aub &!)oms0n's Sctie0. |i (I r Ii A ; T I C A L ' *. \\ ARITHMETIC, f- I KriC ^ TKITINO /iis iDUC" ALSO, JLLTISTRATINO THE -4^^- PRINCIPLES. VTION. FOR SCHOOLS KD ACADE? By JAMES r,. L, JL AUTHOR OF MKVTAl AKnji.HKTir ; t*f< :W-1: nia-aw* AR1T!IMK*' 1 r> NSW YORK: MARK H. NEWMAN & CO., 199 BROADWAY. : o i :; N A T i : w . n . MOORE A c o . CHICAGO: CHICKS, BROSS L ^ V\ . i II. FkAOTICAL ARITHMETIC; Unitih-g the. Inductive with f .he Synthetic mode of Instruction ; also illustrating-^he prin- ciples of CANCELATION. The design of this work is-to make the pupil thoroughly acquainted with the reason of every operation which he is required to perform. It abounds^ in examples, and is emi'nently practical. 1*m.E^TJp\ PRACTICAL 'ARITHMETIC ; Containing tlie" answers^wiin rtumerous suggestions, &c. IV. HIGHER ARITHMETIC; or, the. Science and Applica- tion of Numbers ; For advanced classes. This work is complete in itself^ commencing with^the fundamental rules, and extending to * ^ ' v KEY TO .HIGHER ARITHMETIC ; Containing the ^ highest deparflnent o answers^tb all the examples, with many suggestions, &c. t VI. ELEMENTS OF ALGEBRA >*JBeinffa SjchooAdkion of Day's ^ge AMgebra/>,T.his' wo?k ^ designed to be a ItMd and " 'easy transition frq/n the study of Arithmetic ro^the higher branches .of Mathtjiiatics. The number of examples is much increased ; and Jjj,e worHfcis every way adapted to Schools and Academies. \ VII. KEY TO ELEMENTS OF ALGEBRA ; Containing the answers, the solution of the more difficult problems, &c. VIII. ELEMENTS OF GEOMETRY; Being our abridg- ment of Legendre's Geometry; with practical notes and illus- trations. IX. ELEMENTS OF TRIGONOMETRY, MENSURA- TION. AND LOGARITHMS. X. ELEMENTS OF SURVEYING ;- Adapted both to the wants of the learner and the practical Surveyor. (Published soon.) Entered according to Act of Congress, in the year 1845, BY JEREMIAH DAY and JAMES B. Tuoneaw, in the Clerk's Office of the District Court of Connecticut PREFACE. IT has been well said, that "whoever shortens the road to knowledge, lengthens life." The value of a knowledge of Arithmetic is too generally appreciated to require comment. When properly studied, two impor- tant ends are attained, viz : discipline of mind, and fa- cility in the application of numbers to business calcula- tions. Neither of these results can be secured, unless the pupil thoroughly understands the principle of every operation he performs. There is -no uncertainty in the conclusions of mathematics; there should be no guess- work in its operations. What then is the cause of so much groping and fruitless effort in this department of ed- ucation. Why this aimless, mechanical " ciphering," that is so prevalent in our schools ? The present work was undertaken, and is now offered to the public, with the hope of contributing something toward the removal of these inveterate evils. Its plan is the following : 1. To lead the pupil to a knowledge of each rule by induction ; that is, by the examination and solution of a large number of practical examples which involve the principles of the rule. 2. The operation is then defined, each principle is ana- lyzed separately, and illustrated by other examples. 3. The general rule is now deduced, and put in its proper place, both for convenient reference and review ; thus combining the inductive and synthetic modes of in- struction. 4. The general rule is followed by copious examples for practice, which are drawn from the various departments of business, and are calculated both to call into exercise the different principles of the rule, and to prepare th learner for the active duties of life. IT PREFACE. It is believed that much of this guess-work in " ing," and its concomitant habits of listlessness and cuity of .mind, have arisen from the use, at first, of abstrad numbers and intricate questions, requiring combinations above the capacity of children. Taking his slate and pen- cil, the pupil sits down to the solution of his problem, but soon finds himself involved in an impenetrable maze. He anxiously asks for light, and is directed " to learn the rule." He does it to the letter, but his mind is still in the dark By puzzling and repeated trials, he perhaps finds that certain multiplications and~ divisions produce the answer in the book ; but as to the reasons of the process, he is to- tally ignorant. To require a pupil to learn and understand the rule, before he is permitted to see its principles illus- trated by simple practical examples, places him in the condition of the boy, whose mother charged him never to go into the water till he had learned to swim. These embarrassments are believed to be unnecessary, and are attempted to be removed in the following manner : 1. The examples at the commencement of each rule are all practical, and are adapted to illustrate the particular principle under consideration. Every teacher can bear testimony, that children reason upon practical questions with far greater facility and accuracy than they do upon abstract numbers. 2. The numbers contained in the examples are at first small, so that the learner can solve the question mental ly, and understand the reason of each step in the opera- tion. 3. As the pupil becomes familiar with the more simple combinations, the numbers gradually increase, till the slate becomes necessary for the solution, and its proper use is then explained. 4. Frequent mental exercises are interwoven with ex- ercises upon the slate, for the purpose of strengthening the habit of analyzing and reasoning,&n& thus enable the learn- er to comprehend and solve the more intricate problems. 5. In the arrangement of subjects it has been a cardi- nal point to follow the natural order of the science. No principle is used in the explanation of anpther, until it PREFACE. has itself been demonstrated or explained. Common frac- tions, therefore, are placed immediately after division, for two reasons. First, they arise, from division, and are inseparably connected with it. Second, in Reduction, Compound Addition, &c. it is frequently necessary to use fractions ; consequently fractions must be understood, before it is possible to understand the Compound rules. For the same reason, Federal Money, which is based upon the decimal notation, is placed after Decimal Frac- tions. Interest, Insurance, Commission, Stocks, Duties, &c., are also placed after Percentage, upon whose prin- ciples they are based. 6. In preparing the tables of Weights and Measures, particular pains have been taken to ascertain those that are in present use in our country, and to give the If gal standard of each, as adopted by the General Government.* It is well known that a great difference of weights and measures formerly existed in different parts of the coun- try. More than ten years have elapsed since the Gov ernment wisely undertook to remedy these evils, by adopting uniform standards for the custom-houses and other purposes ; and yet not a single author of arithme- tic, so far as we know, has given these standards to the public. 7. The subject of Analysis is deemed so essential to a thorough knowledge of arithmetic and to business calcu- lations, that a whole section is devoted to its develop- ment and application. The principles of Cancelation have been illustrated, and its most important applications pointed out, in their proper places. The Square and * In the year 1836, Congress directed the Secretary of the Treas- ury to cause to be delivered to the Governor of each State in the Union, or to such person as he should appoint, a complete set of all the Weights and Measures adopted as standards, for the use of the States respectively ; to the end that a uniform standard of Weights and Measures may be established throughout the United States. Most of the States have already received them ; and may we not hope that every member of this great Union will promptly and cor- dially unite in the accomplishment of an object so conducive botK * ttjdividual and publir ; good. VI PREFACE. Cube Roots are illustrated by geometrical figures and cu- bical blocks. Such is a brief outline of the present work. It is not designed to be a book of puzzles, or mathematical anom- alies ; but to present the elements of practical arithmetic in a lucid and systematic manner. It embraces, in a word, all 'he principles and rules which the business man ever has occasion to use, and is particularly adapted to pre- cede the study of Algebra and the higher branches of mathematics. With what success the plan has been executed re- mains for teachers and practical educators to decide. If it should be found to shorten the road to a thorough knovvL edge of arithmetic in any degree, its highest aims will be accomplished. J. B. THOMSON, New Haven, Oct. 3, 1845. SUGGESTIONS ON THE MODE OF TEACHING ARITHMETIC. I. QUALIFICATIONS. The chief qualifications requisite in teaching Arithmetic, as well as other branches are the following : 1. A thorough knowledge of the subject. 2. A love for the employment. 3. An aptitude to teach. These are indispensable to success. II. CLASSIFICATION. Arithmetic, as well as reading, grammar, &c., bhould be taught in classes. 1. This method saves much time, and thus enables the teacher to devote more attention to oral illustrations. "2. The action of mind upon mind, is a powerful stimulant to exer- tion, and can not fail to create a zest for the study. 3. The mode of analyzing and reasoning of one scholar, will often suggest new ideas to the others in the cla-ns. 4. In the classification, those should be put together who possess as nearly equal capacities and attainments as possible. If any of the class leurn quicker than others, they should be allowed to take up an extra study, or be furnished with additional examples to solve, so that the whole class may advance together. 5. The number in a class, if practicable, should not be less than six, nor over twelve or fifteen. If the number is less, the recitation is apt to be deficient in animation ; if greater, the turn to recite does not come round sufficiently often to keep up the interest. III. APPARATUS. The Black-board and Numerical frame are as indispensable to the teacher, as tables and cutlery are to the house- keeper. Not a recitation passes without use for the black-board. If a principle is to be demonstrated or an operation explained, it should be done upon the black-board, so that all may see and understand it at once. To illustrate the increase of numbers, the process of adding, sub- tracting, multiplying, dividing, &c., the Numerical Frame furnishes one of the most simple and convenient methods ever invented.* IV. RECITATIONS. The Jirst object in a recitation, is to secure the attention of the class. This is done chiefly by throwing life and variety into the exercise. Children loathe dullness, while animation and variety are their delight. 2. The teacher should not be too much confined to his text-book, nor depend upon it wholly for illustrations. * Every one who ciphers, will of course have a slate. Indeed, it is desira- ble that every scholar in school, even to the very youngest, should be fur- nished with a small slate, so that when the little fellows have learned their lessons, they may busy themselves in writing and drawing various familiar objects. Idleness in school is the parent of mischief, and employment is the best antidote against disobedience. Geometrical diagrams and solids are also highly useful in illustrating manf points in arithmetic, and no school should hp without them VI PREFACE. Cube Roots are illustrated by geometrical figures and cu- bical blocks. Such is a brief outline of the present work. It is not designed to be a book of puzzles, or mathematical anom- alies ; but to present the elements of practical arithmetic in a lucid and systematic manner. It embraces, in a word, all he principles and rules which the business man ever has occasion to use, and is particularly adapted to pre- cede the study of Algebra and the higher branches of mathematics. With what success the plan has been executed re- mains for teachers and practical educators to decide. If it should be found to shorten the road to a thorough knovvL edge of arithmetic in any degree, its highest aims will be accomplished. J. B. THOMSON, New Haven, Oct. 3, 1845. SUGGESTIONS ON THE MODE OF TEACHING ARITHMETIC. I. QUALIFICATIONS. The chief qualifications requisite in teaching Arithmetic, as well as other branches are the following : 1. A thorough knowledge of the subject. 2. A love for the employment. 3. An aptitude to teach. These are indispensable to success. II. CLASSF ^CATION. Arithmetic, as well as reading, grammar, &c., should be taught in classes. 1. This method saves much time, and thus enables the teacher to devote more attention to oral illustrations. 2. The action of mind upon mind, is a powerful stimulant to exer- tion, and can not fail to create a zest for the study. 3. The mode of analyzing and reasoning of one scholar, will often suggest new ideas to the others in the cla-ss. 4. In the classification, those should be put together who possess as nearly equal capacities and attainments as possible. If any of the class leurn quicker than others, they should be allowed to take up an extra study, or be furnished with additional examples to solve, so that the whole class may advance together. 5. The number in a class, if practicable, should not be less than six, nor over twelve or fifteen. If the number is less, the recitation is apt to be deficient in animation ; if greater, the turn to recite does not come round sufficiently often to keep up the interest. III. APPARATUS. The Black-board and Numerical Frame are as indispensable to the teacher, as tables and cutlery are to the house- keeper. Not a recitation passes without use for the black-board. If a principle is to be demonstrated or an operation explained, it should be done upon the black-board, so that ail may see and understand it at once. To illustrate the increase of numbers, the process of adding, sub- tracting, multiplying, dividing, &c., the Numerical Frame furnishes one of the most simple and convenient methods ever invented.* IV. RECITATIONS. The first object in a recitation, is to secure the attention of the class. This is done chiefly by throwing life and variety into the exercise. Children loathe dullness, while animation and variety are their delight. 2. The teacher should not be too much confined to his text-book, nor depend upon it wholly for illustrations. * Every one who ciphers, will of course have a slate. Indeed, it is desira- ble that every scholar in school, even to the very youngest, should be fur- nished with a small slate, so that when the little fellows have learned their lessons, they may busy themselves in writing and drawing various familiar objects. Illencss in school is the parent of mischief, and employment is the best antidote against disobedience. Geometrical diagrams and solids are also highly useful in illustrating many points in arithmetic, and no school should Iw without them Vlll SUGGEST JONS. 3. Every example should be analyzed, the " why and wherefcre ' of every step in the solution should be required, till each member ot the class becomes perfectly familiar with the process of reasoning and analysis. 4. To ascertain whether each pupil has the right answer to all the examples, it is an excellent method to name a question, then call upon some one to give the answer, and before deciding whether it ia right or wrong, ask how many in the class agree with it. The an- /wer they give by raising their hand, will show at once how many are right. The explanation of the process may now be made. Another method is to let the class exchange slates with each other, and when an answer is decided to be right or wrong, let every one mark it accordingly. After the slates are returned to their owners, each one will correct his errors. V. THOROUGHNESS. The motto of every teacher should be thor- oughness. Without it, the great ends of the study are defeated. 1. In securing this object, much advantage is derived from fre- qutnt reviews. 2. Not a recitation should pass without practical exercises upon the black-board or slates, besides the lesson assigned. 3. After the class have solved the examples under a rule, each one should be required to give an accurate account of its principles with the reason for each step, either in his own language or that of the author. 4. Mental Exercises in arithmetic, either by classes or the whole school together, are exceedingly useful in making ready and accurate arithmeticians, and should be frequently practised. VI. SELF-RELIANCE. The habit of self-reliance in study, is confess- edly invaluable. Its power is proverbial ; I had almost said, omnipo- tent. " Where there is a will, there is a way." 1. To acquire this habit, the pupil, like a child learning to walk, must be taught to depend upon himself. Hence, 2. When assistance in solving an example is required, it should be given indirectly ; not by taking the slate and performing the exam- ple for him, but by explaining the meaning of the question, or illus- trating the principle on which the operation depends, by supposing a more familiar case. Thus the pupil will be able to solve the question himself, and his eye will sparkle with the consciousness of victory. 3. He must learn to perform examples independent of the answer, without seeing or knowing what it is. Without this attainment the pupil receives but little or no discipline from the study, and is unfit to be trusted with business calculations. What though he comes to the recitation with an occasional wrong answer ; it were better to solve ono question understandingly and alone, than to copy a score of answers from the book. What would the study of mental arithmetic be worth, if the pupil had the answers before him 1 What is a young man good for in the counting-room, who has never learned to perform arithmeti- cal operations alone, but is obliged to look to the answer to know what figure to place in the quotient, or what number to place for the third term in proportion, as is too often the case in school ciphering 7 CONTENTS. SECTION I. Page- Suggestions on the mode of teaching Arithmetic, . . , Numbers illustrated and defined, 13 Notation, .......... 14 Roman Notation, ......... 14 Aral)i" Notation, ......... 15 Numeration, ......... 19 Exercises in Numeration, ....... 2t Exercises in Notation, ....... 21 SECTION II. ADDITION, Mental Exercises, 23 Addition Table, 24 Exercises for the Slate, 28 Illustration of the principle of carrying, 32 Proof of Addition, 34 General Rule, 34 Examples for practice, 35 SECTION III. SUBTRACTION, Mental Exercises, .40 Subtraction Table, 41 Exercises for the Slate, 44 Illustration of the principle of borrowing, .... 46 Proof of Subtraction, 49 General Rule, 49 Examples for practice,. 50 SECTION IV. MULTIPLICATION, Mental Exercises, 54 Exercises for the Slate, ... ... GO Illustration of carrying in Multiplication, .... 62 General Rule, 66 Examples for practice, 66 Contractions in Multiplication, 68 X CONTENTS. SECTION V. DIVISION, Mental Exercises, 73 Exercises for the Slate, 78 Short Division, 82 Long Division, 83 Contractions in Division, 87 General principles in Division, ....... 90 CANCELATION illustrated, 93 Greatest Common Divisor, 95 Least Common Multiple, 97 SECTION VI. FRACTIONS, Mental Exercises, . . . . . . 100 Reduction of Fractions, 108 Cancelation applied to reducing Compound Fractious, : 112 Common Denominator, 113 Least Common Denominator, . . .. . , . 114 Addition of Fractions, 115 Subtraction of Fractions, 117 Multiplication of Fractions, 120 Cancelation applied to Multiplication of Fractions, . . 125 Division of Fractions, 128 Cancelation applied to Division of Fractions, . . . 132 Cancelation applied 10 Multiplication of Complex Fractions, 134 SECTION VII. COMPOUND NUMBERS, 136 Tables in Compound Numbers, . 137 The standard unit of Weight of the United States, . . 138 The Avoirdupois Pound of the United States and Great Britain, 139 The standard unit of Length of the United States, . . 140 The standard unit of Liquid Measure of the United States, . 144 The standard unit of Dry Measure of the United States, . 146 REDUCTION, ]49 To find the area of surfaces, 154 To find the solidity of boxes, wood, &c., .... 155 Compound numbers reduced to fractions, . . . . 157 Fractional Compound numbers reduced to whole numbers, . 159 Compound Addition, 160 Compound Subtraction, 163 To find the difference between two Dates, .... 165 Compound Multiplication, 166 Compound Division, 169 OflTENTS. XI SECTION VIII. DECIMAL FRACTIONS, . 171 Exercises in reading and writing Decimals, . . . 174 Addition of Decimals, . . . . . . . 175 Subtraction of Decimals, 177 Multiplication of Decimals, 179 Division of Decimals, . 181 Reduction of Decimals, 184 Decimals reduced to Common Fractions, . Common Fractions reduced to Decimals, '. . . . 185 Circulating Decimals, 186 Compound numbers reduced to Decimals of higher denom., Decimal Compound numbers reduced to whole numbers, FEDERAL MONEY, Reduction of Federal Money, 191 Addition of Federal Money, . . . . . 193 Subtraction of Federal Money, 194 Multiplication of Federal Money, . . . . . 195 When the price of one article, &c., is given, to find the cost of any number of articles, Division of Federal Money 199 When the number of articles, &c., and the cost of the whole are given, to find the price of one, . 199 When the price of one article and the cost of the whole are given, to find the number of articles, 200 Applications of Federal Money to Bills, &c., . . . 202 SECTION IX. PERCENTAGE, 204 Application of Percentage, . 210 Commission, Brokerage and Stocks, 211 INTEREST, 214 General method for computing Interest, .... 220 Second method " .... 223 Partial payments, Rule adopted by the United States, . 227 Connecticut and Vermont Rules, 229 Problems in Interest, 231 Compound Interest, 236 DISCOUNT, 239 Bank Discount, 242 Insurance, 244 Profit and Loss, 247 DUTIES, 256 Specific Duties, 257 Ad valorem Duties, 258 Taxes, assessment of, . 260 Xll CONTENTS. SECTION X. PROPERTIES OP NUMBERS. Proof of Multiplication and Division from the property of 9, Axioms, Deductions from the Fundametal Rules, SECTION XI. Pa** 264 267 268 269 ANALYSIS, Analyticsolut Ditto Ditto Ditto Ditto Ditto Ditto Ditto Ditto ons of qu c I potions in Simple Proportion, Barter, Practice, Partnership, Bankruptcy, General Average, Alligation, Comp'd Proportion, Position, 273 Ex. 1-50, 276 Ex. 51-60, 28-2 Ex. 61 -75, 283 Ex. 76-82, 284 Ex. 83-88, 285 Ex. 89-9 1, 286 Ex. 9-2-1 00, 287 Ex. 101-106, 288 Ex. 107-120. 289 SECTION XII. RATIO, PROPORTION, . . Simple Proportion, or Rule of Three, Compound Proportion, or Double Rule of Three, 290 294 294 307 SECTION XIII. Duodecimals, Multiplication of Duodecimals, SECTION XIV. Involution, Evolution, Square Root, Applications of Square Root, Cube Root, Demonstration of Cube Root by Cubical Blocks, SECTION XV. Equation of Payments, Partnership or Fellowship, Exchange of Currencies, MENSURATION, . Miscellaneous Examples, 306 307 310 313 317 322 324 325 328 331 334 338 344 ARITHMETIC, SECTION I. NOTATION AND NUMERATION. ; ART. 1 Any single thing, as a peach, a rose, a book, is called a unit, or one / if another single thing is put with it, the collection is called two ; if another still, it is called three if another, four if another, Jive, &c. The terms, one, two, three, &c.> by which we express how many single things or units are under consideration, are the names of numbers. Hence, 2. NUMBER signifies a unit, or a collection of units. OES. Numbers have various properties and relations, and are ap- plied to various calculations in the practical concerns of life. These properties and applications are formed into a system, called Arith- metic. Hence, 3* ARITHMETIC is t/ie science of numbers. Numbers are expressed by words, by letters, and by figures. Note. The questions on the observations may be omitted, by be- ginners, till review, if deemed advisable by the Teacher. QUEST. 1. What is a single thing called ? If another ^s put with it, what is the collection called ? If another, what ? What are the terms one, two, three, &c. ? 2. What does number signify ? Obs. To what are numbers applied ? 3. What is Arithmetic ? Ho\\ are numbers ex- pressed ? 14 NOTATION. [SECT. I. NOTATION. 4 The art of expressing numbers by letters or figures, is called NOTATION. There are two methods of notation in use, the Roman and the Arabic. 5 The Roman method employs seven capital letters, viz : I, V 5 X, L, C, D, M. When standing alone, the letter I denotes one', V,five; X, fe#; L, fifty ; C, one hundred; D, five hundred ; M, one thousand. To express the in- tervening numbers from one to a thousand, or any number larger than a thousand, we resort to repetitions and various combinations of these letters. The method of doing this will be easily learned from the following TABLE. I denotes one. XXX denote thirty. II " two. XL " forty. Ill < three. L " fifty. IV ' four. LX " sixty. V ' five. LXX " seventy. VI ( six. LXXX " eighty. VII ' seven. XC " ninety. VIII " eight. C " one hundred. IX ' nine. CI one hundred and ona X ' ten. CX ' one hundred and ten, XI ' eleven. CC ' two hundred. XII xln ' twelve, thirteen. CCC three hundred. CCCC ' four hundred. XIV fourteen. D " five hundred. XV ' fifteen. DC six hundred. XVI ' sixteen. DCC seven hundred. XVII ' seventeen. DCCC eight hundred. XVIII ; eighteen. DCCCC ' nine hundred. XIX ' nineteen. M ' one thousand. XX ' twenty. MM ' two thousand. XXI XXII ' twenty-one. ' twenty-two, &c. MDCCCXLV, one thousand eight hundred and forty-five. QUEST.-4. What is notation ? How many methods are there in use ? What are they t 5. What does the Roman method employ ? Wha . does each of these letters denote when standing alone ? How are thti intervening numbers from one to a thousand expressed? How denote Two? Four? Six? Eight? Nine? Fourteen ? Sixteen? Nineteen? Twenty-four ? Twenty-eight ? What does XL denote ? LX ? XC ? CX ? N. B. Questions on this table should be varied, and continued by the Readier til! the etass becomes perfectly familiar witU it. ARTS. 4 7.] K^TATION. 15 OBS. 1. The learner will perceive from the Table above, that every time a letter is repeated, its value is repeated. Thus I, standing alone, denotes one ; II, two ones or two, &c. So X denotes ten; XX, twenty, &c. 2. When two letters of different value are joined together, if the less is placed before the greater, the value of the greater is dimin- ished; if placed after the greater, the value of the greater is increased, Thus, V denotes fiv ; but IV denotes only four ; and VI, six. So X denotes ten ; IX, nine ; XI, eleven. 3. A line or bar ( ) placed over a_letter, increases its value a thousand times. Thus, V denotes five, V denotes five thousand ; X, ten ; X, ten thousand. 4. This method of expressing numbers was invented by the Ro- mans ; hence it is called the Roman Notation. It is now seldom used, except to denote chapters, sections, and other divisions of books and discourses. 6. The common method of expressing numbers is by the Arabic Notation. The Arabic method employs the following ten characters or figures, viz : 1234567 8 90 one, two, three, four, five, six, seven, eight, nine, zero. The first nine are called significant figures, because each one always has a value, or denotes some number. They are also called digits, from the Latin word digitus, tvhich signifies a finger. The last one is called a cipher, or naught, because when standing alone it has no value, or signifies nothing. OBS. It must not be inferred, however, that the cipher is useless ; for when placed on the right of any of the significant figures, it increases I heir value. It may therefore be regarded as an auxiliary digit, whoso office, it will be seen hereafter, is as important as that of any other figure in the system. Nrte. The pupil must be able to distinguish and to write these characters, before he can make any progress in Arithmetic. 7 It will be seen that nine is the greatest number that QUEST. Ols. What is the effect of repeating a letter? If a let: . is placed before another of greater value, what is the effect ? If placed after, whai ? When a letter has a line placed over it, how is its valu-- affected ? Why is this method of notation called Roman ? To what use is it chiefly applied ? 