is.3. This follows from the rules laid down for the multiplication of decimals. To find the second root of a decimal or of a whole number and a decimal: SECOND OR SQUARE ROOT. 263 Point of periods of two figures each from the decimal point towards the right and left, adding a cipher, or a repetend, if the number of figures be odd. From what has been said, it is plain that every period, except the first on the left, must consist of two digits, and every decimal presupposes something going before, for.5 indicates the half of some unit under consideration, and.5 is equivalent to.50, and not to.05, from which it is obvious that the second root of.5 is not the root of.05, but of.50, and therefore the second root of.5 is not.2+, as the beginner would naturally suppose, but.7+, for.2+ is the approximate root of.05. ADDITIONAL EXERCISES. 11. What is the second root of.7 to five places of decimals? Ans..83666. 12. Find the second root of.07 to six places. Ans. 264575. 13. What is the second root of.05? Ans..2236+. 14. What is the second root of.7? Ans..8819+-. 15. Find the second root of.5. Ans..74535+.. 16. What is the second root of.1? Ans..3162277-+. 17. What is the second root of.i? Ans..3. 18. What is the second root of 1.375? Ans. 1.1726, &c.* 19. What is the second root of.375? Ans. 61237, &c.* 20. What is the second root of 6.4? Ans. 2.52982+. 21. Find to four decimal places 1/3a. Ans. 1.7748. 22. Find i/2 to four decimal places. Ans. 1.4142. 23. Find the value of 1/3271.4207. Ans. 57.196+.. 24. Find the second root of.005 to five places. Ans. 07071.. 25. Find the square root of 4.372594. Ans. 2.09107+-. 26. What is the second root of.01? Ans..1 27. What is the second root of.001? Ans. 03162-. 28. What is the square root of.0001? Ans..01. 29. What is the second root of.000001? Ans..001. 30. What is the second root of 19.0968? * The young student would naturally expect that the decimal figures of 1/1.375 and -/.A375 would be the same, but it is not so. If it were so, 1/1+;/.375 would be equal to 1/1.375. That such is not the case, may be shown by a very simple example. 1/16+ —/9=4+3-.-7, but i/16ti9=1/25=5. Let it be carefully observed, therefore, that the sum of the second roots is not the same as the second root of the sum. 264 ARITHMETIC. OPERATION 4 19.0968 4.37 trial. 83 16 4.36 true. 309 Trial 867 249 6068 Too great by 1 6069 True 866 6068 5196 872 lHcre we find the remainder, 872, is greater than the divisor, 866, which seems inconsistent with ordinary rules; but it must be observed that we are not seeking an exact root, but only the closest possible approximation to it. If the given quantity had been 19.0969, we should have found an exact root 4.37. The remainder 872 being greater than the divisor, shows that the last figure of the root is too small by 9o%, whereas 7 would be too great by Tl1O, and that 866 is not a correct divisor but an approximate one, and that the true root lies between 4.36 and 4.37. When the root of any quantity can be found exactly, it is called a perfect power or rational quantity, but if the root cannot be found exactly, the quantity is called irrational or surd. A number may be rational in regard to one root, and irrational in regard to another.. Thus, 64 is rational as regards 1/64=8, 6 Fy64=4 and /64=2, but it is irrational regarding any other root expressed by a whole number. But 64, with the fractional index ], i. e., 641, is rational, because it has an even root as already shown. We may call 64~ either the second power of the third root of 64, or the third root of the second power. In the former view, the third root of 64 is 4, and the second power of 4 is 16, and according to the second view, 642 is 4096, and the third root of 4096 is 16, the same 4 as before. 1/81=3 is rational, and i/81=9 is rational, but 81 is not'rational regarding any other root; while /25 is rational only regarding the second root, and "8=_2 only regarding the third root. The second root. of an even square may be readily found by resolving the number into its prime factors, and taking each of these THIRD ROOT OR CUBE ROOT. 265 factors once,-the product will be the root. Thus, 441 is 3 X 3 X 7 X 7 and each factor taken once is 3X7=21, the second root. Here let it be observed, that if we used each factor twice we should obtain the second power, but if we use each factor half the number of times that it occurs, we shall have the second root of that power. 64 is 2X2X2X2X2X2=_26, i. e., 2 repeated six times as a factor gives the number 64, and therefore half the number of these factors will give the second root of 64, or 2X2X2=8, and 2X2><2 multiplied by 2X2X2=8X8= 64. As this cannot be considered more than a trial method, though often expeditious, we would observe that the smallest possible divisors should be used in every case, and that if the number cannot be thus resolved into factors, it has'no even root, and must be carried out into a line of decimals, or those decimals may be reduced to common fractions. THIRD ROOT OR CUBE ROOT. As extracting the second root of any quantity is the finding of what two equal factors will produce that quantity, so extracting the third root is the finding of what three equal factors will produce the quantity. By inspecting the table of third powers, it will be seen that no third power has more than three digits for each digit of the first power, nor fewer than two less than three times the number of digits. Hence, if the given quantity be marked off in periods of three digits each, there will be one digit in the first power for each period in the third power. The left hand period may contain only one digit. From the mode of finding the third power from the first, we can deduce, by the converse process, a rule for finding the first power 266 ARITHMETIC. from the third. We know by the rule of involution that the third power of 25 is 15625. If we resolve 25 into 20+5, and perform the multiplication in that form, we have 20 —5 400+100 100+25 400+200+25=(20+5)i 20+5 8000+4000+500 2000+1000+125 8000+6000+1500+125-(20+5) 3-15625 Now, 8000 is the third power of 20, and 125 is the third power of 5; also, 6000 is three times the product of 5, and the second power of 20, and 1500 is three times the product of 20, and the second power of 5. Let a represent 20 and b represent 5, then a3=203 - 8000 3 a2 b =-3X202X5 - 6000 3 a b2 —3X20X5 = 1500 b3 —53 - 125 15625 By using these symbols we obtain the simplest possible method of extracting the third root of any quantity, as exhibited by the subjoined scheme: Given quantity....................... 15625 a==203-20X20X20...........= 8000 Remainder.................... 7625 3 a2 b-3X202 X5...............- 6000 Remainder............................. 1625 3 a b2-3X20X52..............=. 1500 Remainder........................... 125 b3 =53-5X5X5..................- 125 From this and similar examples we see that a number denoted by more than one digit may be resolved into tens and units. Thus, 25 is 2 tens and 5 units, 123 is 12 tens and 3 units, and so of all numbers. THIRD ROOT OR CUBE ROOT. 267 To find the third root of 1860867: As this number consists of three periods, the root will consist of three digits, and the first period from the left will give hundreds; the second tens, and the third units, and so also in case of remainder, each period to the right will give one decimal place, the first being tenths, the second hundredths, &c., &c. We may denote the digits by a, b and c. a=-100 1860867(100+20+3 =123 a3 —=100 —3 1000000 b-3a2 b- 3 a2= 3%-8-~6-= 20+, 860867 remainder. and 30000X20- 600000 260867 remainder..3 a b2 -3X100X400 120000 140867 remainder. b —203=- 8000 Now (a+b)=120.. 3 (a+b)2=-132867 remainder. 43200, which is contained3 times+in 132867,.*. c=3, and 3 (a+-b)c2 -. v 1 9,2 v A-__ 1296i00 268 ARITHMETIC. From the last remainder subtract three times the product of the term last found, and the square of the SUM of the preceding terms, PLUS the product of the square of the last found term by the suMr of the preceding ones, PLUS the third power of the last found term, and so on. EXERCISES. 1. What is the third root of 46656? Ans. 36. 2. What is the third root of 250047? Ans. 63. 3. What is the third root of 2000576? Ans. 126. 4. What is the third root of 5545233? Ans. 177. 5. What is the third root of 10077696? Ans. 216. 6. What is the third root of 46268279? Ans. 359. 7. What is the third root of 85766121? Ans. 441. 8. What is the third root of 125751501? Ans. 501. 9. What is the third root of 153990656? Ans. 536. 10. What is the third root of 250047000? Ans. 630. 11. What is each side of a square box, the solid content of which is 59319? Ans. 39 inches. 12. What is the third root of 926859375? Ans. 975. 13. Find the third root of 44.6. Ans. 3.456+. 14. What is the third root of 9? Ans. 2.08008+. 15. What is the length of each side of a cubic vessel whose solid content is 2936.493568 cubic feet? Ans. 1432 feet. 16. Find the third root of 5. Ans. 1.7099. 17. A store has its length, breadth and height all equal; it can hold 185193 cubic feet of goods; what is each dimension? Ans. 57 feet. 18. How many linear inches must each dimension of a cubic vessel be which can hold 997002999 cubic inches of water? Ans. 999 inches. 19. What is the third root of 1? Ans. 1. 20. What is the third root of 144? Alns. 5.241483. The third root of a fraction is found by extracting the third root of the terms. The result may be expressed either as a dommon fraction, or as a decimal, or the given fraction may be reduced to a decimal, and the root extracted under that form. THRD ROOT OR COBE ROOT. 269 EXERCISES. 1. What is the third root of 2-? Ans. i-=.75, Otherwise: 27=.421875. To find the third root of this we have.42i875(.70+.05=.75 70 3 — 343000 3X70 X5 =73500) 78875 remainder. 3X70 X52=- 5250.53 125 -= 78875...... no remainder. The third root of a mixed quantity will be most readily found by reducing the fractional part to the decimal form, and applying the general rule. It has been already explained that the second root of an even power may be obtained by dividing the given number by the smallest possible divisors in succession, and taking half the number of those divisors as factors. The same principle will apply to any root. If the given quantity is not an even power, it may yet be found approximately. If we take the number 46656, we notice that as the last figure is an even number, it is divisible by 2, and by pursuing the same principle of operation we find six twos as factors, and afterwards six threes; and, as in the case of the second root, we take each factor half the number of times it occurs, so in the case of the third root, we take eachfactor one-third the number of times it occurs. The same principle on which the extraction of the second and third depends may be applied to any root, the line of figures being divided into periods, consisting of as many figures as there are units in the index; for the fourth root, periods of four figures each; for the fifth, five, &c., &c. We may remark, however, that these modes are now superseded by the grand discovery of Logarithmic Computation. 270 AITMETIC. P R O G R E S S IO N. A series is a succession of quantities increasing or decreasing by a Common Difference, or a Common Ratio. Progression by a Common Difference forms a series by the addition or subtraction of the same quantity. Thus 3, 7, 11, 15, 19, 23 forms a series increasing by the constant quantity 4, and 28, 21, 14, 7, forms a series decreasing by the constant quantity 7. Such a progression is also called an equidifferent series.* Progression by a Common Ratio forms a series increasing or decreasing by multiplying or dividing by the same quantity. Thus, 3, 9, 27, 81, 243, is a series increasing by a constant multiplier 3, and 64, 32, 16, 8, 4, 2, is a series decreasing by a constant divisor 2. The quantities forming such a progression are also called Continual Proportionals,* because the ratio of 3 to 9 is the same as the ratio of 9 to 27, &c., &c. From this it is plain that in a progression by ratio, each term is a mean proportional between the two adjacent ones, and also between any two that are equally distant from it. The first and last terms are called the Extremes, and all between them the Means. PROGRESSION BY A COMMON DIFFERENCE. In a series increasing or decreasing by a common difference, the sum of the extremes is always equal to the sum of any two that are equally distant from them. Thus, in the first example 3+23=7+ 19=11+15=26, and in the second 28+7=21+14=35. If the number of terms be odd, the sum of the extremes is equal to twice the middle term. Thus in the series 3, 7, 11, 15, 19, 3+19=2X11=22, and hence the middle term is half the sum of the extremes. * The names Arithmetical Progression and Geometrical Progression are often applied to quantities so related, but these terms are altogether inappropriate. as they would indicate that the one kind belonged solely to arithmetic, and the other solely to geometry, whereas, in reality, each belongs to both these branches of science. PROGRESSION BY A COMMON DIFFERENCE. 271 In treating of progressions by difference or equidifferent series, there are five things to be considered, viz., the first term, the last term, the common difference, the number of terms, and the sum of the series. These are so related to each other that when any three of them are known we can find the other two. Given the first term of a series, and the common difference, to find any other term. Suppose it is required to find the seventh term of the series 2, 5, 8, &c. Here, as the first term is given, no addition is required to find it, and therefore six additions of the common difference will complete the series on to seven terms. In other words, the common difference is to be added to the first term as often as there are units in the number of terms diminished by 1. This gives 7-1-6, and 6X3-18, which added to the first term 2 gives 20 for the seventh term. If we had taken the series, on the descending scale, 20, 17, 14, &c., we should have had to subtract the 18 from the first term 20 to find the seventh term 2. The term thus found is usually designated the last term, not because the series terminates there, for it does'not, but simply because it is the last term considered in each question proposed. From these illustrations we derive the RULE (1.) Subtract 1 from the number of terms, and multiply the remainder by the common difference; then if the series be an increasing one, add the result to the first term, and if the series be a decreasing one, subtract it. E X A P L E S. To find the fifty-fourth term of the increasing'series, the first term of which is 33k, and the common difference 1l. Here 54-1-53, and 53X1I=661, and 661+331=100, the fifty-fourth term. Given 64 the first term of a decreasing series, and 7 the common difference, to find the eighth term. Here 8-1=7, and 7X7=49, and 64-49=15, the eighth term. EXERCISES. 1. Find the eleventh term of the decreasing series, the first term of which is 2481, and the common difference 31. Ans. 2161. 2. The hundredth term of a decreasing series is 392-, and the common difference is 3-, what is the last term? Ans. 36. common ifferene is 3~ 272 ARITHMETIC. 3. What is the one-thousandth term of the series of the odd figures? Ans. 1999. 4. What is the five-hundredth term of the series of even digits? Ans. 1000. 5. What is the sixteenth term of the decreasing series, 100, 96, 92, &c.? Ans. 40. To find the sum of any equidifferent series, when the number of terms, and either the middle term or the extremes, or two terms equidistant from them, are given. We have seen already that in any such series the sum of the extremes is equal to the sum of any two terms that are equidistant from them, and when the number of terms is odd, to twice the middle term. Hence the middle term, or half the sum of any two terms equi-distant from the extremes, will be equal to half the sum of those extremes. Thus, in the series 2+7+12+17+22+27+32, we have 2+32-7=2717, the middle term. It is plain, therefore, that if we take the middle term and half the sum of each equi-distant pair, the series will be equivalent to 17+17+17+17+17+17+17, or 7 times 17, which will give 119, the same as would be found by adding together the original quantities. The same result would be arrived at when the number of terms is even, by taking half the sum of the extremes, or of any two terms that are equi-distant from them. From these explanations we deduce the RULE. (2.) Multiply the middle term, or half the sum of the extremes, or of any two terms that are equidistant from -them, by the number of terms. NOTE.-If the sum of the two terms be an odd number, it is generally more convenient to multiply by the number of terms before dividing by 2. E X A PLES. Given 23, the middle term of a series of 11 numbers, to find the sum. Here we have onlyto multiply 23 by 11, and we find at once the sum of the series to be 253. Given 7 and 73, the extremes of an increasing series of 12 numbers, to find the sum. The sum of the extremes is 80, the half of which is 40, and 40 X 12-480, the sum required. PROGRESSION BY A COMMON DIFFERENCE. 273 Two equidistant terms of a series, 35 and 70, are given in a series of 20 numbers, to find the sum of the series. In this case, we have 35+70-105, and 105X20=2100, and 2100- -2=1050, the sum required. EXERCISES. 1. Find the sum of the series, consisting of 200 terms, the first term being 1 and the last 200. Ans 20100. 2. What is the sum of the series whose first term is 2, and twenty-first 62? Ans. 672. 3. What is the sum of 14 terms of the series, the first term of which is ~ and the last 7? Ans. 52j. 4. Find the sum to 10 terms of the decreasing series, the first term of which is 60 and the ninth 12. Ans. 360. 5. A canvasser was only able to earn $6 during the first month he was in the business, but at the end of two years was able to earn $98 a month; how much did he earn during the two years, supposing the increase to have been at a constant monthly rate? Ans. $1248. 6. If a man begins on the first of January by saving a cent on the first, two on the second, three on the third, four on the fourth, &c., &c., how much will he have saved at the end of the year, not counting the Sabbaths? Ans. $490.41. 7. How many strokes does a clock strike in 13 weeks? Ans. 14196. 8. If 8q is the fourth part of the middle term of a series of 99 numbers, what is the sum? Ans. 3465. 9. In a series of 17 numbers, 53 and 33 are equidistant from the extremes; what is the sum of the series? Ans. 731. 10. In a series of 13 numbers, 33 is the middle term; what is the sum? Ans. 429. To find the number of terms when the extremes and common difference are given: As in the rule (1), we found the difference of the, extremes by multiplying by one less than the number of terms, and addedthe first term to the result, so now we reverse the operation and find the RUL E (3.) Lvivide the difference of the extremes by the common difference and -dd 1 to the result. 274 ARITHMETIC. E X AM P L E. Given the extremes 7 and 109, and the common difference, 3, to find the number of terms. In this case we have 109-7-102, and 102- -3=34, and 34-+1=35, the number of terms. EXERCISES. 1. What is the number of terms when the extremes are 35 and 333, and the common difference 2? Ans. 150. 2. Two equidistant terms are 31 and 329, and the common difference 2; what is the number of terms? Ans. 150. 3. The first term of a series is 7, and the last 142, and the common difference i; what is the number of terms? Ans. 541. 4. The first and last terms of a series are 2- and 35~, and the common difference -; what is the number of terms? Ans. 100. 5. The first term of a series is ~ and last 12,, and the common difference -; what is the number of terms? Ans. 25. Given one extreme, the sum of the series and the number of terms, to find the other extreme. This case may be solved by reversing Rule (2), for in it the data are the same, except that there the second extreme was given to find the sum, and now the sum is given, to find the second extreme. Therefore, as in that rule we multiplied the sum of the extremes by the number of terms and halved the product, so now we must double the sum of the series and divide by the number of terms to -find the sum of the extremes, and from this subtract the given extreme, and the remainder will be the required extreme. This will illustrate the RULE (4.) Divide twice the sum of the series by the number of terms, and from the quotient subtract the given extreme, and the remainder will be the required extreme. E X A MPLE. Given 5050, the sum of a series, 1 the first term, and 100 the number of terms, to find the other extreme. Twice the sum is 10100, which, divided by 100, gives 101, and 101-1-=100, the number of terms. PROGRESSION BY A COMMON DIFFERENCE. 275 E'X E RC ISE S. 1. Given 50, the greater extreme of a decreasing series, 442, the sum, and 17 the number of terms, to find the other extreme. Ans. 2. 2. If 121268 be the sum of a series, 8 the less extreme, and 142 the number of terms; what is the greater extreme? Ans. 1700. 3. The sum of a series of 7 terms is 105, the greater extreme is 21, and the number of terms 7; what is the less extreme? Ans. 9. 4. The sum of a series is 576, the number of terms 24, and the greater extreme is 47; what is the less extreme? Ans. 1. 5. The sum of a series is 30204:, the greater extreme 312, and the number of terms 193; what is the less extreme? Ans. 1. Given the extremes and number of terms, to find the common difference. As explained in the introduction to Rule (1), the number of common differences must be one less than the number of terms. It is obvious also, that the sum of these differences constitutes the difference between the extremes, and that therefore the sum of the differences is the same as 1 less than the number of terms. Therefore the difference of the extremes, divided by the sum of the differences, will give one difference, i. e., the common difference. This gives us the RULE (5.) Subtract 1 from the number of terms, ald divide the difference of the extremes by the remainder. EXA M P L E. If the extremes of an increasing series be 1 and 47, and the number of terms 24, we can find the common difference thus:47-1-46, and 46. —232, the common difference. EXERCISES. 1. If the extremes are 2 and 36, and the number of terms 18; what the common difference? Ans. 2. 2. What is the common difference if the extremes are 58 and 3, and the number of terms 12? Ans. 5. 3. In a decreasing series given 1000 the less extreme, and 1793 the greater, and 367 the number of terms, to find the common difference. Ans. 2-. 276 ARITHMETIC. 4. If 6 and 60 are the extremes in a series of 10 numbers, what is the common difference? Ans. 6. 5. What is the common difference in a decreasing series of 42 terms, the extremes of which are 9 and 50? Ans. 1. There are fifteen other cases, but they may all be deduced from the five here given. We subjoin the Algebraic form as it is more satisfactory and complete, and also more easy to persons acquainted with the symbols of that science. Let a be the first term, d the common difference, n the number of terms, s the sum of the series; the series will be represented by a+(a+d)+(a4+2d)+(a+3d)+&c., to { a+(n-l)d. } By inspecting this series it will be seen that the co-efficient of d is always 1 less than the number of terms, for in the second term where d first appears, its co-efficient is 1, in the third it is 2, and therefore since n represents the number of terms, the co-efficient of d in the last term is n-1, and that term therefore is a+(n-1)d. If the series were a decreasing one, that is, one formed by a succession of subtractions, the last term would be a-(n-l)d. To find the sum of an equidifferent series. We have here s=a+(a+d)+(a+2d)+(a+3d)+ &c......... ~+ {a-(n-l)d. } Since a+(n-l)d is the last term, the last but one will be a+-(n-2)d, and the last but two will be a+-(n —3)d, &c., &c. But the sum of any number of quantities is the same in whatever order they may be written. Let us therefore write this series both as above, and also in reversed order: s=a- (a+d) +-(a+2d) + (a+3d) + (a+4d) +&c............ +a+ (n-3)d+a+ (n-2)d+a+ (n-l)d. s=a+(n-1)d+a+ (n-2)d+a+(n-3)d+&c................ (a+4d) (a+3d) + (a+-2d)+ (a+d) +a. Adding the two members of the second to those of the first, we obtain 2s-{ 2a+(n-l)d + -2a+(n-l)d } + {2a+ (n-1)d } + 2a+(n-1)cd +&c., to n terms. PROGRESSION BY A COMMON DIFFERENCE. 