6. How are numbers commonly expressed ? How many characters does this method employ? What are their names ? What are the first nine called ? Why ? What else are they called? What is the last one called? Why? Obs. In the (ipher useless ? What may U be regarded ? 16 NOTATION. [SECT i can be expressed by any single figure. All numbers larger than nine are expressed by combining together two or more of the ten characters just explained. To express ten for example, we combine the 1 and 0, thus 10 ; eleven is expressed by two Is, thus 11; twelve, thus 12; two tens, or twenty, thus 20 ; one hundred, thus 100, &c, The numbers from one to a thousand are expressed ir the following manner : 1, one. 2, two. 3, three. 4, four. 5, five. 6, six. 7, seven. 8, eight. 9, nine. 10, ten. 11, eleven. 12, twelve. 13, thirteen. 14, fourteen. 15, fifteen. 16 sixteen. 17, seventeen. 18, eighteen. 19, nineteen. 20, twenty. 21, twenty-one, &c. 30, thirty. 31, thirty-one, &c. 40, forty. 41, forty-one, &c. 50, fifty. 51, fifty-one, &c. CO, sixty. 61, sixty-one, &c. 70, seventy. 71, seventy-one, &c. 80, eighty. 81, eighty-one, &c. 90, ninety. 91, ninety-one, &c. 100, one hundred. 101, one hundred and one. 102, one hundred and two. 103, one hundred and three. 110, one hundred and ten. 111, one hundred and eleven. 112, one hundred and twelve. 120, one hundred and twenty. 130, one hundred and thirty. 140, one hundred and forty. 150, one hundred and fifty. 160, one hundred and sixty. 170, one hundred and seventy. 180, one hundred and eighty. 190, one hundred 'and ninety 200, two hundred. 300, three hundred. 400, four hundred. 500, five hundred. 600, six hundred. 700, seven hundred. 800, eight hundred. 900, nine hundred. 990, nine hundred and ninety. 991, nine hundred and ninety- one 992, nine hundred and ninety-two 998, nine hundred & ninety-eight 999, nine hundred & ninety-nine. 1000, one thousand. QUEST. 7. What is the greatest number that can be expressed by one figure ? How are larger numbers expressed ? How express ten t Eleven ? Twelve ? Twenty \ What is the greatest number that car. be expressed by two figures ? How express a hundred ? One hundred and ten ? One hundred and forty-five ? Five hundred and sixty-eight 1 What is the greatest number that can be expressed by three figures '. How express a thousand ? ART. 8.J NOTATION. 17 Note. Questions on the foregoing table should be continued till the rlaw becomes familiar with the mode of expressing any number from 1 tc 1000. They may be answered orally ; but the best way is to let the pupil write the figures denoting the number upon the black- board, and at the same time pronounce the answer audibly. OBS. 1. The terms thirteen, fourteen, fifteen, &c., are obviously derived from three and ten, four and ten, five and ten, which by contraction become thirteen, fourteen, fifteen, &c., and are therefore significant of the numbers which they denote. The terms eleven and twelve, are generally regarded as primitive words ; at all events, there is no perceptible analogy between them and the numbers which they represent. Had the terms onetcen and twoteen been adopted in their stead, the names would then have been significant of the num- bers one and ten, two and ten ; and their etymology would have been similar to that of the succeeding terms. 2. The terms twenty, thirty, forty, &c., were formed from two tens, three tens, four tens, which were contracted into twenty, thirty forty, &c. 3. The terms twenty-one, twenty-two, twenty-three, &c., are com- pounded of twenty and one, twenty and two, &c. All the other numbers as far as ninety-nine are formed in a similar manner. 4. The terms hundred and thousand are primitive words, and bear no analogy to the numbers which they denote. The numbers be- tween a hundred and a thousand are expressed by a repetition of the number below a hundred. Thus we say, one hundred and one, one hundred and two, one hundred and three, &c. 8. It will be perceived from the foregoing- table, that the figures standing in different places have different val- ues. Thus the digits, 1, 2, 3, &c., standing alone or in the right hand place, respectively denote units orj)nes. But when they stand in the second place, they express tens ; thus the 1 in 10, 12, 15, &c., expresses tensor ten ones; that is, its value is ten times as much as when it stands in the first or right hand place, and it is called a unit of the sec- ond order. So the other digits, 2, 3, 4, &c., standing QUEST. Obs. From what is the term thirteen formed ? Fourteen ? Sixteen ? Eighteen ? What is said of the terms eleven and twelve ? How are the terms twenty, thirty, &c., formed? What is said of the terms hundred, and thousand ? How are the numbers between a hundred and a thousand expressed ? 8. Does the same figure always express the Fame value ? What does each of the digits, 1, 2, 3, &c., denote, when standing in the right hand place ? What does the figure 1 Jenote when It stands in the second place 1 What is its value then ? What do the other figures denote when standing in the second place ? 18 NOTATION. [SECT. I. in the second place, denote two tens, three tens, finer tens, &c. When standing 1 in the thicd place, they express hun- dreds: thus the 1 in 100, 102, 123, &c., denotes a hun- dred, or ten tens ; that is, its value is ten times as much as when it stands in the second place, and it is called a unit of the third order. In like manner, 2, 3, 4, &c., standing hi the third place, denote two hundred, three hundred, four hundred, &c. When a digit occupies the fourth place, it expresses thousands: thus the 1 in 1000, 1845, &c., denotes a thou- sand, or ten hundreds; that is, its value is ten times as much as when it stands in the third place, and it is called a unit of the fourth order. Thus, It will be seen that ten units make one ten, ten tens make one hundred, and ten hundreds make one thou- sand ; that is, ten in an inferior order are equal to one in the next superior order. Hence, we may infer universal- ly, that 9. Numbers increase from right to left in a tenfold ratio; that is, each removal of a figure one place towards the left, increases its value ten times. 1 0. The different values which the same figures have, are called simple and local values. The simple value of a figure is the value which it ex- presses when it stands alone, or in the right hand place. The simple value of a figure, therefore, is the number which its name denotes. (Art. 6.) The local value of a figure is the increased value which QUEST. What is a figure called when it occupies the third place \ What is its value then \ What is it called when in the fourth place ? What is its value ? What do the other figures denote when standing in the fourth place ? How many units are required to make one ten ? How many tens make a hundred ? How many hundreds make a thou- sand ? Generally, how many of an inferior order are required to make one of the next superior order ? 9. What is the general law by which numbers increase ? What is the effect upon the value of a figure to remove it one place towards the left ? 10. What are the different va- lues of the same figure called ? What is the simple value of a figure \ What the local value ? Upon what does the local value of a figure de- pend ? Obs. Why is this system of notation called Arabic J WhiU else is it sometimes called ? Why ? ARTS. 9 12.] NOTATION. 19 it expresses by having other figures placed on its right Hence, the local value of a figure depends on its locality, or tbje place which it occupies in relation to other num- bers (with which it is connected. (Art. 8.) Osis. 1. This system of notation is called Arabic, because it is sup- iosed to have been invented by the Arabs. '2. It is also called the decimal system, because numbers increase in a tenfold ratio. The term decimal is derived from the Latin word decem, which signifies ten. XX. The art of reading numbers when expressed by fig- ures, is called NUMERATION. The pupil has already become acquainted with the names of numbers, from one to a thousand. He will now easily learn to read and express the higher numbers in common use, from the following scheme, called the NUMERATION TABLE. <*-i ri Q S* 568, 342, Period VI. Period V. Period IV. Period III. Period II. Period I. Quadrillions. Trillions. Billions. Millions. Thousands. Units. X 2 The different orders of numbers are divided into periods of three figures each, beginning at the right hand. The first, which is occupied by units, tens and hundreds, QUEST. 11. What is numeration ? Repeat the Numeration Table, beginning at the right hand. What is the first place on the right called l The second place ? The third ? Fourth ? Fifth ? Sixth ? Seventh ? Eighth ? Ninth ? Tenth, &c. ? 12. How are the orders of numbers di- vided ? What is the first period called ? By what is it occupied ? What is the second called ? By what occupied ? What is the third called ? By what occupied ? What is the fourth called ? By what occupied * What is the fit'tL caded ? By what occupied 1 20 NUMERATION. [SECT. L is called units 1 period ; the second is occupied by thou. sands, tens of thousands and hundreds of thousands, and is called thousands' period, &c. The figures in the table are read thus : Five hundred and sixty-eight quadrillions, three hundred and forty-two trillions, nine hundred and seventy-five billions, eight hun- dred and ninety-seven millions, six hundred and forty-five thousand, four hundred and thirty-two. 1 3* To read numbers which are expressed by figures. Point them off into periods of three figures each ; then, be- ginning at the left hand, read the figures of each period in the same manner as those of the right hand period are read, and at the end of each period pronounce its name. OBS. 1. The learner must be careful, in pointing of figures, al- ways to begin at the right hand ; and in reading them, to begin at the left hand. 2. Since the figures in the first or right hand period always denote units, the name of the period is not pronounced. Hence, in reading figures, when no period is mentioned, it is always understood to be the right hand, or units' period. EXERCISES IN NUMERATION. Note. At first the pupil should be required to apply to each figure the name of the place which it occupies. Thus, beginning at the right hand, he should say, " Units, tens, hundreds," e seen, that the process of carrying tens is, in effect, simply adding the tens to tens, the hundreds to hundreds, &c., which are con- tained in the given example ; or adding figures of the same order together, which is the only way they can be added. (Art. 22.) For, if the sum of any column ex- ceeds 9, and thus requires two or more figures to express it, (Art. 7,) the right hand figure denotes units of the same order as the column added, arid the left hand figure denotes units of the next higher order ; (Art. 8 ;) conse- quently, it is of the same order as the next column to which it is carried. The result will obviously be the same, whether we add the tens in their proper place, as we proceed in the operation, or reserve them till we have added the respective columns, and then add them to the same orders. The former method is the more convenient and expeditious, and is therefore adopted in practice. 15. What is the sum of 473 and 987? Ans. 1460. 16. 17. 18. 19. 4674 67375 84056 405673 6206 87649 5721 720021 4321 6048 41630 369115 8569 452 163 505181 QUEST. 27. What, in effect, is the process of carrying the tens to the next column ? How does tliis appear \ Does it make any difference with the result, when the tens are added to the next column! When are they commonly added ? Why ? ADDITION. [SECT. II. 28. PROOF. Beginning at the top, add each column downwards, and if the second result is the same as the first, ike work is supposed to be right. OBS. The object of beginning at the top and adding downwards, la that the figures may be taken in a different order from that in which they were added before j otherwise, if a mistake has been made the first time adding, we should be liable to fall into the same again. But the order being reversed, the presumption is, that any mistake which may have been made will thus be detected ; for it can hardly be sup- posed that two mistakes exactly equal will occur. 20. Find the sum of 256, 763, and 894, and prove the operation. 21. Find the sum of 8054. 5730, and 3056, and prove the operation. 22. Find the sum of 74502, 83000, and 62581, and prove the operation. 23. Find the sum of 68056, 31067, 680, and 200, and prove the operation. 24. Find the sum of 50563, 8276, 75009,31, and 856, and prove the operation. 25. Find the sum of 65031,2900, 35221, and 870, and prove the operation. 29. From the preceding illustrations and principles we derive the following GENERAL RULE FOR ADDITION. I. Write the numbers to be added under each other, so that units may stand under units, tens under tens^ fyc. (Art. 21, Ex. 1.) II. Begin at the right hand, and add each column sepa- rately. When the sum of a column does not exceed 9, write it under the column ; but if the sum of a column exceeds 9. write the units' figure under the column added, and carry tht lens to the next column. (Arts. 23, 25.) QUEST. 28. How is addition proved ? Obs. Why add the column* downwards, instead of upwards ? 29. What is the general nils roi ** eition? AETS. 28, 29.] ADDITION. 95 III. Proceed in this manner through all the orders, ana finally set down the, whole sum of the last or the left hand column. (Art. 25.) EXAMPLES FOR PRACTICE. 1. A man bought a quantity of flour for 38 dollars, a ton of hay for 14 dollars, and a firkin of butter for 12 doll'-.i.'s. How much did he give for the whole? 2. A grocer bought three boxes of honey ; the first contained 22 pounds, the second 15, and the third 9 pounds. How many pounds were there in all? 3. A man being asked his age, answered that it was equal to the united ages of his three children, the oldest of whom was 18, the second 16, and the third 14 years old. What was his age ? 4. A man bought 5 hogsheads of molasses for 238 dol- lars, and sold it so as to gain 75 dollars. How much did he sell it for ? 5. A lady purchased - materials for 3 dresses ; for the first she paid 15 dollars, for the second, 9 dollars, and for the third, 7 dollars. How much did she pay for them ail ? 6. A boy bought a cap for 12 shillings, a pair of gloves for 6 shillings, a pair of boots for 16 shillings, and a book for 6 shillings. How much did he give for the whole ? 7. A gentleman owns 3 houses ; for the first he re- ceives a rent of 150 dollars, for the second 175, and for the third 225 dollars. What is the sum of all his rents ? 8. A shopkeeper commenced business with 1530 dol- lars ; after trading some time, he found he had gained 950 dollars. How much >ad he then ? 9. A man bought a horse for 87 dollars, a carriage for 75 dollars, and a harness for 28 dollars. How much did he give for the whole ? 10. What number of dollars are there in four purses; the first containing 25 dollars, the second 73, the third 84 and the fourth 96 dollars ? 1 1. A poor man having lost his house by fire, to help him repair his loss, one man gave him 25 dollars, another 15, ano f her 10, another 5, and another 3. How much ^td ho reativt from all? 86 ADDITION. [SECT. I 12. In a certain school there were three classes in arithmetic ; the first class contained 8 scholars, the second 11, and the third 14. How many scholars were study- ing arithmetic ? 13. A merchant, on closing liis business for the day, found he had received 23 dolldis from one customer, 57 from another, 31 from another, and 25 from various oth- ers. How much did he receive that day ? 14. A laborer, in pursuit of employment, walked 7 miles the first day, 10 the second, 12 the third, 15 the fourth, and 20 the fifth day. How far had he then walked ? 15. A man, owning a large farm, gave to one of hia sons 112 acres, to another 123, to the third 147, and had 200 acres left. How large was his farm at first ? 16. A man bought a barrel of oil for 30 dollars, and sold it so as to gain 15 dollars. How much did he sell it for 1 17. A lad bought a geography for 50 cents, a grammar for 25 cents, an arithmetic for 13 cents, and a slate for 10 cents. How much did he give for them all ? 18. A gentleman purchased a carpet for 38 dollars, a dozen chairs for 36 dollars, a bureau for 15 dollars, and a table for 12 dollars. What did his bill amount to ? 19. A merchant had 4 notes; one for 157 dollars, an- other for 368, another for 576, and another for 1687 dol- lars. What was the whole amount of his notes ? 20. A gentleman bought a cloak for 56 dollars, a coat for 25 dollars, a vest for 9 dollars, a hat for 7 dollars, and a pair of boots for 5 dollars. What did he give for the whole 1 21. A fashionable lady purchased a cashmere shaw for 469 dollars, a watch for 237 dollars, a pocket hand kerchief for 87 dollars, and a bonnet for 53 dollars. Wha< was the amount of her bill ? 22. A farmer had 375 sheep and 168 lambs in one pas- ture, in another 379 sheep and 197 lambs. How many sheep had he ? How many lambs ? How many sheep and lambs together ? 23. Four men entered into partnership ; one furnished 2878 dollars, another 1784 dollars, a third 1265 dollars, and the fourth 894 dollars. What was the amount oj their sto*k? ART. 29.] ADDITION. 37 24. A man sold three house lots ; for one he received 975 dollars, for another 763 dollars, and for the third 586 dollars. What did the whole amount to ? 25. A gentleman purchased a store for 4500 dollars, and paid 75 dollars for repairs, and 150 dollars for having it enlarged. For how mucn must he sell it in order to gain 175 dollars? 26. A gentleman paid 75 dollars for one piece of cloth, 67 dollars for another, 54 dollars for another, and 48 dol- lars for another. How much did he pay for all ? 27. A certain orchard contains 56 apple-trees, 19 peach- trees, 23 plum-trees, and 15 cherry-trees. How many trees are there in the orchard ? 28. The distance from New York to Albany is 150 miles, from Albany to Utica 93 miles, from Utica to Roch ester 158 miles, and from Rochester to Buffalo 75 miles'. How far is it from New York to Buffalo 1 29. A man being asked his age, said he was 17 years old when he left the academy, he spent 4 years in college, 3 years in a law school, practiced law 15 years, was a member of congress 18 years, and it was 16 years since he retired from business. How old was he ? 30. A shopkeeper having a note due, paid 184 dollars at one time, at another 268 dollars, at another 379 dollars, at another 467 dollars, and there were 350 dollars still unpaid. What was the amount of his note ? 31. A gentleman owns a house and lot worth 10800 dollars, a store worth 5450 dollars, a house-lot worth 3700 dollars, and has 15000 dollars in personal property, What is the whole amount of his property ? 32. A man left his estate to his wife, his three sons, and two daughters; to his wife he gave 10350 dollars, to his sons 5450 dollars apiece, and his daughters 3500 dollars apiece. How large was his estate ? 33. A merchant, on looking over his accounts, found he owed one man 750 dollars, another 648, another 597, another 486, another 379, and another 287 dollars. What was the amount of his debts ? 34. A man bought a span of horses for 275 dollars, a 38 ADDITION. [SECT. II carriage for 150 dollars, and a harness for 87 dollars. How much did he give for the whole ? 35. A man bought 268 bushels of wheat for 287 dol- lars, 187 bushels of corn for 98 dollars, and .156 bushels of oats for 128 dollars. How many bushels of grain did he buy ; and how much did he give for the whole ? 36. A man wishing to stock his farm, paid 197 dollars for a span of horses, 86 dollars for a yoke of oxen, 175 dollars for cows, and 169 dollars for sheep. How much did he give for the whole 1 37. A butcher sold to one customer 157 pounds of meat, to another 159, to another 149, to another 97, and to an- other 68. How much did he sell to all ? 38. A carpenter received 879 dollars for one job, for another 786, for another 693, for another 587, for another 476, and for another 368 dollars. How much did he re- ceive in all ? 39. A grocer bought 375 dollars worth of sugar, 287 dollars worth of molasses, 168 dollars worth of tea. 158 dollars worth of coffee, and 137 dollars worth of spices. What was the amount of his bill ? 40. A merchant bought calico to the amount of 568 dollars, silks to the amount of 479 dollars, and broad- cloths to the amount of 784 dollars. He sold them so as to gain 134 dollars on the calico, 178 dollars on the silks, and 242 dollars on the broadcloths. How much did he sell them for ; and what was the amount of his gains ? 41. A merchant pays 560 dollars a year for store rent, 386 dollars to one clerk, 267 to another, and 369 dollars for various other expenses. What does it cost him a year to carry on his business ? 42. A man receives 568 dollars rent for one store, 479 for another, and 276 for another. How much does he re- ceive for them all ? 43. The distance from Boston to Springfield is 98 miles, from Springfield to Pittsfield is 53 miles, from Pittsfield to Albany is 49 miles, from Albany to Auburn is 173 miles, and from Auburn to Buffalo is 152 miles. Ho\r far is it from Boston to Buffalo ? A.RT. 29.] ADDITION. 39 44. A n>9n bought a quantity of oil for 2649 dollars, and candies for 1367 dollars; he afterwards sold them so as to ga;u 568 dollars on the oil, and 346 dollars on the candles. How much did he receive for the whole ? 45. Ii 1840, the state of Maine contained 501793 in- habitants ; New Hampshire, 284574 ; Vermont, 291948 ; Massachusetts, 73/699; Connecticut, 309978; and Rhode [sland, 103830. *What was the population of New Eng- land? 46. In 1840, the state of New York contained 2428921 inhabitants; New Jersey, 373306; Pennsylvania, 1724- 033 ; and Delaware, 78085. What was the population of the Middle States? 47. In 1840, the state of Maryland contained 470019* inhabitants ; Virginia, 1239797 ; North Carolina, 753419 ; South Carolina, 594398; Georgia, 691392; Alabama, 590756; Mississippi, 375651; and Louisiana, 352411. What was the population of the Southern States ? 48. In 1840, the state of Tennessee contained 829210 inhabitants; Kentucky, 779828 ; Ohio, 1519467 ; Michi- gan 212267; Indiana, 685866; Illinois, 476183; Mis- souri, 383702 ; and Arkansas, 97574. What wdf the population of the Western States ? 49. In 1840, the territory of Florida contained 54477 inhabitants; Wisconsin, 30945; Iowa, 43112; and the District of Columbia, 43712 ; on board vessels of war, 6100. What was the population of the Territories and naval service of the United States ? 50. What was the whole population of the United States in 1840? * iccording to Ine Official Revision. 40 SUBTRACTION. [SECT. III. SECTION III. SUBTRACTION. MENTAL EXERCISES. ART. 3O. Ex. 1. Henry having 7 peaches, gave 4 to nis sister : how many had he left ? OBS. To solve this question, consider what number added to 4 makes 7. Now from addition we know that 4 and 3 make 7 ; that is, 7 is composed of the numbers 4 and 3. It is evident, therefore, if one of these numbers be taken from 7, the other number will be left. Hence, 4 peaches from 7 peaches leave 3 peaches. Ans. 3 peaches. 2. James had 7 cents, and spent three of them : how many had he left ? 3. iack has 6 marbles : how many more must ke get to make 10? 4. A farmer having 9 cows, sold 5 of them : how many had he left 1 5. A pound of raisins costs 1 1 cents, and a pound of sugar 8 cents : what is the difference in their prices ? 6. In a stage coach there were 10 passengers, 6 of whom got out at a hotel : how many remained in the coach ? 7. Dick bought a knife for 12 cents, and having but 7 cents in his pocket, agreed to pay the rest to-morrow . how much does he owe for it ? 8. John gathered 8 quarts of chestnuts : how many more must he gather to make 14 quarts? 9. The cost of a cap is 13 shillings, and the cost of a comforter is 3 shillings : what is the difference in theii cost? 10. Susan is 15 years old, and Harriet is only 9: what is the difference in their ages ? ART. SO.] SUBTRACTION. 41 SUBTRACTION TABLE. 2 from 3 from 4 from 5 from 6 from 7 from 8 from 9 from -2 lC ives 3 lea. 4 lea. 5 lea. 6 lea. 7 lea. 8 lea. 9 lea. 3 ' 1 4 1 5 " 1 6 " 1 7 " 1 8 " 1 9 1 10 " 1 4 ' 2 5 o 6 " 2 7 " 2 8 " 2 9 2 10 2 11 " 2 5 ' 3 6 3 7 3 8 3 9 " 3 10 3 11 3 12 3 6 ' 4 7 4 8 4 9 4 10 " 4 11 4 12 4 13 4 7 ' 5 8 5 9 5 10 5 11 " 5 12 5 13 5 14 5 8 1 6 9 6 10 6 11 6112 " 6 13 G 14 6 15 G 9 ' 7 10 7 11 7 12 7 13 " 7 14 7 15 7 16 7 10 ' 8 11 8 12 8 13 8il4 " 8 15 8 16 " 8 17 8 11 1 9 19 g 13 " 9 14 9 15 " 9 16 9 17 " 9 18 9 12 ' 10 13 10 14 " 10 15 " 10 16 " 10 17 " 10 18 " 10 19 10 OBS. This Table is the reverse of Addition Table. Hence, if the pupil has thoroughly learned that, it will cost him but little time or trouble to learn this. (See observations under Addition Table.) 11. 4 from 7 leaves how many? 4 from 9? 4 from 12 ? 4 from 8 ? 4 from 1 1? 4 from 13 ? 12. 6 from 8 leaves how many? 6 from 10? 6 from 13 ? 6 from 1 1 ? 6 from 15 ? 6 from 12? 6 from 16 ? 13.7 from 9 leaves how many ? 7 from 1 1 ? 7 from 14 ? 7 from 15 ? 7 from 16 ? 7 from 13 ? 7 from 17 ? 14. 8 from 11? 8 from 13? 8 from 16? 8 from 12? 8 from 15 ? 8 from 17 ? 8 from 14 ? 8 from 18 ? 15. 9 from 12? 9 from 14? 9 from 11? 9 from 13? 9 from 17 ? 9 from 15 ? 9 from 18 ? 9 from 19 ? 16. 2 from 4 leaves how many? 2 from 14? 2 from 24? 2 from 34? 2 from 44? 2 from 54? 2 from 64? 2 from 74? 2 from 84? 2 from 94 ? 17. 3 from 6 ? 3 from 16 ? 3 from 26 ? 3 from 36 ? 3 from 46 ? 3 from 56 ? 3 from 66 ? 3 from 76 ? 3 from 86 ? 3 from 96 ? 18. 4 from 9 ? 4 from 29 ? 4 from 39 ? 4 from 49 ? 4 from 59 ? 4 from 69 ? 4 from 79 ? 4 from 89 ? 4 from 99 ? 19. 6 from 15 ? 6 from 25 ? 6 from 35 ? 6 from 45 ? 6 from 55 ? 6 from 65 ? 6 from 75 ? 6 from 95 ? 20. 8 from 14? 8 from 24? 8 from 34? 8 from 44? 8 from 54 ? 8 from 64 ? 8 from 74 ? 8 from 84 ? 8 from 94 ? 21. A gentleman bought a coat for 15 dollars, and a hat for 6 dollars : how much more did his coat cost than his hat? 4Q SUBTRACTION. [SECT. Ill, 22. A farmer having sold 6 cords of wood for 18 dol lars, took a barrel of flour at 6 dollars towards his pay and the rest in cash : how much money did he receive 1 23. A lady bought a shawl for 15 dollars, and hand ed the shopkeeper a 20 dollar bill : how much change ought she to receive back ? 24. A man having 25 watermelons in his garden, some wicked boys stole 9 of them : how many had he left ? 25. James is 14 years old, and his sister is 19 : what is the difference in their ages? 26. A merchant had a piece of calico which contained 33 yards ; on measuring the remnant he finds he has but 7 yards left: how many yards has he sold? 27. A hogshead of cider contains 63 gallons : after drawing out 9 gallons, how many will be left ? 28. Henry had 48 silver dollars, and gave 8 to the or- phan asylum : how many dollars did he have left ? 29. A mim bought a piece of cloth containing 39 yards, and sold 6 yards of it : how many yards had he left ? 30. George gave 75 cents for a pair of skates, and sold them for 9 cents less than he gave : how much did he get for his skates 1 31. William had 67 cents; he spent 5 for chestnuts and 2 for apples : how many cents has he left ? 32. A man sold a load of wood for 18 shillings; he laid out 4 shillings for tea and 6 for sugar : how many shillings had he to carry home ? 33. Sarah having 85 cents, gave 10 cents to the Sab- bath School Society, 8 to the Bible Society, and spent 6 for candy : how many cents had she left ? 34. If I pay 27 dollars for a cow and sell it for 18 dol- lars, how much do I lose by the bargain ? 35. Richard had 45 marbles ; he lost 7 and gave away 5 : how many had he left ? 36. A man having 56 dollars in his pocket, bought a hat for 5 dollars, a coat for 10, and a pair of boots for 4 how much money had he left? 37. If I owe a merchant 50 dollars and pay him 20 dollars, how many dollars shall I then owe him ? Ans. 30 dollars. ART. 81.] SUBTRACTION. 43 Suggestion. It is advisable for beginners to analyze the numbers in this question, as in Art. 16, Ex, 31, and then take 2 tens from 5 tens. 38. A farmer having 80 sheep, sold all but 30 : how many did he sell? 39. A man having 90 acres of land, gave 50 acres to his son : how many acres has he left ? 40. George had 70 cents and spent 30 : how many had he left? 41. In a certain orchard there are 100 trees, 60 of them are apple-trees and the rest are peach-trees : how many peach-trees are there ? 42. A grocer bought 150 eggs, and afterwards found that 20 of them were rotten : how many sound ones were there ? 43. In the Centre School there are 150 scholars, 60 of whom are girls : how many boys are there ? 44. A man bought a horse for 90 dollars, and sold it immediately for 130 dollars : how much did he make by his bargain ? 45. A man owing me 200 dollars, turned me out a horse worth 80 dollars, and is to pay the balance in cash : how much money must he pay me ? 46. A boy going to market with 80 cents, bought 20 cents worth of cheese, and 30 cents worth of butter : how much change had he left ? 47. 35 from 42 leaves how many? 63 from 75? 48. 26 from 40 leaves how many ? 35 from 45 ? 49. 65 from 85, how many ? 82 from 94, how many ? 50. 8 from 17, how many? 13 from 26, how many? 6 from 25, how many ? 8 from 94, how many ? 5 from 68, how many? 17 from 34, how many? 7 from 43, how many? 6 from 72, how many? 9 from 75, how many ? 7 from 86, how many ? 3 1 It will be observed that all the preceding exam pies of this section, though expressed in a variety of ways, involve the same principle ; that the object aimed at in each of them, is to find the difference between two numbers; consequently, they are all performed in the 44 SUBTRACTION. [SECT. 1IL same manner. The operation consists in taking a les\ number from a greater, and is called subtraction. Hence, 32S. SUBTRACTION is the process of finding the differ- ence between two numbers. The difference, or the answer to the question, is called the remainder. OBS. 1. The number to be subtracted is often called the subtraherd, and the number from which it is subtracted, the minuend. These terms, however, are calculated to embarrass, rather than assist the earner, and are properly falling into disuse. 2. Subtraction, it will be perceived, is the reverse of addition. Ad dition unites two or more numbers into one single number ; subtrac tion, on the other hand, separates a number into two parts. 3. When the given numbers are of the same denomination, the operation is called Simple Subtraction. (Art. 18. Obs.) 33. Subtraction is often represented by a short hori- zontal line ( ), which is called minus. When placed be tween two numbers, this sign shows that the number after it is to be subtracted from the one before it. Thus the expression- 8 5, signifies that 5 is to be subtracted from 8 ; and is read, " 8 minus 5," or " 8 less 5." Note. The term minus. is a Latin word signifying less. EXERCISES FOR THE SLATE. 34. When we wish to find the difference between two small numbers, it is the most convenient way to per- form the subtraction in the mind. Bat when the num- bers are large, it is difficult to retain them in the mind, and carry on the operation at the same time. By setting them down upon a slate or black-board, however, the process of subtracting large numbers is rendered short and simple. (Art. 21.) Q. What is subtraction? What is the answer called ? Obs. What is the number to be subtracted sometimes called? That from which it is sublrnct/.'d ? Of wiiat is subiraction the reverse ? When the given uumhrrs are of the same denomination, what is the operation called ? 33. Wh:it is the si#7i of subtraction railed ? Of what does it consist ' What does it show? How is ihe expression 8 5, read? Note. What in:;' of til 1 ,' term minus ? 34. What is the most convenient of finding the, difference between two small numbers? WhrU ! two large ones ? ARTS. 32-*84.] SUBTRACTION. 45 Ex. 1. Suppose a man gave 475 dollais for a span oi horses, and 352 dollars for a carriage : how much more did he pay for his horses than for his carriage ? Directions. Write the less Operation. number under the greater, so ^ M that units may stand under units, tens under tens, &c. Now, be- ginning with the units, proceed Horses, 475 Dolls, thus : 2 units from 5 units leave Carriage, 3 5 2 Dolls. 3 units; write the 3 in units' R em . 1 2 3 Dolls, place, under the figure subtract- ed. 5 tens from 7 tens leave 2 tens ; set the 2 in tens' place. 3 hundreds from 4 hundreds leave 1 hundred ; write the 1 in hundreds' place. The remainder is 123 dollars. OBS. It is important for the learner to observe, that we subtract units from units, tens from tens, &c. ; that is, we subtract figures of the same order from each other. This is done for the same reason that we add figures of the same order to each other. (Art. 22.) Hence, in writing numbers for subtraction, great care should be taken to set units under units, &c., in order to prevent the mistake of sub- tracting different orders from each other. 2. A merchant bought 268 barrels of flour ; and on ex- amination, found that only 123 barrels were fit for use" how many were damaged? Ans. 145. Suggestion. Write the less number under the greater, &c., and proceed as above. 3. A traveler having 576 dollars, was robbed of 344 dollars : how many dollars had he left ? 4. What is the difference between 648 and 235 ? 5. What is the difference between 876 and 523 ? 6. What is the difference between 759 and 341 ? 7. What is the difference between 4567 and 1235? 8. What is the difference between 8643 and 5412 ? QUEST. In the 1st example how do you write the numbers for sub- traction ? Where begin to subtract ? Obs. What orders do you sub- tract from each- other ? Why not subtract different orders from each other 1 Why place units under units, &o., in subtraction ? 46 SUBTRACTION. [SECT. Ill 9. 10. 11. 12. From 68476 765274 563181 3286732 Take 36124 152140 32040 135011 35. When the figures in the lower number are all smaller than those directly over them, each lower figure, as we have seen in the preceding examples, must be sub- tracted from that above it, and the remainder must be placed under the figure subtracted. But it often happens that a figure in the lower numbei is larger than that above it, and consequently cannot be taken from it. 13. It is required to find the difference between 75 and 48. It is plain that we cannot take 8 units Operation. from 5 units, for 8 is larger than 5. What 75 then shall we do ? Since 75 is composed 43 of 7 tens and 5 units, we can take 1 ten -^= from the 7 tens, and adding it mentally to the 5 units, it will make 15 units. Then subtracting the 8 units from 15 units, will leave 7 units ; write the 7 un- der the units' column. As we took 1 ten from the 7 tens, we have but 6 tens left ; and 4 tens from 6 tens leave 2 tens : write the 2 under the tens' column. The whole remainder, therefore, is 2 tens and 7 units, 01 27. 36. The process of taking one from a higher order in the upper number, and adding it to the figure froxsi which the subtraction is to be made, is called borrowing ten, and is the reverse of carrying ten. (Art. 24.) OBS. The 1 taken from a higher order, is always equal to 10 ID the next lower order to which it is added. (Art. 8.) 37* The principle of borrowing may be illustrated by the following analytic solution of the last example. QUEST. 35. When the figures in the lower number are each small er than those over them, how proceed ? Where do you place the re- mainder ? Is a figure in the lower number ever larger than that above it ? 36. What is meant by borrowing 10 ? What is the 1 taken from the nighr-r order equal to ? A.RTS. 35-37, a.] SUBTRACTION. 47 75=604-15 Taking 1 ten from 7 tens, and uniting it with the 5 units, we 7, or 27. number. And we simply separate the lower number into the tens and units of which it is composed. Now subtracting, as in the last article, 8 from 15 leaves 7 : 40 from 60 leaves 20. Thus the remain- der is 20-f 7, or 27, the same as before. OBS. It is manifest that this process of borrowing ten, does not change the value of the upper number ; for, it consists simply in transposing a part of one order to another order in the same number, which can no more diminish or increase the number, than it will di- minish or increase the amount of money a man has, if he takes a part from one pocket and puts it into another. It is advisable for the pupil to analyze several examples as above, until the process, of bor- rowing becomes familiar. . A ^ Since 7 units cannot be taken from 2 units > we borrow 10, which added _ to the 2. will make 12 : then 7 units Bern, 3675 f rorn 12 units leave 5. Now hav- ing borrowed 1 of the 4 tens, it becomes 3 tens ; and 6 from 3 is impossible : hence we must borrow again. But the next figure in the upper number, i. e. the figure in the hundreds' place, is a 0, and consequently has nothing to lend. We must therefore borrow 1 from the next order still, i. e. from thousands, and adding it to the 0, it will make 10 hundreds. Then, borrowing 1 of the 10 hun dreds and adding it to the 3 tens, it will make 13 tens, and 6 from 13 leaves 7. Diminishing the 10 hundreds oy 1, (which we borrowed,) it becomes 9, and 3 from 9 leaves 6. Again, diminishing the 6 thousands by 1, (which we borrowed,) it becomes 5, and 2 from 5 leaves 3. The answer is 3675. 3 7 . a. There is another method of borrowing, or rath- er of paying, which the learner will often find more con- QUKBT. How illustrate the principle of borrowing upon the black- board ? Obs. Is the value of the upper number increased by borrow- ing? Is it diminished? How does this appear? 37. a. When w borrow 10, what other way is there to compensate for it ! 48 SUBTRACTION. [SECT, HI venient In practice than the preceding-, and less liable t lead him into mistakes, especially, when the figure in the next higher order is a cipher. When we borrow 10, that is, when we add 10 to the upper figure, instead of considering the next figure in the upper number to be diminished by 1, the result will mani- festly be the same, if we simply add 1 to the next figure in the lower number. Thus, in the last example, instead of diminishing the 4 tens in the upper number by 1, we may add 1 to the 6 tens in the lower number, which will make 7 ; and 7 from 14 leaves 7, the same as 6 from 13. Again, adding 1 to the 3 hundreds (to compensate for the 10 we borrowed) makes 4 hundreds ; and 4 from 10 leaves 6, the same as 3 from 9. Finally, adding 1 to the 2 (because we borrowed) makes 3 ; and 3 from 6 leaves 3. The remainder is 3675, the same as before. *7/t 6 from 4 is impossible : add 10 to 3^6 th 4, and it will make 14; then 6 from 14 leaves 8. Adding 1 to the 2 n OA o makes 3, and 3 from 7 leaves 4. 3 from 5 leaves 2. Ans. 248. Ozs. This method of borrowing depends on the self-evident prin- ciple, that if any two numbers are equally increased, their difference will not be altered. That the two given numbers are equally in- creased by this process, is evident from the fact that the 1 added to the lower number, is of the next superior order to the 10 added to the upper number, and will compensate for it ; for 1 in a superior order, is equal to 10 in an inferior order. (Art. 8.) Hence, 38. When a figure in the lower number is larger than that above it, borrow 10, i. e. add 10 to the upper figure, and from the number thus produced, subtract the lower figure : to compensate this, add 1 to the next figure in the lower number ; or diminish the next figure in the upper number by 1, and proceed as before. 16. 17. 18. From 78562 645630 70430256 Take 24380 520723 4326107 QUEST. -Obs. Upon what does the second method of borrowing de- pend ? How does it appear that you increase the given number* equally ? ARTS. 38-40.] SUBTRACTION. 49 39. PROOF. Add the remainder to the. smaller ber ; and if the sum is equal to the larger number, the work is right. 19. A man bought a horse for 175 dollars, and sold it for 127 dollars: how much did he lose by his bargain? Operation. Proof. Since the sum of the Paid 1 75 dolls. 1 27 Smaller No. im ? 11 f number and rc- Rec'd 127 dolls. 8 Remainder. Lost 48 dolls. 175 Larger No. ration is correct. OBS. This method of proof depends upon the obvious principle, that if the difference between two numbers be added to the less, the sum must be equal to the greater. 20. From 8796 subtract 2675, and prove the operation ? 21. From 6210896 subtract 3456809, and prove the operation. 22. From 1000000 subtract 67583, and prove the ope- ration. 23. From 7834501 subtract 1000000, and prove the operation. 24. From 68436907 subtract 59476012, and prove the operation. 25. From 8006754231 subtract 79756634 17, and prove the operation. 4O. From the preceding illustrations and principles we derive the following GENERAL RULE FOR SUBTRACTION. I. Write the kss number under the greater, so that untt* may stand under units, tens under tens, &c. II. Beginning at the right hand, subtract each figure in (he, lower number from thefiguie above it, and set the remain- der directly under the figure subtracted. (Art 35.) QUEST. 38. How then do you proceed, when a figure in the lower number is larger than the one over it I Why do you add 1 to the next figure in the lower line ? 39. How is subtraction proved ? Obs. Up- on v> hat principle does the proof of subtraction depend ? 40. What ia the general rule for subtraction; 1 50 SUBTRACTION. [SECT. Ill III. Whe?i a figure in the lower number is larger than that above itj add 10 to the upper figure; then subtract a% before, and add 1 to the next figure in the lower number. (Arts. 37, 38.) EXAMPLES FOR PRACTICE. 1. A man bought a piece of cloth containing 37 yards, and sold 24 yards of it. How much had he left ? 2. A merchant had on hand a quantity of flour, for which he asked 245 dollars; but for ready money he made a deduction of 24 dollars. How much did he re- ceive for his flour ? 3. In a certain Academy there were 357 scholars, 168 of whom were young ladies. How many young gentle- men were there 1 4. A farmer raised 4879 bushels of wheat, and sold 3876 bushels. How much had he left ? 5. A man purchased a farm for 4687 dollars, but the times becoming hard he was obliged to sell it for 896 dollars less than he gave for it. How much did he sell it for? 6. A merchant bought 2268 dollars worth of goods, which, in consequence of getting damaged, he sold for 848 dollars less than cost. How much did he sell them for? 7. A merchant sold a lot of silks for 561 dollars, which was 179 dollars more than the cost of them. How much did he give for them ? 8. A man bought an estate for 8796 dollars, and sold it again for 9875 dollars. How much did he gain by his bargain ? 9. A farmer raised 1389 bushels of wheat one year, and 1763 the next. How much more did he raise the second year than the first ? 10. A man bought a house and lot for 5687 dollars The house was worth 3698 dollars .- how much was the lot worth? 11. Suppose a gentleman's income is 3268 dollars a year, and his expenses are 2789 dollars. How much does ne save in a year ? \RT. 40.] SUBTRACTION. 51 12. The United States declared their independence in 1776 : how many years is it since ? 13. Two brothers commenced business at the same time ; one gained 3678 dollars in five years, the other gained 2387 dollars in the same time. How much more lid one gain than the other? 14. The distance from Boston to Springfield is 98 miles, and from Boston to Pittsfieldit is 151 miles. How far is it from Springfield to Pittsfield ? 15. From New York to Utica it is 243 miles, and from New York to Albany it is 150 miles. How far is it from Albany to Utica'? 16. America was discovered by Columbus in 1492: how many years is it since? 17. Dr. Franklin died A. D. 1790, and was 84 years old when he died : in what year was he born ? 18. General Washington was born A. D. 1732, and died in 1799 : how old was he when he died? 19. The first settlement in New England was made at Plymouth in the year 1620 : how many years is it since? 20. A ship sailed having on board a cargo valued ai 100000 dollars, but being overtaken by a storm, 27680 dollars worth of goods were thrown overboard. How much of the cargo was saved ? 21. The population of Massachusetts in 1840, was 737699, and that of Connecticut was 309978. How many more inhabitants were there in Massachusetts than in Connecticut? 22. In 1840. the population of Massachusetts was 737699, and in '1820 it was 523287. How much did the population increase during this period ? 23. In 1840, the population of the state of New York was 2428921, and in 1820 it was 1372812. How much did the population increase during that period ? 24. In 1840, the population of the New England States was 2234822, and that of the State of New York was 242S921. How many more inhabitants were there in the State of New York than in New England? 25. In 1800, the population of the United States was 5305925, and in 1840 it was 17069453. How much d^d it increase irs forty year's?. 52 SUBTRACTION. [SECT. Ill, 26. A farmer having 389 acres of land, sold to onft man 126 acres, and to another 163. How many acres had he left ? 27. A gentleman having 1768 dollars deposited in the bank, gave a check for 175 dollars to one man, to another for 238 dollars, and to another for 369 dollars. How much remained on deposit ? 28. A man bought a horse for 87 dollars, a carriage for 75 dollars, and a harness for 16 dollars, and sold them all together for 200 dollars. How much did he gain by the bargain ? 29. A man bought a quantity of sugar for 25 dollars, a quantity of molasses for 27 dollars, and a quantity of rai- sins for 29 dollars, for which he paid a hundred dollar- bill. How much change ought he to receive back ? 30. An orchard contained 120 apple-trees, 47 peach- trees, and 28 pear-trees. Of the apple-trees 26 were cut down for a Railroad to pass through, 18 of the peach- trees died, and 5 of the pear-trees were blown down. How many trees were left in the orchard ? 31. A gentleman had 2700 dollars which he wished to distribute among his three sons. To the oldest he gave 825 dollars, to the second 785 dollars, and the remain- der to the youngest. How much did the youngest son receive ? 32. A man owing 5648 dollars, paid at one time 536 dollars, at another 378 dollars, and at another 896 dollars. How much did he then owe ? 33. A man having 7689 dollars, invested 689 dollars in Railroad stock, 500 dollars in a woolen factory, and 1250 dollars in bank stock. How much had he left 1 34. A man bought a quantity of oil for 1763 dollars, and a lot of candles for 598 dollars. He afterwards sold them both for 2684 dollars. How much did he gain by the bargain? 35. A man owning 3789 acres of land, gave to one son 869 acres, and to another 987 acres. How much land had he left ? 36. A ship of war sailing with 650 men, lost in one oattle 29 men, in another 37, and by sickness 19 more, How many were still living 1 ART. 40.] SUBTRACTION. 53 37. A merchant owes one man 2684 dollars, another 1786 dollars, another 987 dollars. The whole amount of his property is 4684 dollars. How much more does he owe than he is worth ? 38. A man bought three farms: for the first he gave 4673 dollars, for the second 5674 dollars, and for the third 9287 dollars, He sold them all for 37687 dollars. How much did he gain by the bargain ? 39. A man bought 86 dollars worth of wheat, 48 dol- lars worth of butter, and a fine horse worth 148 dollars. He gave his note for 128 dollars, and paid the rest in :ash. How much money did he pay ? 40. A gentleman left a fortune of 18864 dollars to be divided between his two sons and one daughter ; to one son he gave 6389 dollars, to the other 6984 dollars. How much did the daughter receive ? 41. A man owing 8648 dollars, paid at one time 486, at another 684, at another 729 dollars. How much did he still owe ? 42. Suppose a man gains by one speculation 867 dol- lars, by another 687; another time he gains 563 dollars, and then loses 479 ; still another time he gains 435 dol- lars, and loses 378. How much more has he gained than lost? 43. A man borrowed of a friend 684 dollars at one time, 786 at another, 874 at another, and 976 at another. He has paid 568 dollars. How much does he still owe ? 44. If a man's income is 4586 dollars a year, and he spends 384 dollars for clothing, 568 for house rent, 784 for provisions, 568 for servants, and 369 for traveling, how much will he have left at the end of the year ? 45. A merchant bought a quantity of sugar for 8978 dollars, paid 374 dollars freight, and then sold it for 9684 dollars. How much did he gain by the trade? 46. A merchant had in his storehouse 6384 bushels of wheat, 3752 bushels of corn, 4564 bushels of oats, and 1384 bushelp of rye : it was broken open and 3564 bushels of grain taken out. How many bushels re- r lained ? 47. A man bought a quantity of beef for 5493 dollars. 54 MULTIPLICATION. [SECT. IV a quantity of coffee for 261 dollars, and a quantity of su gar for 157 dollars ; in exchange he gave 3687 dolla worth of flour, 568 dollars worth of oats, and 165 dolla worth of potatoes. How much did he then owe 1 48. A gentleman has real estate valued at 3879 dol iars, and personal property amounting to 9857 dollars, He owes one man 1350 dollars, and another 2687 dollars. How much would he have left if he should pay his debts? 49. A man having property amounting to 30000 dollars, lost by fire a store worth 5000 dollars, and goods to the amount of 3578 dollars. How much property had he left? 50. A man died leaving an estate of 175000 dollars. He gave to his wife 25000 dollars, to his three sons 32000 apiece, to his two daughters, 23000 dollars each, and the rest he gave to a literary institution. How much &d the institution receive ? SECTION IV. MULTIPLICATION. MENTAL EXERCISES. ART. 41* Ex. 1. What will 3 lead pencils cost, at 4 cents apiece ? Solution. Three pencils will evidently cost three times as much as one pencil. Now if 1 pencil costs 4 cents, 3 pencils will cost 3 times 4 cents ; and 3 times 4 cents are 12 cents. Ans. 12 cents. Note. It is highly important for the pupil to give the reason in full for the solution of every example. 2. What will 2 yards of cloth cost, at 8 dollars a yard 1 3. At 6 cents apiece, what will 4 oranges cost ? 4. What cost 5 pounds of ginger, at 7 cents a pound? ART. 41.] MULTIPLICATION. B 5. If 1 pair of gloves cost 6 shillings, what will 6 pair cost? 6. At 9 cents a pound, what will 4 pounds of butter come to ? 7. What wi II 7 barrels of flour cost, at 4 dollars a barrel ? 8. In 1 bughel there are 4 pecks : how many pecks are there in 6 bushels ? 9. What cost 8 pair of boots, at 6 dollars a pair ? 10. At 9 shillings apiece, what will 5 caps cost? 11. What cost 6 pounds of sugar, at 10 cents a pound? 12. What cost 9 inkstands, at 8 cents apiece? MULTIPLICATION TABLE. 2 times 3 times 4 times 5 times 6 times 7 times 1 are 2 1 are 3 1 are 4 1 are 5 1 are 6 1 are 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 6 421 7 14 7 21 7 28 7 35 7 42 7 49 8 16 8 24 8 32 40 8 48 8 56 9 10 18 20 9 10 27 30 9 10 36 40 9 10 45 50 9 10 54 60 9 10 63 70 11 22 11 33 11 44 11 55J11 66|11 77 13 24 12 36 12 48 12 60 12 72il2 84 8 times 9 times 10 times 11 times | 12 times I are 8 I are 9 I are 10 1 are 11 1 are 12 2 1,6 2 " 18 2 20 2 22 2 24 3 24 ' 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 i 9 72 9 81 9 90 9 99 9 108 10 80 10 90 10 100 10 110 I 10 120 11 88 11 99 11 110 11 121 i 11 132 , 12 90 12 108 12 120 12 132 12 144 OBS. The pupil will find assistance in learning this table, by ob- serving the following particulars. 1. The several results of multiplying by 10 are formed by simply adding a cipher to the figure that is to be multiplied. Thus, 10 times 2 are 20- 10 times 3 are 30, &c. 60* MULTIPLICATION. [SECT. IV 2. The results of multiplying by 5, terminate in 5 and 0, alter- nately. Thus, 5 times 1 are 5 ; 5 times 2 are 10 ; 5 times 3 are 1 5, &xx 3. The first nine results of multiplying by 11 are formed by re> pcating the figure to be multiplied. Thus, 11 times 2 are 22; 11 times 3 are 33, lings are there in 5 guineas ? 45. In 1 hogshead there are 63 gallons: how many gallons are there in 4 hogsheads ? 46. What cost 32 pounds of sugar, at 8 cents pei pound ? 47. What cost 85 reams of paper, at 3 dollars per ream ? 48. What cost 90 hats, at 4 dollars apiece ? 49. In 1 week there are 7 days : how many days are there in 70 weeks? 50. In 1 hour there are 60 minutes : how many min- utes are there in 9 hours? Let us now attend to the nature of the preceding operations in this section. Take, for instance, the first example. Since 1 pencil costs 4 cents, 3 pencils will cost 3 times 4 cents. Now 3 times 4 cents is the same as 4 cents added to itself 3 times ; and 4 cents + 4 cents j- 4 cents are 12 cents. Again, in the second example : since 1 yard of cloth costs 6 dollars, 4 yards will cost 4 times 6 dollars : and 4 times 6 dollars is the same as 6 dollars added to itself 4 times ; and 6 dollars -j- 6 dollars + 6 dollars -f- 6 dollars are 24 dollars. 43 This repeated addition of a number or quantity to itself, is called MULTIPLICATION. The number to be repeated or multiplied, is called the multiplicand. The number by which we multiply, or which shows how many times the multiplicand is to be repeated, is called the multiplier. The number produced, or the answer to the question, is called the product. QUEST. 43. What is multiplication ? What is the number to be repeated called ? What the number by which we multiply ? What does the multiplier show ? What is the number produced called ? When we say, 6 times 12 are 72, which is the multiplicand ? Which the multiplier ? Which the product ? ARTS. 42-45.] MULTIPLICATION. 59 Thus, when we say, 6 times 12 are 72, 12 is the mul- tiplicand, 6 the multiplier, and 72 the product. OBS. When the multiplicand denotes things of one denomination only, the operation is called Simple Multiplication. 44. The multiplier and multiplicand together are of- ten called factors } because they make or produce the pro- duct Note. The term factor, is derived from a Latin word which sig- nifies an agent, a doer, or producer. 45. Multiplying by 1, is taking the multiplicand once: thus, 4 multiplied by 1=4. Multiplying by 2, is taking the multiplicand twice : thus, 2 times 4, or 4+4=8. Multiplying by 3, is taking the multiplicand three times : thus 3 times 4, or 4+4+4=12, &c. Hence, Multiplying by any whole number, is taking the multi- plicand as many times, as there are units in the multiplier. Note. The application of this principle to fractional multipliers, will be illustrated under fractions, OBS. 1. From the definition of multiplication, it is manifest that the product is of the same kind or denomination as the multiplicand ; for, repeating a number or quantity does not alter its nature. Thus, if the multiplicand is an abstract number ; that is, a number which does not express money, yards, pounds, bushels, or have reference to any particular object, the product will be an abstract number; if the multiplicand is money, the product will be money ; '^weight, the pro- duct will be weight; if measure, measure, &c. 2. Every multiplier is to be considered an abstract number. In lamiliar language it is sometimes said, that the price multiplied by the weight will give the value of an article ; and it is often asked "how much 25 cents multiplied by 25 cents will produce. But these are abbreviated expressions, and are liable to convey an erroneous idea, or rather no idea at all. If taken literally, they are absurd ; for mul- tiplication is repeating a number or quantity a certain number of times. Now to say that the price is repeated as many times as the given QUEST. When we sav, 6 times 9 are 54, what is the 6 called ? The 9 ? The 54 ? 44. What are the multiplicand and multiplier togethe. called ? Why ? Note. What does the term factor signify ? 45. Whjr, is it to multiply by 1 ? By 2 1 By 3 ? What is it to multiply by -^y whole number? Of what denomination is the product? Ho^< does this appear ? What must every multiplier be considered 1 Can you multiply by a given weight, a measure, or a sum rf moncv * 60 MULTIPLICATION. [SECT. IV. quantity is hazvy, or that 25 cents are repeated 25 cents '\rnes, is non- sense. But we can multiply the price of 1 pound by a number equa to the number of pounds in the -weight of the given article, and the product will be the value of the article. We can also multiply 5 cents by the number 5 ; that is, repeat 5 cents 5 times, and the pro- duct is 25 cents. Construed in this manner, the multiplier becomes an abstract number, and the expressions have a consistent meaning. 