277 In the last expression all the terms are the same, but there are n terms, and therefore. the whole will be 2s=n { 2a —(n-1)d } and therefore s { 2+(n-1)d...... (1.) As we have used no single symbol to represont the last term, wo must now show how it may be obtained from the other data. Wo have seen that the last term is a+(n-l)d, which we may denote by 1, which will give us the formula Z=a+(n-1)d. This formula, in the case of a decreasing series, will become tla-(n-1l)d, and generally l=a — (n-l)d. (2.) This formula is the same as Rule (1.) We may modify (1) by (2) by substituting I for a+(n —l)d. Thus: s=n(a+). (3.) This is a convenient form when the last term is given. Using 1 for the last term, we have five quantities to consider, viz., a, 1, d, n, s, and, as already stated, any three of these being given. the other two can be found from (1) and (2.) To find d when a, 1, n are given: By (2.) 1=a+(n-1)d. -a=(n-l)d _ -a n-1. (4.) This finds the common difference, when the extremes and number of terms are given, and corresponds to Rule (5.) If a, n, s are givcn, we have By (1.) s- {2a+(n-1)d. 2s=2an+n (n-1)d.dn (n —1)=2 (s-an) d 2 (s-an) n (n-1) 19 278 ARITEHMETIC. If n is to be found from a, d, s, we have by(1.) s= 2a+(n-1)d.2s-2an=-dn2-i-d. dn2+n(2a-d)=2s And by solving this quadratic equation, we find nd_-2a ~// 8sds+(2a-d)2 2d EXAMPLES. Given a=6, d —4, n-20, to find s. First by (2) 1=a+(n-l)d -=6+(20-1)4 _-82 20 and hence by (3) s=-(6+82) =880. Given a=3, 1=300, n=33, to find d. By (4) d=ln-1 297 - 332 3 MIXED EXERCISES. 1. Given 70, the less extreme, 10 the common difference, and 44 the number of terms, to find the sum. Ans. 12540. 2. What is the less extreme when the greater is 579, the common difference 9, and the sum of the series 18915? Ans. 3. 3. What is the series when s-143, d=2, n=11? Ans. 3, 5, 7, 9, 11, 13, 15, &c. 4. Given 4 and 49, the extremes, and 6 the number of terms, to find the series. Ans. 4, 13, 22, 31, 40, &c. 5. If 120 stones are laid in a straight line, on level ground, at a regular distance of a yard and a quarter, how far must a person travel to pick them all up one by one and carry them singly and place them in a heap at the distance of 6 yards from the first, and in the same line with the stones? Ans. 10 m. 7 fur., 27 rds., 1~ yds. 6. Insert three means between the extremes 117 and 477. Ans. 207, 297, and 387. + The other variations are left as exercises for the student. PROGRESSIONS BY RATIO. 279 7. A courier agreed to ride 100 miles on condition of being paid 1 cent for the first mile, 5 for the second, 9 for the third, and so on; how much did he get per mile on an average, how much for the last mile, and how much altogether? Ans. $1.99 per mile, $3.97 for the last, and $199 for all. 8. A man performed a journey in 11 days on horseback-the first day he rode 45 miles, but, his horse getting lame, he was forced to slacken the pace at a certain rate per day, so that on the last day he made only five miles; what was the length of the journey, and at what rate did he slacken his speed? Ans. The journey was 275 miles, and the slackening of speed 4 m. per day. 9. Find the series of which 72 is the sum, 17 the first term, and number of terms 6. Ans. 17, 15, 13, 11, 9, 7. 10. The Venetian clocks strike the hours for the whole day; how many strokes will one of these strike in a year. Ans. 109500. 11. An Eastern monarch being threatened with invasion, offered his commander-in-chief a reward equivalent to a mill for the first soldier he would enlist within a month, two for the second, three for the third, and so on; the officer enlisted 999,999 men; what was his reward equal to in our money. Ans. $499,999,500. 12. One hundred sailors were drawn up in line at a distance from each other of 2 yards, including the breadth of the body-the paymaster, seated a distance of two yards from the first, sent a lieutenant to hand to the first a sum of prize money, then back again to the second, and so on to each singly; how far had the lieutenant to walk? Ans. 11 miles, 3 fur., 32 rods, 4 yds. PROGRESSIONS BY RATIO, There are in progression by ratio, as in progression by difference, the same five quantities to be considered, except that in place of a common difference we have a common ratio; that is, instead of increase or decrease by addition and subtraction, we have increase or decrease by multiplication or division. If any three of these are known the other two can be found. We have noticed already that if any quantity, 2, be multiplied by itself, the product, 4, is called the square, or second power of that 280 ARITHMETIC. quantity; if this be again multiplied by 2, the product, 8, is called the cube, or third power of that quantity; if this again be multiplied by 2, the product is called the fourth power of that quantity, and so on to the fifth, sixth, &c., powers. To show the short mode of indicating this, let us take 3X3X3X3X3=-243. For brevity this is written 35, which means that there are 5 factors, all 3, to be continually multiplied together, and 5 is called the index, because it indicates the number of equal factors. Given the first term and the common ratio to find the last proposed term. Let it be required to find the sixth term of the increasing series, of which the first term is 3 and the ratio 4. This may obviously be found by successive multiplications of the first term, 3, by the ratio, 4,-thus:3= —st term. 3X4- 12=2nd term. 12X4= 48=3rd term. 48X4= 192-4th term. 192X4- 768- 5th term. 768X4=3072- 6th term. The series, therefore, is 3, 12, 48, 192, 768, 3072. From this, it is plain, that as to find the last of 6 terms, only 5 multiplications of the first are required, in all cases the number of multiplications will be one less than the number of terms. But to multiply five times by 4 is the same as to multiply by 1024, the fifth power of 4, for 44X4X4X4X4 1024, and 1024X3=3072.* This gives us the general RULE (1.) Multiply the first term by that power of the given ratio which is a unit less than the number of terms. If the series be a decreasing one, divide instead of multiplying. E X AM P L E S. Given in a series of 12 numbers, the first term 4 and the ratio 2, to find the last term. Since 11 is one less than the number of terms, we find the 11th power of 4, which is 2048, and this, multiplied by the first term, 4, gives 8192 for the twelfth term. * For the abbreviated mode see Involution. PROGRESSIONS BY RATIO. 281 Given the ninth term of a decreasing series, 39366, and the ratio 3, to find the first term. As there are 9 terms, we take the 8th power of the ratio, 3, which we find to be 6561, and the first term 39366 —6561=6, the first term. EXERCISES. 1. What is the ninth term of the increasing series of which 5 is the first term and 4 the ratio? Ans. 327680. 2. What is the twelfth term of the increasing series, the first term of which is 1 and the ratio 3? Ans. 177147. 3. In a decreasing series the first term is. 78732, the ratio 3, and the number of.terms, 10; what is the last term? Ans. 4. 4. What is the 20th term of an increasing series, the first of which is 1.06, and also the ratio 1.06? Ans. 3.207135. 5. In a decreasing series the first term is 126.2477, the ratio 1.06; what is the last of 5 terms? Ans. 100. Given the extremes and ratio, to find the sum of the series. It is not easy to give a direct proof of this rule without the aid of Algebra, but the following illustration may be found satisfactory, and, in some sort, be accounted a proof. Let it be required to find the sum of a series of continual proportions, of which the first term is 5, the ratio 3, and the number of terms 4. Since 3 is the common ratio, we can easily find the terms of the series by a succession of multiplications. These are5+15+45+135, and the sum is 200 15+45+135+405 400 Let us now multiply each term by the ratio, 3, and, for convenience and clearness, place each term of the second line below that one of the first to which it is equal. Let us now subtract the upper from the lower line, and we find that there is no remainder, except the difference of the two extreme quantities, viz., 400. Now, it will be seen that this remainder is exactly double of the sum of the series, 200, and consequently 400 divided by 2, will give the sum 200. Also, 405 is the product of the last term by the ratio, and 400 is the difference between that product and the first term, and the divisor, 2, is a unit less than the ratio, 3. Hence the 282 ARITHMLETIC. RULE (2.).1Mutiply the last term by the ratio, from this product subtract the first term, and divide the remainder by the ratio, diminished by inity. EXAMIPLE. Given the first term of an increasing series, equal 4, the ratio 3, and the number of terms 6, to find the sum of the series. By the former rule we find the last term to be 972. This, multiplied by the ratio, gives 2916, and the first extreme, 4, subtracted from this, leaves 2912, and this divided by 2, which is less than the ratio, gives 1456, the sum of the series. EXERCISES. 1. What is the sum of the series, of which the less extreme is 4, the ratio 3, and the number of terms 10? Ans. 118096. 2. What is the sum of the series, of which 1 is the less extreme, 2 the ratio, and 14 the number of terms? Ans. 16383. 3. What is the sum of the series, of which the greater extreme is 18.42015, the less 1, and the ratio 1.06? Ans. 308.755983. 4. A cattle dealer offered a farmer 10 sheep, at the rate of a mill for the first, a cent for the second, a dime for the third, a dollar for the fourth, &c., &c.; in what amount was he " taken in," supposing that each sheep was worth $11.111? Ans. $1111100.00. 5. What is the sum of six terms of the series, of which the greater extreme is i and the ratio.? Ans. 3 2, or 1s 9 9 To find the ratio when the extremes and number of terms are given: Let it be required to find the ratio when the extremes are 3 and 192, and the number of terms 7. This is effected by simply reversing the first rule, and therefore we divide 192 by 3 and find 64, and take the 6th root of 64, which is 2, the ratio. Hence the RULE (3.) Divide the greater extreme by the less, and find that root of the quotient, the index of which is one less than the number of terms. EXAM PLE. If the greater extreme is 1024, and the less 2, and the number of terms 10, we divide 1024 by 2, and find 512, and then by extracting the ninth root of 512, we find the ratio, 2. PROGRESSIONS BY RATIO. 283 EXERCISES. 1. If the first yearly dividend of a joint stock company be $1, and the dividends increase yearly, so as to form a series, of continual proportionals, what will all amount to in 12 years, the last dividend being $2048, and what will be the ratio of the increase? Ans. ratio, 2; sum, $4095. 2. What is the ratio, in the series of which the less extreme is 3 and the greater 98034, and the number of terms 16. Ans..196605. 3. What is the ratio of a series, the extremes of which are 4 and 324, and the number of terms 5? Ans. 3. 4. What is the ratio of a series, the number of terms being 7 and the extremes 3 and 12288? Ans. 4. 5. In a series of 23 terms the extremes are 2 and 8388608; what is the ratio? Ans. 2. To insert any number of means between two given extremes: Find the ratio by Rule (3), and multiply the first extreme by this ratio, and the second will be obtained, and divide the last by the ratio, and the last but one will be obtained; continue this operation until the required term or terms be procured. NOTE.-A mean proportional is found by taking the sguare root of the product of the extremes. EXAMPLE. Let it be required to insert between the extremes 5 and 1280 three terms, so that the numbers constituting the series shall be continual proportionals. The number of terms here is 5, and hence, by Rule (3), we find the ratio to be 4, and 5 multiplied by this will give the second term, 20, and that again multiplied by 4.will give 80, the third, and that again multiplied by 4 will give the fourth term, 320, so that the full series is found to be 5, 20, 80, 320, 1280. The same result would be found by dividing the greater extreme by 4, and so on downwards, thus: 1280, 320, 80, 20, 5. EXERCISES. 1. Between 5 and 405 insert three terms, which shall make the whole a series of continual proportionals. Ans. 5, 15, 45, 135, 405. 2. Insert between ~ and 27 four terms to form a series, and give the ratio. Ratio, 3; series, a, i, 1, 3, 9, 27. 284 ARITHMETIC. 3. What three numbers inserted between 7 and 4375 will form a series of continual proportionals? Ans. 35, 175, 875. 4. What is the mean proportional between 23 and 8464? Ans. 441.2164+. 5. Find a mean proportional between 1 and 4. Ans.. ALGEBRAIC FORM. Let a represent the first term, I the last, r the ratio, n the number of terms, and s the sum. Then s=a+-a-ar ar- -a r + lr3 +a4 -&c...... ar-.2 +ar -'1. Multiplying the whole equation by r, we obtain rs- ar+ar2 -~ars +ar4 +ar5 +&c...... ar'- 1 +arn But s-=a-ar+ar2 +ar3 3-ar4 +ar5 +&c...... arn-1. Subtracting, we obtain rs-s=s(r —1)arn —a, and therefore arn-a = —1......(1.) But we found the last term of the series to be ar7'-1, calling this 1, we have from (1.) srl-a......(2.) If r is a fraction, r" and ar" decrease as n increases, as already shown under the head of fractions, so that if n become indefinitely great, ar" will become unassignably small, compared with any finite quantity, and may be reckoned as nothing. In this case (1) will become s - -- a....(3.) By this formula we can find the sum of any infinite series so closely as to differ from the actual sum by an amount less than any assignable quantity. This is called the limit, an expression more strictly correct than the sum. From the formula s rla, any three of the quantities a, r, 1, s being given, the fourth can be found. Let it be required to find the sum of the series 1-++i+i- + &c., to infinity. Here a —1 and r.. s-1- =- - =-1 X2-2. Therefere, 2 is the number to which the sum of the series continually approaches, by the increase of the number of its terms, and is the limit from which it may be made to differ by a quantity less than any assignable quantity, and is also the limit beyond which it can never pass. PROGRESSIONS BY RATIO. 285 By adding the first two terms, we find 1+- = 2 —I=1-. By adding the first three terms, we find -3+-i-7= 2 —-i=1j. By adding the first'our terms, we find -'r+-15-=-2 — l. By adding the first five terms, we find -+ —- 1 3 G==2 11H~ By adding the first six terms, we find 3 -[ 1 6 23-2 -31 131 It will be observed here that the difference from 2 is continually decreasing. The next term would differ from 2 by X, and the next by T4, &c., &c. Thus, when the series is carried to infinity, 2 may be taken as the sum, because it differs from the actual sum by a quantity less than any assignable quantity. EXAMPLES. To find the sum of the first twelve terms of lhe series 1+3+-9+ 27+&c.: Here a-l, r=3, 11 And sz- _ 33 _-1_ 3X177147-1 265720. -- 1 3 —-1 2 To find the sum of the series 1,-3, 9,-27, &c., to twelve terms, 1 -8X-3 -1 -X-17714 7- 1 132860 S — __- 1 - — 4 — 41 In the case of infinite series, if a is sought, s and r being given, we have from (3) a —s (1-r), and if r is sought, a and s being given, we have r=- — or 1 — EXERCISES. 1. Find the sum of the series 2, 6, 18, 54, &c., to 8 terms. Ans. 6560. 2. Find the sum of the infinite series +- -,. Observe here r —-. Ans.. 3. What is the sum of the series 1, &, C, &c., to infinity? Ans. |. 4. Find the sum of the infinite series 1 —-i —t 28~&c. Ans. 3. 5. What is the sum of nine terms of the series 5, 20, 80, &c.? Ans. 436905. 6. Find the sum of /i4-++'/i+&c., to infinity. Ans. i/-1. 7. What is the limit to which the sum of the infinite series i, i, i, &c., continually approaches? 19 Ans. |. 286 ARITHMETIC. 8. What is the sum often terms of the series 4, 12, 36, &c.? Ans. 118096. 9. Insert three terms between 39 and 3159, so that the whole shall be a series of continual proportionals. Ans. 117,.351 and 1053. 10. Insert four terms between ~ and 27, so that the whole shall form a series of continual proportionals. Ans. i, 1, 3, 9. 11. The sum of a series of continual proportionals is 10k, the first term 3; what is the ratio? Ans. i. 12. The limit of an infinite series is 70, the ratio -; what is the first term? Ans. 40. N N U ITIE S. The word Annuity originally denoted a sum paid annually, and though such payments are often made half-yearly, quarterly, &c., still the term is applied, and quite properly, because the calculations are made for the year, at what time soever the disbursements may be made. By the term annuities certain is indicated such as have a fixed time for their commencement and termination. By the term annuities contingent is meant annuities, the commencement or termination of which depends on some contingent event, most commonly the death of some individual or individuals. By the term annuity in reversion or deferred, is meant that the person entitled to it cannot enter on the enjoyment of it till after the lapse of some specified time, or the occurrence of some event, generally the death of some person or persons. An annuity in prpertuity is one that " lasts for ever," and therefore is a species of hereditary property. An annuity forborne is one the payments of which have not been made when due, but have been allowed to accumulate. By the amount of an annuity is meant the sum that the principal and compound interest will amount to in a given time. The present worth of an annuity is the sum to which it would amount, at compound interest, at the end of a given time, if forborne for that time. Tables have been constructed showing the present and final values per unit for different periods, by which the.value of any annuity may be found according to the following ANNUITIES. 287 RULES. To find either the amount or the present value of an annuity,Multiply the value of the unit, as found in the tables, by the number denoting the annuity. If the annuity be in perpetuity,Divide the annuity by the number denoting the interest of the unit for one year. If the annuity be in reversion,Find the value of the unit up to the date of commencement, and also to the date of termination, and multiply their difference by the number denoting the annuity. To find the annuity, the time, rate and present worth being given. Divide the present worth by the worth of the unit. Tables are appended varying from 20 to 50 years. EXA M PLES. To find what an annuity of $400 will amount to in 30 years, at 3~ per cent. We find by the tables the amount of $1, for 30 years, to be $51.622677, which multiplied by 400 gives $20649.07 nearly. To find the present worth of an annuity of $100 for 45 years, at 3 per cent. By the table we find $24.518713, and this multiplied by 100 gives $2451.88. To find the present worth of a property on lease for ever, yielding $600, at 3~ per cent. The rate per unit for one year is.035, and 600 divided by this gives $17142.86. To find the present worth of an annuity on a lease in reversion, to commence at the end of three years and to last for 5, at 3~ per cent. By the table we find the rate per unit for 3 years to be $2.801637, and for 8 years, the time the lease expires, $6.873956; the difference is $4.072319, which, multiplied by 300, gives $1221.70. Given $207.90, the present worth of an annuity continued for 4 years, at 3 per cent., to find the annuity. By the tables, the value for $1 is $3.717098, and $207.90, divided by this, gives $55.93. 288 ARITHMETIC. TABLE, SHOWING TIE AMOUNT OF AN ANNUITY OF ONE DOLLAR PER ANNUM, IMPROVED AT COMPOUND INTEREST FOR ANY NUMBER OF YEARS NOT EXCEEDING FIFTY. C 3 per cent. 3~ per cent. 4 per cent. 5 per cent. 6 per cent. I per cent, 1 1.000 000 1.000 QO 1.(00 000 1.00 o 00 1.000 000 1.00., 00C 2 2.033 000 2.035 000 2.040 000 2.050 000 2.060 000 2.070 000 3 3.090 900 3.106?25 3.121 600 3.152 500 3.1 83 600 3.t14 900 4 4.183 627 4.214 943 4246 464 4.310 125 4.',74 616 4.439 943 5 5.309 136 5.362 46 5.416 323 5.525 631 5.637 093 5.750 739 6 6.468 410 6.550 152' 6.632 975 6.801 913 6.975 319 7.153 291 7 7.662 462 7.779 408 7.898 294 8.142 00 8.;393 838 8.651 021 8 8.892 336 9.051 687 9.214 226 9.549 109 9.897 468 10.259 803 9 10.159 106 10.368 496.10.582 795 11.026 564 11.491 316 11.977 9891 10 11.463 879 11.731 393 12.006 107 12.577 893 13.180 795 13.816 448 11 12.807 796 13.141 992 13.486 351 14.206 787 14.971 64? 15.783 599. 12 14.192 034! 14.601 962 15.025 805 15.917 127 16.869 941 17.888 45i 13 15.617 790 16.113 030 16.626 838 17.712 983 18.882 138 20.140 643 14 17.086 3241 17.676 98( 18.291 911 19.598 632 21.015 066 22.550 488 15 18.598 914 19.295 681 20.023 588 21.578 564 23.275 970 25.129 022 16 2 156 881 20.971 03d 21.824 531 23.657 492 25.670 528 27..88 054 17 21.761 588 22.705 016 23.697 512 25.840 366 28.212 880 30.840 217 18 23.414 435 24.499 691 25.645 413 28.132 385 30.905 653 33.999 033 19 25.116 868 26.357 180 27.671 229 30.539 004 33.759 992 37.378 965 20 26.870 374 28.279 682 29.778 679 33.065 954 36.785 591 40.995 492 21 28.676i 486 30.269 471 31.969 202 35.719 252 39.992. 727 -44.866 177 22 30.536 780 32.328 902 34.247 970 38.505 214 43.392 290 49.005 739 23 32.452 884 34.460 414 36.617 889 41.430 475 46.995 828 53.436 141 24 34.426 470 36,666 528 39,082 604 44.501 999 50.815 577 58.176 671 25 36.459 264 38.949 857 41.645 908 47.727 099 54.864 512 63.249 030. 26 38.553 042 41.313 102 44.311 745 51.113 454.59.156 383 68.676 470 27 40.709 634 42.759 060 47.084 214 54.669 126 63.705 766 74.483 823 28 42.930 923 46.290 627 49.967 583 58.402 583 68.528 112 80.697 691 29 45.218 850 48.910 799 52.966 286 62.322 712 73.639 798 87.346 529 30 47.575 416 51.622 677 56.084 938 66.438 848 79.058 186 94.460 786 31 50.002 678 54.429 471 59.328 335 70.760 790 84.831 677 102.073 041 32 52.502 759 57.334 502 62.701 469 75.298 829 90.889 778 110.218 154 33 55.077 841 60.341 210 66.209 527 8t).063 771 97.343 165 118.933 425 34 57.730 177 63.453 1521 69.857 909 85.066 959 104.183 755 128.258 765 35 60.432 082 66.674 013 73.652 225 90.320 307 111.434 780138.236 878 36 63.271 944 70.007 603 77.598 314 95.836 323 119.120 867 148.913 460 37 66.174 223 73.457 869 81.702 246 101.628 139 127.268 119 160.337 400 38 69.159 449 77.028 895 85.970 336 107.709 54C 135.904 206 172.561 020 39 72.234 233 80.724 906 90.409 150 114.095 023 145.058 458 185.640 292 40 75.401 260 84.550 278 95.025 516 120.799 774 154.761 966 199.635 112 41 78.663 298 88.509 537 99.826 536 127.839 763 165.047 684 214.609 570 42 82.023 196 92.607 371 104.819 598 135.231 751 175.950 645 230.632 240 43 85.483 892 96.848 629 110.012 382 142.993 339 187.507 577 247.776 496 44 89.048 409 101.238 331 115.412 877 151.143 006 199.758 032 266.120 851 45 92.719 861 105.781 673 121.029 392 159.700 156 212.743 514 285.749 311 46 96.501 457 110.484 031 126.870 568 168.685 164 226.508 125 306.751 763 47 100.396 501 115.350 973 132.945 390 178.119 422 241.098 612 329.224 386 48 104.408 396 120.388 297 139.263 206 1.88.025 393 256.564 529 353.270 093 49 108.540 648 125.601 846 145.833 734 198.426 663 272.958 401 378.999 000 50 112.796 867 130.999 910 152.667 084 209.347 976 290.335 905 406.528 929 ANNUIT s. 289 TABLE, SHOWING THE PRESENT WORTH OF AN ANNUITY OF ONE DOLLAR PER ANNUM, TO CONTINUE FOR ANY NUMBER OF YEARS NOT EXCEEDING FIFTY. 3 per cent. 3~ per cent 4 per cent. 5 per cent. 6 per cent. 7 per cent. o 1 0.970 874 0.966 184 0.961 538 0.952 881 0.943 396 0.934 579 1 2 1.913 470 1.899 694 1.886 095 1.859 410 1.833 393 1.808 017 2 3 2.828 611 2.801 637 2.775 091 2.723 248 2.673 012 2.624 314 3 4 3.717 098 3.673 079 3.629 895 3.545 951 3.465 106 3.387 209 4 6 4.579 707 4.515 052 4.451 825 4.329 477 4.212 364 4.100 195 5 6 5.417 191 5.328 553 5.242 137 5.075 692 4.917 324 4.766 537 6 7 6.230 283 6.114 544 6.002 055 5.786 373 5.582 381 5.389 286 7 8 7.019 692 6.873 956 6.732 745 6.463 213 6.209 744 5.971 295 8 9 7.786 109 7.607 687 7.435 33, 7.107 822 6.801 692 6.515 228 9 10 8530 203 8.316 605 8.110 89( 7.721 735 7.360 087 7.023 577 10 11 9.252 624 9.001 551 8.760 477 8.306 414 7.886 875 7.498 669 11 12 9.954 004 9.663 334 9.385 074 8.863 252 8.383 844 7.942 671 12 13 10.634 955 10.302 738 9.985 648 9.393 573 8.852 683 8.357 635 13 14 11.296 073 10.920 520 10.563 123 9.898 641 9.294 984 8.745 452114 15 11.9o7 935 11.517 411 11.118 387 10.379 658 9.712 24S 9.107 898 15 16 12.561 102 12.094 117 11.652 296 10.837 770 10.105 895 9.446 632 16 17 13.166 118 12.651 321 12.165 669 11.274 066 10.477 266 9.763 206 17 18 13.753 513 13.189 682 12.659 297 11.689 587 10.827 603 10.059 070 18 19 14.323 799 13.709 837 13.133 939 12.085 321 11.158 116 10.335 578 19 20 14.877 475 14.212 403 13.590 326 12.462 210 11.469 421 10.593 997 20 21 15.415 024 14.;97 974 14.029 160 12.821 153 11.764 077 10.835 527 21 22 15.936 917 15.167 125 14.451 115 13.163 003 12.041 582 11.061 241 22 23 16.443 608 15.620 410 14,856 842 13.488 574 12.303 379 11.272 187 23 24 16.935 542 16.058 368 15.246 963 13.798 642 12.550 358 11.469 334 24 25 17.413 148 16.481 515 15.622 08t 14.093 945 12.783 356 11.653 583 25 26 17.876 842 16.890 352 15.982 76S 14.275 185 13003 166 11.825 779 26 27 18.327 031 17.285 365 16.329 58f 14.643 034 13.210 534 11.986 709 27 28 18.764 108 17.667 019 16.663 06S 14.898 127 13.406 164 12.137 111 28 29 19.188 455 18.035 767 16.983 715 15.141 074 13.590 721 12.277 674 29 30 19.600 441 18.392 045 17.292 033 15.372 451 13.764 831 12.409 041 30 31 20.000 428 18.736 276 17.588 494 15.592 811 13.929 086 12.531 814 31 32 20.338 766 19.068 865 17.873 552 15.802 677 14.084 043 12.646 555 32 33 20.765 792 19.390 208 18.147 64C 16.002 549 14.230 230 12.753 790 33 34 21.131 837 19.700 684 18.411 198 16.192 204 14.368 141 12.854 009 34 3) 21.487 220 20.030 661 18.664 613 16.374 194 14.498 246 12.947 672 35 36 21.832 252 20.290 494 18.908 282 16.546 852 14.620 987 13.035 208 36 37 22.167 235 20.570 525 59.142 579 16.711 287 14.736 780 13.117 017 37 38 22.492 42 20.841 087 19.367 864 16.867 893 14.846 019 13.193 473 38 39 22.808 215 21.102 500 19.584 485 17.017-041 14.949 075 13.264 928 39 40 23 114 772 21.355 07f 19.792 774 117.159 086 15.046 297 13.331 709 40 41 23.412 400 21.599 1 4 19.993 052 17.294 368 15.138 016 13.394 120 41 42 23.701 359 21.834 883 20.185 627 17.423 208 15.224 543 13.452 449 42 43 23.981 902 22.062 689 20.370 795 17.545 912 15.306 173 13.506 962 43 44 24.254 274 22.282 791 20.548 841 17.662 773 15.383 182 13.557 908 44 45 24.518 713 22.495 450 20.720 040 17.774 070 15.455 832 13.605 522 45 46 24.775 449 22.700 918 20.884 654 17.880 067 15.524 370 13.650 020 46 47 25.0?4 708 22.899 438 21.042 936 17.981 016 15.589 028 13.691 608 47 48 25.266 707 23.091 244 21.195 131 18.077 158 15.650 027 13.730 474 48 49 25.513 657 23.276 564 21.341 472 18.168' 722 15.707 572 13.766 799 49 50 25.729 764 23.455 618 21.482 185 18.255 925 15.761 861 13.80 746 50 7. 290 ARITHMETIC. PARTNERSHIP SETTLEMENTS. The circumstances under which partnerships are formed, the conditions on which they are made, and the causes that lead to their dissolution, are so varied that it is impossible to do more than give general directions deduced from the cases of most common occurrence. In forming a partnership, the great requisite is to have the terms of agreement expressed in the most clear and yet concise language possible, setting forth the sum invested by each, the duratioA of partnership, the share of gains or losses that fall to each, the sum that each may draw from time to time for private purposes, and any other circumstances arising out of the peculiarities of each case. The ease and satisfaction of making an equitable settlement, in case of dissolution, depends mainly on the clearness of the original agreement, and hence the necessity for its being distinct and explicit. Even when no dissolution is contemplated, settlements should be frequently made, in order that the parties may know how they stand to each other, and how the business is succeeding. This is of great importance in preserving unanimity and securing vigour and regularity in all the transactions of a mercantile house. A dissolution may take place from various causes.. If the partnership is formed for a term of years, the expiration of those years necessarily involves either a dissolution or a new agreement. The death of one of the partners may or may not cause dissolution, for the deceased partner may have, by his will, left his share in the business to his son, or some other relative or friend. In no case, however, can an equitable settlement be made, except by the mutual consent of the parties, or else in exact accordance with the terms of agreement. It is also necessary that when a dissolution takes place public notice should be given thereof, in order that all parties having dealings with the firm may be apprized of the change, and have their accounts arranged. For the same reason, it is necessary that some one of the partners, or some trustworthy accountant appointed by them, should be authorized to collect all debts due to the firm, and pay all accounts owing by it. Partnerships are sometimes formed for a specific speculation, and therefore, of course, cease with the completion of the transaction, and a settlement must necessarily be then made. No matter for what PARTNERSHIP SETTLEMENTS. 291 time the partnership has been made, any partner is at liberty, at any time, to withdraw, on showing sufficient cause and giving proper notice. This is a just provision, for the circumstances of any partner may so change, from various causes, as to make it undesirable for him to remain in the business. If one partner is deputed to settle the accounts of the house, it would be reckoned fraudulent for any other partner to collect any moneys due, except that on receipt of them he hands them directly over to the person so deputed. The resources and liabilities, with the net investment on colmnmencing business, being given, to find the net gain or loss. 1. W. Smith and R. Evans are partners in business, and invested when commencing $1000 each. On dissolving the partnership, the assets and liabilities are as follows:-Merchandise valued at $1295; cash, $344; notes against sundry individuals, $790; W. II. Monroe owes on account $86.40; E. R. Carpenter owes $132.85, and C. F. Musgrove owes $67.50. They owe on sundry notes, as per bill book, $212.40; E. G. Conklin, on account, $29.45, and H. C. Wright, on account, $41.30. What has been the net gaia? SOLUTION. Assets. Liabilities. Merchandise on hand... $1295.00 Bills Payable..............$212.40 Cash on hand............ 344.00 Amt. due E. G. Conklin. 29.45 Bills Receivable.......... 790.00 Amt. due H. C. Wright. 41.30 Amt. due from W. H. W. Smith's investment...1000.00 Monroe............... 86.40 R. Evan's investment... 1000.00 Amt. due from E. R. Carpenter.............. 132.85 $2283.15 Amt. due from C. F. Musgrove............. 67.50 Total amount Assets....$2715.75 ( i" Liabilities, 2283.15 Net gain..........$432.60 RULE. Find the sum of the assets and liabilities;from the assets subtract the liabilities, (including the net amount invested) and the difference will be the net gain; or, if the liabilities be the larger, subtract the assets from the liabilities, and the difference will be the net loss. 292 ARITHMETIC. 2. Harvey Miller and James Carey are partners in a dry goods business; Harvey Miller investing $1400, and James Carey $1250. When closing the books, they have on hand-cash, $1125.30; merchandise as per inventory book, $1855.75; amount deposited in First National Bank, $1200; amount invested in oil lands, $963; a site of land for building purposes, valued at $1600; Adam Dudgeon owes them, on account, $104.92; William Fleming owes $243.80, a noto against Alfred Mills for $89.43, and a due bill for $3), drawn by James Laing. They owe W. S. tHope & Co., on account, $849.21; R. J. King & Co., $609.12, and on notes, $1326.14. What has been the net gain or loss? Ais. $1761.73 gain. 3. James HIenning and Adam Manning have formed a co-partnership for the purpose of conducting a general dry goods and grocery business, each to share gains or losses equally. At the end of one year they close the books, having $1280 worth of merchandise on hand; cash, $714.27; Girard Bank stock, $500; deposited in Merchants' Bank, $320.60; store and fixtures valued at $3100; amount due on notes and book accounts, $3-71.49. The firm owes on notes $3400, and on open accounts $747.10. James Henning invested $1200, and Adam Manning, $1000; what is each partner's interest in the business at closing? Ans. James Henning's interest, $2719.63. Adam Manning's interest, $2519.63. NoTE.-Where the interest of each partner at closing is required, the gain or loss is first found, as' in former examples, than the share of gain or loss is added to or subtracted from each partner's investmen., and the slm, or difference, is the interest of each partner. If a partner has withdrawn anything from the business, the amount thus withdrawn must be deducted from the sum of his investment, plus his share of the gain, or minus his sharks of the loss, and the remainder will be his net capital Mr interest. 4. F. A. Clarke, W. H. Marsden, and J. M. Musgrove, are conducting business in partnership; F. A. Clarke is to be i gain or loss, W. 11. Marsden and J. M. Musgrove, each i. On dissolving the partnership, they have cash on hand $712.90; merchandise as per Inventory Book, $4360; bills receivable, as per Bill Book, $1450.75; amount deposited in Third National Bank of Syracuse $3475; merchandise shipped to Richmond, to be sold on own account and risk, valued at $995; debts due from individuals on book account, $2644.67.' They owe on notes $3760, and to Manning and Munson, $1312.60. PARTNERSHIP SETTLEMENTS. 293 -. A. Clarke invested $5750, and has drawn out $875; W. H. Marsden invested $2500, and has drawn out $500; J. M. Musgrove invested $3030, and has drawn out $750. What has been the net gain or loss, and what is each pirtner's interest in the business? Ans. Net loss, $559.28; F. A. Clarke's interest, $4595.36; AW. II. Marsden's interest, $1860.18; J. M. Musgrove's interest, $2110.18. NoTE.-In this and succeeding examples, no interest is to be allowed on anvestment) or charged on amounts withdrawn, unless so specified. 5. A, and C are partners. The gains and losses are to be shared as follows: A,'d; B, -; and C, y. A invested $3000, and has withdrawn $2500, with the consent of B. and C, upon which no interest is to be charged; B invested $2700, and has,withdrawn $1150; C invested $2500, and has withdrawn $420. After doing business 14 months, C retires. Their assets consist of bills receivable, $2937.20, merchandise, $1970; cash, $1240.80; 50 shares of the Chicago Permanent Building and Savings' Society Stock, the par value of which is $50 per share; cash deposited in the Third National Bank, $1850; store and furniture, $3130; amount due from W. Smith, $360.80; G. S. Brown, $246.40; and E. R. Carpenter, $97.12. Their liabilities are as follows: Amount due Samuel Harris, $1675; unpaid on store and furniture, $933; and notes unredeemed, $3388.76. The Savings' Society stock is valued at 10 per cent. premium, and C in retiring takes it as part payment. What is the amount due C, and what is A's, and what is B's interest in the business? Ans. Due (C, $815.52; A's interest, $2356.90; B's interest, $2664.14. 6. E, F, G and H are partners in business, each to share i of profit and losses. The business is carried on for one year, when E and F purchase from G and H their interest in the business,. allowing each $100 for his good will. Upon examination, their resources are found to be as follows: Cash deposited in Girard Bank, $3645; cash on hand, $1422; bills receivable, $1685; bonds and mortgages, $2746, upon which there is interest due $106, Metropolitan Bank stock, $1000; Girard Bank stock, $500; store and fixtures, $3500; house and lot, $1800; span of horses, carriages, harness, &c., $495; outstanding book debts due the firm. $4780. Their liabilities are: Notes payable, $2345; upon which there is interest due. $57; due on book debts, $1560. E invested $5000; F $4500; 294 ARITHMETIC. G, $4000; and H, $3000. E has drawn from the business $1200, upon which he owes interest $32; F has drawn $1000-owes interest, $24.50; G has drawn $950-owes interest $12; and H has drawn nothing. In the settlement a discount of 10 per cent., for bad debts, is allowed, on the book debts due the firm and on the bills receivable. G takes the Metropolitan Bank stock, allowing on the same a premium of 5 per cent.; and H takes the Girard Bank stock, at a premium of 8 per cent.; E and F take the assets and assume the liabilities, as above stated. What has been the net gain or loss, the balances due G and H, and what are E and F each worth after the settlement? Ans. Due G, $3057.75; due H1, $3529.75; E's net capital, $4637.75; F's net capital, $4345.25. 7. H. C. Wright, W. S. Samuels, and E. P. Hall, are doing business together-H. C. W. to have - gain or loss; W. S. S. and E. P. H. each i. After doing business one year, W. S. S. and E. P. H. retire from the firm. On closing the books and taking stock, the following is found to be the result: merchandise on hand, $3216.50; cash deposited in Sixth National Bank, $1627.35; cash in till $134.16; bills receivable, $940.60; G. Brown owes, on account, $112.40; Thos. A. Bryce owes $94.12; W. McKee owes $143.95; J. Anderson owes $54,20; R. H. Hill owes $43.60; and S. Grahanm owes $260.13. They owe on notes not redeemed $1864; H. T. Collins, on account, $124.45; and W. E. Curtis, $79.40. 11. C. Wright invested $3200, and has drawn from the business $350. W. S. Samuels invested $2455, and has drawn $140; E. P. Hall invested $2100, and has drawn $2000. A discount of 10 per cent. is to be allowed on the bills receivable and book accounts due the firm for bad debts. H. C. Wright takes the assets and assumes the liabilities as above stated. What has been the net gain or loss, and what does H. C. Wright pay W. S. Samuels and E. P. Hall on retiring? 8. T. P. Wolfe, J. P. Towler and E. R. Carpenter have been doing business in partnership, sharing the gains and losses equally. After dissolution and settlement of all their liabilities'they make a division of the remaining effects without regard to the proper proportion each should take. The following is the result according to their ledger:-T. P. Wolfe invested $3495, and has drawn $2941; J, P. Towler invested $2900, and has drawn $2200; E. R. Carpenter PARTI'RBSHIP SETTLEMENTS. 295 invested $3150, and las drawn $3000. How will tne partners settle with each other? Ans. E. R. Carpenter pays T. P. Wolfe $86, and J.P. Towler $232. 9. I, J, K, L and MI have entered into co-partnership, agreeing to share the gains and losses in the following proportion:-I, T45; J, -3; K, -,; L, -; and M, -1. When dissolving the partnership the resources consisted of cash $4700; merchandise, $9855; notes on hand $7680; debentures of the city of Albany valued at $6780, on which there is interest due, $123; horses, waggons, &c., $1280; Merchant's bank stock, $5000; First National bank stock, $5000; mortgages and bonds, $3600; interest due on mortgages, $345.80; store and fixtures, $8000; amount due from W. P. Campbell & Co., $2418; due from R. B. Smith, $712.60; due from J. W. Jones, $1000. The liabilities are:-Mortgage on store and fixtures, $5000; interest due on the same, $212.25; due the estate of R. M. Evans, $14675; notes and acceptances, $11940, on which interest is due, $85; sundry other book debts, $7500; I invested $7800, interest. on his investment to date of dissolution, $702; J invested $6400, interest on investment, $576; K invested $6100, interest on investment, $549; L invested $5800, interest on investment, $522; M invested $5000, interest on investment, $450. I has withdrawn from the firm at different times, $2425, upon which the interest calculated to time of dissolution is $183.40; J has drawn $2960, interest,, $267.85; K has drawn $1850, interest $37.30; L has drawn $3000, interest, $460; AI has drawn $895, interest, $63.45. What is the net gain or loss of each partner, and what is the net capital of each partner? Ans. I's net loss, $1233.29; I's net capital, $4660.31. J's net loss, $924.97; J's net capital, $2823.18. K's net loss, $616.65; K's net capital, $4095.05. L's net loss, $1541.62; L's net capital, $1320.38. M's net loss, $308.32; AI's net capital, $4183.23. 10. A, B, C and D are partners. At the time of dissolution, and after the liabilities are all cancelled, they make a division of the effects, and upon examination of their ledger it shows the following result:-A has drawn from the business $3465, and invested on commencement of business, $4240; B has drawn $4595, and invested $3800; C has drawn $5000, and invested $3200; D has drawn $2200, and invested $2800. The profit or loss was to be divided in 296 ARITHMETIC. proportion to the original investment. What has been each partner's gain or loss, and how do the partners settle with each other? Ans. A's net gain, $368.43; B's net gain, $330.20; C's net gain, $278.06; D's net gain, $243.31. B has to pay in $464.80; C has to pay in $1521.94. A receives $1143.43; D receives $843.31. 11. Three mechanics, A. W. Smith, James Walker and P. Ranton, are equal partners in their business, with the understanding that each is to be charged $1.25 per day for lost time. At the close of their business, in the settlement it was found that A. W. Smith had lost 14 days, James Walker 21 days, and P. Ranton 30 days. How shall the partners properly adjust the matter between them? Ans. P. Ranton pays A. W. Smith, $9.58i, and James Walker, 83~ cents. 12. There are 5 mechanics on a certain piece of work in the following proportions:-A is Bv; B, 2; C, D4; D I, andE, 3.. A is to pay $1.25 per day for all lost time; B, $1; C, $1.50; D, $1.75, and E, $t.62J. At settlement it is found that A has lost 24; B, 19; C, 34; D, 12; and E, 45 days. They receive in payment for their joint work, $2500. What is each. partner's share of this amount according to the above regulations? Ans. A's share, $374.12; B's, $250.41; C's, $487.83; D's, $787.24; E's, $600.40. 13. A. B. Smith and T. C. Musgrove commenced business in partnership January 1st, 1864. A. B. Smith invested, on commencement, $9000; May 1st, $2400; June 1st, he drew out $1800; September 1st, $2000, and October 1st, he invested $800 more. T. C. Musgrove invested on commencing, $3000; March 1st, he drew out $1600; May 1st, $1200; June 1st, he invested $1500 more, and October 1st, $8000 more. At the time of settlement, on the 31st December, 1864, their merchandise account was-Dr. $32000; Cr. $27000; balance of merchandise on hand, as per inventory, $10500; cash on hand, $4900; bills receivable, $12400; R. Draper owes on account, $2450. They owe on their notes, $1890, and G. Roe on account, $840. Their profit and loss account is, Dr. $866; Cr. $1520. Expense account is, Dr. $2420. Commission account is, Cr. $2760. Interest account is Dr. $480; Cr. $950. The gain or loss is to be divided in proportion to each partner's capital, and in proportion to the time it was invested. Reauired each partner's share of the gain or loss, the net balance PROPERTIES OF NUMBERS. 297 due each, and a ledger specification exhibiting the closing of all the accounts, and the balance sheet. Ans. A. B. S.'s net gain, $6671.73; his net balance, $15071.73. T. C. MI.'s net gain, $2748.27; his net balance, $12448.27. PROPERTIES OF NUMBERS. The term Integer, or Whole Number, is used in contradistinction to the term Fraction. All numbers expressed by the natural series 1, 2, 3...10...20...100, &c., are called integers, so that 3 and 4 are integers, but i is a fraction. All numbers in the natural series 1, 2 3, &c., that can be resolved into factors, are called Composite, while those that cannot be so resolved are called Prime. Since 4-2X2, it is called composite, and so 6, 8, 9, 10, &c., but 1, 2, 3, 5, 7, 11, &c., are called prime because they cannot be resolved into factors. Thus, 11 can only be resolved into 11X 1, or 1 X11, and these aru not factors in the strict meaning of the word. A Prime Factor is a prime number, which is a factor of a composite number. The factors of 10 are 2 and 5, both prime numbers. A composite number may have composite factors, as 36, which has 4 and 9 as factors, and both of these are composite. When any number will divide two or more others, it is called a Common Factor. Thus, 3 is called a common factor of 6, 9,12,15, &c. Numbers that have no common factor, as 4, 5, 9, are said to be prime to each other. To resolve a composite number into its prime factors, divide it by the least possible factor that it contains, and repeat the process till a prime number is obtained. EXAMPLES. 2)96 2)48 2)24 2)12 2) 6 3 so that the prime factors of 96 are 2X2X2X2X2X3. 298 ARITHMETIC. Also, because 5X7Xll=385, we see that 5, 7 and 11 are the prime factors of 385. EXERCISES. 1. What are the prime factors of 2310? Ans. 2, 3, 5, 7, 11. 2. What are the prime factors of 1764? Ans. 2, 2, 3, 3, 7, 7. 3. What are the prime factors of 180642? Ans. 2, 3, 7, 11, 17, 23. 4. What are the prime factors of 95? Ans. 5, 19. 5. What are the prime factors of 51? Ans. 3-17. 6. What are the prime factors of 99? Ans. 3, 3, 11. 7. What are the prime factors of 651? Ans. 3, 7, 31. 8. What are the prime factors of 362880? Ans. 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 5, 7. 9. What factors are common to 84, 105, and 147? Ans. 3, 7. 10. What are the prime factors of 308? Ans. 4, 7, 11. Whether a number is prime or composite can only be found by trial. The only even prime number is 2; for 4, 6, 8, 10, &c., are all multiples of 2. The only prime number ending in the digit 5 is 5 units, and all other numbers ending in either 5 or 0 are multiples of 5. ADDITIONAL EXERCISES. 11. Is 101 prime or composite? Ans. Prime. 12. Is 198 prime or composite? Ans. It his the factors 2, 3, 3,11. 13. Is 171 prime or composite? Ans. It has the factors 3, 3, 19. 14. Is 473 prime or composite? Ans. Prime. 15. Is 477 prime or composite? Ans. Composite. 16. Is 549353259 prime or composite? Ans. Composite. 17. Is 674041 prime or composite? Ans. Composite. 18. Is 199 prime or composite? Ans. Prime. 19. What are the prime factors of 210? Ans. 5, 6, 7. 20. What are the prime factors of 51051? Ans. 3, 7, 11, 13, 17. NoTE. —We have thought it sufficient under this head to give only the leading and most useful principles, QUESTIONS FOP COMMERCIAL STUDENTS. 299 QUESTIONS FOR COMMERCIAL STUDENTS. 1. The following questions may be found interesting and instructive to young men preparing for the practical duties of accountants. On the 1st of May I purchased for cash, on a commission of 2per cent., and consigned to Ross, Winans & Co., commission merchants, Baltimore, Md.; 380 bbls. of mess pork, at $27.50 per bbl., to be sold on joint account of himself and myself, each one half. Paid shipping expenses, $7.40. July 7th, I received from Ross, Winans & Co. an account sales showing my net proceeds to be $5319,79, due as per average, August 12th. August 8th, I draw on them at sight for the full amount of their account, which I sell at i per cent. discount for cash; interest 7 per cent. What amount of money do I receive and what are the journal entries? 2. B. Empey, a merchant doing business in Montreal, Canada East, purchased from A. T. Stewart, of New York'city, on a credit of three months, the following invoice of goods: 845 yds. Fancy Tweed,....................1.90 per yd. 1712 " Amer. black broadcloth,;....... 3.85 " " 423 " Blue pilot, @........................ 275 " " 700 " Black Cassimere.................. 2.10 " When the above goods were passed through the custom-house, a discount of 27j- per cent. was allowed on American invoices; duty 25 per cent., freight charges paid in gold, $29.35. What must each piece be marked at, per yd., to sell at a net profit of 15 per cent. on full cost? What would be the gain or loss by exchange, if at the expiration of the three months B. Empey remitted A. T. Stewart, to balance account, a draft on Adams, Kimball & Moore, bankers, New York city, purchased at 32{ per cent. discount, and what are the journal entries? 3. I purchased for cash, per the order of J. P. Fowler, 70 boxes C. C. bacon, containing on an average 400 lbs. each, at 13- cents 300 ARITHMETIC. per lb., and 140 firkins butter, 8312 lbs., at 17~ cents per ID., on a. commission of 21 per cent; paid shipping and sundry expenses in cash $13.40. For reimbursement I draw on J. P. Fowler at sight, which I sell to the bank at - per cent. discount; what is the face of draft, and what are the journal entries? Ans. Face of draft $5479.05. 4. Sept. 27th, I received from James Watson, Leeds, England a consignment of 1243 yards black broadcloth, invoiced at 13s. 6d. per yard, to be sold on joint account of conpignor and myself, each one half, my half to be as cash, invoice dated Sept. 16th. Oct 5th, I sold R. Duncan, for cash, 207 yards, at $6.10 per yard; Oct 24th, sold 317 yards to James Grant, at $6.25 per yard, on a credit of 90 days; Nov. 18th, sold E. G. Conklin, for his note at 4 months, 400 yards, at $6.30 per yard; Dec. 12th, sold the remainder to J. A. Musgrove at $6.00 per yard, half cash and a credit of 30 days for balance; charges for storage, advertising, &c., $13.40; my commission, with guarantee of sales, 5 per cent. What would be the average time of sales; the average time of James Watson's account; and what would be the face of a sterling bill, dated Dec. 15th, at 60 days after date, remitted James Watson to balance account purchased at $1081, money being worth 7 percent, and gold being 70 per cent. premium? 5. Buchanan & Harris of Milwaukee, Wis., are owing W. A. Murray & Co. of Washington, $1742.75, being proceeds of consignment of tobacco sold for them, and Simpson & Co. of Washington, are owing Buchanan & Harris $2000 payable in Washington. Buchanan & Harris wish to remit W. A. Murray & Co. the proceeds of their consignment and they do so by draft on Simpson & Co., but Washington funds are 2 per cent. premium over those of Chicago. Required the face of the draft and the journal entries. 