46 . Multiplication is often denoted by two oblique lines crossing each other (x), called the sign of multiplication. It shows that the numbers between which it is placed, are to be multiplied together. Thus the expression 9x6, signifies that 9 and 6 are to be multiplied together, and is read, " 9 multiplied by 6," or simply, 9 into 6." OBS. The product will be the same, whether we multiply 9 by 6, or 6 by 9 ; for, by the table, 6 times 9 are 54, also 9 times 6 are 51 So 6X4=4X6; 5X3-3X5; 8X7-7X8, &c. To illustrate this point ; suppose there is a certain orchard which contains 4 rows of trees, and each row has 6 trees. Let the number of rows be represented by the num- ^ % ber of horizontal rows of stars in the margin, and the number of trees in each row by the number of stars in a row. Now it is evident, that the whole number of trees in the orchard is equal either to the number of stars in a horizontal row repeated four times, or to the number of stars in a perpendicular row repeated six times; that is, equal to 6X4, or 4X6. Hence, 47 . The. product of any two numbers will be the same, whichever factor is taken for the multiplier. EXERCISES FOR THE SLATE. Ex. 1. What will 3 house-lots cost, at 231 dollars each ? Suggestion. If 1 house-lot costs 23 1 dollars, 3 lots will cost 3 times 231 dollars; that is, three lots will cosl 231+231+231, or 693 dollars. QUEST. 46. How is multiplication sometimes denoted ? What doe* the sign of multiplication show ? How is the expression 9X6, read ? How 6X7=42? 47. Does it make any difference in the prodtwt, Wliteh factor Is made lhe multiplier ! How illustrate this ! ARTS. 46, 47.] MULTIPLICATION. 61 Having written the numbers Operation. upon the slate, as in the margin, 2 3 1 Multiplicand, we proceed thus : 3 times 1 unit 3 Multiplier, are 3 units. Set the 3 in units' ^ n .. no , J place under the multiplier. 3 Dolk 693 Product - times 3 tens are 9 tens ; set the 9 in tens' place. 3 times 2 hundreds are 6 hundreds ; set the 6 in hundreds' place. The product is 693 dollars. 2. What will 4 horses cost, at 120 dollars apiece? Suggestion. Write the less number under the greater, and proceed as before. Ans. 480 dolls. 3. What is the product of 312 multiplied by 3 ? Ans. 936. 4. What is the product of 121 multiplied by 4 ? Ans. 484. 5. In 1 mile there are 320 rods : how many rods are there in 3 miles? 6. If a man travels 110 miles in 1 day, how far can he travel in 8 days ? 7. 8. 9. 10. Multiplicand, 3032 22120 101101 3012302 Multiplier, 3453 11. What will 6 stage-coaches cost, at 783 dollars n piece ? Proceeding as before, 6 times 3 Operation, units are 18 units, or simply say, 6 733 times 3 are 18. Now 1 8 requires two 6 figures to express it; hence, we set A -777^ j i, the 8 under the figure multiplied, and Ans ' 4698 dolls ' reserving the 1, carry it to the next product, as in addi- tion. (Art. 25.) 6 times 8 are 48, and 1 (to carry) makes 49. Set the 9 under the figure multiplied, and carry the 4 to the next product, as before. 6 times 7 are 42, and 4 (to carry) make 46. Since there are no more figures to be multiplied, set down the 46 in full. The product ia 4698 dollars. Hence, 62 MULTIPLICATION. [SECT. IV 4Sf, When the multiplier contains but one figure. Write the multiplier under the multiplicand ; then, be* ginning at the right hand, multiply each figure of the mul tiplicand by the multiplier separately. If the product of an y figure of the multiplicand into the multiplier docs not exceed 9, set it in its proper place wider the figure multiplied; but if it does exceed 9, write the U7iits > figure under the figure multiplied, and carry tJie lens to the next product on the left, as in addition. (Art. 25.) 5 O. The principle- of carrying the tens in multiplica- tion is the same as in addition, and may be illustrated in a similar manner. (Art. 26.) ' Take, for instance, the last exa* .pie, and set the pro- duct of each figure in a separate line. Thus. 783 ' Or, separate the multiplicand into 6 the orders of which it is composed ~T8 units, ^us, 783-700+80+3 48* tens, Now 700x6=4200 hund. 42** hunds. 80x6=- 480 tens. 4698 Prod. 3x6= 18 units. Adding these results, we have' 4698 Product. In this analytic solution it will be seen that the tens'" figure in each product which exceeds 9, is added to the next product on the left, the same as in the common meth- od of solving this and similar examples. The only dif- ference between the two operations is, that in one case we add the tens as we proceed in the multiplication ; in the other we reserve them till each figure is multiplied, and then add them to the same orders as before : consequently, the result must be the same in both. (Art. 27.) QUEST. 49. How do you write the numbers for multiplication Where begin to multiply ? When the product of a figure in the ^nul tiplicand does not exceed 9, where is it written ? When it exceeds 9 what is to be done with it ? 50. How illustrate the principle of car tying in multiplioaticm \ ARTS. 49-51.] MULTIPLICATION. 63 51* From this and the preceding- illustrations, the earner will perceive, that units multiplied by units pro- luce units ; tens into units produce tens ; hundreds into units produce hundreds, &c. Hence, When the multiplier is units, the product will always be of the same order as the figure multiplied. 12. What cost 83 pounds of opium, at 8 dollars per pound ? 13. At 9 shillings per day, how much can a man earn in 213 days? 14. If 1 sofa costs 78 dollars, Mbw much will 8 sofa* cost? 1 5. What cost 879 barrels of flour, at 7 dollars a barrel? 16. At 8 shillings apiece, what will a drove of 650 lambs come to ? 17. 18. 19. 20. Multiply 8006 76030 10906 4608790 By 5 8 7 9 21. What will 26 horses cost, at 113 dollars apiece? Suggestion. Reasoning as before, if 1 horse costs 113 dollars, 26 horses will cost 26 times as much. Since it is not conven- Operation. lent to multiply by 26 at 113 Multiplicand, once, we first multiply 26 Multiplier, oy the 6 uftits, then by -g78 cost of 6 horses, the 2 tens and add the ggS* cost of 20 " two results together. Thus 6 times 3 are 18; Ans - 2938 cost of 26 " set doAvn the 8 and carry the 1, as above. 6 times 1 are 6, and 1 to carry makes 7. .6 times 1 are 6. Next, mul- tiply by the 2 tens thus : 20 "times 3 unitb are 60 units or 6 tens ; or we may simply say, 2 times 3 are 6. Now the 6 must denote tens ; for units into tens, or what is QUEST. 51. What do units multiplied into units produce ? Tana into units ? Of what order is the product universally, when the multl- olier is units ? 64 MULTIPLICATION. [SECT. IV the same thing, (Art. 47,) tens into units, produces tens . consequently the 6 must be written in tens' place in the product; that is, under the figure 2 by which we are multiplying. 20 times 1 ten are 20 tens or 200 ; or sim- ply say, 2 times 1 are 2 : and since the 2 denotes hun- dreds, as we have just seen, set it on the left of the 6 in hundreds' place. 20 times 1 hundred are 20 hundred 01 2000 ; or simply say, 2 times 1 are 2 : and since the 2 denotes thousands, set it in the thousands' place on the left of the last figure in the product. Finally, adding these two results together as they stand, units to units, tens to tens, &c., we have 2938 dollars, which is the whole pro- duct required. Note. The several products of the multiplicand into the separata figures of the multiplier, are called partial products. Hence, 52. When the multiplier contains more than one figure. Multiply each figure of the, multiplicand by each figure of the multiplier separately, and write, each partial product in a separate line, placing the first figure of each line directly under that by which you multiply; finally, add the several partial products together, and the sum will be the true pro- duct or answer required. 53. PROOF. Multiply the multiplier by the multipli- cand, and if the product thus obtained is the same as the other product, the work is supposed to be right. OBS 1. This method of proof depends upon the principle, that the product of any two numbers is the same, whichever is* taken for the multiplier. (Art. 47.) 2. When the multiplier is small, we may add the multiplicand to itself as many times as there are units in the multiplier, and if the sum is equal to the product, the work is right. Thus 78X3=234. Proof. 78-|-78-(-78=:234, which is the same as the product. 3. Multiplication may also be proved by division, and by casting yitt the nines ; but neither of these methods can be explained her* QUEST. Note. What is meant by partial products ? 52. How da you proceed when the multiplier contains more than one figure ? How should the partial products be written ? Where write the first figure of each line ? What do you finally do with the partial products ? 53. How is multiplication proved ? Obs. On what principle c method of proof depend ? When the multiplier is small, hew )le does tliii may we prove it I ARTS. 52, 53.] MULTIPLICATION. 65 without anticipating principles belonging to division, with which the learner is supposed as yet to be unacquainted. 22. What will 45 cows cost, at 27 dollars a head? Operation. Proof. 45 Multiplicand, 27 27 Multiplier, 45 315 135 90 108 1215 Product. 1215 Product. 23. What cost 63 hats, at 36 shillings apiece ? 24. How much corn can a man raise on 87 acres, at 45 bushels per acre ? 25. How many pounds of sugar will 75 boxes contain, if each box holds 256 pounds ? 26. What cost 278 hogsheads of molasses, at 23 dol tars per hogshead ? 27. What is the product of 347 multiplied by 256? Operation. Suggestion. Proceed in the same 347 manner as when the multiplier con- 256 tains but two figures, remembering to 2082 place the right hand figure of each 1735 partial product directly under the fig- 594 ure by which you multiply. . 28. What is the product of 569 into 308 ? After multiplying by the 8 units, Operation. ve must next -multiply by the 3 hun- 569 dreds, since there are no tens in the 308 multiplier, and place the first figure of this partial product directly under the figure 3 by which we are multi- plying. 29. What is the product of 67025 into 4005 ? AM. 268435125. 80. What is the product of 841072 mto 603 2 6 MULTIPLICATION. [SECT. IV 54. From the preceding illustrations and principles we derive the following GENERAL RULE FOR MULTIPLICATION. I. Write the multiplier under the multiplicand, units under units, tens under tens, fyc. II. When the multiplier contains but one figure. Begin with, the units, and multiply each figure of the multiplicand by the multiplier, setting down the result and carrying as in addition. (Art. 49.) III. When the multiplier contains more than one figure. Multiply each figure of the multiplicand by each figure of the multiplier separately, beginning at the right hand, and write the partial products in separate lines, placing the first figure of each line directly under the figure by which you multiply. (Art. 52.) Finally, add the several partial product* together, and- the sum will be the whole product. OBS. It is immaterial as to the result which of the factors is taken for the multiplier. (Art. 47.) But it is more convenient and therefore customary to place the larger for the multiplicand and the smaller for the multiplier. Thus, it is easier to multiply 254672381 by 7, than it is to multiply 7 by 254672381, but the product will be the same. EXAMPLES FOR PRACTICE. 1. What will 465 hats cost, at 6 dollars apiece? 2. What will 638 sheep cost, at 4 dollars a head? 3. What will 1360 yards of cloth cost, at 7 dollars a yard? 4. What cost 169 bushels of potatoes, at 4 shillings per bushel ? 5. What cost 279 barrels of salt, at 9 shillings a barrel ? 6. At 12 dollars a suit, how much will it cost to fur- nish 1161 soldiers with a suit of clothes apiece? 7. What cost 1565 acres of wild land, at 7 dollars pel acre? QUEST. 54. What is the general rule for multiplication? Obs. Which number is usually taken for the multiplicand 1 A.E.T. 54.] MULTIPLICATION. 67 8. What will 758 baskets of peaches cost, at 5 dollars per basket? 9. What cost 25650 pounds of opium, at 6 dollars a pound ? 10. How much can a man earn in 12 months, at 15 dollars per month ? 11. What will 23 loads of hay come to, at 18 dollars a load? 12. What will 45 cows come to, at 21 dollars apiece? 13. What will 56 hogsheads of molasses cost, at 32 dollars a hogshead ? 14. What cost 128 firkins of butter, at 13 dollars a firkin ? 1 5. What cost 97 kegs of tobacco, at 26 dollars per keg ? 16. What cost 110 barrels of pork, at 19 dollars per barrel ? 17. How much will 235 sheep come to, at 21 shillings ahead? 18. How many bushels of corn will grow on 83 acres, at the average rate of 37 bushels to an acre ? 19. In one bushel there are 32 quarts: how many quarts are there in 92 bushels ? 20. What will a drove of 463 cattle come to, at 48 dol- lars per head ? 21. How much will 78 thousand of boards cost, at 19 dollars per thousand ? 22. What cost 243 chests of tea, at 37 dollars per chest? 23. A man bought 168 horses, at 63 dollars apiece : what did they come to ? 24. What cost 256 barrels of beef, at 16 dollars a barrel ? 25. If 376 men can build a fortification in 95 days, how long would it take 1 man to build it ? 26. Allowing 365 days to a year, how many days has a man lived who is 45 years old? 27. If a garrison consume 725 pounds of beef in one day, how many pounds will they consume in 125 days? 28. How many pounds will the same garrison con- eurae in 243 days 2 MULTIPLICATION. [SECT. IV. 29. How far will a ship sail in 365 days, at 215 miles per day? 30. What costs 678 tons of Railroad iron, at 115 dol- lars per ton ? CONTRACTIONS IN MULTIPLICATION. 55. The general rule is adequate to the solution of all examples that occur in multiplication. In many in- stances, however, by the exercise of judgment in apply- ing the preceding principles, the operation may be very much abridged. CASE I. When t/ie multiplier is a composite number. Ex. 1. What will 14 hats cost, at 8 dollars apiece? Analysis. Since 14 is twice as much as 7 ; that is, 14=7x2, it is manifest that 14 hats will cost twice as much as 7 hats. Instead of multiplying by 7 14, we may first find the cost of 7 hats, and then 56 cost of 7 hats. multiply that product by 2, which will give the cost of Dolls. 112 cost of 14 hats. 14 hats. In other words, we may first multiply by the factor 7. and that product by 2, the other factor of 14. Proof. 14x8=112, the same as before. 2. What will 27 horses cost, at 85 dollars apiece ? Suggestion. Find the factors of 27 ; that is, find two numbers, which being multiplied together, produce 27, and multiply first by one of these factors, and the product thus arising by the other. OBS. 1. Any number which may be produced by multiplying tw or more numbers together, is called a composite number, and the fac- tors, which being multiplied together, produce the composite number, are sometimes called the component parts of the number. Thus, 14 27, 32, &c., are composite numbers, and the factors 7 and 2, 9 and 3 8 and 4, are their component parts. ARTS. 55-57.] MULTIPLICATION. 69 2. The process of finding the factors of which a given number is composed, is called resolving the number into factors. 56. Some numbers may be resolved into wore than two factors ; and also into different sets of factors. Thus, the factors of 24 are 3, 2, 2 and 2 ; or 4, 3 and 2 ; or 6, 2 arid 2 ; or 8 and 3 ; or 6 and 4 ; or 12 and 2. OBS. We have seen that the product of any two numbers is the same, whichever factor is taken for the multiplier. (Art. 47.) In like manner, the product of any tkree or more factors is the same, in whatever order they are multiplied. For, the product of two factors, may be considered as one number, and this may be taken either for the multiplicand, or the multiplier. Again, the product of three fac- tors may be considered as one number, and be taken for the multipli- cand, or the multiplier, &c. Thus, 24=3X2X2X2=0X2X2=12 X 2=6X4=4X2X.3=8X3. 3. What will 24 hogsheads of molasses cost, at 37 dol- lars per hogshead? Ans. 838 dollars. Suggestion. Resolve 24 into any two or mare factors, and proceed as before. Hence, 57. To multiply by a composite number. Resolve the multiplier into two or more factors; multi- ply the multiplicand by one of these factors, and this pro- duct by another factor, and so on till you have multiplied by all the factors. The last product will be t/ie product required. OBS. The factors into which a number may be resolved, must not be confounded with the parts into which it may be separated. (Art. 26.) The former have reference to multiplication, the latter to ad- lition ; that is, factors must be multiplied together, but parts must be idded together to produce the given number. Thus, 56 may be re- solved into two factors, 8 and 7 ; it may be separated into two parts, 5 tens or 50, and 6. Now 8Xf 56, and 50-j-6=56. 4. What will 36 cows cost, at 19 dollars a head? QUEST. Obs. What is a composite number? What are the factors which produce it, sometimes called ? What is meant by resolving a number into factors ? 56. Are numbers ever composed of more than two factors ? What are the factors of 24 ? 32 ? 36 ? 40 ? 42 ? 60 ? 64 ? 72 ? 108 ? Obs. When three or more factors are to be multiplied to- gether, does it make any difference in what order they are taken ? 57. When the multiplier'is a composite number, how do you proceed ? Obs. What is the difference between the factors into which a number may be resolved, and the parts into which it may bo separated ? TO MULTIPLICATION. [SECT. IV, 5. What cost 45 acres of land, at 110 dollars per acre? 6. At 36 shillings per week, how much will it cost a person to board 52 weeks ? 7. If a man travels at the rate of 42 miles a day, how far can he travel in 205 days ? 8. At the rate of 56 bushels per acre, how much corn can be raised on 460 acres of land 1 9. What cost 672 yards of broadcloth, at 24 shillings per yard ? 10. What cost 1265 yoke of oxen, at 72 dollars per yoke ? CASE II. When the multiplier is 1 with ciphers annex- ed to it. 58. It is a fundamental principle of notation, that each removal of a figure one place towards the left, increases its value ten times; (Art. 9 ;) consequently, annexing a cipher to a number will increase its value ten times, or multiply it by 10; annexing two ciphers, will increase its value a hundred times, or multiply it by 100; annex- ing three ciphers will increase it a thousand times, or multiply it by 1000, &c. ; for each cipher annexed, re- moves each figure in the number one place towards the left. Thus, 12 with a cipher annexed, becomes 120, and is the same as 12x10; 12 with two ciphers an- nexed, becomes 1200, and is the same as 12x100; 12 with Hire* ciphers annexed, becomes 12000, and is the same as 12x1000, &c. Hence, 59. To multiply by 10, 100. 1000, &c. Annex as many ciphers to the multiplicand as there are ciphers in the multiplier, and the number thus formed will he the product required. Note. To annex means to place after, or at the right hand. 1 1. What will 10 drums of figs weigh, at 28 pounds a drum 1 Ans. 280 pounds. QUEST. 58. What effect does it have to remove a figure one place towards the left hand ? Two places ? 59. How do you proceed when the multiplier is 10, 100, 1000, &c. I Note. What is the meaning of ?b* term ami** 7 \RTS. 58-60.] MULTIPLICATION. 71 12. How many pages are there in 100 booics, each book having 352 pages'? 13. Multiply 476 by 1000. 14. Multiply 53486 by 10000. 15. Multiply 12046708 by 100000. 16. Multiply 26900785 by 1000000. 17. Multiply 89063457 by 10000000. 18. Multiply 9460305068 by 100000. 19. Multiply 78312065073 by 10000. CASE III. When the multiplier has ciphers on ike right. 20. What will 20 acres of land cost, at 32 dollars per acre? Note. Any number with ciphers on its right hand, is obviously a romposite number ; the significant figure or figures being one factor, and 1 with the given ciphers annexed to it, the other factor. Thus 20 may be resolved into the factors '2 and 1 0. We may therefore first multiply by 2 and then by 10, by annexing a cipher as above. Solwtwn. 32x2=64, and 64x10-640 dolls. Ans. 21. If the expenses of an army are 2000 dollars per day, what will it cost to support the same army 365 days? n . 2000 may be resolved into the factors ^ and 365 1 ? 00 ' Then 2 times 3( ? 5 are 73 5 now ? d - o ding three ciphers to this product, multiplies it by 1000, (Art. 59,) and we have 730000 730000 (10^^ f or the answer. Hence, 6O. When there are ciphers on the right hand of the multiplier. Multiply the multiplicand by the significant figures of the multiplier, and to this product annex as many ciplters as are found on the right hand of the multiplier. OF,S. It will be perceived that this case combines the principles of the two preceding cases; for, the multiplier is a composite number and one of its factors is 1 with ciphers annexed to it. QUEST. GO. When there are ciphers on the riaht of the multiplier, ho'.v do you proceed ! O.V. What principles a~e. combine*! in this r,ase t 72 MULTIPLICATION. [SECT. IV 22. How many days are there in 36 months, reckon ing 30 days to a month ? 23. If 1 barrel of flour weighs 192 pounds, how much will 200 barrels weigh ? 24. Multiply 4376 by 2500. 25. Multiply 50634 by 4 1000. 26. Multiply 630125 by 620000. CASE Ju When the multiplicand has cipJiers on tfo. right. 27. Multiply 12000 by 31. Suggestion. 12000 is a composite Operation. number, the factors of \vhich are 12 and 12000 1000. But the product of two or more 31 numbers is the same in whatever order j2 they are multiplied ; (Art. 47 ;) conse- gg quently multiplying the factor 12 by 31, and this product by 1000, will give the A - 3 ' 20UO same result as 12000x31. Thus, 31 times 12 are 372; then annexing three ciphers, we have 372000, which is the same as 12000x31. Hence, 6 1 . When there are ciphers on the right of the mul- tiplicand. Multiply the significant figures of the multiplicand by the multiplier, and to the product annex as many ciphers as are found on the right of the multiplicand. OBS. When the multiplier and multiplicand both have ciphers on the right, multiply the significant figures together, and to their pro- duct annex as many ciphers as are found on the right of both factors. 28. Multiply 370000 by 32. 29. Multiply 8120000 by 46. 30. Multiply 56300000 by 64. 31. Multiply 623000000 by 89. 32. Multiply 54000 by 700. Ans. 37800000. QUEST. 61. When there are ciphers on the right hand of the multi- plicand, how proceed ? Obs. How, when there are ciphers en th right both of the multiplier and multiplicand ? ARTS. 61-63.] DIVISION. 73 33. Multiply 4300 by 600. Ans. 2580000. 34. Multiply 563800 by 7200. 35. Multiply 1230000 by 12000. 36. Multiply 310200 by 20000. 37. Multiply 2065000 by 810000. 38. Multiply 2109090 by 510000. SECTION V. DIVISION. MENTAL EXERCISES. ART. 63. Ex. 1. How many oranges, at 3 cents apiece, can you buy for 12 cents'? Suggestion. If 3 cents buy one orange, 12 cents will buy as many oranges as there are 3 cents in 12 cents; that is, as many as 3 is contained times in 12. Now 3 is contained in 12, 4 times. .Arcs. 4 oranges. 2. How many lemons, at 4 cents apiece, can you bu> for 20 cents ? Suggestion. To find how many times 4 cents are contained in 20 cents, think how many times 4 make 20, or what number multiplied by 4, produces 20. 3. At 3 dollars per yard, how many yards of cloth can be bought for 15 dollars? 4. How many hats, at 5 dollars apiece, can you buy for 30 dollars ? 5. How many barrels of flour will 36 bushels of wheat make, allowing 4 bushels to one barrel ? 6. If you pay 6 cents a mile for riding in a stage, h j k v far can you ride for 48 cents? 7. If a pound of sugar cost 7 cents, how many pounds can you buy for 56 cents. 74 DIVISION. [SECT. V 8. How many slates, at 8 cents apiece, can you buy for 40 cents ? 9. Four quarts make one gallon : how many gallons are there in 48 quarts ? 10. At 7 dollars a ton, how many tons of coal can b bought for 63 dollars ? DIVISION TABLE. 2 in 3in 4 in 5 in 6 in 7 in 8 in 9 in | 2, once 3,once 4, once 5, once 6,once 7, once 8, once 9, once) 4, 2 6, 2 8, 2 10, 2 12, 2 14, 2 ]6, 2 18, 2 6, 3 9, 3 12, 3 15, 3 18, 3 21, 3 24, 3 27, 3. 8, 4 12, 4 16, 4 20, 4 24, 4 28, 4 32, 4 36, 4 10, 5 15, 5 20, 5 25, 5 30, 5 35, 5 40, 5 45, 5 12, 6 18, 6 24, 6 30, 6 36, 6 42, 6 48, 6 54, 6 14, 7 21, 7 28, 7 35, 7 42, 7 49, 7 56, 7 63, 7) 16, 8 24, 8 32, 8 40, 8 48, 8 56, 8 64, 8 72, s; 18, 9 27, 9 36, 9 45, 9 54, 9 63, 9 72, 9 81, 9: 11. How many pair of boots, at 2 dollars a pair, can be bought for 24 dollars? for 22? 20? 18? 16?' 14? 12? 10? 12. How many barrels of cider, at 3 dollars a barrel, can you buy for 36 dollars? for 30? 27? 24? 21? 18? 15? 12? 13. How many quarts of milk, at 4 cents a quart, can you buy for 48 cents? for 44? 40? 36? 32? 28? 24? 20? 16? 14. At 5 cents an ounce, how many ounces of wafers can you buy for 60 cents ? for 55 ? 50 ? 45 ? 40 ? 35 ? 30? 25? 15. At 6 shillings a pair, how many pair of gloves can be bought for 60 shillings? for 54? 48? 42? 36? 30? 24? 18? 16. How many pounds of butter, at 7 cents a pound, can be purchased for 63 cents? 56? 49? 42? 35? 28? 21? 14? 17. How many cloaks will 72 yards of cloth make, allowing 8 yards to a cloak ? how many 64 ? 56 ? 48 ? 40? 32? 24? 18. How many cows, at 9 dollars apiece, can be ART. 63.] DIVISION. 75 bought for 81 dollars? for 72? 63? 54? 45? 36? 27? 18? 9? 19. How many times is 4 contained in 36 ? 48 ? 40 ? 20. How many times is 8 contained in 40? 56? 48? 64? 72? 21. In 25, how many-'times 4, and how many over? Ans. 6 times and 1 over. 22. In 34, how many times 5, and how many over ? In 43? 45? 37? 28? 39? 23. In 23, how many times 3, and how many over ? How many times 4 ? 2 ? 10 ? 6 ? 24. In 24, how many times 7, and how many over ? 6? 5? 9? 12? 2? 25. In 36, how many times 6? 7? 3? 8? 12? 5? 9? 26. In 32, how many times 6? 4? 3? 16? 27. How many hats, at 6 dollars apiece, can be bought for 60 dollars? 28. How many tons of hay, at 9 dollars per ton, can you buy for 81 dollars. 29. If you travel 7 miles an hour, how long will it fake to travel 70 miles ? 30. If you pay 10 cents apiece for slates, how many can you buy for 95 cents, and how many cents over ? 31. George bought 12 oranges, Avhich he wishes to di- vide equally between his 2 brothers : how many can he give to each? Suggestion. Since there are 12 oranges to be divided equally between 2 boys, each boy must receive 1 orange as often as 2 oranges are contained in 12 oranges; that is. each must receive as many oranges as 2 is contained times in 12. But 2 is contained in 12, 6 times; for 6 times 2 make 12. Ans. 6 oranges. 32. Henry has 15 apples, which he wishes to divide equally among 3 01 his companions : how many can he give to each? 33. A gentleman sent 20 peaches to be divided equally among 4 boys : how many did each boy receive ? 34. A dairy-woman having 30 pounds of butter, wish- 76 DIVISION. [SECT. V. es to pack it in 5 boxes, so that each box shall have an equal number of pounds : how many pounds must she put in each box ? 35. I have 21 acres of land, which I wish to fence into 7 equal lots : how many acres must I put into each lot? 36. A boy having 28 marbles, wished to divide them into 4 equal piles : how many must he put in a pile ? 37. I have 40 peach-trees, which I wish to set out in 5 3qual rows : how many must I sgt in a row ? 38. There were 45 scholars in a certain school, and the teacher divided them into 5 equal classes : how many did he put in a class ? 39. If 50 dollars were divided equally among- 10 men, how many dollars would each man receive ? 40. A company of 8 boys buying a boat for 32 dollars, agreed to share the expense equally: how much must each one pay? 41. In a certain orchard there are 54 apple-trees, and 5 trees in each row : how many rows are there in the orchard ? 42. If 63 quills are divided equally among 7 pupils, how many will each receive ? 43. If you divide 36 into 4 equal parts, how many will there be in a part ? 44. If you divide 56 into 8 equal parts, how many will each part contain ? 45. If you divide 48 into 6 equal parts, how many will each part contain ? 46. A gentleman distributed 40 dollars equally among 8 beggars : how many dollars did he give to each ? 47. A company of 6 boys found a pocket-book, and on returning it to its owner, he handed them 60 dollars to be shared equally among them : what was each one's share ? 48. A merchant received 72 dollars for 6 coats of equal value : how much was that apiece 1 49. A man paid 81 cents for the use of a horse and ouggy to ride 9 miles : how much was that a mile ? 50. If you divide 90 dollars into 10 equal paits, how many dollars will there be in each part ? . 64-o7.] DIVISION. 77 OBS. The object in each of the last twenty questions, is to divide a given number into several equal parts, and ascertain the value of these parts ; but the method of solving them is precisely the same as that of the preceding ones. 64. The process by which the foregoing examples are solved, is called DIVISION. It consists in finding how many times one given number is contained in another. The number to be divided, is called the dividend. The number by which we divide, is called the divisor. The number obtained by division, or the answer to the question, is called the quotient. It shows how many times the dividend contains the divisor. Hence, it may be said 6 5 Division is finding a quotient, which multiplied in- to the divisor, will produce the dividend. Note. The term quotient is derived from the Latin word quatiet, which signifies how often, or how many times. 66. The number which is sometimes left after divis- ion, is called the remainder. Thus, in the twenty-first ex- ample, when we say 4 is contained in 25, 6 times and 1 over, 4 is the divisor, 25 the dividend, 6 the quotient, and 1 the remainder. OBS. 1. The remainder is always less than the divisor; for if it vere equal to, or greater than the divisor, the divisor could be con- fined once more in the dividend. 2. The remainder is also of the same denomination as the divi- dend ; for it is a part of it. 67. Division is denoted in two ways : QUEST. 64. In what does division consist? What is the number to be divided, called ? The number by which we divide ? What is the number obtained, called ? What does the quotient show ? 65. What then may division be said to be? 66. What is the number called which is sometimes left after division ? When we say 4 is in 25, 6 times and 1 over, what is the 4 called ? The 25 ? The 6 ? The 1 ? When we say 6 is in 45, 7 times and 3 over, which is the divisor? The dividend ? The quotient? The remainder. Obs. Is the remain- der greater or less than the divisor ? Why ? Of what denomination is it 1 Why ? 67. How many ways is division denoted ? 78 DIVISION. [SECT. V. First, by a horizontal line between two dots (-*-), called the sign of division, which shows that the number be- fore it, is to be divided by the number after it. Thus the expression 24-J-6, signifies that 24 is to be divided by 6. Second, division is often expressed by placing the di visor under the dividend with a short line* between them Thus the expression 3 T 5 , shows that 35 is to be divided by 7, and is equivalent to 35-*-7. OES. It will be perceived that division is similar in principle to sub- traction, and may be performed by it. For instance, to find how many times 3 is contained in 12, as in the first example, subtract 3 (the divisor) continually from 12 (the dividend) until the latter is ex- hausted ; then counting these repeated subtractions, we shall have the true quotient. Thus, 3 from 12 leaves 9 ; 3 from 9 leaves 6 ; 3 from 6 leaves 3 ; 3 from 3 leaves 0. Now by counting, we find that 3 can be taken from 12, 4 times; or that 3 is contained in 12, 4 times. Hence, 6 7 a. Division is sometimes defined to be a short way of 'performing repeated subtractions of the same number. OBS. 1. It will also be observed that division is the reverse of mul- tiplication. Multiplication is the repeated addition of the same num- ber ; division is the repeated subtraction, of the same number. Tko product of the one answers to the dividend of the other : but the lat- ter is always given, while the former is required. 2. When the dividend denotes things of one denomination only, the operation is called Simple Division. EXERCISES FOR THE SLATE. Ex. 1. How many barrels of cider, at 2 dollars a bar. rel. can you buy for 648 dollars ? Suggestlm. Since 2 dollars will buy 1 barrel, 648 dol lars will buy as many barrels as 2 is contained times in 648. QUEST. What is the first ? What does this sign show ? What ia the second way of denoting division ? Obs. To what rule is division similar in principle ? How is division sometimes defined ? Of what is division the reverse ? How does this appear ? When the dividend denotes things of one denomination only, what is the operation called ? ARTS. 68, 69.] DIVISION. 79 Having written the numbers upon the Operation. slate, as in the margin, we proceed thus : Divig0 r. Dividend. 2 is contained in 6, 3 times. Now as the 2)648 6 denotes hundreds, the 3 must also be ^ ' f hundreds. We therefore write it in hun- ^ uot " * * * dreds' place ; that is, under the figure which we are di- viding. 2 in 4, 2 times. Since the 4 is tens, the 2 must also be tens, and we write it in tens' place. 2 in 8, 4 times. The 8 is units ; hence the 4 must be units, and we write it in units' place. The answer is 324 barrels. 2. Divide 63 by 7. Ans. 9. 3. Divide 56 by 8. 4. Divide 42 by 7. 5. Divide 54 by 9. 6. Divide 72 by 8. 7. How many hats, at 2 dollars apiece, can be bought for 468 dollars ? Ans. 234 hats. 8. How many sheep, at 3 dollars a head, can be bought for 369 dollars? 9. A man wishes to divide 248 acres of land equally etween his two sons : how many acres will each receive 1 10. How many times is 4 contained in 488 ? 68. Hence, when the divisor contains but one figure, Write the divisor on the left, hand of the dividend with a curve line between them ; then, beginning at the left hand, divide each figure of the dividend by the divisor, and set each quotient figure directly under the figure from which it arose. 11. A farmer bought 96 dollars worth of dry goods, and agreed to pay in wood at 3 dollars a cord : how many cords will it take to pay his bill ? Ans. 32 cords. 12. In 963 feet, how many yards are there, allowing 3 feet to a yard ? 13. Divide 63936 by 3. 14. Divide 48848 by 4. 15. Divide 55555 by 5. 16. Divide 2486286 by 2 69. When the divisor is not contained in the first QUEST. 68. How do you write the numbers for division ? Where begin to divide ? Where place each quotient figure ? 69 When the divisor is not contained in the first figure of the dividend, what must he done ? 80 DIVISION. [SECT. V, figure of the dividend, we must find how many times i 1 is contained in the first two figures. 1 7. How many hats, at 3 dollars apiece, can be bough for 249 dollars ? Operation. Since the divisor 3, is not contained in 3)249 ^ the first figure of the dividend, we say 3 is in 24, 8 times, and write the 8 under , , B , the 4 3 in 9j 3 times AnSm 83 hats 18. Divide 124 by 4. 19. Divide 366 by 6. 20. Divide 255 by 5. 21. Divide 1248 by 4. 22. Divide 24693 by 3. 23. Divide 4266 by 6. 24. Divide 35555 by 5. 25. Divide 5677 by 7. 26. Divide 64888 by 8. 27. Divide 8199 by 9. 7 O. After dividing any figure of the dividend, if there is a remainder, prefix it mentally to the next figure of the dividend, and then divide this number as before. Note. To prefix means to place before, or at the left hand. 28. A man bought 741 acres of land, which he divi- ded equally among his 3 sons : how many acres did each receive ? Operation. When \ve divide 7 by 3, there is 1 3)741 remainder. This we prefix mentally A - 9A7 to me next % ure f tne dividend. We then say, 3 in 14, 4 times, and 2 over. Prefixing the remainder 2 to the next figure, as be fore, we say, 3 in 21, 7 times. 29. If a man travel at the rate of 5 miles an hour, how long will it take him to travel 345 miles ? Ans. 69 hours. 30. If 192 pounds of flour were equally divided among 4 persons, how many pounds would each receive ? 31. Divide 45690 by 6. 32. Divide 52584 by 8. 33. Divide 81670 by 5. 34. Divide 28296 by 9. 35. When flour is 6 dollars a barrel, how much cas. be bought for 642 dollars ? QUEST. 70. If there is a remainder after dividing a figure, of th' IIAH hrn solved by short division. (Art. 70. Ex. 28.) We have hitnxkuvd it here for the purpose of illustrating a different mode of dividing. Having written the divisor on the Operation. luft of the dividend as before, we find Divisor. Divid. Quot. ;) is contained in 7, 2 times, and place 3) 741 (247 the 2 on the right of the dividend, 5' with a curve line between them. We -r\ next multiply the (''visor bv this quo- .^ lient figure 2 times 3 are 6 and, placing the product under the 7, the figure divided, subtract it therefrom. We now bring down the next figure of the dividend, and placing it on the right of the remain- der I, we have 14. And 3 is in 14, 4 times. Set the 4 QUEST. 74. How is multiplication proved by division ? Ols. Upon what principle does this proof depend ? How are the numbers written for long division ? Where begin to divide ? Where is the quotient placed 1 84 DIVISION. [SECT. \ on the right hand of the last quotient figure, and multi ply the divisor by it: 4 times 3 are 12. Write the pro- duct under 14, and subtract as before. Finally, bringing down the last figure of the dividend to the right of the last remainder, we have 21 ; and 3 is in 21, 7 times. Set the 7 in the quotient, then multiply and subtract as before. The quotient is 247, the same as in short division. 75. This method of dividing is called Long Division. It is the same in principle as Short Division. The only difference between them is, that in Long Division the result o'f each step in the operation is written down, while in Short Division we carry on the process in the mind, and simply write the quotient. Note. To prevent mistakes, it is advisable to put a dot under each figure of the dividend, when it is brought down. The following questions are designed to be performed by long division, and each operation should be proved. 2. How many times is 2 contained in 578 ? Ans. 289. 3. How many times is 5 contained in 7560 ? Ans. 1512. 4. How many times is 4 contained in 126332 ? Ans. 31583. 5. How many times is 6 contained in 763251 ? 6. How many times is 3 contained in 4026942 ? 7. How many times is 8 contained in 2612488? 8. How many times is 5 contained in 1682840? 9. How many times is 7 contained in 45063284 ? 10. How many times is 9 contained in 650031507? 11. Divide 2234 by 21. Operation, 21 is contained in 22 once. 21)2234(106- 8 -. Ans. Write the 1 in the quotient. Then 2i multiplying and subtracting, the -r^7 remainder is 1. Bringing down the next figure, we have 1 3 to be divided by 21. But 21 is not con- 8 rem - tained in 13, therefore we put a QUEST. 75. What is the difference between long and short divisio*. ' ARTS. 75-77. J DIVISION. 85 cipher in the quotient, (Art. 70. Obs.) and bring 1 down the next figure. Then, 21 in 134, 6 times, and 8 remain- der. Write the 8 over the divisor, and annex it to the quotient. (Art. 71.) 76. After the first quotient figure is obtained, formed figure of the dividend which is brought down, either a sig- nificant figure or a cipher must be put in the quotient. 12. Divide 345 by 15. Ans. 23. 13. Divide 5378 by 25. Ans. 215 JL. 14. Divide 7840 by 32. 16. Divide 59690 by 45. 1G. Divide 81229 by 67. 17. Divide 99435 by 81. 18. How many times is 131 contained in 18602? Ans. 142. OBS. When the divisor is not contained in the first two figures of the dividend, find how many times it is contained in the first three ; and, generally, find how many times it is contained in the fewest fig- ures which will contain it, and proceed as before. 19. How many times is 93 contained in 100469 ? 20. How many times is 156 contained in 140672? 77. From the preceding principles we derive the fol- lowing RULE FOR LONG DIVISION. Begin on the left of the dividend, find how. many times the divisor is contained in the fewest figures thai, will con- tain it, and 'place the quotient figure on the right of the dividend with a curve line between tJiem. Then multiply the divisor by this figure and subtract the product from the fig- ures divided ; to the right of the remainder bring down the next figure of the dividend and divide this number as before. Proceed in this manner till all the figures of the dividend are diviflcd. When there is a remainder after dividing the last figure, write it over the divisor and annex it to the quotient, as in thort division. (Art. 71.) QUEST. 76. What is placed in the quotient, on bringing down each figure of the dividend ? Obs. When the divisor is not contained in the first two figures of the dividend, what is to be dune ? 77. What is the rule for long division ? 86 DIVISION. [SECT. V. OBS. When the divisor contains? but one figure, the operation by Short Division is the most expeditious, and should therefore be prac- ticed ; but when the divisor contains two or more figures, it will ge- nerally be the most convenient to divide by Long Division, EXAMPLES FOR PRACTICE. 1. If a man travel at the rate of 8 miles an hour, how long will it take him to travel 192 miles? 2. How many yards of broadcloth, at 9 dollars a yard, ;an be bought for 324 dollars 1 3. A farmer bought a lot of young cattle, at 1 1 dollars per head, and paid 473 dollars for them : how many did he buy 1 4. How many tons of coal, at 7 dollars a ton, can be bought for 756 dollars ? 5. At 12 dollars a month, how long will it take a man to earn 156 dollars? 6. In one day there are 24 hours : how many days are there in 480 hours ? 7. A man traveled 215 miles in 21 hours : how many miles did he travel per hour ? 8. At 16 dollars a ton, how many tons of hay can be bought for 176 dollars? 9. How many casks of wine, at 25 dollars a cask, can be bought for 275 dollars ? 10. The ship George Washington was 25 days in cross- ing the Atlantic Ocean, a distance of 3000 miles. How many miles did the ship sail per day ? 11. The steamer Great Western crossed it in 15 days. How many miles did she sail per day ? 12. The steamer Caledonia crossed it in 12 days. How many miles did she sail per day? 13. If a man can earn 32 dollars a month, how long will it take him to earn 420 dollars ? 14. If 63 gallons make a hogshead, how many hogs- heads will 1260 gallons make? 15. If a ship can sail 264 miles per day, how far can "she sail in an hour? QURST. Obs. When should short division be used* When long division ? ART. 77. a.] DIVISION. 87 16. How many tiriles 12 in 172, and how many over ? 17. How many times 15 in 630, and how many over? 18. How many times 22 in 865, and how many over? 19. 1236 is how many times 17, and how many over ? 20. 7652 is how many times 13, and how many over? 21. 3061 is how many times 125, and how many over? 22. 1861 is how many times 231, and how many over? 23. 8 times 256 is how many times 9 ? 24. 12 times 157 is how many times 7? 25. 15 times 2251 is how many times 12 ? 26. 19 times 136 is how many times 75 ? 27. 63 times 102 is how many times 37 ? 28. 78 times 276 is how many times 136? 29. 115 times 321 is how many times 95? 30. 144 times 137 is how many times 312? CONTRACTIONS IN DIVISION. 7 7 a. The operations in division, as well as in mul implication, may often be shortened by a careful attention ,o the application of the preceding principles. CASE I. When the. divisor is a composite number. Ex. 1. A gentleman divided 168 oranges equally among 14 grandchildren who belonged to 2 families, each family containing 7 children : how many oranges did he give to each child ? Suggestion. First find how many each family received, I hen how many each child received. If 2 families receive 168 oranges, 1 fami- ^ . y will receive as many oranges, as 2 is ? er contained times in 168, viz: 84. But there 2)168 are 7 children in each family. If then 7 7)84 children receive 84 oranges, 1 child will ~^ Ans receive as many, as 7 is contained times in 84, viz : 12. He therefore gave 12 oranges to each child. NOTE. This operation is exactly the reverse of that in Ex. 1. Art. 55. The divisor 14 being a composite number, we divide first by one of its factors, and the quotient thus found by the other. The final result would have been the same, if we had divided by 7 first, then by 3. Hence, 88 DIVISION. [SECT. V, 78. To divide by a composite riUmber. Divide the dividend by one of the factors of the divisor find the quotient thus obtained by the other factor. The. last quotient will be the answer required. To find the true remainder, should there he any. Multiply the last remainder by the first divisor, and to the. product add the first remainder. OBS. 1. If the divisor can be resolved into more than two factors, we may divide by them successively, as above. *2. To find the true remainder when more than two factors are em- ployed, multiply each remainder by all the preceding divisors, and to the sum of the products add the iirst remainder. 2. Divide 465 by 35. 1 last remainder. Tfirstdiviso, - 7 product. J* first rem. added. 10 true rem. Ans. 3. A teacher having 36 scholars arranged in 4 equal classes, wishes to distribute 216 pears among thein equally : how many can he give to each scholar ? 4. How many cows, at 27 dollars a head, can be bought for 945 dollars 'I 5. How many times is 64 contained in 453 ? 6. How many times is- 72 contained in 237 ? CASE II. When the divisor is \ iwth ciphers annexed to it. 7 9. It has been shown that annexing a cipher to a number, increases its value ten times, or multiplies it by 10. (Art. 58.) Reversing this process; that is, remo ring a cipher from the right hand of a number, will evi- dently diminish its value ten times, or divide it by 10 ; for, QUEST. 78. How proceed when the divisor is a composite number 1 How find the true remainder I Ols. How proceed when the divisor can be resolved into more than two factors ? How find the remaindei in this case ? 79. What is the effect of annexing a cipher to a num- ber ? What is the effect of removing a cipher from the right of a number ? ARTS. 78-80.] DIVISION. 89 each figure in the number is thus restored to its original place, and consequently to its original value. Thus, an- nexing a cipher to 12, it becomes 120, which is the same as 12x10. On the other hand, removing the cipher from 120, it becomes 12, which is the same as 120-*-10. In the same manner it may be shown, that removing two ciphers from the right of a number, divides it by 100 removing three, divides it by 1000 ; removing four, di vides it by 10000, &c. Hence, 8O. To divide by 10, 100, 1000, &c. Cut of as many figures from the right hand of the divi- dend as there are ciphers in the divisor. The remaining figures of tlie dividend will be the quotient, and those cut off the remainder. 7. How many times is 10 contained in 120? Ans. 12. 8. In one dime there are 10 cents : how many dimes are there in 100 cents? In 250 cents? In 380 cents? 9. In one dollar there are 100 cents: how many dol- lars are there in 6500 cents? In 76500 cents? In 432000 cents ? 10. Divide 675000 by 10000. Ans. 67 and 5000 rem. 11. Divide 44360791 by 1000000. 12. Divide 82367180309 by 10000000. CASE III. When the divisor has ciphers on the right. 13. How many acres of land, at 20 dollars per acre, can you buy for 645 dollars ? Analysis. The divisor 20 is a composite number, the factors of which are 2 and 10. (Art. 55. Obs. 1.) We may, therefore, divide first by one factor, and the quo- tient thence arising by the other. (Art. 78.) Now cutting off the right hand figure of the dividend, divides it by 10 ; (Art. 80 ;) consequently, dividing the remaining QWEST. 80. How proceed when the divisor is 10, 100, 1000, & c . *0 DIVISION. [SECT. V figures of the dividend by 2, the other factor of the di visor, will give the true quotient. sy . Cut off the cipher on the right of the divisor ; also cut off the right hand figure of the dividend ; then divide the 64 by 2. The 5 which we cut off, is the re- 32-5 rem. mainder. Ans. 32/ 7 acres. Hence, 8 1 . When there are ciphers on the right hand of the divisor. Cut off the ciphers, also cut off as many figures from the right of the dividend. Then divide the other figures of the dividend by t/ie remaining figures of the divisor, and annex the figures cut off from the dividend to the re- 14. How many horses, at 80 dollars apiece, can you buy for 640 dollars ? 15. How many barrels will 6800 pounds of beef make, allowing 200 pounds to the barrel? 16. How many regiments of 4000 each, can be formed from 840000? 17. Divide 143900 by 2100. 18. Divide 4314670 by 24000. 8 1 a. The four preceding rules, viz : Addition, Sub- traction, Multiplication, and Division, are usually called the FUNDAMENTAL RULES of Arithmetic, because they are the foundation or basis of all arithmetical calculations. GENERAL PRINCIPLES IN DIVISION. 82. From the nature of division, it is evident, that the value of the quotient depends both on the divisor and the dividend. If a given divisor is contained in a given dividend a QUEST. 81. When there are ciphers on the right of the divisor, how proceed ? What is to be done with figures cut off from the dividend ! 81. a. What are the four preceding rules called? Why? 82. Upon what does the value of the quotient depend ? ARTS. 81-85.] DIVISION. 91 certain number of times, the same divisor will obviously be contained, In double that dividend, twice as many times ; In three times tir-it dividend, thrice as many times ; &c. Thus, 4 is contained in 12, 3 times; in 2 times 12 or 24, 4 is contained 6 times ; (i. e. twice 3 times ;) in 3 times 12 or 36, 4 is contained 9 times; (i. e. thrice 3 times ;) &c. Hence, 83 If the divisor remains the same, multiplying the dividend by any number, is in effect multiplying the quotient by that number. Again, if a given divisor is contained in a given divi- dend a certain number of times, the same divisor is con- tained, In half that dividend, half as many times ; In a third of that dividend, a third as many times, &c. Thus, 4 is contained in 24, 6 times ; in 24-*-2 or 12, (rnlf of 24,) 4 is contained 3 times ; (i. e. half of 6 times ;) in 24-*-3 or 8, (a third of 24,) 4 is contained 2 times ; (i. e. a third of 6 times ;) &c. Hence, 4-- If the divisor remains the same, dividing the divi- dend by any number, is in effect dividing the. quotient by that number. If a given divisor is contained in a given dividend a certain number of times, then, in the same dividend, Twice that divisor is contained only half as many times ; Three times that divisor, a third as many times, &c. Thus, 2 is contained in 12, 6 times ; 2 times 2 or 4, is contained in 12, 3 times; (i. e. half of 6 times ;) 3 times 2 or 6, is contained in 12, 2 times ; (i. e. a third of 6 times ;) &c. Hence, 85. If the dividend remains the same, multiplying the divisor by any number, is in effect dividing the quotient by tJiat number. QUEST. 83. If the divisor remains the same, what effect has it on the quotient to multiply the dividend? 84. What is the effect of divi- ding the dividend by any given number ? 85. If the dividend remains the same, what is the effect of multiplying; the divisor by any given number ? 92 DIVISION. [SECT. V If a given divisor is contained in a giver dividend a certain number of times, then, in the same dividend, Half^ that divisor is contained twice as many times ; A third of that divisor, three times a? many times, &c. Thus, 6 is contained in 24, 4 times : 6-*-2 or 3, (half of 6,) is contained in 24, 8 times ; (i. e. twice 4 times ;) 6-*-3 or 2, (a third of 6,) is contained in 24, 12 times ; (i. e. three times 4 times ;) &c. Hence, 86 If the dividend remains the same, dividing the di- visor by any number, is in effect multiplying the quotient by that number. 87. From the preceding articles, it is evident that any given divisor is contained in any given dividend, just a? many times, as twice that divisor is contained in twice thai dividend ; three times that divisor in three times that div- idend, &c. Conversely, any given divisor is contained in any given dividend just as many times, as half that divisor is con- tained in half that dividend ; a third of that divisor, in a third of that "dividend, &c. Thus, 4 is contained in 12, 3 times ; 2 times 4 is contained in 2 times 12, 3 times ; 3 times 4 is contained in 3 times 12, 3 times, &c. Again, 6 is contained in 24, 4 times ; 6-^-2 is contained in 24-J-2, 4 times ; 6-s-3 is contained in 24-5-3, 4 times, &c. Hence 88. If the divisor and dividend are both multiplied, or both divided by the same number, the quotient will not bt altered. 89. If any given number is multiplied and the product divided by the same number, its value will not be altered. rnus, 12x5=60; and 60-7-5=12, the given number. QUEST. 86. What of dividing the divisor ? 88. What is the effect upon the quotient if the divisor and dividend are both multiplied or both divided by the same number \ 89. What is the effect of mlli- plying and dividing any given number by the same number ? ARTS. 86-91.] DIVISION. 93 CANCELATION.* 90. We have seen that division is finding a quotient, which multiplied into the divisor will produce the divi- dend. (Art. 65.) If, therefore, the dividend is resolved into two such factors that one of them is the divisor, the other factor will, of course, be the quotient. Suppose, for example, 42 is divided by 6. Now the factors of 42 are 6 and 7, the first of which being the divisor, the other must be the quotient. Therefore, Canceling a factor of any number, divides the number by that factor. Hence, 91. When the dividend is the product of two or more factors, one of which is the same as the divisor, the division way be performed by CANCELING that factor in the divisor and dividend. (Art. 88.) Note. The term cancel, means to erase or reject. 21. Divide the product of 19 into 25 by 19. Common Method. 19 ne By Cancelation. ^ 10)10X25 38 25 Ans. io\A7^/9c; A Cancel the factor 19, which is com 38 mon both to the divisor and dividend > and 25, the other factor of the dividend, is the quotient. (Art 90.) y o 22. Divide 85x31 by 85. Ans. 31. 23. Divide 76x58 by 58. 24. Divide 75x40 by 40. 25. Divide 63x28 by 7. Analysis. 28=4x7. We may therefore contract the division by canceling the 7, which is a factor both of the dividend and the divisor. (Arts. 88, 90.) QUEST. 90. What is the effect of canceling a factor of any number ? Note. What is meant by the term cancel ? 91. When the divisor is a factor of the dividend, how may the division be performed ? * Birk'a Arithmetical Collections, London, 1 764. 94 DIVISION. [SECT. Operation. #)63x4x# The product of 63x4, the other factors of 252 Ans. tne dividend, is the answer required. 26. In 32 times 84, how many times 8 1 Ans. 336. 27. In 35 times 95, how many times 7 ? 28. In 48 times 133, how many times 8 ? 29. In 96 times 156, how many times 12? 30. Divide 168x2x7 by 7x3. Operation. /?X3)168x2xff We cancel the factor 7, which is corn- 3)336 mon to the divisor and dividend, then 112 Ans. divide the product of 168 into 2 by 3. 31. Divide the product of 8, 6, and 12 by the product of 2, 6, and 8. Solution. 2x0X$)$X6x-*2=6. Ans. Note. We cancel the factors 2, 6 and 8 in the divisor, and the 12 and 8 in the dividend. Canceling the same or equal factors, both in the divisor and dividend, is dividing them both by the same number, and consequently does not affect the quotient. (Arts. 88, 90.) Hence, 91. a. When the divisor and dividend have factors common to both, the division may be performed by canceling the common factors, and then dividing those that are left as before. 32. Divide the product of 7, 9, 15, and 8 by the pro- duct of 5, 7, and 8. 33. Divide the product of 6, 3, 7, and 4 by the product of 12 and 6. 34. Divide the product of 2, 28, and 15 by 30. 35. Divide the product of 5, 6, and 56 by 7x3. 92. The method of contracting arithmetical opera- tions, by rejecting equal factors, is called CANCELATION. It applies with great advantage to that class of examples and problems which involve both multiplication and division ; that is, when the product of two or more numbers is to be divided by another number, or by the product of two or more numbers. Note. Its farther developments and application may be seen in re- ARTS. 91. a.-94] DIVISION. 95 d action of compound fractions to simple ones ; in multiplication and division of fractions ; in simple and compound proportion, &c., &. GREATEST COMMON DIVISOR. 92. a. A Common Divisor of two or more numbers, is a number which will divide them without a remainder. Thus, 2 is a common divisor of 4, 6, 8, 12, 16. 