6. A. Cummings, of London, England, is owing me a certain sum, payable there, and I am owing Charles Massey, of the same place, $1985.42, being proceeds of consignment of broadcloth sold for him. I remit C. Massey in full of account, after allowing him $21,12 for inserest, my bill of exchange on A. Cummings at 60 days' sight; exchange 1091, gold 42 per eent. premium. What is the face of the draft, and what arc the journal entries? QUESTIONS FOR COMMERCIAL STUDENTS. 301 7. March 10, I shipped per steamer Vandcrbilt ana consigned to Samue! Vestry, Liverpool, England, to be sold on joint account of consignee and conrsignor, each one half, (consignee's half as cash), 27894 lbs. prime American Cheese at 15{ cents per pound. Paid shipping expenses $18.30. Insurante It per cent. and insured for such an amount that, in case the cheese was lost, the total cost would be recoverable. Aay 19, I received from Samuel Vestry an account sales, showing my net proceeds to be ~298 14 93-, due as per average August 21. June 1, I drew on Samuel Vestry at the number of days after date that it would take to make the bill fall due at the properly equated time of his account. Sold the above bill to R. Ramldsey, broker, at 108{. Required the number of days I drew the bill at, its face, gold being at a premium of 43J per cent., the amount of money in greenbacks I received, and the journal entries. 8. I am doing a commission business in New York, and on Sept. 14, I received from A. J. Rice, of Hudson, to be sold on joint account of himself ~, A. H. Peatman, of Newburg, -, and myself i; merchandise invoiced at $1262.40, paid freight $14.20. The same day, I received from A. H. Peatman to be sold on joint account of himself 2, A. J. Rice i,'and myself i, merchandise invoiced at $1102.12; paid freight $10.00. I also invest to be sold on joint account of A. J. Rice -, A. H. Peatman l, and myself -, merchandise valued at $745.35. The shares of each are subject to average sales. October 29th, I sold i- of the merchandise received from A. J. Rice to S. King at an advance of 20 per cent., on a credit of 90 days. November 9th, sold for cash one half of the remainder at 15 per cent. advance, closed the company, and rendered account sales; storage $3.50, commission 2' per cent. November 12, sold to A. M. Spafford, on a credit of 30 days one half of goods received from A. H. Peatman, at an advance of 25 per cent. November 23, sold for cash the merchandise that I invested at an advance of 15 per cent.; closed the company and rendered account sales; storage $2.75, commission 2~ per cent. December 4th, sold i of the remainder of mnerchandize received from A. H. Peatman to G. W. Wright, on a credit of 60 days, at 33- per cent. advance. December 12th, sold the balance of Peatman's merchandise for cash at 25 per cent. advance; closed the company and rendered account sales, storage $5.00 commission 2~ per cent. December 23rd, I wish to settle with A. J. Rice, and A. H. Peatman, in full; I take to my own account, 302 ARITHMETIC as cash, the balance of merchandise unsold at an advance of 8 per cent.. What is the average time of sales of each Mdse. Co., the average time of A. J. R: and A. H. P's. accounts, the amount of money I shall have to pay them on December 29, how do A. J. R. and A. H. P. stand with each other, and what are the journal entries? 9. E. R. Carpenter, S. Northrup and Levi Williams, commenced business together as partners under the name and style of E. R. Carpenter & Co., on January 1st, 1865, with the following effects: merchandise, $7844; cash, $5000; store and fixtures, $3984; bills receivable, $1732.50; of this amount there belonged to E. R. Carpenter, as capital, $8000; S. Northrup, $6000; Levi Williams, $4561.50. The firm assumed the liability of Levi Williams, which was a note to the amount of $425.80; this note was paid on March 10th. The loss or gain is to be shared equally by the partners, but interest at the rate of 7 per cent. per annum is to be allowed on investments, and charged on amounts withdrawn. E. R. Carpenter is to manage the business on a salary of $1000 a year, payable half yearly (the time of the other partners not being required in the business). March 14th, S. Northrup draw's cash, $300; Levi Williams, $200; April 19th, E. I. Carpenter draws cash, $500; S. Northrup, $100. On the 1st of May, they admit Geo. Smith aas a partner, under the original agreement, with a cash capital of $4000. The books not being closed, he pays each partner for a participation in the profits to this time $450, which they invest in the business. May 14th, E. I. Carpenter draws cash, $160; May 24th, Levi Williams draws cash, $100; June 12th, S. North'rup draws cash, $250, and E. R. Carpenter, $200; July 1st, Levi Willianis draws cash, $300, and S. Northrup, $450; July 21st, Levi Williams diaws cash, $180; August 1st, Levi Williams retires from the partn-ership, the firm allowing him $500 for his profits and good-will in thie biiiness, this amount, together with his capital, thas been paid in cash. Oct. 14th, Geo. Smith draws cash, $340; E. R. Carpenter, $725. November 1st, with the consent of the firm, S. Northrup disposes of his right, title and interest in the business to J. K. White, who is admitted as a partner under the original agreement. J. K. White is to allow S. Northrup $600 for his share of the profits to date, and his good-will in the business. J. K. White not receiving funds anticipated, is unable to pay S. Northrup but $1500, the firm therefore assumes the balance as a liability. December 10th, received from QUESTIONS FOR COMMrERCIOAL STUDENTS. 303 J. K. White, and paid over to S. Northrup, the full amount due him (S. N.) to date. December 31st, the books are closed, and the following effects are on hand:-Mdse, $11943.75; cash, $2110.12; bills receivable, $6400; store and fixtures, $3850; personal accounts Dr. $14987.50; personal accounts Cr. $10711; Bills Payable unredeemed, $4000. What has been the net gain or loss, the net capital of each partner at the end of the year, and what are the double entry journal entries on commencing business, when Levi Williams retires, when Geo. Smith is admitted, when S. Northrup sells his interest and right to J. K. White, for E. R. Carpenter's salary, and the interest due from and to each partner? The student may also, in the above example, after finding the interest on the partners investments, and on the amounts withdrawn, give a journal entry that will adjust the matter of interest between the partners without opening any profit and loss account. 304: ARITHMETIC. MENSURATION.0 We have already observed that no solid body can have more than three dimensions, viz.: length, breadth, and thickness, or depth, and that a line is length, or breadth, or depth, or it is a line or unit repeated a certain number of times. A foot in length is a line measured by repeating the linear unit called an inch 12 times, and a yard is the linear unit called a foot, repeated 3 times, and so on. Thus, 1 ft. 1 ft. 1 ft. — Thus, 1 ft. 1 -— 3 feet. But there may be two such lines drawn at right angles to each other, and each three feet long, and if the figure be completed it is a square...i...... - Also, if lines be drawn, each an inch a b c apart from the other, and parallel to the KfMi _ __ _:_ two first-mentioned lines, we shall find 3 that there are three small figures, each an I_ -__ _ inch square, between the two upper horizontal lines, and 3 of the same extent between the two intermediate lines, and 3 between the two lowerlines, making 9 in all, or 3 times 3. This is the origin of the expression that 9 is the square of 3. Let the learner mark the difference between 3 square feet and 3 feet square. a, b and c are 3 square feet, but the whole figure is 3 feet square, and therefore three feet square must be equal to 9 square feet.- Three feet square, then^ is a square, each of whose tides measures 3 linear feet; but'3 square feet would denote 3 squares, each side of'each Ineasuring one linear foot. The space thus inclosed is called the area. This is the principle on which surfaces are measured. PROBLEM I. To find the area of a paralellogram: RU LE. Multiply the length by the perpendicular breadth. If the figure be rectangular, one of the sides will be the perpendicular breadth. * We have taken for granted that those studying mensuration have learned, at least, the elementary principles of geometry. We have, therefore, only given the rules, as our space would not admit of our giving demonstrations as this would require a separate treatise MENSURATION. 305 T' the fijure be not rectangular, either the perpendicular breadth btust be given or data from which to find it. EXERCISES. 1. How many acres are there in a square, each side of which is 24 rods? Ans. 3 acres, 2 roods, 16 rods. 2. What is the area of a square picture frame, each side of which.is 5 ft. 9 in.? Ans. 33 ft. 9 in. 3. How many acres are there in a rectangular field, the length of which is 131- chains, and the breadth 9?? Ans. 130.625 square chains, or 13 acres, 0 roods, 10 rods. 4. What is the area of a rectangle, whose sides are 14 ft. 6 in. and 4 ft. 9 in.? Ans. 68 ft., 126 sq. in. 5. What does the surface of a plank measure, which is 12 ft. 6 in. long and 9 in. broad? Ans. 9 sq. ft. 54 sq. in. 6. What is the area of a rhomboidal field, the length of which is 10.52 chains and the perpendicular breath 7.63 chains? Ans. 8 acres, 0 roods, 4.2816 rods. 7. What is the area of a rhomboidal field, the length of which is 24 rods and the perpendicular breadth 24 rods? Ans. 3 acres, 2 roods, 16 rods. 8. What is the length of each side of a square field, the area of which is 788544 square yards? Ans. 888 yards. 9. The area of a rectangular garden is 1848 square yards, and one side is 56 yards; what is the other? Ans. 33 yards. 10. The area of a rhomboidal pavement is 205, and the length is 20 feet; what is the perpendicular breadth? Ans. 10 feet. PROBLEM II. To find the area of a triangle. 1. If the base and perpendicular, or data to find them, be given, we have the RULE. Multiply the base by the perpendicular, and take half the product; or, multiply half the one by the other. 2. If the three sides are given RULE. From half the sum of the sides subtract each side successively, and the square root of the continual product of the half sum, and these three remainders will be the area. 306 ARITHMETIC. Expressed algebraically this area=lVs(s-a)(sb))(s —c). EXERCISES. 11. What is the area of a triangle, the base of which is 17 inches, and the altitude 12 inches? Ans. 102 square inches. 12. What is the area of a triangular garden, the length of which is 46 rods, and the breadth 19 rods? Ans. 437 square rods. 13. Find how many acres, &c., are in a triangular field, the length of which is 49.75 rods, and the breadth 34i rods. Ans. 5 acres, 1 rood, 18 3 rods. 14. The area of a triangular inclosure is 156 square rods, and the base is 30 linear rods; what is the altitude? Ans. 10 rods. 15. The area of a triangle is 400 rods, and the altitude 40 rods, what is the base? Ans. 20 rods. 16. Three trees are so planted that the lines joining them form a right angled triangle; the two sides containing the right angle are 33 and 56 yards; what is the area in square yards? Ans. 924. 17. Let the position of the trees, as in the last example, be represented by the triangle A B C, and let the distance from A to B be 50 rods, and from B to C 30 rodg. Bj ____c_ - Required the area.-(See EuclidI. 47.) Ans. 600 square rods. 18. In the figure annexed to 17, suppose A B to represent the pitch of a gallery in a church, inclined to the ground at an angle of 45~; how many more persons will the gallery contain than if the seats were made on the flat B 0, supposing B C to be 20 feet and the frontage 60 feet in length? Ans. None. We have introduced this question and the next to correct a common misapprehension on this point. A. Because the distance fr6m B to A is greater than the distance from B to C, it is commonly supposed that more persons can be accommodated on the slant A B, than on the flat B C. / By inspecting the annexed diagram it will be seen that the seats are not perpendicular to A B, but to B C, and that precisely the same number of seats can B be made, and the same number of persons accommodated on B C as on A B. IMENSURATION. 307 19. If B C be half the base of a hill, and A B one of its sloping sides, and B C=-30 yards, and A B=50 yards; how many more roWs of trees can be planted on A B, than on B C, at 1 yard apart? Ans. None, because the trpes being all perpendicular to the horizon, are parallel to each other as represented by the vertical lines in the last figure. 20. How many acres, &c., are there in a triangular field of which the perpendicular length and breadth are 12 chains, 76 links and 9 chains, 43 links? Ans. 6 acres, 0 roods, 2- rods. 21. A ship was stranded at a distance of 40 yards'from the base of a cliff 30 yards high; what was the length of a cable which reached from the top of the cliff to the ship? Ans. 50 yds. 22. A cable 100 yards log was passed from the bow to the stern of a ship through the cradle of a mast placed in midships at the height of 30 yards; what was the length of the ship? Ans. 80 yards. 23. A man attempts to row a boat directly across a river 200 yards broad, but is carried 80 yards down the stream by the current; through how many yards was he carried? Ans. 215.4-yards. 24. Let the three sides of a triangle be 30, 40, 20; to find the area in square feet. Ans. 290.4737 square feet. 25. What is the area of an isosceles triangle, each of the equal sides being 15 feet, and the base 20 feet?* Ans. 111.803 sq. feet. 26. What is the area of a triangular space, of which the base is 56, and the hypotermse 65 yards? Ans. 924 square yards. 27. What is the area of a triangular clearing, each side of which is 25 chains? Ans. 27.0632 acres. 28. What is the area of a triangular clearing, of which the three sides are 380, 420 and 765? Ans. 9 acres, 37~ perches. 29. A lot of ground is represented by the three sides of a right,angled triangle, of which the hypotenuse is 100 rods, and the base 60 rods; what is the area? Ans. 15 acres. 30. What is the area of a triangular field, of which the sides are 49, 34 and 27 rods respectively? Ans. 2 acres, 3 roods —. 31. What is the area of a triangular orchard, the sides of which are 13, 14 and 15 yards? Ans. 84 square yards. 32. Three divisions of an army are placed so as to be represented * This question, and some others may be solved by either rule, and it will be found a good exercise to solve by both. 308 RITMETIC. by three sides of a triangle, 12, 18 and 24; how many square miles do they guard within their lines? Ans. Between 104 and 105 square miles. 33. A ladder, 50 feet long, was placed in a street, and reached to a parapet 28 feet high, and on being turned over reached a parapet on the other side 30 feet high; what was the breadth of the street? Ans. 76.123+feet. PROBLEM III. To find the area of a regular Polygon. 1. When one of the equal sides, and the perpendicular on it from the centre, are given. Multiply the perimeter by the perpendicular on it from its cen tre, and take half the product; or, multiply either by half the other. 2. When a side only is given. Multiply the square of the side by the number found opposite the number of sides in the subjoined table. NOTE.-This table shows the area when the side is unity; or, which is the same thing, the square is the unit. SIDES. REGULAR FIGURES. 3 Triangle........................................ 0.4330127. 4 Square.................................... 1.0000000. 5 Pentagon....................................... 1.7234774. 6 Pexagon.................................. 2.5980762. 7 Heptagon....................................... 3.6339125. 8 Octagon.......................................... 4.8284272. 9 Nonagon.................................. 6.1818241. 10 Decagon....................................... 7.6942088. 11 Heredecagon.................................. 9.3656395. 12 Dodecagon........................ 11.1961524. 34. If the side of a pentagon is 6 feet and the perpendicular 3 feet, what is the area? Ans. 45 feet. 35. What is the area of a regular polygons each side of which is 15 yards? Ans. 387.107325 sq. yds. 36. If each side of a hexagon be 6 feet, and a line drawn from the centre to any angle be 5 feet, what is the area? Ans. 72 sq. feet. MENSURATION. 309 37. The side of a decagon is 20.5 rods; what is the area? Ans. 20 acres, 0 roods, 33.5 rods, nearly. 38. A hexagonal table has each side 60 inches, and a line from the centre to any corner is 50 inches; how many square feet in the surface of the table? Ans. 38 feet, 128 inches. 39. What is the area of a regular heptagon, the side being 19 —9 and the perpendicular 10? Ans. 678.3. 40. An octagonal enclosure has each side 6 yards, what is its area? Ans. 3 acres, 2 roods, 14 rods. 19 yards. 41. Five divisions of an army guard a certain tract of countryeach line is 20 miles; how many square miles are guarded? Ans. 688.2, nearly. 42. Find the same if there are 6 divisions, and each line extends 5 miles? Ans. 64.95+ miles. 43. The area of a hexagonal table is 73J feet; what is each side? Ans. 5I feet. PROBLEM IV. To find the area of an irregular polygon. Divide it into triangles by a perpendicular on each diagonal from the opposite angle. Find the area of each triangle separately, and the sum of these areas will be the area of the trapezium. NOTE.-Either the diagonals and perpendiculars must be given, or data from which to find them. 44. The diagonal extent of a four-sided field is 65 rods, and the perpendiculars on it from the opposite corners are 28 and 33.5 rods; what is the area? Ans. 1 acre, 1 rood, 22.083 rods. 45. A quadrangle having two sides parallel, and the one is 20.5 feet long and the other 12.25 feet, and the perpendicular distance between them is 10.75 feet; what is the area? Ans. 176.03125 sq. feet. 46. Required the area of a six-sided figure, the, diagonals of which are as follows: the two extreme ones, 20.75 yards and 18.5, and the intermediate 27.48; the perpendicular on the first is 8.6, on the second 12.8, and those on the intermediate one 14.25 and 9.35? Ans. 531.889 yards. 47. If the two sides of a hexagon be parallel, and the diagonal parallel to them be 30.15 feet, and the perpendiculars on it from 21 310 ARITHMETIC. the opposite angles are, on the left, 10.56, and on the right 12.24, and the part of the diagonal cut off to the left by the first perpendicular, 8.26, and to the right by the second, 10.14; on the other side, the perpendicular and segment of the diagonal to the left are 8.56 and 4.54, and on the right 9.26 and 3.93; what is the area? Ans. 470.4155 sq. feet. PROBLEM V. To find the area of a figure, the boundaries of which are partly right lines and partly curves or salients. Find the average breadth by taking several perpendiculars from the nearest and most remote points, from a fixed base, axd dividing the sum of these by their number, the quotient, multiplied by the length, will be a close approximation to the area. Let the perpendiculars 9.2, 10.5, 8.3, 9.4, 10.7, their sum is 48.1, then 48.1 —5=9.62, and if the base is 20, we have 9.62X20= 192.4, the area. When practicable, as large a portion of the space as possible should be laid off, so as to form a regular figure, and the rest found as above. A field is to be measured, and the greater part of it can be laid off in the form of a rectangle, the sides of which are 20.5 and 10.5, and therefore its area is 215.25, and the offsets of the irregular part are 10.2, 8.7, 10.9, and 8.5, the sum of which, divided by their number, is 7.66, and 7.66X20.5=157.03, the area of the irregular part, and this, added so the area of the rectangles, gives 372.28, the whole area. 48. The length of an irregular clearing is 47 rods, and the breadths at 6 equal distances are 5.7, 4.8, 7.5, 5.1, 8.4 and 6.5; what is the area? Ans. 1 acre, 1 rood, 29.86 rods. PROBLEMI VI. To find the circumference of a circle when the diameter is known, or the diameter when the circumference is known.* The most accurate rule is the well-known theorem that the.diameter is to the circumference in the ratio of 113 to 355,and * In strictness the circumference and diameter are not like quantities, but we may suppose that a cord is stretched round the circumference, and then drawn out into a straight line, and its linear units compared with those of the diameter. MENSURATION. 311 consequently the circumference to the diameter as 355 to 113. Now, 355 —113-3.1416 nearly, and for general purposes, sufficient accuracy will be attained by this RULE. To find the circumference from a given diameter, multiply the diameter by 3.1416; and to find the diameter from a given circumference, divide by 3.1416. 49. What is the length round the equator of a 15-inch globe? Ans. 47.124 inches. 50. If a round log has a circumference of 6 feet, 10 inches;'what is its diameter? Ans. 2 feet, 2-1- inches nearly. 51. If we take the distance from the centre of the earth to the equator to be 3979; what is the number of miles round the equator? Ans, 25001 nearly. PROBLEM VII. To find the area of a circle. 1. If the circumference and diameter are known,Multiply the circumference by the diameter, and take one-fourth of the product. 2. If the diameter alone is given,Multiply the square of the diameter by.7854. 3. If the circumference alone be given,Multiply the square of the number denoting the circumference by.07958. 52. If the diameter of a circle is 7, and the circumference 22; what is the area? Ans. 38k. 53. What is the area of a circle, the radius of which is 3- yds? Ans. 35 square yardst 54. If a semicircular arc be denoted by 10.05; what is the area of the circle? Ans. 289.36. 55. If the diameter of a grinding stone be 20 inches; what superficial area is left when it is ground down to 15 inches diameter, and what superficial area has been worn away? Ans. 176.715 sqr. inches left, and 137.445 worn away. 56. If the chord of an arc be 24 inches, and the perpendicular on it from the centre 11.9; what is the area of the circle? Ans. 2.689804. 312 ARITHMETIC. MENSURATION OF SOLIDS. To find the solid contents of a parallelopiped, or any regularly box-shaped body: Let it be required to find the number of cubic feet in a box 8 feet long, 4~ feet broad, and 6| feet deep. In the first place, the length being 8 feet and the breadth 41, the area of the base is 8X4 =36 square feet, and therefore every foot of altitude, or depth, or thickness, will give 36 cubic feet, and as there are 6: feet of depth, the whole solid content will be 36 times 61, or 243 cubic feet. Hence the RULE. Take the continual product of the length, breadth, and depth. NOTE.-Let it be carefully observed that the unit of measure in the case of solids is to be taken as a cube, the base of which is a superficial unit used in the measurement of surfaces. The solid content is indicated by the repetition of this unit a certain number of times. If the body is of uniform breadth the rule needs no modification, but if it is rounded or tapering, as a globe, cone, or pyramid, the calculation becomes virtually to find how much the rounded or tapering body differs from the one of uniform breadth. Suppose, for example, we take a piece of wood 6 feet high, in the form of a pyramid, and having the length and breadth of the base each 6 feet, then the area of the base is 36; but if, at the height of 1 foot, the dimensions have each diminished by 1 foot, the area is 25; at another foot higher it is 16; at the,next 9; at the next 4; at the next 1; and at the 6th 0, i. e., it has come to a point, and the calculation is, how much remains from the solid cube after so:nuch has been cut off each side as to give it this form. This gives rise to the following varieties: I. To find the solid contents of a cone or pyramid: Multiply the area of the base by the perpendicular height, and take one-third of the product. II. To find the solid contents of. a cylinder or prism: Multiply the area of the base by the perpendicular height. III. To find the surface of a sphere: Mlultiply the square of the diameter by 3.1416. IV. To find the solid contents of a globe or sphere: Multiplu the third power of the diameter by.5236. MENSURATION OF SOLIDS. 313 V. To find the volume of a spheroid, the axes being given: Multiply the square of the axis of revolution by the fixed axis, ceid the product by.5236. EXERCISES. 