93. The Greatest Common Divisor of two or more numbers, is the greatest number which will divide them without a remainder. Thus, 6 is the greatest common divisor of 12, 18, and 24. OBS. 1. One number is said to be a measure of another, when the former is contained in the latter any number of times without a re- mainder. Hence, a Com. divisor is often called a Common Measure. 2. It will be seen that a common divisor of two or more numbers, is simply a factor which is common to those numbers, and the greatest common divisor is the greatest factor common to them. Hence, 94. To find a common divisor of two or more num- bers. Resolve each number into two or more factor s, one of which shall be common to all the given numbers. Ex. 1. Find a common divisor of 8, 10, and 12. Analysis. 8 may be resolved into the factors 2 and 4 ; that is, 8=2x4 ; 10=2x5 ; and 12=2x6. Now the fac- tor 2 is common to each number and is therelbre a com- mon divisor of them. 2. Find a common divisor of 9, 15, 18, and 24. OBS. The following facts may assist the learner in finding common divisors : 1. Any number ending in 0, or an even number, as 2, 4, 6, &c. may be divided by 2. 2. Any number ending in 5 or 0, may be divided by 5. 3. Any number ending in 0, may be divided by 10. 4. When the two right hand figures are divisible by 4, the whole number may by divided by 4. 3. Find a common divisor of 16, 20, and 36. QUEST. 92. a. What is a common divisor of two or more numbers ? 93. What is the greatest common divisor of two or more numbers? Obs. When is one number said to be a measure of another ? What is a common divisor sometimes called ? 94. How do you find a com- mon divisor of two or more numbers ? DIVISION. [SECT. V, 4. Find a common divisor of 35, 50, 75, and 80. 5. Find a common divisor of 148 and 184. 6. Find a common divisor of 126 and 4653. 95. No two numbers can have a common divisor greater than a unit, unless they have a common factor Thus, the factors of 8 are 2 and 4 ; the factors of 15 an 3 and 5 ; hence, 8 and 15 have no common divisor. 96. To find the greatest common divisor of two num bers. Divide the greater number by the less; then the prece- ding divisor by the last remainder, and so on, till nothing remains. The last divisor ivill be the greatest common di- visor. 7. What is the greatest common divisor of 70 and 84 ? Operation. Dividing 84 by 70, the remainder is 14 ; 70)84(1 then dividing 70 (the preceding divisor) 70 by 14, (the last remainder,) nothing re- T4\70(5 mains. Hence, 14 the last divisor, is the 70 greatest common divisor. 8. What is the greatest common divisor of 63 and 147 1 9. What is the greatest common divisor of 91 and 117? 10. What is the greatest common divisor of 247 and 323? 11. What is the greatest common divisor of 285 and 465? 12. What is the greatest common divisor of 2145 and 3471? 97. To find the greatest common divisor of more than two numbers. First find the greatest common divisor of any two of them; then, that of the common divisor thus obtained and of another given number, and so on through all the given QUEST. 95. If two numbers have not a common factor, what is true as to a common divisor ? 96. plow find the greatest common divisoi of two numbers ? 97. Of more, than two ? ARTS. 95-100.] DIVISION. 97 numbers. The last common divisor found j loiH be the one re- 13. What is the greatest common divisor of 63, 105, and 140? Ans.7. Suggestion. Find "the greatest common divisor of 63 and 105, which is 21. Then, that of 21 and 140. 14. What is the greatest common divisor of 16, 24, rind 100? 15. What is the greatest common divisor of 492, 744, and 1044? LEAST COMMON MULTIPLE. 98. One number is said to be a multiple of another when the former can be divided by the latter without a remainder. Thus, 4 is a multiple of 2; 10 is a mul- tiple of 5. OBS. A multiple is therefore a composite number, and the num- ber thus contained in it, is always one of its factors. 99. A common multiple of two or more numbers, is a number which can be divided by each of them without a remainder. Thus, 12 is a common multiple of 2, 3, and 4 ; 15 is a common multiple of 3 and 5. OBS. A common multiple is also a composite number, of which each of the given numbers must be a factor ; otherwise it could not be divided by them. 100. The continued product of two or more given numbers, will always form a common multiple of those numbers. The same numbers, therefore, may have an unlimited number of common multiples ; for, multiplying their con- tinued product by any number, will form a new common multiple. (Art. 99. Obs.) QUEST. 98. What is a multiple of a number ? Obs. What kind of a number is a multiple ? 99. What is a common multiple I Obs. What kind of a number is a common multiple ? 100. How may a common multiple of two or more numbers be found 1 How many cow- mon multiples may there be of any given numbers I 98 DIVISION. [SECT. V, 101. The hast common multiple of two or more num- bers, is the least number which can be divided by each of them without a remainder. Thus, 12 is the least com' mon multiple of 4 and 6, for it is the least number which can be exactly divided by them. 15. Find the least common multiple of 6 and 10. Analysis. 6=2x3 ; and 10=2x5. Now it is evident that the number required must contain all the different factors which are in each of the given numbers ; other- wise it will not be a common multiple of them. (Art. 99. Obs.) The continued product of the factors 2x3x2x 5=60, is exactly divisible by 6 and 10, but it will be seen that 60 is twice as large as is necessary to be a common multiple of them. We also perceive that the factor 2 is common to both the given numbers ; hence it is that the continued product is twice too large. If, therefore, we retain this factor only once, the continued product of 2x3x 5=30, which is the smallest number that is exactly di- visible by 6 and 10, and is therefore the least common multiple of them. Operation. We divide both numbers by 2. This 2\g /; ,Q resolves them into factors, and the divisor '* and quotients contain all the different fac- 3 " 5 tors found in each of the given numbers 2x3x5=30 once, and only once. Then we multiply the divisor ant! quotients together and the pro duct is 30,whichis the least common multiple required. Hence, 102. To find the least common multiple of two or more numbers. Write the given numbers in a line with two points be- tween them. Divide by the smallest number which will di- vide any two or more of them without a remainder, and set the quotients and the numbers not divided in a line belmo. Divide this line and set down the results as before; thus QUEST. 101. What is the least common multiple of two or more numbers ? 102. How is the least common multiple of two or mor numbers found ? ARTS. "01, 102.1 DIVISION. 99 continue the operation till tJiere are no two numbers which can be divided by any number greater than 1. The contin- ued product of the divisors into the numbers in the last tine, mil be the least common multiple required. 16. Find the least common multiple of 6, 8, and 12. First Operation. . Second Operation. 2)6 " 8 " 12 6)6 /' 8 " 12 2)3 " 4 " 6 2)1 " 8 /' 2 3)3 " 2 ' 3 1 a 4 " T * 1 " 2 " 1 Now 6x2x4=48. 2x2x3x2=24 Ans. OBS. 1. In the first operation, we divide by the smallest num- bers which will divide any two of the given numbers without a re- mainder, and the product of the divisors, and the numbers in the last line, is 24, which is the answer required. In the second operation, we divide by 6, then by 2. But 6 is not the smallest number that will exactly divide two of the given num- bers, and the continued product of the divisors into the figures in the last line is 48, which is not the least common multiple. Hence, 2. We must divide, in all cases, by the smallest number that will divide any two of the given numbers exactly ; otherwise, the divisor may contain a factor common to it and some one of the quotients, or undivided numbers in the last line, and consequently the continued product of them will be too large for the least common multiple. Thus in the 2d operation, the 6 and 4 contain a common factor 2, which must be rejected from them, in order that the product of the divisors Wid quotients may be the least common multiple. 17. Find the least common multiple of 4, 9, and 12. 18. F'-id the least common multiple of 16, 12, and 24 19. Find the least common multiple of 15, 9, 6, and 5. 20. Find the least common multiple of 10, 6, 18, 15. 21. Find the least common multiple of 24, 16, 15, 20. 22. Find the least common multiple of 25, 60, 72, 35. 23. Find the least common multiple of 63, 12, 84, 72. 24. Find the least common multiple of 54, 81, 14, 63. 25. Find the least common multiple of 12, 72, 36, 144. QUEST. Obs. Why do you divide by the smallest number that will iivide two or more without a remainder ? 100 FRACTIONS. [SECT. V] SECTION VI. FRACTIONS. MENTAL EXERCISES. ART. 1O3. When a number or thing is divided into two equal parts, one of these parts is called one half. If the number or thing is divided into three equal parts, one of the parts is called one third ; if it is divided into four equal parts, one of the parts is called one fourth, or one quarter; two of the parts, two fourths; three, three fourths , if divided into five equal parts, the parts are called fifths ; if into six equal parts, sixths ; if into ten, tenths ; if into a hundred, hundredths, &c. That is, When a number or thing is divided into equal parts, the parts always take their name from the number of parts into which the thing or number is divided. 1O4. The value of one of these equal parts mani- festly depends upon the number of parts into which the given number or thing is divided. Thus, if an orange is successively divided into 2, 3, 4, 5, 6, &c., equal parts, the thirds will be less than the halves ; the fourths, than the thirds ; the fifths, than the fourths, &c. Ex. 1. What is one half of 2 cents ? Of 4 cents ? 6 1 8? 16? 18? 20? 24? 30? 40? 50? 60? 70? 80? 100? 2. What is one third of 6 cents ? Of 9? 12? 15? QUEST. 103. What is meant by one half? How many halves make a whole one ? What is meant by one third ? How many thirds make a whole one ? What is meant by a fourth ? 3 fourths ? What are fourths sometimes called? How many fourths make a whole one? What is meant by fifths ? By sixths ? Eighths ? How many sevenths make a whole one ? How many tenths ? What is meant by twenti eths ? By hundredths ? When a number or thing is divided into equa. parts, from what do the parts take their name ? 104. Upon what does the value of one of these equal parts depend ? Which is the greater, a half or a third I A sixth or a fourth ? A seventh or a tenth 1 ARTS. 103, 104.] FRACTIONS. 101 OBS. A half of any number, it will be perceived, is equal to as many units as 2 is contained times in that number; a third of a number is equal to as many units, as 3 is contained times in the given number; Q fourth is equal to as many, as 4 is contained in it, &c. 3. What is a third of 12 ? Of 15 ? 18 ? 21 1 24 ? 27 ? 30? 36? 39? 45? 60? 4. What is a fourth of 8 dollars? Of 12? 16? 20? 24? 28? 32? 36? 40? 44? 48? 5. What is a fifth of 5 ? 10? 15? 20? 25? 30? 35? 40? 45? 50? 55? 60? 100? 6. What is a sixth of 12? 18? 24? 36? 30? 48? 60? 54? 42? 72? 7. What is a seventh of 14? 28? 35? 21? 42? 56? 4$? 63? 8. What is an eighth of 16? 24? 40? 32? 64? 48? 56? 72? 88? 9. What is a ninth of 9 ? 18? 36? 27? 45? 54? 72? 63? 81? 99? 10. What is a tenth of 20 ? 40? 60? 50? 30? 100? 90? 120? 1 1 . What part of 2 is 1 ? Ans. One half. 12. What part of 3 is 1 ? Of 4? 5? 7? 10? 15? 19? 37? 200? 13. What part of 3 is 2? Suggestion. Since 1 is 1 third part of 3, 2 must be two limes the third part'of 3, or two thirds of 3. 14. What part of 5 is 2? is 3? is 4? is 5? is 6? is 8? is9? 15. What part of 8 is 3 ? is 7? is 6? is 9? is 8? 12? ? 16. What part of 17 is 5? 8? 9? 13? 15? 16? 20? 17. What part of 100 is 13? 29? 63? 75? 92? 18. If 1 half an orange cost 2 cents, what will a whole orange cost? Analysis. If 1 half of an orange cost 2 cents, 2 halves or a whole orange, will cost twice as much ; and 2 times 2 cents are 4 cents. Ans. 4 cents. 15? 102 FRACTIONS. [SECT. VL 19. If 1 third of a pie cost 4 cents, what will 2 thirds cost ? What will a whole pie cost ? 20. If 1 fourth of a pound of ginger cost 3 cents, whtt will 2 fourths of a pound cost 1 3 fourths 1 What will a whole pound cost ? 21. If 1 eighth of a yard of cloth cost 2 shillings, what will 3 eighths cost? 5 eighths? 7 eighths'? What will a whole yard cost ? 22. If 1 third of a barrel of flour cost 3 dollars, how much will a whole barrel cost ? How much will 5 bar- rels cost ? 8 barrels 1 23. If 1 sixth of a hogshead of molasses cost 5 dollars, what will be the cost of a hogshead ? Of 4 hogsheads ? Of 10 hogsheads? 24. If 1 pound of sugar cost 12 cents, what will 1 half a pound cost ? Suggestion. If 1 pound cost 12 cents, it is plain that 1 half of a pound will cost 1 half of 12 cents ; and 1 half of 12 cents is 6 cents. Ans, 6 cents. 25. If one yard of ribbon ^ost 15 cents, how much will 1 third of a yard cost ? 26. If one pound of tea cost 4 shillings, how much will 1 fourth of a pound cost ? How much will 2 fourths cost? 27. If a ton of hay cost 15 dollars, how much will 1 fifth of a ton cost ? How much 2 fifths ? 3 fifths ? 28. What will 1 tenth of an acre of land cost, at 30 dol lars per acre? 2 tenths ? 6 tenths ? 29. What will 1 eighth of a ton of iron cost, at 48 dol- lars per ton ? 3 eighths ? 5 eighths ? 7 eighths ? 30. If 1 bushel of corn cost 1 half a dollar, what wiF 2 bushels cost ? 4 bushels ? Suggestion. If 1 bushel cost 1 half a dollar, 2 bushel* will cost twice as much. 2 times 1 half are 2 halves, or a whole dollar. 4 bushels will cost 4 times 1 half, or 2 whole dollars. 31. If one man eats 1 half of a loaf of bread at a meal, how many loaves will 3 men eat ? ART. 104.] FRACTIONS. 103 32. How many whole ones are 4 halves equal to ? 5 halves ? 6 halves ? 8 halves 1 9 halves ? 33. If I burn 1 third of a ton of coal in a week, how much shall I burn in 3 weeks 1 4 weeks 1 6 weeks ? 10 weeks? 12 weeks? 34. How many whole ones in 4 thirds, and how many over? In 6 thirds? 8 thirds? 11 thirds? 14 thirds? 35. If a horse eat 1 fourth of a bushel of oats a day, how many will he eat in 6 days? In 8 ? In 10 ? In 12? 36. If a boy can saw 1 eighth of a cord of wood in a day, how much can he saw in 6 days? In 12 days? In 15 days? In 24 days? 37. If 12 oranges were divided equally among 4 boys, what part of them would each boy receive; and how many oranges would each have ? Analysis. 1 is 1 fourth of 4 ; hence, 1 boy must re- ceive 1 fourth part of the oranges. 1 fourth of 12 oran- ges is 3 oranges. 38. A builder employed 6 men to do a job of work, for which he gave them 24 dollars : what part of the money did 1 man receive ? What part did 2 receive ? What part did 3 receive? What part did 4 receive? How many dollars did one man receive ? How many did two ? Three ? Four ? 39. If 5 yards of cloth cost 40 dollars, what part of 40 dollars will 1 yard cost ? 2 yards ? 3 yards ? 4 yards ? How many dollars will 1 yard cost ? 2 yards ? 3 yards ? 4 yards ? 40. 2 is 1 third of what number ? Solution. If 2 is 1 third of a number, 3 thirds or the whole number, must be 3 times as many. Or thus, 2 is a third of 3 times 2 ; and 3 times 2 are 6. 41. 4 is 1 fifth of what number? 1 sixth of what num oer ? 1 third ? 1 eighth ? 1 fourth ? 1 seventh ? 42. 6 is 1 third of what number ? 1 fourth ? 1 seventh ? I tenth? 1 ninth? 1 twelfth? 43. 5 is 1 fourth of what number ? 1 sixth ? 1 eighth ? eleventh ? 1 twelfth ? 104 FRACTIONS. [SECT. VI, 44. 8 is 1 seventh of what number ? 1 sixth ? 1 tenth ? 1 ninth ? 1 twelfth ? 45. 4 is 2 thirds of what number ? Suggestion. First find 1 third. Now if 4 is 2 thirds, 1 third is 1 half of 4, which is 2 ; and 3 thirds is 3 times 2, or 6. Ans. 6. 46. 9 is 3 fourths of what number ? 47. 8 is 4 fifths of what number 1 48. 16 is 4 ninths of what number ? 49. 20 is 5 eighths of what number ? 50. 32 is 8 twelfths of what number ? 105. When a number or thing is divided into equal parts, as halves, thirds, fourths, &c., these parts are called FRACTIONS. A whole number is called an Integer. 106. Fractions are divided into two classes, Com- mon and Decimal. (For the illustration of Decimal Frac tions, see Section VIII.) 1O7 Common Fractions are expressed by two num.' bers, one placed over the other, with a line between them. One half is written thus \ ; one third, i ; one fourth, \ , nine tenths, -^ ; thirteen forty-fifths, if, &c. The number below the line is called the denominator, and shows into how many parts the number or thing is divided. The number above the line is called the numerator, and shows how many parts are expressed by the fraction, Thus in the fraction f , the denominator 3, shows that the number is divided into three equal parts ; the numerator 2, shows that two of those parts are expressed by the the fraction. The denominator and numerator together, are called the terms of the fraction. QUEST. 105. What are fractions ? What is an integei 106. Of cow many kinds are fractions ? 107. How are common fractions ex- pressed ? What is the number below the line called ? What does it show ? What is the number above the line called ? What does it show What are the denominator and numerator, taken together, called ? ARTS. 105-110.] FRACTIONS. 105 OBS. 1. The term fraction is of a Latin origin, and signifies brok- en, or separated into parts. Hence fractions are sometimes called broken numbers. 2. Common fractions are often called vulgar fractions. This term, however, is very properly falling into disuse. 3. The number below the line is called the denominator, because it gives the name or denomination to the fraction ; as. halves, thirds, fifths, &c. The number above the line is called the numerator, because it num- bers the parts, or shows how many parts are expressed by the fraction. 108. A proper fraction is a fraction whose numer- ator is less than its denominator ; as, 1, f , |. An improper fraction is one whose numerator is equal to, or greater than its denominator ; as, f , |. A mixed number is a whole number and a fraction ex- pressed together ; as, 4|, 25-H-. A simple fraction is a fraction which has but one nu- merator and one denominator, and may be proper, or improper ; as, f , -f-. A compound fraction is a fraction of a fraction ; as, - of foff. ^ A complex fraction is one which has a fraction in its 2-1. 4 2^ numerator or denominator, or in both ; as, > 5 0-3 84 109. Fractions, it will be seen, both from the defini- tion and the mode of expressing them, arise from division, and may be treated as expressions of unexecuted divis- ion, the numerator answering to the dividend, and the denominator to the divisor. (Arts. 67, 105.) Hence, 110. The value of a fraction is the quotient of the numerator divided by the denominator. Thus the value of f is two ; of -f- is one ; of -^ is one third ; &c. Hence, QUEST. Obs. What is the meaning of the term fraction ? What Ere common fractions sometimes called ? Why is the lower number called the. denominator? Why is the upper one called the numerator ? 108. What, is a prope- fraction? An improper fraction? A mixed number? A simp'j fraction? A con/pound fraction? A complex fraction? 109. From what do fractions arise ? 110. What is the val- ue cf a fraction ! 1 06 FRACTIONS. [SECT. VI. 111. If the denominator remains the same, multiplying he numerator by any number, multiplies the value of the fraction by that number. For, the numerator and denom- inator answer to the dividend and divisor ; therefore, multiplying the numerator is the same as multiplying the dividend. Now multiplying the dividend, we have seen, multiplies the quotient, (Art. 83,) which is the same as the value of the fraction. (Art. 110.) Thus, the value of f =2. Multiplying the numerator by 3, the fraction be- comes -^-, whose value is 6, and is the same as 2x3. 112* Dividing the numerator by any number, divide* the value of the fraction by that number. For, dividing the dividend divides the quotient. (Art. 84.) Thus, f=2. Now dividing the numerator by 2, the fraction becomes f, whose value is 1, and is the same as 2-^-2. Hence, OBS. With a given denominator, the greater the numerator, the greater will be the value of the fraction. 113* If the numerator remains the same, multiplying the denominator by any number, divides the value of the fraction by that number. For, multiplying the divisor divides the quotient. (Art. 85.) Thus, ^=4. Now multiplying the denominator by 2, the fraction becomes f, whose value is 2, and is the same as 4-J-2. 114. Dividing the denominator by any number, mul- tiplies the 'value of the fraction by that number. For, divi- ding the divisor, multiplies the quotient. (Art. 86.) Thus, ^ L =4. Now dividing the denominator by 2, the fraction becomes -^ whose value is 8, and is the same as 4x2. Hence, OBS. With a given numerator, the greater the denominator, the less will be the value of the fraction. QUKST. 111. What is the effect of multiplying the numerator, while the denominator remains the same 1 Explain the reason. 112. What is the effect of dividing the numerator \ Obs. With a given denomin- ator, what is the effect of increasing the numerator? 113. What is the effect of multiplying the denominator ? Why I 1 14. What is the effect of dividing the denominator ? Why? Obs. With a given nu> merator, what is tho effect of increasing the denominator ? ARTS. 111-118.] FRACTIONS. 107 115. It is evident from the preceding articles, that multiplying the numerator by any number, has the same effect on the value of the fraction, as dividing the denomi- nator by that number. (Arts. Ill, 114.) Dividing the numerator has the .same effect, as multi plying the denominator. (Arts. 112, Il3.) 116. If the numerator and denominator are both multiplied or both divided by the same number, the value of the fraction will not be altered. (Arts. 88, 109.) Thus, ^=3. Now if the numerator and denominator are both multiplied by 2, the fraction becomes * ; whose value is 3. If both terms are divided by 2, the fraction becomes f , whose value is 3 ; that is, Y=^-=f =3. 117. Since the value of a fraction is the quotient of the numerator divided by the denominator, it follows, that If the numerator and denominator are equal, the value is a unit or one. Thus, f=l, -f=l, &c. If the numerator is greater than the denominator, the value is greater than one. Thus, f =2, f =l-f. If the numerator is less than the denominator, the value is less than one. Thus, ^-=1 third of 1, -=4 fifths of 1. 118. It will be seen from the preceding exercises, that fractions may be added, subtracted, multiplied, and di- vided, as well as whole numbers. OBS. 1. In order to perform these operations, it is often necessary to make certain changes in the terms of the fractions. QUEST. 115. What maybe done to the denominator to produce the earne effect on the value of the fraction, as multiplying the numerator by any given number ? What, to produce the same effect as dividing the numerator by any given number ? 1 16. What is the effect if the numerator and denominator are both multiplied, or both divided by the same number ? 117. When the numerator and denominator are equal, what is the value of the fraction ? When the numerator is the larger, ivhat ? When smaller, what 1 108 REDUCTION OF [SECT. VI, 2. It is evident that any changes may be made in the terms of fraction, which do not alter the quotient of the numerator divideq by the denominator; for, if the quotient is not altered, the val- ue remains the same. (Art. 1 10.^ Thus, the terms of the fraction ^ may be changed into -2, -S., JUI, &c., without altering its value ; for in each case the quotient of the numerator divided by the denomin- ator is 2. Hence, for any given fraction, we may substitute any other fraction, which will give the same quotient. REDUCTION OF FRACTIONS. 119* The process of changing the terms of a fraction into others, without altering its value, is called REDUCTION OF FRACTIONS. EXERCISES FOR THE SLATE. CASE I. Ex. 1. Reduce ~fa to its lowest terms. Dividing 1 both terms of the fraction by 2, it be- =f: again, 3)f =. Ans. comes A. a g a i nj dividing both by 3, we obtain , whose terms are the lowest tc which the given fraction can be reduced. If we divide both terms by 6, their Second Operation. common \- Ans. th e given fraction will be reduced to its lowest terms by a single division. Hence, 1 2O. To reduce a fraction to its lowest terms. Divide the numerator and denominator by any number which will divide them both without a remainder ; and thus continue the operation, till there is no number greater than 1 that will divide tJiem exactly. Or, divide both the numerator and denominator by thei? greatest common divisor ; and the two quotients thus atising will be the lowest terms to which the given fraction can be re duced. (Art. 96.) QUEST. Obs. What changes may be made in the terms of a frac lion 1 119. What is meant by reduction of fractions? 120. How is q fraction reduced to its lowest terms ? ARTS. 119-121. ] FRACTIONS. 109 OBS. 1. A fraction is said to be reduced to its lowest terms, when Hs numerator and denominator are expressed in tne smallest num- bers possible. 2. The value of a fraction is not altered by reducing it to its lowest terms. (Art. 116.) 3. Wnen the terms of the fraction are small, the former method will generally be found to be the shorter and more convenient; but when the terms are large, it is often difficult to determine whether the frac- tion is in its simplest form, without finding their greatest common divisor. 2. Reduce -ft- to its lowest terms. A?is. . 3. Reduce -ft-. 4. Reduce f . 5. Reduce -'if. 6. Reduce ft. 7. Reduce if. 8. Reduce f 9. Reduce -ftfe. 10. Reduce 11. Reduce -fff- 12. Reduce 13. Reduce f 14. Reduce 15. Reduce-] 3 ^. 16- Reduce 17. Reduce -.. 18. Reduce CASE II. 19. Reduce -Y 1 to a whole or mixed number. Suggestion. The object in this example, is to Operation. find a whole or mixed number, whose value is equal to the given fraction. But the value of a 5)17 fraction is the quotient of the numerator divided ~ q a by the denominator. (Art. 110.) Hence, 6 * To reduce an improper fraction to a whole, or mixed number. Dityide the numerator by the denominator, and & '^ iient will be the whole, or mixed number required. 20. Reduce -^ to a whole or mixed number. Ans. 9|. QUEST. Obs. What is meant by lowest terms t Is the value of A fraction altered by reducing it to its lowest terms ? 121. How if an improper fraction reduced to a whole or mixed number ? HO REDUCTION OF [SECT. VI. Reduce the following fractions to whole or mixed numbers : 21. Reduce *. 22. Reduce ty. 23. Reduce *-. 24. Reduce A. 25. Reduce -H-. 26. Reduce 27. Reduce ^. 28. Reduce 29. Reduce *&. 30. Reduce CASE III. 31. Reduce the mixed number 15f to an improper fraction. Operation. 15_a OBS. In 1 there are 4 fourths, and in 15, there are 15 times as many. 4X15=60, and 3 fourths make __! 63 fourths. Hence, ^ Ans. 122* To reduce a mixed number to an improper fraction. Multiply the whole number by the denominator of the frac- tion ; to the product add the given numerator. The sum placed over the given denominator, ivill form the improper fraction required. OBS. 1. Any whole number may be expressed in the form of a frac- tion without altering its value, by making \ the denominator. 2. A whole number may also" be reduced to a fraction of any de- nomination, by multiplying the given number by the proposed denom- inator; the product will be the numerator of the fraction required. Thus 25 may be expressed by -*-, -"-f -, or *?-, &c., foi 25=^^=4^=4^1, & c . So 12=V e ===-; fo^ the quotient of each of these numerators divided by its de- nominator, is 12. 32. Reduce 8- to an improper fraction. Ans. ^-. QUEST. 122. How reduce a mixed number to an improper r raction ? Obs. How express a whole number in the form of a fraction I How re- duce a whole number to a fraction of a given denominator ? AETS. 122, 123.] FRACTIONS. Ill Reduce the following numbers to improper fractions: 33. Reduce 9-f. 34. Reduce 16*. 35. Reduce 23f 36. Reduce 45^. 37. Reduce 64^. 38. Reduce 56ff. 39. Reduce 304^ 40. Reduce 725. . 41. Reduce 45 to fifths. 42. Reduce 72 to eighths. 43. Reduce 830 to sixths. 