57. If the diameter of the base of a cylinder be 2 feet, and its height 5 feet, what is the solid content? Ans. 25.708 feet: 58. If the diameter of the base of a cone be 1 foot 6 inches, and the altitude 15 feet, what are the solid contents? Ans. 8 feet, 120 inches. 59. If the diameter of the base of a cylinder be 7 feet, and the height 5 feet, what is the solid content? Ans. 245 cubic feet. 60. What are the solid contents of a hexagonal prism, each side of the base being 16 inches and the height 15 feet? Ans. 69.282 cubic feet; 61. A triangular pyramid is 30 feet high, and each side of the base is 3 feet; required the solid contents. Ans. 39.98 cubic feet. 62. What are the solid contents of the earth, the diameter being taken as 7918.7 miles? Ans. 259992732079.87. 63. In a spheroid the less axis is 70 and the greater 90; what are the solid contents? Ans. 230907.6. PILING OF BALLS AND SHELLS. Balls are usually piled on a base which is either a triangle, or square, or rectangle, each side of each course containing one ball less than the one below it. If the base is an equilateral figure, the vertex of a complete pile will be a single ball; but if one side of the base be greater, than the contiguous one, the vertex will be a row of balls. Hence, if the base be an equilateral figure, the pile will be a pyramid, and as the side of each layer contains one layer less than the one below it, the number of balls in height will be the same as the number of balls in one side of the lowest layer. If the pile be rectangular, each layer must also be rectangular, and the number of balls in height will be the same as the number in the less side of the base. If the base be triangular, we have the 314 ARITHMETIC. RULE. Multiply the number on one side of the bottom row by itself PLUS one, and the product by the same base row PLUS two, and divide the result by six. For a complete square pile we have the RULE. Multiply the number of balls in one side of the lowest course by itself PLUS one, and this product by double the first multiplier PLUS one, and take one-sixth of the result. If the pile be rectangular, we have the RULE. From three times the number of balls in the length of the lowest course subtract one less than the number in the breadth of the same course; multiply the remainder by the breadth, and this product by one-sixth the breadth PLUS one. If the pile be incomplete, find what it would be if complete; find also what the incomplete one would be as a separate pile, and subtract the latter from the former. EXERCISES. 64. In a complete triangular pile each side of the base is 40; how many balls are there? Ans. 11480. 65. In each side of the base of a square pile there are 20 shells; how many in the whole pile? Ans. 2870. 66. In a rectangular pile there are 59 balls in the length, and 20 in the breadth of the base; how many are in all? Ans. 11060. 67. In an incomplete triangular pile, each side of the lowest layer consists of 40 balls, and the side of the upper course of 20; what is the number of balls? Ans. 10150. NOTE.-Since the upper course is 20, the first row in the wanting part would be 19. MEASUREMENT OF TIMBER. 315 MEASUREMENT OF TIMBER. Timber is measured sometimes by the square foot, and sometimes by the cubic foot. Cleared timber, such as planks, beams, &c., are usually measured by the square foot. What is called board measure is a certain length and breadth, and a uniform thickness of one inch. Large quantities of round timber are often estimated by the ton. To find either the superficial extent or board measure of a plank, &c. RULE. Multiply the length in feet by the breadth in inches, and divide by 12. NOTE.-The thickness being taken uniformly as one inch, the rule for finding the contents in square feet becomes the same as that for finding surface. If the thickness be not an inch,Multiply the board measure by the thickness. If the board be a tapering one, take half the sum of the two extreme widths for the average width. If a one-inch plank be 24 feet long, and 8 inches thick, then we have 8 inches equal i of a foot, and i of 24 feet=16 feet. A board 30 feet long is 26 inches wide at the one end, and 14 inches at the other, hence 20 is the mean width, i. e., 1~ feet, and 30X 1==50; or, 30X20-600, and 600 -12=-50. To find the solid contents of a round log when the girt is known. RULE. Multiply the square of the quarter girt in inches by the length in feet, and divide the product by 144. If a log is 40 inches in girt, and 30 feet long, the solid contents will be found by taking the square of 10, the quarter girt in inches, which is 100, and 100X30-3000, and 3000 -144=201. To find the number of square feet in round timber, when the mean diameter is given. 316 ARITHMETIC. RULE. llultiplg the diameter in inches by half the diameter in inches, and the product by the length in feet, and divide the result by 12. If a log is 30 feet long, and 56 inches mean diameter, the number of square feet is 56X28X30- -12=3920 feet. To find the solid contents of a log when the length and mean diameter are given. RULE. Mfultiply the square of half the diameter in inches by 3.1416, and this product by the length in feet, and divide by 144. 68. How many cubic feet are there in a piece of timber 14X18, and 28 feet long? Ans. 49 —cubic feet. 69. How many cubic feet are there in a round log 21 inches in diameter, and 40 feet in length? 70. What are the solid contents of a log 24 inches in diameter, and 34 feet in length? Ans. 106.81+-cubi6 feet. 71. How many feet, board measure, are there in a log 23 inches in diameter, and 12 feet long? Ans. 2641. 72. How many feet, board measure, are there in a log, the diameter of which is 27 inches, and the length 16 feet. Ans. 486. 73. What are the solid contents of a round log 36 feet long, 18 inches diameter at one end, and 9 at the other? 74. Ilow many feet of square timber will a round log 36 inches in diameter and 10 feet long yield? Ans. 540 solid feet. 75. IIow many solid feet are there in a board 15 feet long, 5 inches wide, and 3 inches thick? Ans. 1 9 cubic feet. 76. What are the solid contents of a board 20 feet long, 20 inches broad, and 10 inches thick? Ans. 27V feet. 77. What is the solid content of a piece of timber 12 feet long, 16 inches broad, and 12 inches thick? Ans. 16 feet. 78. How many cubic feet are there in a log that is 25 inches in diameter, and 32 feet long? 79. How many feet, board measure, does a log 28 inches in diameter, and 14 feet in length contain? Ans. 4573. 80. How many cubic feet are contained in a piece of squared timber that is 12 by 16 inches, and 47 feet in length? Ans. 62~. MEASUREMENT OF TIMBER. 317 81. How many feet, board measure, are there in 22 one-inch boards, each being 13 inches in width, and 16 feet in length? Ans. 3814. BALES, BINS, &C. As bales are usually of the same form as boxes, the same rule applies. 82. Hence, a bale measuring 4~ inches in length, 33 in width, and 3~ in depth, is, in solid content, 371 feet. 83. A crate is 5 feet long, 41 broad, and 3-7 deep, what is the. solid content? Ans. 85 4. To find how many bushels are in a bin of grain: RULE. Find the product of the length, breadth and depth, and divide by 5150.4. 84. A bin consists of 12 compartments; each measures 6 feet 3 inches in length, 4 feet 8 inches in width, and 3 feet 9 inches in depth; how many bushels of grain will it hold? Ans. 1055, nearly. Tofind how many bushels of grain are in a conical heap in the middle of a floor: RULE. Multiply the area of the base by one-third the height. The base of such a pile is 8 feet diameter and 4 feet high; what is the content?. The area oi the base is 64X.7854-83.777, and 83.777X== 67.02, the number of bushels. If it be heaped against a wall take half the above result. If it be heaped in a corner, take one-fourth the above result. 21 318 ARITMll TIC. MISCELLANEOUS EXERCISES. 1. What number is that - and 3- of which make 255? Ans. 201-. 2. What must be added to 217~, that the sum may be 171 times 19? Ans. 118X. 3. What sum of money must be lent, at 7 per cent., to accumulate to $455 interest in 3 months? Ans. $26000. 4. Divide $1000 among A, B and C, so that A may have $156 more than B, and B $62 less than C. Ans. At. $416j; B, $260%; C, $322~. 5. Where shall a pole 60 feet high be broken, that the top may rest on the ground 20 feet from the stump? Ans. 26j feet. 6. A man bought a horse for $68, which was 3 as much again as he sold it for, lacking $1; how much did he gain by the bargain? Ans. $12.50. 7. A fox is 120 leaps before a hound, and takes 5 leaps to the hound's 2; but 4 of the hound's leaps equal 12 of the fox's; how many leaps must the hound take to catch the fox? Ans. 240. S. A, B and C can do a certain piece of work in 10 days; how long will it take each to do it separately, if A does 1~ times as much as B, and B does i as much as C? Ans. A, 30 days; B, 45; C, 22}. 9. At what time between five and six o'clock, are the hour and minute hands of a clock exactly together? Ans. 27 min., 164- sec. past 5. 10. A courier has advanced 35 miles with despatches, when a second starts with additional'instructions, and hurries to overtake the first, tavelling 25 miles for 18 that the first travels; how far will both have travelled when the second overtakes the first? Ans. 125 miles. 11. What is the sum of the series 2 - -4~ s - G36 _+43 &c.? 5 1 5 y4 5, 51. Ans. 6 12. If a man earn $2 more each month than he did the month before, and finds at the end of 18 months that the rate of increase will enable him to earn the same sum in 14 months; how much did he earn in the whole time? Ans. $4032. 13. How long would it take a body, moving at the rate of 50 MISCELLANEOUS EXERCISES. 319 miles an hour, to pass over a space equal to the distance of the earth from the sun, i. e., 95 millions of miles, a year being 365 days? Ans. 216 years, 326 days, 16 hours. 14. Two soldiers start together for a certain fort, and one travels 18 miles a day, and after travelling 9 days, turns back as far as the second had travelled during those 9 days, he then turns, and in 22~ days from the time they started, arrives at the fort at the same time as his comrade; at what rate did the second travel? Ans. 18 miles a day. 15. What quantity must be subtracted from the square of 48, so that the remainder may be the product of 54 by 16? Ans. 1440. 16. A father gave 3 of his farm to his son, the son sold ~ of his share for $1260; what was the value of the whole farm? Ans. $5040. 17. There were 5 of a flock of sheep stolen, and 672 were left; how many were there in all? Ans. 1792. 18. A boy gave 2 cents each for a number of pears, and had 42 cents left, but if he had given 5 cents for each, he would have had nothing left. Required the number of pears. Ans. 14. 1 19. Simplify + I A ~1+^. Ansle 1Ans. 20. A man contracted to perform a piece of work in 60 days, he employed 30 men, and at the end of 48 days it was only half finished; how many additional hands had to be employed to finish it in the stipulated time? 21. A gentleman gave his eldest daughter twice as much as his second, and the second: three times as much as the third, and the third got $1573; how much did he give to all? Ans. $15730. 22. The sum of two numbers is 5643, and their'-difference 125; what are the numbers? Ans. 2884 and 2759. 23. How often will all the four wheels of a carriage turn round in going 7 miles, 1 furlong, and 8 rods, the hind wheels being each 7 feet 6 inches in circumference, and the fore wheels 5 feet 7~ inches? Ans. 23716. 24. What is the area of a right angled triangular field, of which the hypotenuse is 100 rods and the base 60? Ans. 2400 sq. rds. 54 —2' 4of~1 o- 2 3 Ans1 1 25. Simplify s fof of' I. Ans. 15 3j+ 9 4 1 719-21 1 320 ARITHMETIC. 26. Find the value of 1+l Ans. A. 27. If ~ of A's age is J of Bs', and A is 37k, what age is B? Ans. 40. 28. What is the excess of 1 1 - above - - 1 -? Ans. T-'0 29.;The sum of two numbers is 5330 and their difference 1999; what are the numbers? Ans. 3664- and 16651. 30. A person being asked the hour of the day, replied that the time past noon was equal to one-fifth of the time past midnight; what was the time? Ans. 3 P.M. 31. A snail, in getting up a pole 20 feet high, climbed up 8 feet every day, but slipped back 4 feet every night; in what time did he reach the top? Ans 4 days. 32. What number is that whose ~,,'and 4 parts make 48? Ans. 44-4. 33. A merchant sold goods to a certain amount, on a commission of 4 per cent., and, having remitted the net proceeds to the owner, received i per cent. for immediate payment, which amounted to $15.60; what was the amount of his commission? Ans. $260. 34. A criminal has 40 miles the start of the detective, but the detective makes 7 miles for 5 that the fugitive makes; how far will the detective have travelled before he overtakes the criminal? Ans. 140 miles. 35. A man sold 17 stoves for $153; for the largest size he received $19, for the middle size $7, and for the small size $6; how many did he sell of each size? Ans. 3 of the large size, 12 of the middle, 2 of the small. 36. A merchant bought goods.to the amount of $12400; $4060 of which was on a credit of 3 months, $4160 on a credit of 8 months and the remainder on a credit of 9 months; how much ready money would discharge the debt, money being worth 6 per cent.? Ans. $12000. 37. If a regiment of soldiers, consisting of 1000 men, are to be clothed, each suit to contain 3. yards of cloth that is 1 — yards wide, and to be lined with flannel 11 yards wide; how many yards will it take to line the whole? Ans. 5625. 38. Taking the moon's diameter at 2180 miles, what are the solid contents? Ans. 5424617475+ sq. miles. MISCELLANEOUS EXERCISES. 321 39. A certain island is 73 miles in circumference, and if two men start out from the same point, in the same direction, the one walking at the rate of 5 and the other at the rate of 3 -miles an hour; in what time will they come together? Ans. 36 hcprs, 30 minutes. 40. A circular pond measures half an acre; what length of cord will be required to reach from the edge of the pond to the centre? Ans. 83263+ feet. 41. A gentleman has deposited $450 for the benefit of his son, in a Savings' Bank, at compound interest at a half-yearly rate of 3j per cent. He is to receive the amount as soon as it becomes $1781.66~. Allowing that the deposit was made when the son was 1 year old, what will be his age when he can come in possession of the money? Ans. 21 years. 42. The select men of a certain town appointed a liquor agent, and furnished him with liquor to the amount of $825.60, and cash, $215. The agent received cash for liquor sold, $1323.40. He paid for liquor bought, $937; to the town treasurer, $300; sundry expenses, $29; his own salary, $265; he delivered to indigent persons, by order of the town, liquor to the amount of $13.50. Upon taking stock at the end of the year, the liquor on hand amounted to $616.50. Did the town gain or lose by the agency, and how much; has the agent any money in his hands belonging to the town; or does the town owe the agent, and. how much in either case? Ans. The town lost $103.20; the agent owes the town $7.40. 43. A holds a note for $575 against B, dated July 13th, payable in 4 months from date. On the 9th August, A received in advance $62; and on the 5th September, $45 more. According to the terms of agreement it will be due, adding 3 days of grace, on the 16th November, but on the 3rd of Ootober B proposes to pay a sum which, in addition to the sums previously paid, shall extend the pay day to forty days beyond the 16th of November; how much must B pay on the 3rd of October? Ans. $111.43. 44. A accepted an agency from B to buy and sell grain for him. A received from B grain in store, valued at $135.60, and cash, $222.10; he bought grain to the value of $1346.40, and sold grain to the amount of $1171.97. At the end of four months B wished to close the agency, and A returned him grain unsold, valued at $437.95; A was to receive for services, $48.12. Did A owe B, or B owe A, and how much? Ans. B owed A 45 cents. 322 APITHMETIC. 45. A general ranging his men in the form of a square, had 59 men over, but having increased the side of the square by one man, he lacked 84 of completing the square; how many men had he? Ans. 5100. 46. What portion, expressed as a common fraction, is a pound and a half troy weight of three pounds avoirdupois? Ans. 7. 47. What would the last fraction be if we reckoned by the ounces instead of grains according to the standards? Ans. -. 48. If 4 men can reap 6 J acres of wheat in 2A~ days, by working 8{ hours per day, how many acres will 15 men, working equally, reap in 3M days, working 9 hours per day? Ans. 40}~ days. 49. Out of a certain quantity of wheat, 3 was sold at a certain gain per cent., 1 at twice that gain, and the remainder at three times the gain on the first lot; what was the gain on each, the gain on the whole being 20 per cent.? Ans. 9-, 191 and 284 per cent. 50. If a man by travelling 6 hours a day, and at the rate of 4miles an hour, can accomplish a journey of 540 miles in 20 days; how many days, at the rate of 4% miles an hour, will he require to accomplish a journey of 600 miles? Ans. 21-. 51. Smith in Montreal, and Jones in Toronto, agree to exchange operations, Jones chiefly making the purchases, and Smith the sales, the profits to be equally divided; Smith remitted to Jones a draft for $8000 after Jones had made purchases to the amount of $13682.24;-Jones had sent merchandise to Smith, of which the latter had made sales to the value of $9241.18; Jones had also made sales to the worth of $2836.24; Smith has paid $364.16 and Jones $239.14 for expenses. At the end of the year Jones has on hands goods worth $2327.34. and Smith goods worth $3123.42. The term of the agreement having now expired, a settlement is made, what has been the gain or loss? What is each partner's share of gain or loss? What is the cash balance, and in favor of which partner? 52. In a certain factory a number of men, boys and girls are employed, the men work 12 hours a day, the boys 9 hours and the girls 8 hours; for the same number of hours each man receives a half more than each boy, and each boy a third more than each girl; the sum paid each day to all the boys is double the sum paid to all the girls, and for every five shillings earned by all the boys each day, twelve shillings are earned by all the men; it MISCELLANEOUS EXERCISES. 323 is required to find the number of men, the number of boys and the number of girls, the whole number being 59. Ans. 24 men, 20 boys and 15 girls. 53. A holds B's note for $575, payable at the end of 4 months from the 13th July; on the 9th August, A received $62 in advance, as part payment, and on the 5th September $45 more; according to agreement the note will not be due till 16th November, three days of grace being added to the term; but on the 3rd October B tenders such a sum as will, together with the payments already made, extend time of payment forty days forward; how much must B pay on the 3rd of October? Ans. $111.43. 54. If a man commence business with a capital of $5000 and realises, above expenses, so much as to increase his capital each year by one tenth of itself less $100, what will his capital amount to in twenty years? Ans. $27910. 55. A note for $100 was to come due on the 1st October, but on the 11th of August, the acceptor proposes to pay as much in advance as will allow him 60 days after the 1st of October to pay the balance; how much must he pay on the 11th of August? Ans. $54. 56. A person contributed a certain sum in dollars to four charities;-to one he gave one half of the whole and half a dollar; to a second half the remainder and half a dollar; to a third half the remainder and half a dollar; and also to the fourth half the remainder and half a dollar, together with one dollar that was left; how much did he give to each? Ans. To the first, $16; to the second, $8; to the third, $4; to the fourth, $3. 57. A farmer being asked how many sheep he had, replied that he had them in four different fields, and that two-thirds of the number in the first field was equal to three-fourths of the number in the second field; and that two-thirds of the number in the second field was equal to three-fourths of the number in the third field; and that two-thirds of the number in the third fiold was equal to fourfifths of the number in the fourth field; also that there were thirtytwo sheep more in the third field than in the fourth; how many sheep were in each field and how many altogether? Ans. First field, 243; second field, 216; third field, 1921 fourth field, 160. Total. 811. 324 ARITHMETIC. 58. How many hours per day must 217 men work tor 5k clays to dig a trench 231 yards long, 3J yards wide, and 2- deep, if 24 men working equally can dig one 33; yards long, 53 wide, and 3j deep, in 189 days of 14 hours each. Ans. 16 hours. 59. A man bequeathed one-fourth of his property to his eldest son;-to the second son one-fourth of the remainder, and $350 besides; to the third one-fourth of the remainder, together with $975; to the youngest one-fourth of the remainder and $1400; he gives his wife a life interest in the remainder, and her share is found to be one-fifth of the whole; what was the amount of the property? Ans. $20,000. 60. Five men formed a partnership which was dissolved after four years' continuance; the first contributed $60 at first and $800 more at the end of five months, and again $1500 at the end of a year and eight months; the second contributed $600 and $1800 more at the end of six months; the third gave at first $400 and $500 every six months; the fourth did not contribute till the end of eight months; he then gave $900, and the same sum every six months; the fifth, having no capital, contributed by his labor in keeping the books at a salary of $1.25 per day; at the expiration of the partnership what was the share of each, the whole gain having been $20000? Ans. 1st, $2019.65 nearly; 2nd, $4871.81 nearly; 3rd, $4815.81 nearly; 4th, $646.74 nearly; 5th, $1825.00. 61. Four men, A, B, 0, and D, bought a stack of hay containing 8 tons, for $100. A is to have 12 per cent. more of the hay than B, B is to have 10 per cent. more than C, and C is to have 8 per cent. more than D. Each man is to pay in proportion to the quantity he receives. The stack is 20 feet high, and 12 feet square at its base, it being an exact pyramid; and it is agreed that A shall take his share first from the top of the stack, B is to take his share the next, and then C and D. How many feet of the perpendicular height of the stack shall each take, and what sum shall each pay? Ans. A. takes 13.22+ft., and pays $28.93~; B takes 3.14+ft., and pays $25.83-; C takes 2.06+ft., and pays $23.48I; D takes 1.58+ft., and pays $21.74-. 62. A merchant bought 500 bushels of wheat and sold one half of it at 80 cents per bushel which was 10 per cent more than it MISCELLANEOUS EXLRCISES. 325. _ 1 _ _ _ 1..,L _ 1 -. _1 L - _ 1 1 326 ARITHMETIC. FOREIGN GOLD COINS. MINT VALUE. COUNTRY. DENOMINATIONS. WEIGHT. VFALUE. Dedutr NESS. Deduction. Oz. DEC. THOUS. Australia...... Pound of 1852......... 0.281 916.5 $5.32.37 $5.29.71."..... Sovereign 1855-60....... 0.256.5 916 4.85.58 4.83.16 Austria........ Ducat..................0.112 986 2.28.28 2.27.04........ Souverain.....,...... 0.363 900 6.75.35 6.71.98.......... NewUnion Crown (assumed) 0.357 900 6.64.19 6.60.87 Celgium...... Twenty-five francs........ 0.254 899 4.72.03 4.69.67 olivia...... Doubloon........... 0.867 870 15.59.25 15.51.46 Brazil........ 20 Milreis........... 0.575 917.5 10.90.57 10.85.12 Centr 1 America Two escudos.......... 0.209 853.5 3.68.75 3.66.91 Chili........ Old doubloon.......... 0.867 870 15.59.26 15.51.47..........Ten Pesos........... 0.492 900 9.15.35 9.10.78 Denmark...... Ten thaler.......0.427 895 7.90.01 7.86.0( Equador...... Four eseudos.......... 433 844.55.4 7.51.69 Enland...... Pound or Sovereign, new. 0.256.7 916.5 4.86.34 4.83.91.".....Pound or Sverign,average 0.256.2 916 4.84.92 4.82.50 Franc........ Twenty francs, new....... 0.207.5 899.5 3.85.83 3.83.91......Twenty francs, average... 0.207 899 3.84.69 3.82.77 Germany, North Ten thaler............... 0.427 895 7.90.01 7.86.06...'.Ten thaler, Prussian..... 0.427 903 7.97.07 7.93.09.;. Krone [crown].........0.357 900 6.64.20 6.60.88 Germany, Sontl Ducat..................0.112 986 2.28.28 2.27.14 Greece..... Twenty drachms....... 0.185 900 3.44.19 3.42.47 I1indostan..... Mohur............... 0.374 916 7.08.18 7.04.64 Italy........ 20 lire................. 0.207 898 3.84.26 3.82.34 Japan.......Old Cobang.......... 0.362 568 4.44.0 4.41.8 ";....... New Cobang............. 0.289 572 3.57.6 3.55.8 Mexico......... Doubloon, average.......0.867.5 866 15.52.98 15.45.22 44.......; nnew.......... 0.867.5 870.5 15.61.05 15.53.25 Naples...... Six ducati, new........... 0.245 996 5.04.43 5.01.91 Netherlands.... Ten guilders... 