44. Reduce 743 to fifteenths CASE IV. 45. Reduce f of f to a simple fraction. Analysis. f of f is 2 times as much as 1 third of -f-. Now i of -f- is -fa ] for, multiplying the denominator di vides the value of the fraction. (Art. 1 13.) And 2 thirds is 2 times -/v, which is equal to it, or f (Art. 120.) The answer is ^. OBS. This operation consists in simply multiplying the two nu iterators together and the two denominators. Hence, 123* To reduce compound fractions to simple ones. Multiply all the numerators together for a new numera- tor, arid all the denominators together for a new denomv nator. 46. Reduce -f of f of -f to a simple fraction. Ans. jVV, or ^. 47. Reduce | of if of W to a simple frnction. 48. Reduce of f of $ to a simple fraction. 49. Reduce f of -^ of -fi to a simple fraction. 50. Reduce of of f of -f- to a simple fraction. Operation. Since the product of the numera- ^ A * 5 5 tors is to be divided by the product - of -of -of-= - of the denominators, we may can- $ 4 7 28 ce [ t j ie factors 2 and 3, which are common to both ; for, this is dividing the terms of the new fraction by the same number, (Art. 90,) and therefore aloes not alter its value. (Art. 116.) Multiplying the re- QUEST. 123. How are compound fractions reduced to simple ones 1 113 BEDUCTIOW OP [SECT. VL maining 1 factors together, we have -fa, which is the ai> swer required. Hence, 124* To reduce compound fractions to simple ones by CANCELATION. Cancel all the factors which are common to the numer- ators and denominators ; then multiply the remaining terms together as before. (Art. 123.) OBS. This method not only shortens the operation of multiplying but at the same time reduces the answer to its lowest terms. A little practice will give the learner great facility in its application. 51. Reduce i of -ff of f to a simple fraction. Operation. First, we cancel the 4 and 3 2 in the numerator, then the 12 in ^ & 2 t ^ ie denominator, which is equal -rof -^of -;=== A. to the factors 4 and 3. Final- ly, we cancel the 5 in the de- nominator, and the factor 5 in the numerator 10, placing the other factor 2 above. We have 2 left in the numera- tor and 7 in the denominator. Ans. -f 52. Reduce f off of if to a simple fraction. 53. Reduce $ of -J- of -fa of to a simple fraction. 54. Reduce -f of -f- of of -^ to a simple fraction. 55. Reduce -ft- of $ of -ff to a simple fraction. 56. Reduce -fa of -ft of -f- of ^ to a simple fraction 57. Reduce -f of -ff- of -ff of -^ to a simple fraction. 58. Reduce -fa of -fa of -f- of -f to a simple fraction. Note. For the method of reducing complex fractions to simple onea see Art. 143. CASE V. Ex. 1. Reduce and to a common denominator. Note. Two or more fractions are said to have a common, denon* inatw, when they have the same denominator. QUEST 124. How by cancelation ? How does it appear that tlu# method does not alter the value of the fraction ? Obs. What is tha advantage of tliis method ? Note. What is meant by a common de- nominator ? ART. 124, 125.] FRACTIONS. 113 Suggestion,. The object of this example is to find two ether fractions, which have the same denominator, and whose values are respectively equal to the values of the given fractions, and . Now, if both terms of the first fraction --, are multiplied by the denominator of the sec- ond, it becomes , and if both terms of the second fraction -, are multiplied by the denominator of the first, it be- comes -f. But the fractions f- and -f have a common de- nominator, and are respectively equal to the given fractions, viz: f=, andf=i. (Art. 116.) Hence, 125. To reduce fractions to a common denominator. Multiply each numerator into all the denominators except its own for a new numerator, and all the denominators together for a common denominator. 2. Reduce , -f- , and -J- to a common denominator. Operation. I x4x6=24 } 3x2x6=36 > the three numerators. 5x2x4=40 ) 2x4x6=48, the common denominator. The fractions required are -f^-, -f-f-, and . OBS. It is manifest that the process of reducing fractions to a com- mon denominator, does not change their value ; for, it is simply mijl- tiplying each numerator and denominator of the given fractions by the game number. (Art. 116.) 3. Reduce -f, f , and -f- to a common denominator. Ans. -Mr, tW, -tffr. 4. Reduce -f , -f, and f to a common denominator. Reduce the following fractions to a common denomi- nator : 5. Reduce |, i , and f . 6. Reduce -f, -*-, f, and f. 7. Reduce -,f,^V, and -A. *. Reduce -ft, f, if, and*. QUEST. 125. How are fractions reduced to a common denominator ? 06s. Does the process of reducing fractions to a common denominator alter their value ? Why not ? 114 REDUCTION OP [SECT. VI 9. Reduce f, fjf-, and 10. Reduce &, -fifr, and ff 1 1. Reduce /r, -f-fr, and if. 12. Reduce ^V, * and iff. CASE VI. 13. Reduce f, -, and to the least common denomi- nator. Operation. We first find the least common 2)4 " 6 " 8 multiple of all the given denomi- 2)2 " 3 " 4 nators, which is 24; (Art. 102;) 7 and this is the least common de- nominator required. The next 24, the least ig tQ re( ce the iyen frac tions to twenty-fourths without al- tering their value. This may evidently be done, by mul- tiplying both terms of each fraction by the number of times its denominator is contained in 24. (Art. 116.) Thus 4, the denominator of the first fraction, is contained in 24, 6 times ; now multiplying both terms of the fraction -f by 6, it becomes -^f. The denominator 6 is contained in 24, 4 times ; and multiplying the second fraction -f- by 4, it be- comes -^4-. The denominator 8 is contained in 24, 3 times ; and multiplying the third fraction by 3, it becomes -J-f-. Therefore -J-f, -tfa, and -J-f- are the fractions required. Hence. ' 126. To reduce fractions to their least common de- nominator. I. Find the least common multiple of all the denominator? of the given fractions, and it will be the least common denomi- nator. (Art. 102.) II. Divide the least common denominator by the denomi- nator of each of the given fractions, and multiply the quotient by the numerator ; the products icill be the numerators re- quired. QUEST- 126. How ?ire fractions reduced to the least common de nominator ? ART. 126.] FRACTIONS. 115 OBS. Multiplying each numerator into the number of tunes its de- nominator is contained in the least common denominator, is in effect multiplying both terms of the given fractions by the same number. For, if we multiply each denominator by the number of times it is contained in the least common denominator, the product will be equal to the least common denominator. Hence, the new fractions must be of the same value as the given fractions. (Art. 116.) 14. Reduce f, f, and to the least com. denominator. Operation, 2x3x2=12, the least com. denominator. 2^3 " 4 " 6 ^ T w (12-f-3)x2=8, numerator of 1st. o o 3 (12-r4) X 3=9, of2d. 3 ) 3 * 6 (12-T-6)X5=10, of 3d. 1 " 2 " 1 Ans. A, A, and if. 15. Reduce and -ft to the least common denominator. Ans. -fi and -f$. Reduce the following fractions to the least common de- nominator : 16. i, , and f 17. -f, f , and -ft. 1 8. f , -I, A, and Tft-. 19. *, f , -ft, and A- 20. A, i, *, i, and f 21. -^, ^-, and &. 22. Tft, A, and -rib. 23. -Ji, f , and -ft. 24. f , A, and if. 25. -ft, A, and ffr. ADDITION OF FRACTIONS. MENTAL EXERCISES. Ex. 1. What is the sum of i, , f, and -f? Suggestion. Since all these fractions have the same de- nominator, it is plain their numerators may be added as well as so many pounds or bushels, and their sum placed over the common denominator, will be the answer re- quired. Thus, 1 eighth and 2 eighths are 3 eighths, and 3 are 6 eighths, and 5 are 1 1 eighths. Ans. -V", or 1-f . QUEST. Obs. Does this process alter the value of the given frac- tions? Why not? 116 ADDITION OF [SECT. VL 2. What is the sum of i, -f-, f , and f ? 3. What is the sum of f , f , -, and -f ? 4. What is the sum of -ft, ii, -ft, and 5. What is the sum of -ft-, -ft-, -ft-, -ft, and 6. What is the sum of -&, &, ^, if, and 7. What is the sum of if, -ft, J f , -ft-, and 8. What is the sum of , fj-, ^ A, and - 6 \ ? 9. What is the sum of if, if, ^y, A, and & ? 10. What is the sum of -fifty, ifo, iW, and -rfjr ? EXERCISES FOR THE SLATE. 1 1. What is the sum of -f, , and f ? Solution. f+i+f =f, or If ^ws. 12. What is the sum of , and i ? Suggestion. A difficulty here presents itself; for it 13 manifest that 1 half added to 1 third will make neither 2 halves nor 2 thirds. (Art. 22.) This difficulty may be removed by reducing the given fractions to a common da nominator. (Art. 125.) Thus, 1x3=3 } 1x2=2 \ ^ e new numerators - 2x3=6, the common denominator. The fractions reduced are -f- and . and may now b< added. Thus, f -ff = . Ans. 127* From these illustrations we deduce the follow- ing general RULE FOR ADDITION OF FRACTIONS. Reduce the fractions to a common denominator ; &dd their numerators, and place the sum over the common denomi- nator. OBS. 1. Compound fractions must, of course, be reduced to simple ones, before attempting to reduce the given fractions to a common de nominator. (Art. 123.) QUEST. 127. How are fractions added ? Obs. What must be dona with compound fractions ? ART. 127.] FRACTIONS. 117 2. Mixed numbert may be reduced to improper fractions, then added according to the rule ; or, we may add the whole numbers and frae tional parts separately, and then unite their sums. 13. What is the sum off, and f ? Ans. =! , or 1, 14 What is the sum o f, and ? 15. What is the sum off, , and -f ? 16. What is the sum off -H, and ? 17. What is the sum of -&, f, and -^ ? 18. What is the sum of f , -ft, and T^ ? 19. What is the sum of ^f, f , and f ? 20. What is the sum of ^ -f, and -^ ? 21. What is the sum of f , f, -f, and -f ? 22. What is the sum of -J-, f , f , and -f- ? 23. What is the sum of f , -f of , and -& ? 24. What is the sum of f , -f , -f of f , and f ? 25. What is the sum of i of 3, f of f, and f ? 26. What is the sum of 2, 6i, and f ? 27. What is the sum of f of 2, 3, and 5f ? 28. What is the sum of If, ff , and -^ ? 29. What is the sum of 351, -fi, and -f of f ? 30. What is the sum of -^-, &J-. If, and f ? SUBTRACTION OF FRACTIONS. MENTAL EXERCISES. Ex. 1. Henry had -f- of a watermelon, and gave away of it : how much had he left ? Solution. 3 sevenths from 5 sevenths leaves 2 sevenths. Ans. f 2. John had - of a bushel of chestnuts, and gave away f- : how many had he left ? 3. If I own f of an acre of land, and sell f of it, how much shall I have left 1 QUEST. Obs. How are mixed numbers added ? 118 SUBTRACTION OF [SECT. YL 4. A man owning of a ship, sold -f : what part of the ship had he left ? 5. William had -ft of a dollar, and spent -fa : how many tenths had he left ? 6. What is the difference between -fc and 7. What is the difference between and 8. What is the difference between -JHr and ft ? 9. What is the difference between ii and fg- ? 10. What is the difference between T 3 ^ and EXERCISES FOR THE SLATE. 11. From-j^- take -fa. Solution. $3 &*=-$[ Ans. 12. From -f- take f. Suggestion. A difficulty here meets the learner, simi lar to that which occurred in the 12th example of addi- tion of fractions, viz : that of subtracting a fraction of one denominator from a fraction of a different denomina^ tor. He must therefore reduce the fractions to a com mon denominator, before the subtraction can be per formed. Also 6x4=24, the common denominator. The fractions are f and if. Now -^ &=& Ans 12S. From these illustrations we deduce the follow ing general RULE FOR SUBTRACTION OF FRACTIONS. Reduce the given fractions to a common denominator ; sub- tract the less numerator from the greater, and place the remain- der over the common denominator. OBS. Compound fractions must be reduced to simple ones, as in ad- dition of fractions. (Art. 123.) QUEST. 128. How is one fraction subtracted from another ? O5* What Is to be done with compound fractions \ ARTS. 128, 129.] FRACTIONS. 119 13 From -f talre i. Ans. . 14. From f take f. 15. From -f take -ft. 16. From it take f. 17. From if take \. 18. From $f take -&&. 19. From if take &. 20. From & take ^ 21. From f take -ft. 22. From ff take ff. 23. From -ff- take -&. 129. Mixed numbers may be reduced to improper fractions, then to a common denominator and subtracted ; or, the fractional part of the less number may be taken from the fractional part of the greater, a^d the less whole number from the greater. 24. From 8$- take 5|. Operation. 17 thirds from 25 thirds leaves 8 thirds, which are equal to 2f. A f QJL Note. Since we cannot take 2 thirds from 1 >r tims, o- thir(J) we b orrow a un i tj w hich, reduced to thirds 5f and added to 1 third, makes 4 thirds. Now 2 j-wc Oi thirds from 4 thirds leaves 2 thirds : 1 to carry to 4 8 5 makes 6, and 6 from 8 leaves 2. 25. From 12-f take 7-J-. Ans. 5-f. !26. From 1 5-f- take 9. 27. From 25f take 17f. 28. From 37-$- take 19-f-. 29. From 2 take f . Suggestion. Since 5 fifths make a whole one, in 2 whole ones there are 10 fifths ; now 3 fifths from 10 fifths leaves 7 fifths. Ans. , or If. Hence, QUEST. 129. How are mixed numbers subtracted ? 130. How is a fraction subtracted from a whole number ? 120 MULTIPLICATION OF [SECT. VI, 13O* To subtract a fraction from a whole number. Change the whole number to a fraction having the samt denominator as the fraction to be subtracted, and proceed a\ before. (Art. 128.) OBS. If the fraction to be subtracted is a proper fraction, we may simply borrow a unit and take the fraction from this, remembering to diminish the whole number by 1. (Art. 36.) 30. From 6 take -f. Ans. 5 31. From 65 take 25 W. 32. From-f off take i off. 33. From i off take i of T\. 34. From f of 10 take f of 6. 35. From f of 24 take f of 27. MULTIPLICATION OF FRACTIONS. MENTAL EXERCISES. 1. It a man spends of a dollar for rum in 1 day, how much will he spend in 7 days ? Suggestion. If he spends i in 1 day, in 7 days he will spend 7 times i ; and ix7 is . Ans. i of a dollar. 2. If a man spends of a dollar for rum in 1 week, how much will he spe-nd in 4 weeks. Ans. * or 3 dolls. 3. If 1 man drinks -f of a barrel of beer in a month, how much will 10 men drink in the same time ? 4. What cost 4 yards of cloth, at 2 dollars per yard ? Solution. 4 yards will cost 4 times as much as 1 yard ; and 4 times i is 4 halves, equal to two whole ones : 4 times 2 dollars are 8 dollars, and 2 i?>*?Ve. 10 dollars. Ans. 4 yards will cost 10 aoiiars. 5. What cost 5i bushels of peanuts, at 3 dolls, a bushel ? 6. What cost 10-f pounds of tea, at 4 shillings a pound 1 7. If 1 drum of figs costs 16 shillings, what will 3 fourths of a drum cost ? Suggestion. First find what 1 fourth will cost. Then 3 fourths will cost 3 times as much. ARTS. 130-132.] FRACTIONS. 121 8. If an acre of land produces 40 bushels of corn, how many bushels will 3 eighths of an acre produce ? 9. If a man can travel 50 miles in a day, how far can he travel in 2 fifths of a day? 3 fifths? 4 fifths? 10. Henry's kite line was 90 feet long, but getting en- tangled in a tree, he lost 3 ninths of it : how many feet did he lose ? 131. We have seen that multiplying by a whole number is taking the multiplicand as many times as there are units in the multiplier. (Art. 45.) On the other hand, If the multiplier is only a part of a unit, it is plain we must take only a part of the multiplicand. That is, 132* Multiplying by a fraction is taking a certain PORTION of the multiplicand as many times as there are like portions of a unit in the multiplier. Multiplying by -J-, is taking 1 half of the multiplicand once. Thus, 6xi=3. (Art. 104. Obs:) Multiplying by -J-, is taking 1 third of the multiplicand once. Thus, 6xi=2. Multiplying by -f, is taking 1 third of the multiplicand twee. Thus, 6x-f=4. * OBS. If the multiplier is a unit, the product is equal to the multi- plicand; if the multiplier is greater than a unit, the product is greater than the multiplicand; (Art. 45;) and if the multiplier is less than a unit, the product is less than the multiplicand. EXERCISES FOR THE SLATE. CASE I. 11. If a bushel of corn is worth of a dollar, how much as 5 bushels worth ? QUEST. 131. What is meant by multiplying by a whole number ? 132. By a fraction? Byi? By i? By -ft Byf? By!? 06s. If the multiplier is a unit or 1, what is the product equal to ? When the multiplier is greater than 1, how is the product, compared with the multiplicand ? When less, how ? 122 MULTIPLICATION OF [SECT. VL Solution. 5 bushels will cost 5 times as much as 1 bushel. Now ix5=-f, or 2-J- ; that is, 5 times i are 5 halves, equal to 2 and 1 half. Ans. 2-J- dollars. 12. Multiply -f- by 5. Ans^ or 3f. 13. Multiply T^ by 8. 14. Multiply - by 12. 15. Multiply A by 18. 16. Multiply if by 10. 17. If a pound of tea cost 6 shillings, how much will f of a pound cost ? Solution. Multiplying by -f , is taking 1 third oi the multiplicand twice. (Art. 132.) Now 1 third of 6 is the same as 6 thirds of 1, or f ; and 2 thirds of 6 must be 2 times as much ; that is, f x2=V 5 an d J a =4. A/is. Note. Since the product of any two numbers will be the same, whichever is taken for the multiplier, (Art. 47,) the fraction may bo taken for the multiplicand, and the whole number for the multiplier, when it is more convenient. Thus, -f X6=Y, or 4 ; and 6x1=4. 18. Multiply 12 by -J-. Ans. 3. 19. Multiply .10 by f. 20. Multiply 15 by f. 21. Multiply -f by 2. Ans. fx2= J ^ L , or 1|. Suggestion. Dividing the denominator of a fraction b* any number, multiplies the value of the fraction by thai number. (Art. 114.) Now, if we divide the denominator 8 by 2, the fraction will become -f, which is equal to 1-J-, the same as before. Hence, 133. To multiply a fraction and a whole number together. Multiply the numerator of the fraction by the whole number, a-nd write the product over the denominator. Or, divide the denominator by the whole number, when this vin be done without a remainder. (Art. 1 14.) QUEST. 133. How multiply a fraction and a whole number together! ARTS. 133-134. a.] FRACTIONS. 123 OBS. 1. A fraction is multiplied into a number equal to its denomi- nator by canceling the denominator. (Arts. 89, 91.) Thus -f-X7= 4. 2. On the same principle, a fraction is multiplied into any factor in its denominator, by canceling that factor. (Arts. 91, 114.) Thus, 22. Multiply if- by 5. Ans. *-, or 3. 23. Multiply f by 9. 24. Multiply if by 25. 25. Multiply 36 by ff. 26. Multiply 120 by if. 27. Multiply fff by 25. 28. Multiply ff by 50. 29. Multiply 9i by 5. Operation. 5 times i are , which are equal to 2 9i and . Set down the . 5 times 9 are 45, and 2 (which arose from the fraction) make Ans. 47 47. Hence, 134. To multiply a mixed number by a whole one. Multiply the fractional part and the whole number sepa- ately, and unite the products. 30. Multiply 15f by 7. Ans. 110J-. 31. Multiply 25-f- by 10. 32. Multiply 48-Lg- by 8. 33. Multiply 24 by 3. Operation. We first multiply 24 by 3, then by -, 2)24 and the sum of the products is 84. Mul- 3^ tiplying by is taking one half of the mul- 72 tipiicand once. (Art. 132.) But to find a 12 half of any numbe" we divide the num- ber by 2. (Art. 104. Obs.) Hence, /ins. o 4 134* a. To multiply a whole by & mixed number. Multiply first by the integer, then by the fraction, &nd add \*e products together. 34. Multiply 27 by 3. Ans. 90. 35. Multiply 63 by lOf 36. Multiply 75 by !2f QUEST. Obs. How is a fraction multiplied by a number equal to Its denominator ? How by any factor in its denominator ? 134. How i a mixed number multiplied by a whole one ? 134. a. How i& a whole number multiplied by a mired number ? 124 MULTIPLICATION OF [SECT. VI, CASE II. 37. A man owning of a ship, sold -f of what he owned. What part of the ship did he sell ? Analysis. J- of -f is -ft- ; for, multiplying the denomi* nator by any number, divides the value of the fraction. (Art. 1 13.) Now 2 thirds of f is twice as much ; that is, =-i 6 5, which, reduced to its lowest terms, is . Ans. Or, we may reason thus : Since he owned f , and sold f of what he owned, he must have sold -J of -f of the ship. Now -f of f is a compound fraction, whose value is found by multiplying the numerators together for a new numerator, and the denominators for a new denomi- nator. (Art. 123.) Solution. f xi=Aj or f An*- Hence, 135* To multiply a fraction by a fraction. Multiply the numerators together for a new numerator and the denominators together for a new denominator. OBS. It will be seen that the process of multiplying one fraction by another, is precisely the same as that of reducing compound fractions to simple ones. 38. Multiply i by f . Ans. &=. 39. Multiply f by f . 40. Multiply f by f . 41. Multiply & by f. 42. Multiply H by -f. 43. Multiply -f and -f and -f and together. Operation. 123^2 Since the factors 3 and 4 are common -X-X-X-= both to the numerators and denomina- 7 $ ^ 5 35 tors, we may cancel them, and multiply the remaining factors together as in reducing compound fractions to simple ones. (Art. 124.) Hence, QUEST. 135. How is a fraction multiplied by a fraction ? Obs. To what is the process of multiplying one fraction by another cimilar I 136- How multiply fractions together by cancelation ? ARTS. 135-137.] FRACTIONS. 125 136* To multiply fractions by CANCELATION. Cancel all the factors common both to the numerators and denominators ; then multiply the remaining factors in the numerators together for a new numerator ', and those remain- ing in the denominators for a new denominator^ as in reduc- tion of compound fractions. (Art. 124.) OBS. 1. This process, in effect, is dividing the product of the nu- merators and that of the denominators by the same number, and therefore does not alter the value of the answer. (Art. 116.) 2. Care must be taken that the factors canceled in the numerators are exactly equal to those canceled in the denominators. 44. Multij / -f- by . Ans. f . 45. Multij y -f by i and f. Ans. rV 46. Multi] y -fe by -f- and f. 47. Multiply f by -^ and -J- and -}-f . 48. Multiply -f^ by if and and f and f . 49. Multiply f-f- by -ft- and -fa and -^ and f . 50. Multiply 7 by 3^. Solution. 7-J-, reduced to an improper fraction, be- comes J^, and 3i becomes -^ Now -^X^-^- 1 ^, or 25. 137* Hence, when the multiplier and multiplicand are both mixed numbers, they should be reduced to im- proper fractions, and then be multiplied according to the rule above. EXAMPLES FOR PRACTICE. 1. What will 12 apples cost, at of a cent apiece ? 2. If a bushel of wheat weighs f of a hundred weight, how much will 10 bushels weigh ? 3. If a man earns -f- of a dollar per day, how much can he earn in 12 days ? QUEST. Ohs. How does it appear that this process will give the true answer ? What is necessary to be observed with regard to can- celing factors ? 137. When the multiplier and multiplicand are mixed numbers, how proceed ? 126 MULTIPLICATION OP [SECT, VL 4. If a family consume f of a barrel of flour in a week, how much wiil they consume in 15 weeks? 5. If I burn of a cord of wood in a month, how much shall I bum in 12 months? 6. If a man can reap -^ of an acre of grain in a day how many acres can he reap in 9 days? 7. If a pound of powder is worth 6 shillings how much is -f of a pound worth ? 8. If a gallon of oil is worth 7 shillings, how much is 5- of a gallon worth ? 9. When beeric 1 10 dollars a barrel, how much will i of a barrel cost ? 10. What will i of a firkin of butter cost, 15 dollars a firkin ? 11. At f of a dollar a cord, how much wii'. the sawing of 20 cords of wood amount to ? 12. What will 16 pounds of cheese cost, at 8 cents per pound ? 13. Wh- 1 cost 9 dozen of eggs, at 12 cents per dozen? 14. What cost 15-f yards of cambric, at 15 pence per yard? 15. What cost 1 ! cords of wood, at 1 dollar per cord ? 16. At 12 cents a pound, what cost 2-f pounds of pep- per? 17. At 5 shillings a pound, what cost 12f pounds of tea ? 18. What will 6 pounds of starch come to, at J2- cents per pound ? 19. What will 18 ounces of nutmegs come to, at 6i cents an ounce? 20. At 12-f cents a yard, what will 17 yards of cotton come to ? 21. At 3^ dollars a yard, what cost 15 yards of broad- cloth ? 22. What cost 15f- yards of ribbon, at 10 cents per yard? 23. What cost 22 pocket handkerchiefs, at if of a dollar apiece? 24. At -fo of a dollar a yard, what will -f- of a yard of lac cost? ART. 137.] FRACTIONS. 127 25. At f of a dollar a yard, what will -f of a yard of muslin come to ? 26. At f of a dollar a bushel, what cost T 9 7 of a bushel of wheat ? 27. What will f of a pound of tea cost, at -f of a dollar a pound f 28. What cost 66 bushels of apples, at 18f cents a bushel ? 29. At 62 cents a yard, what cost 12 yards of balzo- rine? 30. What cost 18 yards of tape, at 6-J- cents per yard ? 31. What cost 13 bushels of oats, at 18f cents per bushel? 32. What cost 31 yards of sheeting, at ^ of a dollar per yard ? 33. At T ^ of a dollar a quart, what cost 8$ quarts of cherries ? 34. At 3-f- shillings a yard, what cost 7$ yards of ging- ham? 35. What cost 14f bushels of potatoes, at 18^- cents a bushel ? 36. At 7-f shillings a yard, what cost 8-f yards of silk 1 37. At -I of a dollar a bushel, what cost 47-f- bushels of peaches ? 38. What cost 63^- pounds of sugar, at 9-f- cents per pound ? 39. What cost 2f yards of velvet, at 3-f dolls. a yard? 40. What cost 9f yards of calico, at 1-f shillings a yard ? 41. WKat cost 25-|- pounds of figs, at 15 cents a pound ? 42. What cost 35f cords of wood, at 18-J- shillings per cord? 43. What cost 175 bushels of corn, at -f of a dollar a bushel? 44. What cost 8f tons of hay, at 15 dollars a ton ? 45. If a man can travel 42 miles in one day, how fnr ean he travel in 17 days? 128 DIVISION OP [SECT. VL DIVISION OF FRACTIONS. MENTAL EXERCISES. Ex. 1. A man divided f of a pound of honey equally among his 3 children : what part of a pound did each receive ? Analysis. 1 is one third of 3 ; therefore 1 child must have received 1 third of 6 sevenths. 1 third of 6 sevenths is 2 sevenths. Ans. Each child received f of a pound. 2. If 4 pounds of loaf sugar cost -f of a dollar, how much will 1 pound cost ? 3. A father gave his 2 sons if of a dollar : how many twelfths did each receive ? 4. A little girl bought 5 lead pencils for if of a shil ling : how much did she give apiece for them ? 5. A father gave parts of a vessel to his 6 sons what part of the vessel did each receive? 6. At -J- dollar a yard, how many yards of French mus- lin can you buy for 4 dollars ? Suggestion. 4 dollars will buy as many yards as 1 half is contained times in 4, or as there are halves in 4 dollars. Now since there are 2 halves in 1 dollar, in 4 dollars there are 4 times 2 halves ; and 4 times 2 halves are 8 halves. Ans. 4 dollars will buy 8 yards. 7. At cent apiece, how many apples can I buy for 6 cents? 8. At of a dollar a pound, how many pounds of aimonds can you buy for 12 dollars? 9. How many quills, at f of a penny apiece, can you buy for f of a, penny ? Suggestion. f of a penny will buy as many quills as is contained times in f ; and -f is contained in f , 3 times. Ans. 3 quills. 10. How many yards of cloth can I buy for of a cord of wood, if I give \ of a cord for a yard of cloth ? ART. 138.] FRACTIONS. 129 EXERCISES FOR THE SLATE. CASE I. 11. If 3 bushels of oats cost f of a dollar, what will 1 bushel cost? Analysis. 1 is 1 third of 3 ; therefore, 1 bushel will cost 1 third part as much as 3 bushels. 1 third of f is . Ans. i of a dollar. We divide the numerator of the frac- Operation. tion -f , whicl^ is the whole cost, by 3 the |-s-3=|. Ans. whole number of bushels, and place the quotient 2 over the given denominator. 12. If 4 yards of calico cost -f- of a dollar, what will 1 yard cost ? Operation. In this case we cannot divide f -*-4=*&r, or A- Ans. the numerator of the dividend by 4 the given divisor, without a remainder. We therefore multiply the denominator by the 4, which is in effect dividing the fraction. (Art. 113.) Hence, 138* To divide a fraction by a whole number Divide the numerator by the whole number, when it can be done without a remainder; but when this cannot be done, multiply the denominator by the whole number. 13. Divide f by 3. First Method. Second Method. -5-3=1, or \. Ans. f -*-3=^r, or \. Ans. 14. Divide H by 6. 15. Divide if by 8. 16. Divide -V s - by 7. 17. Divide H by 12. QVEST.-138. How is a fraclioa divided by a whole number t 5 130 DIVISION OF [SECT. VL 18. Pivide fg- by 9. 19. Divide If by 8. 20, Divide Hi by 25. 21. Divide -Hi by 30. CASE II. 22. At -J- of a dollar a pound, how many pounds of honey can be bought for f of a dollar ? Suggestion. Since } of a dollar will buy 1 pound, f oi a dollar will buy as many pounds as \ is contained times in -f. Now \ is contained in f , 3 times. Ans. 3 pounds. 