0.215 899 3.99.56 3.97.57 New Granada.. Old Doubloon, Bogota 0.868 870 15.61.06 15.53.26;" " Old Doubloon, Popayan.. 0.867 858 15.37.75 15.30.07 " "( Ten pesos, new.......... 0.525 891.5 9.67.51 9.62.68 Peru..........Old doubloon..... 0.867 868 15.55.67 15.47.90 Portugal...... (old crown............. 0.308 912 5.80.66 5.77.76 Prussia..... NewUnion Crown [assumed] 0.357 900 6.64:19 6.60.87 Rome...... 2 scudi, new.......... 0140 900 2.60.47 2.59.17 Russia........ Five roubles....... 0.210 916 3.97.64 3.95.66 Spain.... 100 reals................ 0.268 896 4.96.39 4.93.91 "..... 80 reals.............215 869.5 3.86.44 3.84.51 Sweden...... Ducat......... 0.111 975 2.23.72 2.22.61 Tunis........ 25 piastres....... 0.161 900 2.99.54 2.98.05 Turkey....... 100 piastres.............0.231 915 4.36.93 4.34.75 Tuscany..... Sequin................. 0.112 999 2.31.29 2.30.14 FOREIGN SILVER CO)INS. 327 FOREIGN SILVER COINS. MINT VALUE. COUNTRY. DENOMINATIONS. WEIGHT. FINENESS. VALUE. Oz. DEC. THOUS. Austria........... Old rix dollar......... 0.902 833 $1.02.27...... Old scudo.............. 0.836 902 1.02.64 "........... Florin before 1858...... 0.451 833 51.14........... New florin............. 0.397 900 48.63...........New Union dollar..... 0.596 900 73.01........... Maria Theresa dol'r,1780 0.895 838 1.02.12 Belgium........... Five francs...........0.803 897 8.04 Bolivia.......... New dollar............ 0.643 903.5 79.07r........... Half dollar............ 0.432 667 39.22Brazil............. Double Milreis......... 0.820 918.5 1.02.53: Canada........... 20 cents.............. 0.150 925 18.87 Ceniral America... Dollar.............. 0.866 850 1.00.19 Chili.............Old Dollar............. 0.864 908 1.06.79 ".......... New Dollar... 0.801 900.5 98.17 Denmark.'.... Two rigsdaler......... 0.927 877 1.10.65 England........... Shilling, new..........0.182.5 924.5 22.9 Shilling, average....... 0.178 925 22.41 France.......... Five franc, average.... 0.800 900 98.00 Germany, North... Thaler, before 1857.. 0.712 750 72.67." New thaler....... 0.595 900 72.89 Germany, South.... Florin, before 1857..... 0.340 900 41.65........ New florin [assumed]... 0.340 903 41.65 Greece. Five drachms.......... 0.719 900 88.08 Hindostan......... Rupee................ 0.374 916 46.62 Japan............. Itzebu................ 0.279 991 37.63............ New Itzebu............ 0.279 890 33.80 Mexico........... Dollar, new............ 0.867.5 903 1.06.62 "......... Dollar, average........ 0.866 901 1.06.20 Naples........... Scudo................. 0.844 830 95.34 Netherlands....... 2 guild............... 0.804 944 1.03.31' Norway.......... Specie daler........... 0.927 877 1.10.65 New Granada..... Dollar of 1857......... 0.803 896 97.92 Peru............ Old dollar.... 0.866 901 1.06.20'............. Dollarof 1858... 0.766 909 94.77............ Half dollar, 1835-38.... 0.433 650 38.31 Prussia............ Thaler before 1857.... 0.712' 750 72.68 "............New thaler............0.595 900 72.89 Rome............. Scudo...0.864 900 1.05.84 Russia........... Rouble........... 0.667 875 79.44 Sardinia........... Five lire.............. 0.800 900 98.00 Spain............ New pistareen...... 0.166 899 20.31 Sweden.......... Rix dollar.......... 1.092 750 1.11.48 Switzerland........ Two francs............0.323 899 39.52 Tunis............. Five piastres.......... 0.511 898.5 62.49 Turkey............ rwenty piastres........0.770 830 86.98 Tuscany........... Florin................. 0.220 925 27.60 LAWS OF THE UNITED STATES RELATING TO INTEREST, DAMAGES ON BILLS, AND THE COLLECTION OF DEBTS The following brief sketches of the laws of the different States of the Union, will be found useful, not only to business me: but also to private individuals. The inforrmtion on which they are founded, has been derived from authentic sources and condensed into a convenient epitome, which may be relied upon as correct as regards the present state f the law, which is all that any one can be answerable for, as alterations may hereafter be made on some points. ALABAMA. Interest.-The rate of interest in Albama is eight per cent. per annum. Penalty for Violation of the Usury Laws -All contracts made at a. higher rate of interest than eight per cent. are usurious, and cannot be enforced except as to the principal. Damages on Bills.-Damages on inlan I bills of ex's,ange protested for non-payment, are 5 per cent.; on foreign bills of exchange 10 per cent. on the sum drawn for. All bills drawn and payable within this State are termed inland bills; those drawn in this State and payable elsewhere, are considered foreign bills. Sight Bills.-Grace is allowed on bills, drafts, etc., payable at sight. Collection of Debts.-Original attachments, foreign and domes ic, are issued by judges of the circuit or county courts, or justices'of the peace. An attachment may issue, although the,debt or demand of the plaintiff be not due; and shall be a lien on the property attached until the debt or demand becomes due, when judgment shall be rendered and exe. ution issued. A:non-resident plaintiff may have an attachment against the property of a non-resident defendant, provided he gives good and sufficient resident security in the required bond, mraking oath that the defendant has not sufficient property within the State of defendant's residence to satisfy the debt or demand. AR I AN SAS. Interest.-The legal rate of interest in Arkansas is six per cent. Special contracts in writing will admit an interest not to exceed ten per cen,. All judgments or decrees upon contracts bearing more than six per cent. shall bear the same rate of interest originally agreed upon.(Gould's Digest, chap. 92, sec. 1, 2, &c., 1858.) Penalty for Violation of the Usury Laws.-All contracts for reservation of a greater rate of interest' than ten per cent are void. The excesstaken or charged beyond ten per cent. may be recovered back, provided the action for recovery shall be brought within o:e year after payment. (Ib. secs.- 6 & 7.) Damages on Bills -The damages on Bills of Exchange drawn or negotiated in Arkansas, expresses to be for value received, and protested for non-acceptance, or for non-payment after non-acceptance, are as follows.-(Ib. chap. 25.) 1. If payable within the State, 2 per cent. 2. If payable in Alabama, Louisiana, Mississippi, Tennessee, Kentucky, Ohio, Indiana, Illinois or Missouri, or at any point on the Ohio River, 4 per cent. 3. If payable in any other Ltaze or territory, 5 per cent. 4. If payable withia either of the United States, and protested for non-payment, after' acceptance, 6 per cent. Foreign Bzlls.-The damages on bills of exchange, expressed for value received, and payable beyond the limits of the United States (lb. chap. 25), are 10 per cent. Sight Bills.-ihere is no statute in f.i'ce in Arkansas in ref.rcnce to grace or sight bills. Section 15, Gould's Digest, says' Foreign and inland bills shall be governe.l by the lawmerchant as to days of grace, protest and notices." Collection of Debts.-An at.aclment may be issued against the property of a non-resident, and also against a resident of the tate when the latter is about to remove ut of the State; or is about to remove his goods or effects, or about to secrete himself, so that the ordinary prJcess of law cannot b)o served on him. 329 CALIFORNIA. Interest.-The legal rate of interest in California is, oy statute, fixed at TEN per cent. On special contracts any rate of interest may be agreed upon or paid. Penalty for Violation of the Interest Lawz.-There is no law in California fixing any penalty for charging any rate ot interest above ten per cent. The matter is thus left entirely free between the contracting parties. Damages on Bills. —The damages on bills of exchange drawn or negotiated in California payable in any State east of the Rocky Mountains, and returned under protest for non-acceptance or non-payment, are uniformly, 15 per cent. Foreign Bills.-The damages on foreign of exchange returned under protest, ar~0 per cent. Sight Bills. —Grace is not allowed by the bankers on bills, checks, drafts, etc., payable at sight. The notarial fees for protesting a bill of exchange or promissory note are $5 or more, according to the number of notices sent. Act March 13, 1850. Collection of Debts.-1. Creditors may proceed by attachment when the defendant has absconded, or is about to abscond from the State, or is conicealed therein to the injury of his creditors. 2. When the defendant has removed or is about to remove any of his p:operty out of the State, with intent to defraud his creditors. 3. When the defendant fraudulently contracted the debt or incurred the obligation, respecting to which the suit is brought. 4. When the defendant is a non-resident. 5, When he has fraudulently conveyed, disposed of or concealed his property, or a part of it, or intends to convey the same to defraud his creditors. In Calitbrnia the real estate shall be bound, and the attachment shall be a lien thereon, although the debt or demand due the plaintiff be not due-in case the defendant is about to remove himself or his property from the State. The law of attachment applies in California when the contract has been made in that State, or when made payable in that State. CONNECTICUT. Intcrest.-The legal rate of interest in Connecticut is six per cent., and no higher rate is allowed on special contracts. Banks are torbidden, under penalty of $500, from taking directly or indirectly over 6 per cent. Law passed May, 1854. Penalty for Violation of the Usury Laws.-Forfeiture of all the interest received. In suits on usurious contracts, judgment is to be rendered for the amount lent, without interest. Damages on Bills.-The damages on bills of exchange negotiated in Connecticut, payable in other States, and returned under protest, are as follows: 1. Maine, New Hampshire, Vermont, Massachusetts, Rhode Island, New York (interior), New Jersey, Pennsylvania, Delaware, Maryland, Virginia, District of Columbia, 3 per cent. 2. New York City, 2 per cent. 3. North Carolina, South Carolina, Georgia and Ohio, 5 per cent. 4. All the other States and Territories, 8 per cent. Foreign Bills.-There is no statute in'orce in Connecticut in reference to damages on foreign bills of exchange. Sight Bills.-Grace is not allowed by statute or usage on checks, bills, ect., payable at sight. Collection af Debts.-Attachment may be granted against the goods and chattels and land of the defendant;and likewise against his person when not exempted from imprisonment on the execution in the suit. The plaintiff to give bonds to prosecute his claim to effect. DELAWARE. Interest.-The legal rate of interest is six per cent, and no more is allowed on direct or indirect contracts. Penalty for Violation of the Uusury Lwas.-Forfeiture of the money tLi other things lent, one half to the Governor for the support of government, the other half payable to the person sueing for the same. Damages on Bills.-There is ho statute in force in Delaware is reference to damages on domestic or inland bills of exchange. Foreign Bills.-The damages upon bills of exchange drawn upon any person in England, or other parts of Europe, or beyond the seas, and returned under protest, are 20 per cent, Sight Bills.-There is no statute with reference to bills, drafts, etc., at sight. They are not, by usage, entitled to grace. Collection of Debts.-A writ of domestic attachment issues against an inhabitant of Delaware when the defendant cannot be found, or has absconded with intent to defraud his creditors; and a writ of foreign attachment when the defendant is not an inhabitant of this State. This attachment is dissolved by the defendant's appearing and putting in special bail at amnv time before judgment. 330 FLORIDA. Interest.-The legal rate oftert interest is six per cent. On special contracts eight per cent. may be chargedl. Penally Jbr Violation of the Usury Lawzs.-Forfeiturs of the whole interest paid. Damages on Bills.-The damages on bills of exchange, negotiated in Florida, payable in other States, and returned under protest for non-payment, are uniformly 5 per cent. Foreign Bills.-Damages on foreign bills of exchange 5 per cent. Sight Bills.-Grace is not allowed on bills, drafts, etc., payable at sight. There is no statute in Florida upon this subject. Collection of Debts.-Ani attachment issues when the amount is actually due, and the defendant is actually moving o-it of the State, or absconds or conceals himself. GEORGIA. Interest.-The legal rate of interest in Georgia is seven per cent., and no higher rate is allowed on special contracts. Open accounts, unliquidated, do not bear interest. Penalty Jbr Violation of the Usury LaZvs.-Foifeiture cf only the excess of interest over seven per cent. Principal and legal interest are recoverable. (Acts of 1855-6, page 259.) Damages on Bills.-The damages on bills of exchange, negotiated in Georgia, payable in other States, and returned under protest, are uniformly 5 per cent. Foreign Bills.-The damages on foreign bills of exchange, returned under protest, are 10 per cent. Sight Bills. — Three days, commonly called the three days' of grace, shall not be allowed upcn any sight drafts or bills of exchange drawn payable at sight, after the passage of this Act; but the same shall be payable on presentation thereof; subject to the provisions of the first section of this Act. The first section designates the holidays." Act passed Feb. 8,1850. (, ee Cobb's New Digest of the luws of Georgia, pp. 519, 522.) Endorsers.-Endorsers are not entitled to notice of dishonour, except upon notes and bills payable at bank, or negotiated in bank, or placed in bank in collection. Collection of Debts.-A judge of the Superior Court, or a justice of the inferior court, or a justice of the peace, may grant an attachment against a debtor whether the debt be matured or not, when the latter is removing without the limits of the State, or any county, or conceals himself. The remedy by attachment may be resorted to by non-residentaswellasby resident creditors. The necessary affidavit may be made before any commissioner appointed by the State to take affidavits. Indorsers of notes, obligations and all other instruments In writing, are entitled to theosame remedy as provided for securities. In all cases the attachment first served shall be first satisfied. No lien shall be created by the levying of an attachment, to the exclusion of any judgment obtained by any creditor, before judgment is obtained by the attaching creditor. ILLINOIS. Interest.-The legiglature, in 1857, passed the following act: SECTIOX 1. That from and after the passage of this act, the rate of interest upon all contract and agreements, written or verbal, express or implied, for the payment of money, shall be six per cent. per annum upon every one hundred dollars, unless otherwise provided by law. SECTION 2. That in all contracts hereafter to be made, whether written or verbal, it shall be lawful for the parties to stipulate or agree that ten per cent. per annum, or any less sum of interest, shall be taken and paid upon every one hundred dollars of money loaned, or in any manner due and owing from any person or corporation to any person or corporation in this State. Penalty for Violation of the Usury Laws.-If any person or corporation in this State shall contract to receive a gre iter rate of interest than ten per cent. upon any contract verbal or written. such person or corporation shall forfeit the whole of said interest so contracted to be receivedl and shall be entitled only to recover the principal sum due to such person or corporation. (Act of 1857). Damages on Bills.-The- damages on bills of exchange negotiated in Illinois, payable in other States or Territories, and returned under protest for non-payment, are uniformily (by act of March 3, 1845) 5 per cent. in addition to the interest. Foreign Bills.-The damages payable on foreign bills of exchange, returned under protest, are [by act of March 3, 1845] 10 per cent. in addition to the interest. Sight Bills.-Heretofore there has been no statute in force regarding bills or drafts at sight, but by an act of the legislature, approved February 22d, 1861, it is enacted that "no note, check, draft, bill of exchange, order or. other negotiable or commercial investments payable at sight or on demand, or on presentation, shall be entitled to days of grace, but shall be absolutely payable on presentment. All other notes, drafts or bills of exchange, shall be entitled to the usual days grace. This act is in force from its passage. Collection of Debts.-Attachments are issued by the clerks of the Circuit Court, when affidavit is tiled that the delendant has departed, or is about to depart, out of the State, or conceals himself so that the process cannot be servcd upon hb'" 331 INDIANA. Interest.-The legal interest in Indiana is six per cent., which may be taken in. advance, if so expressly agreed. Penalty fibr Violation of the Usury Laws.-If a greater rate of interest than as above shall be contracted for, received or reserved, the contract shall not therefore, be void; but if it is proved in any action that a greater rate than six lper cent. per annum has been contracted for, the plaintiff shall only recover his principal with six per cent. interest and costs; and if the defendant has paid thereon over six per cent, interest, such excess of interest shall be deducted from the plaintiff's recovery. If any action for a recovery of a debt, it is proved that previous to the commencement of the suit the defendant has tendered the amount due, with legal interest, the defendant shall recover costs, and the plaintiff shall only recover the amount tendered. Damages on Bill-s-Damages, payable on protest for non-payment or non-accptance of a bill of exchange, drawn or negotiated within the State of Indiana, if drawn upon any person at any place out of this State, are at 5 per cent. Beyond such damages no interest or charges accruing prior to protest shall be allowed, and the rate of exchange shall not be taken into account. Foreign Bills. —_ he damages payable on protest for non-payment or non-acceptance of a bill of exchange, drawn on any place not in the United States, are, on the principal of such bill, 10 per cent. No damages beyond the cost of protest are chargeable against the drawer or the endorser of either species of bill, if upon notice of protest and demand of the principal sum, the same is paid. Sight Bills.-Grace is allowed on all bills of exchange payable in Indiana, whether sight or time bills. Collection of Debts.-The property of an inhabitant of the State nlay be attached, whenever he is secretly leaving the State, or shall have left the State with intent to defraud his creditors. The property of a non-resident is liable to attachment as in other States. IOWA. Interest.-The legal rate ef interest in Iowa is six per cent. Ten per cent. may be charged *on special contracts. On judgments, interest is chargeable as on the contract. Penalty for Violation of the Usury Laws.-Forfeiture of the excess of interest paid for the bene..t of the School Fund. The borrower is by law a competent witness to prove usury. Damages on Bills.-The rates of damages allowed on non-acceptance or non-payment of bills drawn or indorsed in this State, are as follows: If drawn upon a person at a place out of the United States, or in California, or in the Territories of Oregon, Utah, or New Mexico, ten per cent. upon principal, expressed in the bill, with interest from time of protest. If drawn upon a person at a place in Iowa, Missouri, Illinois, Wisconsin, or in Minnesota, three per cent.. with interest. If upon a person at a place in Arkansas, Louisiana, Mississippi, Tennessee, Kentucky, Indiana, Ohio, Virginia, District of Columbia, Pennsylvania, Maryland, New Jersey, New York, Massachusetts, Rhode Island, or Connecticut, five per cent., with interest. If drawn upon a person at a place in any other State, 8 per cent., with interest. (Code, ~965.) Sight Bills. -Grace is allowed on bills and notes, c:ccording to principles of the law merchant, and notice to indorsers, etc., according to the rules of the commercial law. (Laws, 1852-3.) Collection of Debts.-The plaintiff may cause any property of the defendant, which is not subject to execution, to be attached at the commencement, or during the progress of the proceedings, whether the claim be matured or not; provided that an affidavit is filed to the effict that the defendant is a foreign corporation, or acting as such, or that he is a non-resi. dent of the State, or (if a resident) that he is in some manner about to dispose of or remove his property out of the State. KENTUCKY. Interest.-The legal rate of interest in Kentucky is six per cent. No higher rate of interest is allowed even on special contracts. All contracts made, directly or indirectly, for the loan, or forbearance of money, or other thing, at a greater rate than legal interest (6 per cent. per annum), shall be void for the excess of legal interest. Jtenalty for Violation of the Usury Laws.-If any discount or interest greater than the legal interest or discount is taken by any bank, or otbr' corporation, authorized to loan money, the whole contract for interest shall be void, and any thing paid thereon for interest may be recovered back by the person paying the same; or any creditor of his may recover the same by bill in equity. Banks, or other monied corporations, or individuals, are not prevented, in discounting bills of exchange, from taking a fair rate of exchange between the place where it is bought,,and the placJ where it is payable, in addition to the discount for interest. But such privilege of buying bills of exchange at less than par value, shall not be used to disguise a loan of money.at a greater rate of discount than the legal interest or discount. Damag.s on Bills.-No statute is in force in Kentucky upon the subject of damages on.inland bills uf excihangie. 332 Foreign Bills.-Where any bill of excnange, drawn on any person out of the United States,. shall be protested for non-payment or non-acceptance, it shall bear ten per cent. per year interest from the day of protest, for not longer than eighteen months, unless payment be sooner demanded from the party to be charged. Such interest shall be recovered up to the time ot thejudgment, and the judgment shall bear legal interest thereafter. Damages on all other bills are disallowed. [Revised Statutes, pages 193 and 194.] Sight Bills.-Grace is allowed, by some banks, on bills, drafts, etc., payable at sight, but the point is not yet fully settled in this State. Collection of Debts.-1. The plaintiff may have an attachment against the property of the defendant when the latter is a foreign corporation, or a non-resident of this State; or, 2, who has been absent therefrom four months; or, 3, has departed from the State with intent to defraud his creditors; or, 4. has left the county of his residence to avoid the service of a sum.mons, or conceals himself that a summons cannot reach him; or, 5, is about to remove his property, or a material part thereof, out of the State; or, 6, has sold or conveyed his propertywith the intent to defraud his creditors, or is about so to sell or convey. Such attachment isbinding up.on the defendant's property in the county from the time of the delivery of theorder to the Sheriff. LOUISIANA. Interest.-1. All debts shall bear interest at the rate of FIVE per cent, from the time they become due, unless otherwise stipulated. (Act March 15, 1855.) 2. Conventional interest not exceeding eight per cent. per annum may be contracted foi. -Ibid. 3. The owner of any promissory note, bond, or written obligation, for the payment of money to order or to bearer, or transferable by assignment, shall have the right to collect the whole amount of such promissory note, bond, or written obligation, notwithstanding such promissory note, bond, or written obligation may include a greater rate of interest or discount than eight per cent interest per annum. Provided that such obligations shall not bear morethan eight per cent. interest per annum after their maturity until paid. (Act of March 2d, 1860.) Damages on Bills. —The damages on bills of exchange, negotiated in Louisiana, paya':le in other States, are uniformly 5 per cent. Foreign Bills.-The damages on foreign bills of exchange, returned under protest, are uniformly (Statute of 1838).... - 10 per cent. Sight Bills.-There is no statute upon this subject in Louisiana. A decision has been made in one of the inferior courts allowing three days' grace on sight bills, but the usage is to pay on presentation. "Collectios of Debts-A creditor may obtain an attachment against the property of his debtor upon affidavit: 1, when the latter is about leaving permanently the State before obtaining or eecuting judgment against him; 2, when the debtor resides out of the State; 3, when he conceals himself to avoid being cited to answer to a suit, and provided the term of payment has arrived. In the absence of the creditor, the oath may be made by his agent or attorney, to the best of his knowledge and belief. MAINE. Interest.-The legal rate of interest in Maine is six per cent., and no higher rate is allowed on special contracts. [R. S. 322. Cap. 45, sec. 2.] Penalty for Violation of the Usury Laws.-Excess of interest not recoverable, nor costs where excess of interest has been taken; but the defendant may recover costs of the party taking the excess. Excess of interest may be recovered back by the party having paid it. The provisions do not extend to bona fide holders of negotiable paper for~value without notice. [R. S 323. Cap. 45, sees. 2 and 3. Laws of 1862, ch. 136.] Damages on Bils.-Tlie damages on bills of exchange negotiated in Maine, payable in otherStates, and returned under protest, are as follows: [R. S. 519. Cap. 82, sec. 35.] 1. New Hampshire, Vermont, Massachusetts, Rhode Island, Connecticut, NewYork, 3 per cent2. New Jersey, Pennsylvania, Delaware, Maryland, Virginia, District of Columbia, South Carolina, Georgia, 6 per cent. 3. All others, namely, North Carolina, Alabama, Arkansas, Florida, Illinois, Indiana, Iowa, Kentucky, Louisiana, Michigan, Mississipp: Missouri, thio, Tennessee, Texas, Wisconconsin, California, 9 per cent. Sight Bills.-Grace is allowed on bills, drafts, checks, &c., payable in this State at a future day or at sight, but not on those payable on demand. [R. S. 264.] Collection of Debts: -In this State an original writ may be framed either to attach the goods or estate of the defendant, or for.want thereof to take his body. All goods and chattels may be attached by the creditor and held as security pending anys nit against the debtor. Such a writ will authorize an attachment of goods and estate of the principal defendant, in his own hands, as well as in the hands of trustees. Real estate, liable to be taken in execution may beattached 333 MARYLAND. Interest.-The revised constitution of Maryland provides that the rate of interest in the State shall not exceed six per cent. per annum, and no higher rate shall be taken or dem inded. And the legislature shall provide by law all necessary forfeitures and penalties against usury. Penaltie_.-Any person guilty of usury shall forfeit all the excess above the real sum or value ( f t;.e goods or chattels actually lent or advanced and the l.gal interest on such sum or value, which forfeiture shall enure to the benefit o' any defendant who shall plead u-ury, and prove the same. The plea must, however, state the sum or amount ofthe debt, and the plainLifl'shall have judgment for that amount and legal interest only. Md. Code, vol. 1, p. 697. Damages on Bills.-The damages on bills of exchange negotiated in Maryland, payable in other States, and.returned under protest, are uniformly 8 per cent. The claimant is entitled to receive a sum sufficient to buy another bill of the same tenor, and eight per cent. damages on the value of the principal sum mentioned in the bill, and interest from the time of protest, and costs. The prote.t of an inTand bill must be made according to the law or usage of the State where it is payable. Practice includes the District of Columbia inl this law of damages [Act of Ass mbly, 1785, ch. 38); but it is questionable whether the District be within the law, which provides only for States. Foreign Bills.-The damages on foreign bills of exchange returned under protest are 15 per cent. The claimant is to receive a sum sufficient to buy another bill of the same tenor, and 15 per cent. damages On the value of the principal sum mentioned in tile bill, and interest from time of protest, and costs. Sight Bills.-Grace is not allowed by the Banks on bills, drafts, checks, etc., payable at sight. Collect'on of Debts. — creditor may obtain an attachment, whether he be a citizen of Maryland or not, against his debtor, who is not a citizen of this State, and not residing therein. If any c.tizen of the State, being indebted to another citizen thereof, shall actually run away or abscond, or secretly remove himself from his place of abode, with intent to evade the payment of his just debts, an attachment may be obtained against him. An attachment may be laid upon debts due the defendant upon judgments or decrees rendered or paqsed by any court of this State, and judgment of condemnation thereof may be had, as upon other debts due the defendant. MASSACHUSETTS. Interest.-The legal rate of interest in Massachusetts is six per cent., and no higher rate is allowed on special contracts. Penalty for Violation of the Usury Laws.-No contract for the payment of money with interest greaer than six per cent. shall be void; but in an action on such contract the defendint shall rec;ver h s full costs, and the plaintiff shall forfeit three-fold the amount of the whole intere.st reserved or taken. Damages on Bills of Exchange.-The damages'on bills of exchange negotiated in Mlassachusetts, payable in other States, and returned under protest, are as follows: 1. Bills payable in Maine, New Hampshire, Vermont, Rhode Island, Connecticut, or News York, 2 per cent. 2. Bil's payable in New Jersey, Pennsylvania, Maryland, or Delaware. 8 per cent. 3. Bills payable in Virginia, District of Columbia, North Carolina, South Carolina, or Georgia 4 p-r cent. 4. Bills payable elsewhere within the United States or the Territories, 5 per cent. 5. Bills for one hundred dollars or more, payable at any place in Massachusetts, not within seventy-five miles of the place where drawn, 1 per cent. Foreign Bills.-The damages on foreign bills of exchange, returned under protest, are as follows: 1. Bills payable beyond the limits of the United States (excepting places in Africa, beyond the Cape of Good Hope, and places in Asia and the islands thereof) shall pay the current rate of exchange when due, and five per cent. additional. 2. Bills payable at any place in Africa, beyond the Cape of Good Hope, or any place in Asia or the islands thereof, shall pay damages, 20 per eent. Sight Bills.-Bills of exchange, drafts, etc., payable at sight, or at a future day certain. within this State, are entitled to three days' grace. But not bills, notes, drafts, etc., payablk on demand. Notes on Demand.-In order to charge an indorser, payment must be demanded within Sixty days from its date, without grace, on any note payable on demand. Collection of Debts.-Original writs may be framed, either to attach the goods or estate of the defendant, or for want thereot to take his body; or there may be an original summons, either with or without an order to attach the goods or estate. All real estate, or goods and chattles that are liable to be taken in execution, may be attached upon the original writ, in any action in which any debt or damages are recoverable, and may be held as security to satisfy such judgment as the plaintiff may recover. 8384 MIGHIG-A N. )nterest.-The legal rate of interest in Michigan is seven per cent. But it is lawful for parties to stipulate in writing for any sum not exceeding tea per cent. Penalty bor Violation of the Usury Laws.-Parties suing upon contracts reserving over ten per cent. interest, may recover judgment for the principal and legal rate of interest. There is no provision for recovering back illegal interest paid, and no penalty for receiving it. Bona fide holders of usulrious negotiable paper taken before maturity, without notice of usury, may recover the full amount of its face. Damages on Bills.-Damages on'bills drawn or negotiated in Michigan and payable elsewhere and protested arc as follows: 1. If payable out of the United States, 5 per cent. i. If payable in Wisconsin, Illinois, Indiana, Ohio, Pennsylvanla, or New York, 3 per cent. 3. If payable in Missouri, Kentucky, New England, New Jersey, Delaware,'Maryland, Virginia, or District of Columbia, 5 per cent. 4. If payable in any other State or Territory, 10 per cent. Sight Bills.-Grace is allowed on all paper not payable on demand. Collection of Debts. —The grounds of attachment in this State are: 1, that the defendant nas absconded, or is about to abscond, or has concealed himself; 2, that he has assigned or concealed, or is about to remove his property with a view to defraud: 3, that he fraudulently contracted the debt,'or incurred the obligation about which the suit is brought; 4, that he is not a resident of the State, or has not resided there three months immediately preceding the suit; 5, that the defendant is a foreign corporation. MINNESOTA. Interest.-Interest for any legal indebtedness shall be at the rate of $7 for $100 for a year unless a diflerent rate be contracted for in writing, but no agreement or contract f.r a greate rate of interest than $12 for every $100 for a year shall be valid for the excess of interes' over twelve per cent.; and all agreements and contracts shall bear the same rate of interest after they become due. as before, if the rate be clearly expressed therein. Provided, the same shall not exceed, t^lve per cent. per annum. All judgment's cr decrees, made by any court in this State, shall draw interest at the rate of six (6) per cent. per annum. [Laws of 1860, p. 226.] Pelnaltyfor Violation of Interest Law. —Excess ot interest over 12 per cent. forfeited. Days of Grace.-On all bills of exchange payable at sight, or at a future day certain- within cbis State, and on all negotiable promi.ssory notes, orders and drafts, payable at a future day certain within this State, in which there is not an express stipulation to the contrary. lWhen Grace not allowed.-On bills of exchange, note or draft, payable on demand. Wlten presentedfor Paymnent, &c. —Bills of exchange, bank checks and promissory notes falling due, or the presentment for acceptance or payment whereofshould be made on the 1st day of January, the 4th day of July, the 25th day of December, the 22d day of February, and every day appointed by the President of the United States or the Governor of the State as a day of fasting or thanksgiving, shall be presented for acceptance or payment on the day preceding. Such days [above enumerated] shall be treated and considered as the first day of the week, commonly called Sunday. [Col. Laws, 376.] Acceptance of Bills ofExct.e;inge.-No persont within this State shall be charged as an acceptor on a bill of exchange, unless his acceptance shall be in writing, signed by himself or his lawful agent. Damages on Bills iC Exchange.-On any bill of exchange drawn or endorsed within thijs State, and payable without the limits of the United States, which shall be duly protested fior non-acceptance or non-payment, the party liable for the contents of such bill shall, on due notice and demand thereof, pay the same at th3 current rate of exchange, at the time of the demand, and damages at the rate of ten per cent. upon the'contents thereof; together with interest on said contents to be computed from the date of the protest; and said amount of contents, damages and interest shall be in full of all damages. charges and expenses. On all bills drawn onany person, body politic or corporation out of this State, but within some State or Territory of the United States, and protested for non-acceptance or non-payment live per cent. damages and interest, and costand charges of protest. Collection of Debts.-A warrant of attachment may be issued against the property of a defendant when a foreign corporation;- o, when not a resident of this Territory; or, has left the Territory with intent to defraud his creditors. Thus it will be seen that in all the States the property of non-residents and foreign corporatio!s is liable to attachments at the suit of creditors, before judgment is rendered; likewise against domestic debtors when they have absconded from the State, or have fraudulently conveyed, or are about to convey, sell, assign or secrete their effects. In some few States, how' ever, even this condition is not essential before a writ of attachment will issue. In the States of Alabama, Massachusetts, Connecticut, Maine, New Hampshire, Vermont ancd Rhode Island, the creditor may have a writ of attachment against the property of the debtor at the first institution of a suit-and without any ground of fraud or fraudulent intentsrch property being held by the attachment until the termination of the suit, or until judgment: the plaintiff in such cases giving bond or security to indemnify the defendant for any loss or damage sustained, should the case be decided in favor of the latter. Generally, the property is liable only when actually levied upon; but in the State of Kentucky the property is liable fromn the time of delivery of the order to the sheriffi 335 MISSISSIPPI. Interest.-Ine legal rate of interest in Mississippi is six per cent. by the act passed in March, 1856. Damnages on Bills.-No damages are allowed for default in the payment of any bill of exchange drawn by any person or persons within the State on any person or persons in any other Ltate. On all domestic or inland bills [drawn on persons within the State], and protested bfr non-payment, five per cent. [See act of May 11, 1837.] Foreign Bills.-The damages on bills of exchange drawn on persons without the United States, returned under protest, are 10 per cent., with all incidental charges and lawful interest. Sight Bills.-Grace is not allowed on bills of exchange, drafts, etc, payable at sight. Collection of Debts.-An attachment against the estate, including real estate, goods, chattels, &c., ctfa debtcr, when it is shown that he has removed, or is about removing or absconding from the State, or privately conceals himself Attachment also lies against the property of non-resident descedents. It may be obtained before the debt is due for which it issues, when the creditor has ground to believe that the debtor will remove with his effects out of the State, or has removed. MISSOURI. Interest.-The legal rate of interest in Missouri is six per cent. when no other rate is agreed upon. Parties may agree in writing for any larger rate, not exceeding ten per cent. Parties may so contract as to compound the interest annually. Penaltybor Violation of the Usury Laws.-Forfeiture of the entire interest; but judgment to be rendered for the principal with ten per cent. interest, the interest to be appropriated to the school fund. The damages allowed on bills of exchange payable in other States or Territories of the United States returned under protest, are uniformly 10 per cent. On bills of exchange payable within the State, 4 per cent. On negotiable notes, if actually negotiated, 4 per cent. In these last two cases no damages can be recovered, if payment is made or tendered within twenty days after demand or notice of dishonour. Foreign Bills.-The damages allowed on boreign bills of exchange, protested for non-payment, are 20 per cent. The damages allowed in all of the above cases are in l'eu of interest, charges of protest and other expenses incurred previous to or at the time of giving notice of dishonour, or maturity of note or bill when notice is required; but after protest the interest will be allowed on the aggregate sum of principal and damages. Sight Bills.-A statute of 1853-4 provides, that on bills of exchange, payable at sight, grace shall not be allowed. Collection of Debts.-An attachment may be issued here when the debtor is not a resident of the State; or if a resident, when he absconds, absents or conceals himself, or is about to remove his property or fraudulently convey it, with a view to hinder or delay his creditors, or when the debt was contracted out of the State, and the debtor has secretly removed his effects into this State with intent to defrand. NEW HAMPSHIRE. Interest.-The legal rate of interest in New Hampshire is six per cent., and no more is allowed on contracts, direct or indirect. Penaltyfor Violation of the Usury Laws.-The person receiving interest at a higher than the legal rate, shall forfeit for every such offence three times the sum so received. Damages on Bills.-No statute in force in New Hampshire. IForeign Bills.- No statute in force in New Hampshire allowing damages on foreign bills returned under protest. Sight Bills.-No bill of exchange, negotiable promissory note, order or draft, except such as are payable on demand, shall be payable until days of grace have been allowed thereon, unless it appear in the instrument that it was the intention of the parties that days of grace should not be allowed. [Revised St. 389, ~ 10.] Collection of Debts.-In this State a writ of attachment may be issued upon the institution of any personal action; and will hold real and personal property, shares of stock in corporations, pews in churches, and the franchise of any corporation authorized to receive tolls, until the period of thirty days from the time of rendering the judgment. NEW JERSEY. Interest.-The legal rate of interest in New Jersey is six per cent., and no higher rate of interest is allowable on special contracts, except as provided in the following acts: The legislature of New Jersey passed the following special act in March, 1852, supplementary to an act against usury, approved April 10, 1846, the provisions of which act now apply also to the counties of Hudson, Bergen and Essex, and to the town of Paterson, in Passaic County: Be it enacted, etc., That upon all contracts hereafter made in the city of Jersey City, and in the township of Hoboken, in the county of Hudson, in this State, for the loan of or forbearance, or giving day of payment, for any money, wares, merchandise, goods or chatte s, it shall be lawful for any person to take the value of seven dollars for the forbearance of one hundred dollars 336 for a year, and after that rate fur a greater or lets sum, or for a longer or shorter period, any thing contained in the act, to which -this is a supplement, to the contrary notwithstanding. Provided, such contract be made by and between persons actually located in either said city or township, or by persons not residing in this State'. April 6, 1855. The latter proviso was amended, "Provided the contracting parties, or either of them reside in eitherof said places, or out of the State." The following changes have since been made so as to make it legal to charge 7 per cent interest: Act, February 21, 1860, Acquackanonnde, Passaic County. Act, February 6 1858, Bergen County. Act, Februa.yl8, 1858, Uni:n County. Act, Siacch 18, 1858, City of lRahway. Act, March 20, 1857, to all Savings Institutions in the State. By.Act of March 28,1862, the legislature authorized contracts at s ven per cent. i.lterest by parties residing in Middlesex County. Penaltyfor Violation of the Usury Laws.-The coatract is void, and the whole sum is forfeited. Damages on Bills of Exchange.-There is no statute in force in reference to damages on bills of exchange. Foreign Bills.-There is likewise no statute in force in reference to damages on protested foreign bills of exchange. Sight Bills.-That all bills of exchange or drafts drawn payable at sight, at any place within this State, other than those upon banks or banking associations, shal be deemed due and payable at the expiration of three days' grace after the same shall Le presented for acceptance. Collection of Debts.-An attachment may issue at the instance of a creditor (or, in his absence, of his agent or attorney), against the property of a debtor when the latter is about to abscond from the State, or is not a resident of the State, or is a foreign corporation. NEW YORK. Interest.-The legal rate of interest in New York is seven per cent., and no higher rate is allowed on special contracts. Penalty for Violation of the Usury Laws.-Forfeiture of the contract in civil actions. In criminal actions, a fine not exceeding one thousand dollars; or imprisonment notexceeding six months; or both. All bonds, bills, notes, assurances, conveyances, all other contracts or securities whatsoever (except bottomry and respondentia bonds and contracts), and all deposits of goods, or other things whatsoever, whereupon or whereby there shall be reserved or taken, or secured, or agreed to be reserved or taken, any greater sum, o: greater value for the loan or forbearance of any money, goods or other things in action than seven per cent. shall be void. (Rev. Stat. Vol. 1I., p. 182). For the purpose of calculating interest, a month shall be considered the twelfth part of a year, and as consisting of thirty days; and interest for any number of days less than a month shall be estimated by the proportion which such number of days shall bear to thirty. Damages on Bills.-The damages cn bills of exchange, negotiated in New York and payable in other State;, and returnned under protest for non-acceptance or non-payment, are as follows: 1. Maine, New Hampshire, Vermont, Massachusetts, IlRhode'Island, Connecticut, New Je;sey, Pennsylvania, Delaware, Maryland, Virginia, District of Columbia, and Ohio, 3 per cent. 2. North Carolina, South CarolinL-, Georgia, Kentucky, and Tennes ee, 5 per cent. 3. If drawn upon parties in any other State, 10 por cent. The following days, namely, the first day of.lanuary, commonly called New Year's day; the fourth day of July; the twenty-fifth day of Dece:rber, commonly called Christmas day; and any day appointed or recommended by the Governor of the State, or the President of the United States, as a day of fast or thanksgiving, shal, for all purposes whatsoever, as regards the presenting for payment or acceptance, and of the protesting-and giving notice of the dishonour of bills of exchange, bank checks and promissory notes, made after the passage of this act, be treated and considered as is the first day of the week, commonly called Sunday. [.849, ch. 261.) Foreign Bills.-The damages on foreign bills of exchange, returned under protest, are 10 per cent. Sight Bills. —Grace is not allowed by the banks of the city of New York and of ihe interior upon bills, drafts, checks, &c., payable at sight. Collection of Debts.-Any creditor to the amount of $25 may compel the assignment of their estates by debtors imprisoned on execution in civil causes for more than 60 years. If the debtor refuses to be examined, and to disclose his affairs, he is liable to be comlhitted to close confinement. If he refuses to render an account inventory, and make an assignment, hle will not be entitled to his discharge; though the officer having jurisdiction in the case is authorized to make the assignment for him. The proceedings and the effect of the discharge, when duly obtained, and the duties of the debtor, and the rights of the creditors, are essentially the same as in the case of proceedings with the assent of two-thirds of the creditors. Every insolvent debtor may also pietition the proper officers for leave voluntarily to assign his estate for the benefit of his creditors; and the same proceedings and checks are substantially prescribed as in other cases of insolvency. His discharge, obtained in such a case, exempts hill from imprisonment, as to debts due at the time of the assignment, or previously contracted, and as to liabiiliies incurred by making or indorsing any promissory note or bill of exchange. But thl discharge does not affect or impair any debt, demand, payment, or decree against the insolvent; and they remati good against his property acquired after the execution of the as. signmeant and the lien ofj Idggment and decree is not affected by the discharge. 337 NORTH CAROLINA. Interest.-The legal rate of interest in North Carolina is six per cent., and no higher rate is allowed on special contracts. Penalty for Violaion of the Usury Laws.-A forfeiture of the principal and interest; and if usurious interest is collected, a liability to pay double the amount of principal and interest paid-one half of the amount recovered for the use of the State, the other half for the claimant. Damages on Bills.-The damages on bills of exchange negotiated in JNorth Carolina, payable in other States, and returned under protest, are uniformly 3 per cent. Foreign Bills.-The damages on foreign bills of exchange returned under protest, are as follows: 1. Bills payable in any part of North America, except the Northwest Coast and the West Indies, 10 per cent. 2. Bills payable in Maderia, the Canaries, the Azores, Cape de Verde Islands, Europe and South America, 15 per cent. 3. Bills payable elsewhere, 20 per cent. Sight Bills.-By virtue of an act of the Legislature, passed in January, 1849, grace is allowed on bills at sight, unless there is a stipulation to the contrary. Prior to that-date the usage was, not to allow grace on such bills. Collection of Debts.-An attachment may issue on the complaint of a creditor, his agent, attorney or factor, against the property of a debtor when he has removed or is about-to remove, privately from the State, so that the ordinary process of lawwill not reach him. OHIO. Interest.-The law allows interest at six per cent. per annum on all money due, and no more. (The law allowing 10 per cent. on special contracts was repealed April 1st, 1859, but the repeal does not affect contracts entered into prior to this date.) Railroad Companies are authorized to borrow at the rate of 7 per cent. Penalties.-There are no penalties ordinarily for Usury. Contracts for greater rates are void as to the excess only; and if interest beyond six per cent. has been paid, the debtor has a right to have such excess applied as payment on the principal. An excess of interest taken BY BANKS invalidates the debt. Bills of Exchange.-" Damages on protested bills of exchange, drawn by a person or corporation in Ohio, are not recoverable on any contract entered into after the passage of this act." (Passed and took ellect April 4th 1859.) A check is not entitled to grace; but a check "p payable on a future specified day is a bill of exchange,ib and entitled to grace. (5 Ohio State Rep. 13.) " The usage of banks in any particular place, to regardfdrafts upon them, payable at a day certain after date, as checks, and not entitled to days of grace, is inadmissible to control the rules of law in relation to such paper."'-(lb.) Sight Bills.-By an act of the legislature, approved February 22nd, 1861, it is enacted that " no note, check, draft, bill of exchange, order or other negotiab'e or commercial instrument, payable at sight or on demand, or on presentation, shall be entitled to days of grace but shall be absolutely payable on presentment.' All other notes, drafts or bills of exchange shall be entitled to the usual days of grace. This act is in forco from its passage. No graca is allowed on bank checks payable at sight. A statute is in force providing that' all bonds, notes or bills, negotiable by this act, shall be entitled to three days' grace in the time of payment." Collection of Debts.-A creditor may procure, before or after the maturity of the claim, an attachment against the property of the debtor, where the latter is a foreign corporation or a nonresident; or, i a resident, when he has absconded, or left the county of his residence, or conceals himself, or is about to remove or convert his property, with a view to defraud his creditors. 2. When the debtor fraudulently contracted the debt or incurred the obligation. PENNSYLVANIA. Interest -The legal rate of interest in Pennsylvania is six per cent., except as provided in the fol.owing acts: SEO. 1. le it enacted, etc., That commission merchants and agents of parties not residing in this commonwealth be, and they are hereby authorized to enter into an agreement to retain the balances of money in their hands, and pay for the same a rate of interest not exceeding seven per centum per annum, and receive a rate of interest, not exceeding that amount, for any advance of money ma:.e by them on goods or merchandise consigned to them for sale or disposal: Provided, that this act shall only apply to moneys received from or held on account of a y advances made upo i goods consigned from importers, manufacturers, and others, living and transacting business in places beyond the limits ( f the State. Act of1857. In investments by building associations, in loans to members thereof, the premium given for i reference or priority of loan shall not be deemed usurious. Act of 8 Xay, 1855, ~ 1, P. L 519. Loans to railroads or canal companies, and bonds taken for a larger sum than the amount of money advalnced, not usurious. Act of July 26, 1842, ~ 11, P. L. 434 338 There is now no penalty for usury in l'ennsylvania, but the principal sun and legal interest can only be recovered. If a person voluntarily pays greater than legal interest, he may recover back the excess if sued for within six months. Act May 28th, 1858. Damages on Bills.-The damages on bills of exchange negotiated in Pennsylvania, payable in other States, and returned under protest, are as follows [May 13, 1850]: 1. Upper and Lower California, New Mexico, and Oregon, 10 per cent. 2. All other States, 5 per cent. Foreign Bills.-The damages on foreign bills of exchange, returned under protest, are as follows [May 13, 1850]: 1. Payable in China, India, or other parts of Asia, Africa, or Islands in the Pacific Ocean, 20 per cent. 2. Mexico, Spanish Main, West Indies, or other Atlantic islands, East Coast of South America, Great Britain, or other parts of Europe, 10 per cent. 3. West Coast of South America, 15 per cent. 4. All other parts of the world, 10 per cent. Sight Bills. —y a law passed May 21, 1857, all drafts and bills of exchange, payable at sight, "shall be and become due on presentation, without grace; and shall and may, if dishonored, be protested on and immediately after such presentation." Collection of Debts.-In this State the writ of domestic attachment issues against any debtor, being an inhabitant of the State, if he has absconded from his usual place of abod, or shall have remained absent from the State, or shall have confined himself in his own house, or concealed himself elsewhere, to defraud his creditors. No second attachment will be issued against the same property, unless the first be not executed or be dissolved by the court. A writ of attachment may be also issued against the property of a foreign corporation or a nonresident. In the latter cas?, the attachment inures to the benefit of the attaching creditor only. In the former case, it is for the benefit of creditors at large. RHODE ISLAND. Interest.-The legal rate of interest in Rhode Island is six per cent., and no higher rate is allowed on special contracts. Penalty for Violation of the Usury Laws..-Forfeiture of the excess taken above six per cent. Damages on Bills.-The damages on bills of exchange, payable in other States, and returned under protest, are uniformly 5 per cent. Foreign Bills.-The damages on foreign bills of exchange, returned under protest, are 10 per cent. Sight Bills.-By statute it is provided that "all bills of exchange drawn at sight, which shall be due and payable in this State, (Rhode Island), shall be deemed to bo due and payable on the day of presentation, without grace." Collection of Debts.-In this St-te a writ of attachment is first levied against the body of the defendant, and if he cannot be found, then against his goods and chatties. The property of foreign corporations and debtors is also liable to attachment at the suit of a creditor. SOUTH CAROLINA. Interest.-The legal rate of interest in South Carolina is seven per cent., and no higher rate is allowed on special contracts. Penalty for Violation of the Usury Laws.-Loss of all the interest taken. Damages on Bills.-The damages on bills of exchapge negotiated in South Carolina, payable in other States, and protested for non-payment, are uniformly 10 per cent. together with costs of protest. A bill drawn in South Carolina, payable in another State, is deemed a foreign bill, and damages may be claimed, although such bill be not actually returned after protest. Foreign Bills.-The damages on foreign bills of exchange, negotiated in South Carolina, are as follows: 1. On bills on any part of North America other than the United States and on the West In dies, 12', per cent. 2. On bills drawn on any other part of the world, 15 per cent. Sight Bills.-The statute of 1848 enacts that "bills of exchange, foreign or domestic, pay able at sight, shall be entitled to the same days of grace as now allowed by law on bills of exchange payable on time." By a statute passed in 1831, it is enacted that if money or other commodity be lent or advanced upon unlawful interest, the plaintiff'shall be allowed to recover the amount or value actually lent, but without interest or cost. By an act passed in 1839, it is enacted that a debtor by bond, note, or otherwise, about to leave the State, the debt not being yet due, may be sued and held to bail. The plaintiff must swear to the debt, and that he d d not know the debtor meant to remove at the time the contract was made. But the writ must be made returnable to the term next succeeding the maturity of the note, etc. Collection of Debts.-A writ of attachment will issue at the instance of the creditor wherever residing, against a debtor when he is a non-resident-or against a citizen who has been absent more than a year and a day; or when he absconds or is removing out of the county; or conceals himselfrso that the ordinary process of law cannot reach him. 339 TENNESSEE. Interest.-The legal rate af interest in Tennessee is six per cent., and no higher rate can be recovered at law. Contracts,at a greater rate of interest are vo'd as to the excess, and the lender is liable to a fine of $10 to $1,000. Penaltyfor Violation of the Usury Laws.-Liable to an indictment for misdemeanor. II convicted, to be fined a sum not less than the whole usurious interest taken and received, and no fine to be less than ten dollars. The borrower and his judgment cr:ditors may also, at any time within six years after usury paid, recover it back from the lender. Damages on Btlls.-The damages on bills of exchange negotiated in Tennessee, payable in other States, and prbtested for non-payment, are 3 per cent. Foreign Bills.-The damages allowed on foreign bills of-exchange, returned under protest, are as follows: 1. If upon any person out of the United States, and in North. America, bordering upon the Gulf of Mexico, or in any part of the West India islands, 15 per cent. 2. If payable in any other part of the world, 20 per cent. Sight Bills.-The legislature has passed an act providing that bills at sight shall sOT be entitled to days of grace. By law, all negotiable paper due July 4, December 25, January 1, or on any day appointed by the Governor as a day of Thanksgiving, or as a public holiday, shall be payable the day preceding either of those days. Collection of Debts.-When a debtor has removed, or is about to remove out of the county privately, or absconds or conceals himself, an attachment may be obtained against his property at the suit of a creditor, or his agent, attorney or factor. In the case of non-resident debtors, having any real or personal property in the State, it is required, in order to obtain an attachment,'to file a bill in chancery. TEXAS. Interests.-On all written contracts ascertaining the sums due, when no rate of interest is expressed, interest may be recovered at the rate of eight per cent, per annum. The parties to any written contract may stipulate ftr any rate of interest, not exceeding twelve per cent. per annum. Judgments bear eight per cent. interest, except where they are recovered on a contract in writing which stipulated for more, not exceeding twelve, in which case, they bear the rate contracted for. No interest on accounts, unless there be an express contract; but only eight per cent. can be recovered on a verbal contract. Contracts to pay interest on account will not 1b presumed from previous course of dealing. Penaltyfor Violation of the Usury Laws.-Forfeiture of all the interest paid or charged. Damages on Bills.-An act giving damages upon protested drafts and bills of exchange drawn upon persons living out of the limits of the State, passed December, 1851. SECTION 1. Be it enacted by the Legislature of the State of Texas, That the holder of any protested draft or bill of exchange, drawn within the limits of this State, upon any person or persons living beyond the limits of this Stat, shall, after having fixed the liability of the drawer or endorser of any such draft or bill of exchange, as provided for in the act of March 20, 1848, be entitled to recover and receive 10 per cent. on the amount of such dralt or-bill, as damages, together with interest and cost of suit thereon accruing. Provided, that the provisions of this act shall not be so construed as to embrace drafts drawn by persons other than merchants upon their agents or factors. Sight Bills.-By usage, grace is not generally allowed on bills, drafts. etc., payable at sight, but the rule is not invariable in this State. Bills of Exchange.-The general rule is that the holder of any bill of exchange may fix the liability of the drawer (where bill has been accepted} or any endorser, without protest or notice, by instituting suit against the acceptor beforo the first term of the district court to which suit can be brought, (or, if the amount do not exceed $100, exclusiveof interest, by instituting suit before Justice of the Peace, within sixty days) after the right of action accrues; or by instituting suit before the second term of said court, and showing good cause why the suit was not instituted before the first term. The drawer of any bill of exchange which shall not be accepted when presented for acceptance, shall be immediately liable for the payment thereof. Collection of Debts.-Original attachments are issued against the property of a debtor when he is not to be found in the county; and the property attached shall remain in custody until final judgment. Attachment will also lie when the defendant is a non-resident; or when a resident is about to remove out of the State, and whether the debt, be matured or not. VERMONT. Interest.-The legal rate of interest in Vermont is six per cent, and no higher rate of interest is allowed on special contracts, except upon railroad notes or bonds, which may bear seven per cent. Penalty for Violation of the Usury Laws.-The excess of interest received beyond six per cent. may be recovered by action of assumpsit. Damages on Bills of Exchange. —There is no statute in force in Vermont in reference tc damages on protested bills ot exchange. 340 Foreign Bills.-There is no statute in force in Vermont in reference to damages on protested foreign bills of exchange. Sight Bills.-Grace is not allowed on bills, drafts, checks, etc., payable at sight, or on bills and notes made and payable within the State. JR. S. xxiii. ~ 1.] Collection of Debts.-Writs of attachment may issue against the goods, chatties or estate of the defendant, or for want thereof, against his body, before or after the maturity of a claim. Actions against non-residents, or when the defendant has absconded from the State, may be commenced by trustee process. VIRGINIA. Interest.-The legal rate of interest in Virginia is six per cent., and no higher late is allowed on special contracts. Penaltyfor Violation of the Usury Laws.-All contracts for a greater rate of interest than six per cent. per annum are void. Damages on Bills.-The damages on bills of exchange negotiated in Virginia, payable in other States, and returned under protest, are uniformly 3 per cent. Foreign Bills.-The damages on foreign bills of exchange, returned under protest, are uniformly, 10 per cent. Sight Bills.-Grace is not allowed by statute or by usage on bills, etc., payable at sight. Collection of Debts.-The property cf the defendant, if a non-resident, or a resident who is abcut to remove himself or effects from the State, is liable to attachment. An attachment in such cases will hold before the claim is due and payable. WISCONSIN. After January, 1863, the legal rate of interest, by an act of the legislature, is seven per cent. An usurious contract is void, and the party loaning the money is liable to a penalty of three times the usury in addition. Penalty for Violation of the Usury Laws.-Whenever any person shall apply to any court in this State to be relieved in case of a usurious contract or security, or when any person shall set up the plea atf usury in any action or suit instituted against him, such person, to be entitled to such relief or the benefit of such plea, shall prove a tender of the principal sum of money or thing loaned, to the party entitled to receive the same. Act March 29, 1856. Damages on Bills of Exchange.-The damages on bills ot exchange, drawn or indorsed in Wisconsin, payable in either of the States adjoining that State, and protested for non-acceptance or non-payment, are 5 per cent. If drawn upon a person, or body politic or corporate, within either of the United States, and not adjoining to that State, the damages are 10 per cent. Foreign Bills.-The damages on bills of exchange, drawn or endorsed in Wisconsin, payable beyond the limits of the United States, and protested for non-acceptance or non-payment, are {R. S1849, p. 2C3], 5 per cent, together with, the current rate of exchange at the time of demand. Sight Bills.-On all bills of exchanre, payable at sight, or at a future day certain, grace shall be allowed [R. S. 1849, p. 263], but not on bills of exchange or notes payable on demand. Collection of Debts.-An attachment will hold against the property of a debtor when h-e has absconded, or is about to abscond from the State; or has fraudulentil assigned, disposed of or concealed his effects; or removed his property from the State; or when the defendant is a nRnresident or a foreign corporation. UPPER AND LOWER CANADA. Interest.-Six per cent. is the legal rate of interest, but any rate agreed upon can be recovered. Judgments bear six per centum perannum int'e:-est from the dato oftentry. Banbksare not allowe a higher rate than seven per cent. Corporations and associations authorized by law to borrow and lend money, unless specially allowed by some Act of Parliament, are prohibited from taking a ligher rata of interest than six per cent. lsurancC Companies, however) are authorized to take eight per cent. Bills of Exchange and Promissory Notes.-Three days of grace are allowed on all bills and notes payable within Upper or Lower Canada, except when drawn on demand. When the last day of grace falls on Sunday, or a legal holiday, it is payable the following day. Acceptances must be in writing. No person or corporation in Upper Canada can issceo notes for less than one dollar. Protest may be made, and the parties to the bill (r note notified on the same day the bill or note is dishonoured; but, in c.:se of non-payment in Upper Canada, not before three o'clock, p.m., and in Lower Canada any time after the forenoon of the lat ( ay of grace. Dishonoured inland bills or notes in Upper Canada, when protested, and in lower Canada without protest, bear interest at the rate of six per cent. from date of protest, or in Lower Canada f om maturity to time of l'ayment; but if interest is expressed to le payable from a particu'ar period, then from the time of such period to the time of payment. The damages allowe'd upon protested foreign bills drawn, sold or negotiated within Upper or Lower Canada, 341 if drawn upon any person in Europe, West Indies. or in any part of America not within the the Province or asy other British North American colony and not within the territory of the United t t tes of Amenca, Ten percent. upon the principal sum speeificd in the bill. If drawn in Lower Canad l on persons in Upper Canada, or if drawn in either Upper or Lower Canada on any person in any other of the Brilish North American colonies or United States of America four per cent. on the principal sum speciiled in the bill. The above for ign billh are also u )ject to six per c ntum per annum of interest on the amount for which the bill was drawn, to be reckoned from the date of protest to day of repayment, together with the current rate of exchange of the day when repayment is demanded, and the expenses of noting and protesting the Lill. Promissory notes made in Upper Canada, payable in the United States of America or Bridis' North American Colonies, not being Canada, and not otherwise or elsewhere, and protested, in addition to the principal sum, are liable to damages at the rate of four per cent. on such lrinc pal sum, and interest at the rate of six per centum per annum, to be reckoned from the day of protest to the day of repayment, together with the current rate of exchange of the day when repayment is demanded and the expenses of protesting the note. The Statute of Limitations bars the right of action on bills of exchange and premissory notes, in Upper Canada in six years, and in Lower Canada in five years. Collection of Debts. —Debts may be recovered in Lower Canada by actions at law, an d in Upper Canada by actions at law or suits in equity. Debtors may be arrested and held to bail in Upper Canada upon a i affidavit of the creditor, or of some other indiuidual, shewing that he has a cause cf action to the amount of $100, or upwards, and has suffered damages to that amount, and shows facts a:d circumstances to satisfy the judge that thr re is good and probable cause for believing that such person unless he is forthwith apprehended, is about to quit Canada with intent to defrau l his creditors generally or the deponent in particular. In Lower Canada debtors may be arrestedand held to bail upon affidavit of the Plaintiff, his bookkeeper, clerk, or legal attorney, that the Defendant is personally indebted to the Plaintiff for a sum amounting to or exceeding $40, and that deponent believes, upon gro nds set forth in affidvit, that Pe'e:dant is immediately abo':t to leave the Province with intent to defraud his creditors generally, or the Plaintiff in particular, a:d that such departure would deprive. the Plaintiff of his remedy agai:Lst the Defendant, or that the I)efendant has secreted or is about to secrete his property with such in'ent. A resident of Upper Canada, cannot in Lower Canada, be arrested at the suit cf any person residing in Upper Canada, unless, in addition to the ab:)ve, the Plaintiff or somo other person, makes oath before a Judge or s~me other authorized officer that the Defendant is immed atelv about to resort to some country or place without the limits of the Province, and hath not within Upper Canada any lands or other real estate out of which the Plaintiff can reasonably expect to be paid the amount of his debt. In Lower Canada any debtor impris)ned or held to bail, in a cause wherein judgment for a sum of $80 or upwards is rendered, is obliged to make a statement under oath, and a declaration of abandonment of all his property, for the benefit of his creditors, according to the rules, and subject to the penalvy of imprisonment in certain cases. When such statement and declaration are made without s'aud, the debtor is exempt from arrest and imprisonment by reason of any cause of action existing before the making of such statement and declaration. In Upper Canada the property credits and ef'ects of an absconding debtor, that is to say,-any person resident in Up-er Canada indebted to any other person departing from Upper Canada, with inte-.t to defraud his creditors, and at the time of his so departing is possessed toThis own use and benefit of any real or personal effects therein-may be seized by a writ of attachment, provided the debt exceeds $100. Judgment debtors may be examined as to what debts are due to them, and such debts may be attached upon affidavit, showint that a judgment was recovered and is still unsatisfied. In Lower Canada a writ of attachment may issue before judgment upon proof on oath that Defendant is indebted to the Plaintiff in a sum exceeding $40 and is about to secrete the same or doth abscond or doth suddenly intend to depart from Lower Canada with i tent to defraud his creditors, and that the deponent believes without the benefit of such attachment the Plaintiff would lose his debt or sustain damage. A trader's goods may be at' ached in Lower Canada, (ad if the suit be brought in the Superior Court he may be arrested,) if, in addition to the allegation, the Defendant is indebted to thoPlaintiffin the sum required, it is all ged, that he is a trader, that he is notoriously insolvent, and has refused to compromise or arrange with his creditors but still continues his trade. The estate of insolvent debtors may be also attached by creditors for sums of rot less than $200 in both Upper and Lower Canada. There is no homestead law in either Province, but certain articles are protected from seizure under execution. In all matters not specially provided lor by the Provincial Legislature, recourse is had to the law of England. 2 3 AJ1UTT A2:R3BO01 AND TELEGRAPHIC INSTITUTE, CORNER MAIN AND WASHINGTON STREETS, ANN ARBOR, MICHIGAN. - -vow - The most Practical & Thorough Business College IN THE N ORTI-WEST. BRANCHES AT ADRIAN, GRAND RAPIDS, JACKSON, & STURGIS, MICH. BOOK-KEEPING, PENMANSHIP, COMMERCIAL LAW, ARITHMETIC, AND TELEGRAPHING TAUGHT BY EXPERIENCED TEACHERS. A. 0. PARSONS, A.M., - - PRESIDENT. W. F. PARSONS, Resident Principal. JUDGE T. M. COOLEY,'Professor of Law in University of Michigan, Lecturer on Commercial Law. TRACY BARNUM, Professor of Business and Ornamental Penmanship. A. PARSONS, Jr., City Operator, Sup't of Telegraphic Department.