23. At of a dollar a bushel, how much barley can be bought for f of a dollar ? We first reduce the given First Operation. fractions to a common denomi- f=-jHJ- nator; (Art. 125;) then divide i=-fo the numerator of the dividend -^B-f.^=l^.. Ans, by the numerator of the divisor, as above. OBS. 1. After the fractions are reduced to a common denominator, it will be perceived that no use is made of the common denominator itself. In practice, therefore, it is simply necessary to multiply the numerator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor, in the same manner as two fractions are reduced to a common denominator; or, what is the same in effect, invert the divisor, and proceed as in multiplication of fractions. (Art. 135.) Note. To invert a fraction is to put the numerator in the place of the denominator, and the denominator in the place of the numerator, Thus, in the example above, inverting the divisor -f , it becomes f j and fXf *-, or 1^, which is the same as before. Again, we may also illustrate the principle thus : Second Operation. Dividing the dividend f by 2, the f~t-2=-f quotient is f. (Art. 113.) But it is |X5= -^ required to divide- it by only of 2 ; And-^-=l-. Ans. consequently the -f is 5 times too small for the true quotient. There- fore -f multiplied by 5 will be the quotient required. Now $xo ^, or 1-;, which is the same result as before, ART. i39.J FRACTIONS. 131 OBS. 2. By examination the learner will perceive that this process is precisely the same in effect as the preceding ; for in both cases the denominator of the dividend is multiplied by the numerator of the di- visor, and the numerator of the dividend, by the denominator of the divisor. Hence, 139. To divide a fraction by a fraction. I. If the given fractions have a common denominator; Divide the numerator of the dividend by the numerator of the divisor. II. When the fractions have not a common denominator ; Invert the divisor, and proceed as in multiplication of frac- tions. (Art 135.) OBS. 1 . Compound fractions occurring in the divisor or dividend, must be reduced to simple ones, and mixed numbers to improper fractions. 2. The method of dividing a fraction by a fraction depends upon the obvious principle, that if two fractions have a common denomi- nator, the numerator of the dividend, divided by the numerator of the divisor, will give the true quotient. Now multiplying the numerator of the dividend by the denominator of the divisor, and the denomi- nator of the dividend by the numerator of the divisor, is in effect re- ducing the two fractions to a common denominator. The object of inverting the divisor, is simply for convenience in multiplying. 24. Divide | off by 1. Solution. | of $=*&, and l|=f Now or f . Ans. 25. Divide 7-J- by 2|. Ans. 3*. 26. Divide 13*- by f. 27. Divide -f- by 1-f. 28. Divide ff- by -J-f 29. Divide -f-f by QUEST. 139. How is one fraction divided by another when they have a common denominator ? How, when they have not common denominators ? Ohs. How proceed when the divisor or dividend are compound fractions, or mixed numbers ? Upon what principle doea the method of dividing a fraction by a fraction, depend ? Why mul- tiply the numerator of the dividend by the denominator of the divisor &c. ? Why invert the divisor ? 182 DIVISION OF -[SECT. VI, 30. Divide i of $ by off. Operation. For convenience we arrange th* numerators, (which answer to divi- * dends,) on the right of a perpendic- ular line, and the denominators, (which answer to divisors,) on the le left ; then canceling the factors 3 8 I 5=-|. Ans. and 2, which are common to both sides, (Art. 91. a,) we multiply the remaining factors in the numerators together, and those remaining in the denominators, as in the rule above. Hence, 1 4O* To divide fractions by CANCELATION. Having inverted the divisor, cancel all the factors common both to the numerators and denominators, a?id proceed as in multiplication of fractions. (Art. 136.) OBS. Before arranging the terms of the divisor for cancelation, it is always necessary to invert them, or auppose them to be inverted. 31. Divide 4 by 2i. Ans. 2. 32. Divide f of 6 by | of 4. 33. Divide 4| by i of ^. 34. Divide-f offfby^of f 35. Divide -& of f by f 36. Divide i of 15-f- by 4-f. 37. Divide i by ff of -fr. 38. Divide f by -ft of 2f 39. Divide 25i by i of 26, CASE III. 40. A merchant sent 12 barrels of flour to supply some destitute people, allowing -f of a barrel to each family. How many families shared in his bounty ? Solution. If -f of a barrel supplied 1 family, 12 barrels will supply as many families as -f is contained times in 12. Reducing the dividend 12 to the form of a fraction, it be- comes -V*; now inverting the divisor, we have -^X^ 3 /- or 18. Ans. 18 families. QUEST.- 140. How divide fractions by cancelation ? How arranga the terms of the given fractions ? Oba. What must be done to the divi- or before arranging its terms ? ART. 140-143.] FRACTIONS. 133 Or, we may reason thus : - is contained in 12, as many times as there are thirds in 1 2, viz : 36 times. Now 2 thirds are contained in 12, only half as many times as 1 third; and 36-=-2=18. Ans. Hence, 141* To divide a whole number by a fraction. Reduce the whole number to the form of a ft -action , (Art. 122. Obs. 1,) and then proceed according to the rule for di- viding a fraction by a fraction. (Art. 139.) Or, multiply the whole number by the denominator, ana divide the product by the numerator. OBS. When the divisor is a mixed number, it must be reduced to an improper fraction, then proceed as above. 41. Divide 120 by 3f. Ans. 33i. 42. Divide 35 by . 43. Divide 47 by f 44. Divide 165 by f 45. Divide 237 by 4 \. 142. From the definition of complex fractions, and the manner of expressing them, it will be seen that they arise from division of fractions. Thus the complex frac- 41 tion |, is the same as -f-s-f- ; for, the numerator 4=-f, and the denominator l-J-=f; but the numerator of a frac- tion is a dividend, and the denominator a divisor. (Art. 109.) Now -f-HHf $, which is a simple fraction. Hence, 143* To reduce a complex fraction to a simple one. Consider the denominator as a divisor, and proceed as in division of fractions. (Art. 139.) 2_L 46. Reduce to a simple fraction. 5f Operation. Now f^/H^fa or ff . Ans. k \ QUEST. 141.' How is a whole number divided by a fraction? Oba, How by a mixed number? 142. From what do complex fraction* irise ? 143. How reduce them to simple fractions ? 134 DIVISION OP [SECT. VI 47. Reduce to a simple fraction. Ans. J A 3i 48. Reduce _I to a simple fraction. A?is. ft. 6 f 49. Reduce ^ to a simple fraction. Ans. -ft. 50. Reduce the following complex fractions to simple ones. 4-| 8 9-J- 12-J- 18-J- 20-i- T 5f 7i "6f T2i 25l 144* To multiply complex fractions together. First reduce the complex fractions to simple ones ; (Art, 143 ;) then arrange the terms, and cancel the commo?i factors as in multiplication of simple fractions. (Art. 136.) OBS. 1 . The terms of the complex . /actions may be arranged for reducing them to simple ones, and for multiplication at the same time. 2. To divide one complex fraction by another, reduce them to sim pie fractions, then proceed as in Art. 139. 51. Multiply^ by || Operation. The numerator 2- f. (Art. 122.) Place the 7 on the right Q * (Art. 143;) i. e. place the 4 on the 3 | 8=2|. Ans. right and the 9 on the left of the line. 4=, and lf=f, both oi which must be arranged in the same manner as the terms of the multiplicand. Now, canceling the common fac- tors, we divide the product of those remaining on the right of the line by the product of those on the left, and the quotient is 2|. (Art. 136.) QUEST. 144. How are complex fractions multiplied together I Ob* How is one complex fraction divided by another ? hand and 3 on the left of the per- 4 pendicular line. The denomina- tor 2f=f 5 which must be inverted ; ART. 144.] FRACTIONS. 135 52 Multiply ?iby?t 53. Multiply fUby??. 2-f A 6i *21 54. Multiply xbyl. 65. Multiply xb EXAMPLES FOR PRACTICE. 1. At % dollar per bushel, how many bushels of pears can be bought for 5 dollars ? 2. At -f- of a penny apiece, how many apples can be bought for 18 pence? 3. At f of a dollar a pound, how many pounds of tea will 7 dollars buy ? 4. How many bushels of pears, at 1 dollar a bushel, can be purchased for 15 dollars? 5. How many gallons of molasses, at 2 dimes per gallon, will 10 dimes buy? 6. How many yards of satinet, at If of a dollar per yard, can be purchased for 20 dollars ? 7. At 4-f dollars per yard, how many yards of cloth can be obtained for 25 dollars ? 8. At 6f cents a mile, how far can you ride for 62 cents ? 9. At 12 cents a pound, how many pounds of flax will 67-f- cents buy ? 10. At 16-J- cents per pound, how many pounds of figs can you buy for 87^- cents ? 11. How many cords of wood, at 6 dollars per cord, will it take to pay a debt of Q7$ dollars ? 12. How many barrels of beer, at llf dollars per bar- rel, can be obtained for 95-J- dollars? 13. A man bought 15-| barrels of beef for 124f dollars, how much did he give per barrel ? 14. A man bought 13 pounds of sugar for 94 cents: Aow much did his sugar cost him a pound ? 15. A lady bought 15-f yards of silk for 145A shil- lings : how much did she pay per yard ? 16. Bought 151 baskets of peaches for 24| dollars : how much was the cost per basket ? 136 , COMPOUND [SECT. VII 17. Bought 30i yards of broadcloth, for 181- dollars, what was the price per yard ? 18. Paid 375 dollars for 125| pounds of indigo : what was the cost per pound ? 19. How many tons of hay, at 16- dollars per ton, can be bought for 196^- dollars ? 20. How many sacks of wool, at 17 i dollars per sack, can be purchased for 1500 dollars ? 21. How many bales of cotton, at 15-f dollars per bale, can be bought for 2500 dollars ? 22. Divide 145^ by 16. 23. Divide 16ft by 25. 24. Divide 8526 by 45^. 25. Divide 12563 by 68^- 26. Divide 85ff by 18$. 27. Divide 105^- by 82-&. 28. Divide f of -fa by 6. 29. Divide -f of 16 by -f off. 30. Divide -ft of 30 by 19. 31. Divide f of -f by 21. 32. Divide T\of f by f of 31. 33. Divide -^of SECTION VII. COMPOUND NUMBERS. AUT. 146. Numbers which express things of the same kind or denomination, as 3 pears, 7 rose?, 15 horses, are called simple numbers. Numbers which express things of different kinds or de- nominations, as the divisions of money, weight, and mea- sure, are called compound numbers. Thus 6 shillings 7 pence ; 5 pounds 2 ounces ; 7 feet 3 inches, &c., are compound nnmbers. OBS. Compound Numbers, by some late authors, are called De- nominate Numbers. QUEST. 146. What are simple numbers ? What are compound numbers * ARTS, 146, 147.] NUMBERS, 137 / STERLING MONEY. 147. Sterling Money is the national currency of England. 4 farthings (qr, or far.) make 1 penny, marked d. 12 pence " 1 shilling, " s. 20 shillings " 1 pound, or sovereign,. 21 shillings " 1 guinea. OBS. 1. It is customary, at the present day, to express farthings in fractions of a penny. Thus, 1 qr. is written i d.; 2 qrs.,-1- d.; 3 qrs. ad. 2. The Pound Sterling is represented by a gold coin, called a Sovereign. According to Act of Congress, 1842, it is equal to 4 Dollars and 8-1 cents. MENTAL EXERCISES. 1. In 5 pence, how many farthings? Solution. Since there are 4 farthings in 1 penny, in 5 pence there are 5 times as many ; and 5 times 4 are 20. Ans. 20 farthings. 2. In 8 pence, how many farthings ? In 10d.? In 12d. ? 3. How many shillings are there in 3 pounds? In 5? In 8? In 10? 4. How many pence are there in 8 farthings? Solution. Since 4 farthings make 1 penny, 8 farthings will make as many pence, as 4 is contained times in 8 ; and 4 is contained in 8, 2 times. Ans. 2 pence. 5. How many pence in 12 farthings? In 15 qrs. ? In 20 qrs. ? In 25 qrs. ? In 33 qrs. ? In 36 qrs. ? 6. How many shillings in 15 pence? In 24d. ? In 30d. ? In 36d. ? In 60d. ? In 68d. ? In 75d. ? 7. How many pounds in 25 shillings ? In 30s. ? In 40s. ? In 65s. ? In 80s. ? In 89s. ? QUEST. 147. What is Sterling Money ? Repeat the Table. Obs. How are farthings usually expressed ? How is a pound sterling repre- lented ? What is its value in dollars and centa ? 198 COMPOUND [SECT. VIL TROY WEIGHT. Note. Most children have very erroneous or indistinct ideas of th weights and measures in common use. It is, therefore, strongly re- commended for teachers to illustrate them practically, by referring to some visible object of equal magnitude, or by exhibiting the ounce; the pound; the linear inch, foot, yard, and rod; also a square and cubic inch, foot,and yard; the pint, quart, gallon, peck, bushel, &c. / 148. Troy Weight is used in weighing gold, silver jewels, liquors, &c., and is generally adopted in philo sophical experiments. 24 grains (gr.) make 1 pennyweight, marked pwt. 20 pennyweights " 1 ounce, " oz. 12 ounces " 1 pound, " Ib. OBS. 1. The standard of Weights end Measures is different in dif- ferent States of the Union. In 1834, the Government of the United States adopted a uniform standard, for the use of the several custoir houses arid other purposes. 2. The standard unit of Weight adopted by the Government, is the Tray Pound of the United States Mint, which is identical with the Imperial Troy pound of England, established by Act of Parliament. A. D. 1826.* 3. Troy Weight was formerly used in weighing articles of every suppose think it was derived from Troy-novant, the former name o f London.t 8. How many grains in 2 pennyweights? In 3 pwts? In 4 pwts ? 9. How many pennyweights in 2 ounces ? In 3 oz. ? In 4 oz. ? In 5 oz. ? 1U. How many ounces in 2 pounds ? In 3 Ibs. ? In 4 Ibs. ? In 5 Ibs. ? In 6 Ibs. ? In 7 Ibs. ? In 10 Ibs. ? QUEST. 148. In what is Troy Weight used ? Repeat the Table, Obs. When was Troy Weight introduced into Europe ? From what was its name derived ? Do all the States have the same standard of weights and measures ? What is the standard unit of weight adopted by the Government of the United States ? * Hassler on Weights and Measures, p. 10. Also, Reports of the Secretary ol the Treasury, March 3, 1831 ; and June 20, 1832. t Hind's Arithmetic, Art. 224. Also, North America? Review, VoL XLV ARTS. 148, 149.] NUMBERS. 139 AVOIRDUPOIS WEIGHT. 149* Avoirdupois Weight is used in weighing gro- ceries and all coarse articles ; as, sugar, tea, coffee, butter, cheese, flour, hay, &c., and all metals, except gold and silver. 16 drams (dr.) make 1 ounce, marked oz. 16 ounces " 1 pound, " Ib. 25 pounds " 1 quarter, " qr. 4 quarters, or 100 Ibs. " 1 hundredweight, cwt. 20 hundred weight " 1 ton, marked T. OBS. 1. The Avoirdupois Pound of the United Stales is determfced from the standard Troy Pound, and is in the ratio of 5760 to 7000 ;* that is, 1 pound Troy contains 5760 grains. 1 pound Avoirdupois " 7000 " Troy. 1 ounce " " 437i " Idram " " 27-T 2. The British Imperial Pound Avoirdupms is defined to be the weight of 27i 7 u 2 u 7 o 4 o cubic inches of distilled water, at the tempera- ture of 61 Fahrenheit, when the barometer stands at 30 .t 3. Gross weight is the weight of goods with the boxes, casks, or bags which contain them. Net weight is the weight of the goods only. 4. Formerly it was the custom to allow 112 pounds for a hundred weight, and 28 pounds for a quarter ; but this practice has become nearly or quite obsolete. In buying and selling all articles of com- merce estimated by weight, the laws of most of the States as wU as general usage, call 100 pounds a hundred weight, and 25 pounds a quarter. 11. How many drams are there in 2 ounces? In 3 oz. ? In 4 oz. ? In 5 oz. ? 12. How many ounces in 2 pounds? In 3 Ibs.? In 4 Ibs. ? In 5 Ibs. ? 13. How many pounds in 2 quarters? QUEST. 149. In what is Avoirdupois Weight used \ Repeat the Table. Point to an object that weighs an ounce. A pound. Obs. How is the Avoirdupois pound of the United States determined 1 What is gross weight ? Net weight I How many pounds were for- merly allowed for a hundred weight ? For a quarter 1 * Reports of Secretary of Treasury, March 3, 1832 : June, 90, 1832. Also, Congressional Documents of 1833. t Hind's Arithmetic, Art. 223 140 COMPOUND [SECT. VII, 14. How many quarters in 2 hundred weight? In 3 cwt. ? In 5 cwt. ? In 6 cwt. ? APOTHECARIES' WEIGHT. 15O. Apothecaries 1 Weight is used by apothecaries and physicians in mixing medicines. 20 grains (gr.) make 1 scruple, marked sc., or 3. 3 scruples " 1 dram, " dr., or 3. 8 drams " 1 ounce, " oz. : or . 12 ounces " 1 pound, " ft>. OBS. I. The pound and ounce in this weight are the same, as thi Troy pound and ounce ; the other denominations are different. 2. Drugs and medicines are bought and sold by avoirdupois weight 15. In 2 scruples, how many grains? In 3 sc. ? 16. In 3 drams, how many scruples? In 4 dr.? In 5 dr.? In 7 dr.?* 17. In 2 pounds, how many ounces ? In 3 as. ? LONG MEASURE. 151* Long Measure is used in measuring distances or length only, without regard to breadth or depth. 12 inches (in.} make 1 foot, marked ft. 1 yard, " yd. 1 rod, perch, or pole, " r. or p. 1 furlong, " fur. 1 mile, m. I league, " / 3 feet 51 yards, or 16 feet 40 rods 8 furlongs, or 320 rods 3 miles 1 360 degrees make a great circle, or the circumference of the earth. Note. 4 inches make 1 hand ; 9 inches, 1 span ; 18 inches, J cu- bit; 6 feet, 1 fathom. QUEST. 150. In what is Apothecaries' Weight used ? Recite the Table. Obs. To what are the apothecaries' ounce and pound equal f How are drugs and medicines bought and sold ? 151. In what i* Long Measure used ? Recite the Table. ARTS. 150-152.] NUMBERS. 141 OBS. 1. The standard untt of Length adopted by the General Go- "ernment, is the Yard of 3 feet, or 36 inches, and is identical with the imperial Yard of England. It is made of brass, and is determined from the scale of Troughton* at the temperature of 62 Fahrenheit. 2. Long measure is frequently called linear, or lineal measure. Formerly the inch was divided into 3 barleycorns ; but the barleycorn is not employed as a measure at the present day. The inch is com- monly divided either into eighths or tentlis; sometimes, however, it ia divided into twelfths, which are called lines. 19. In 3 feet, how many inches? In 3 feet and 4 in., how many inches ? In 4 feet and 7 in. ? 20. How many furlongs in 3 miles and 2 furlongs ? How many in 4 m. and 5 fur. 1 In 6 m. and 7 fur. ? 21. How many yards in 6 feet ? In 1 2 ft. ? In 1 6 ft. ? In 23 ft. ? 22. How many feet in 27 inches? In 36 in. ? In 41 in.? In 64 in.? 23. How many yards in 12 feet ? In 17 ft ? In 25 ft ? In 30 ft? CLOTH MEASURE. 152* Cloth Measure is used in measuring cloth, lace, and all kinds of goods which are bought and sold by the yard. 24 inches (in.) make 1 nail, marked no. 4 nails, or 9 in. 4 quarters 3 quarters, or f of a yard 5 quarters, or 1 4 yard 6 quarters, or 1 1 yards 1 quarter of a yard, ' qr. 1 yard, " yd. 1 Flemish ell, Fl. e. 1 English ell, " E. e. 1 French ell, " F. e. OBS. Cloth measure is a species of long measure. The yard is th *ame in both. Cloths, laces, &c., are bought and sold by the linear an d farthings. Hence. 2 3 6 2 wnen tne pounds are reduced to 20s in 1 shillings, the given shillings (3) must be added mentally to the pro- 43 shillino-s. duct. In like manner when the 12d. in Is. shillings are reduced to pence, the given pence (6) must be added ; 522 pence. an d when the pence are reduced to 4 * ar - m W? farthings, the given farthings (2) 2090 far. Ans. must be added. OBS. In these examples it is required to reduce higher denomina- tions to lower; as pounds to shillings, shillings to pence, &c. This s done by successive multiplications. QUEST. 160. What is Reduction ? How are pounds reduced to ihillingi 1 Why multiply by 20 ? How are shillings reduced to pence? Why ? How, pence to farthings ? Why ? 150 REDUCTION. [SECT. VII. I6O. a. It often happens that we wish to reduce '.ower denominations to higher, as farthings to pence, pence to shillings, and shillings to pounds. Thus, 3. In 2880 farthings, how many pounds ? First, we reduce the given farthings Operation. to pence? which is the next higher de- 4)2880 far. nomination. This is done by dividing l9V79ffH tnem by" 4. For, since 4 far. make Id., (Art. 147,) in 2880 far. there are as 20)60s. many pence, as 4 is contained times in 3 Ans. 2880 ; and 4 is contained in 2880, 720 times. We now reduce the 720 pence to shillings, by dividing them by 12, because 12d. make Is. Finally, we reduce the shillings (60) to pounds, by dividing by 20, because 20s. make 1. Thus, 2880 far. =3, which is the answer required. 4. How many pounds in 2090 farthings? Operatien. In dividing by 4 there is a 4)2090 fa,r. remainder of 2 far. ; in dividing 12)522d:2far.over. b Y ^V f h ere is a remainder of 6d. ; in dividing by 20, the quo- 2) 43s - 6d. over. tient is 2 and 3s over . The 2, 3s, over. answer, therefore, is 2, 3s. 6d. Ans. 2, 3s. 6d. 2 far. 2 far. That is, 2090 far.=2, 3s. 6d. 2 far. OBS. 1. The last two examples are exactly the reverse of the first two ; that is, lower denominations are required to be reduced to high- er, which is done by successive divisions. 2. Reducing compound numbers to lower denominations is usually called Reduction Descending; reducing them to higher denomina- tions, Reduction Ascending. The former employs multiplication ; the latter division. They mutually prove each other. QUEST. Ex. 3. How are farthings reduced to pence ? Why divide by 4 ? How reduce pence to shillings 1 Why ? How shillings to pounds ? Why ? Obs. What is reducing compound numbers to lower denominations usually called ? To higher denominations ? Which of the fundamental rules is employed by the former ? Which by the latter 1 ARTS. 161 162.) REDUCTION. 151 161* From the preceding illustrations we derive the following GENERAL RULE FOR REDUCTION. t. To reduce compound Nos. to lower denominations. Multiply tJie highest denomination given, by that number which it takes of the next lower denomination to make ONE of this higher ; to the product, add the number expressed in this lower denomination in the given example. Proceed in this manner with each successive denomination, till you come to the one required. II. To reduce compound Nos. to higher denominations. Divide the given denomination by that number which it takes of this denomination to make ONE of the next higher. Proceed in this manner with each successive denomination, till you come to the one required: The last quotient, with th& several remainders, will be the answer sought. 162. PROOF. Reverse the operation; that is, reduce, back the answer to the original denominations, and if the result correspond with the numbers given, the work is right. OBS. Each remainder is of the same denomination as the dividend from which it arose. (Art. 66. Obs. 2.) STERLING MONEY. (ART. 147.) 5. In 35, 4s. 6d. how many pence ? Operation. Proof. s. d. 12)8454 pence. 35 4 6 20 20)704s. 6d. 35, 4s. 6d. 704 12 AJU. 8454d. QUEST. 161. How are compound numbers reduced to lower denom- inations ? How reduced to higher denominations ? 162. How is Re- duction proved ? Obi. Of what denomination is each remainder \ 152 REDUCTION. [SECT. VIL 6. In 57600 farthings, how many pounds? Operation. Proof. 4)57600 far. 60 12)14400 d. 20 20) 1200s. 1200 s. 60 Ans. 12 14400 d. 4 57600 far. 7. In 43, 12s., how many shillings ? 8. In 1 7 shillings, how many farthings ? 9. In 1 1 76 pence, how many pounds ? 10. In 12356 farthings, how many shillings? 11. In 175 pounds, how many farthings? 12. In 84, 16s. 7-d., how many farthings ? 13. In 25256 pence, how many pounds? 14. In 56237 farthings, how many pounds? 15. In 25, 9s. 7-d., how many farthings? TROY WEIGHT. (ART. 148.) 16. In 11 Ibs., how many pennyweights? Ans. 2640 pwts. 17. In 15 ounces, how many grains ? 18. In 10 Ibs. 5 oz. 6 pwts., how many grains ? 19. In 512 pwts., how many pounds? 20. In 2156 grains, how many ounces? 21. In 35210 grains, how many pounds? AVOIRDUPOIS WEIGHT. (ART. 149.) 22. Reduce 25 pounds to dran^ Ans. 6400 drams. 23. Reduce 36 cwt. 2 qrs. to pounds. 24. Reduce 5 tons, 7 cwt. 15 Ibs. to ounces. 25. Reduce 3 quarters, 15 Ibs. 10 oz. to drams. 26. Reduce 875 ounces to pounds. 27. Reduce 1565 pounds to hundred weight 28 Reduce 1728 drams to pounds. ART. 162.J REDUCTION. 153 29. Reduce 5672 ounces to hundred weight. 30. Reduce 15285 pounds to tons. 31. Reduce 26720 drams to hundred weight. APOTHECARIES' WEIGHT. (ART. 150.) 32. How many drams are there in 70 pounds ? Ans. 6720 drams. 33. How many scruples in 156 pounds ? 34. How many ounces in 726 scruples ? 35. How many pounds in 1260 drams ? LONG MEASURE. (ART. 151.) 36. In 96 rods, how many feet ? 2)96 5^- yds. in 1 r. We .first multiply by 5, then "^QQ by i, and unite the two results. 43 (Art. 134. a.) But to multi- = , ply by i, we take half of the 3 ifnfi d multiplicand once. (Art. 132.) Ans. 1584 feet. 37. In 45 furlongs, how many inches ? 38. In 1584 feet, how many rods? 3)1584 We first reduce the feet to yards, 528 fry dividing by 3 ; next, reduce the 2 yards to rods, by dividing by 5. Divide % ^ 2-5 we reduce it to 1 T \inqfi 2-5 ) 1U5b halves, and also reduce the dividend Ans. 96 (528 yds.) to halves, then divide 1056 by 11. (Art. 139,1.) 39. In 1728 inches, how many rods? 40. In 26400 feet, how many miles ? 41. In 25 leagues, how many inches? 42. In 40 leagues, 6 furlongs, 2 in., how many inches? 43. In 750324 inches, how many miles ? 44. How many inches in the circumference of the arth 1 154 REDUCTION. [SECT. VII CLOTH MEASURE. (ART. 152.) 45. How many quarters in 45 yards ? 46. How many nails in 53 Flemish ells ? 47. How many nails in 8 1 English ells ? 48. Reduce 563 quarters to yards. 49. Reduce 1824 nails to French ells. 50. Reduce 5208 nails to English ells. SQUARE MEASURE. (ART. 153.) 51. In 1766 square rods and 19 yards, how many feet 1 52. In 56 acres and 3 roods, how many square teet? 53. In 1275 square miles, how many acres? 54. How many square rods in 25640 feet ? 55. How many acres in 1865 roods? 56. How many acres in 2118165^ yards'? 1G3. The area of a floor, .a piece of land, or any surface which has four sides and four right angles, is found by multiplying its length and, breadth together. Note. The area of a figure is the superficial contents or space con- tained within the line or lines, by which the figure is bounded. It ia reckoned in square inches, feet, yards, rods, &c. 57. How many square feet are there in a table which is 4 feet long and 3 feet wide? Suggestion. Let the given table be represented by the subjoined figure, the length of which is divided into 4 equal parts, and the breadth into 3 equal parts, which we will call linear feet. Now it is plain that the table will contain as many sq. feet as there are squares in the given figure. But QUEST. 163. How do you find the area or superficial contents of a surface having four sides and four right angles ? Note. What is meant by the terra area ? How is it reckoned ? Obs. What is a figure which nas four siJss and four right angles, called ? ARTS. 163, 164.] REDUCTION. 155 the number of squares in the figure is equal to the number of equal parts (linear feet) which its length contains, re- peated as many times as ihere are equal parts (lineai feet) in its breadth; that is, equal to 4x3, or 12. The fable therefore contains 12 square feet. OBS. A figure which has four sides and four right angles, like the preceding, is called a Rectangle, or Parallelogram. 58. What is the area of a garden, which is 8 rods long and 5 rods wide ? Ans. 40 square rods. 59. How many square feet in a floor, 18 feet Ions: and 17 feet wide? 60. How many square yards in a ceiling, 20 feet long and 18 feet wide? 61. What is the area of a field, which is 36 rods long and 25 rods wide ? 62. How many acres are there in a piece of land, 80 rods long and 48 rods wide ? CUBIC MEASURE. (ART. 154.) 63. In 75 cubic feet, how many inches ? 64. In 37 tons of round timber, how many inches ? 65. In 28124 cubic feet, how many tons of hewn timber ? 66. In 16568 cubic feet of wood, how many cords? 67. In 65 cords of wood, how many cubic feet ? 164. The solidity ; or cubical contents of boxes ol goods, piles of wood, &c., are found by multiplying the length, breadth, and thickness together. 68. How many cubic inches are there in a box, wnose length is 30 inches, its breadth 18, and its depth 15 inches? Ans. 8100 cu. in. 69. How many cubic inches in a block of marble, 43 inches long, 18 inches broad, and 12 inches thick? QUEST. 164. How are the cubical contents of a box of goods, a pile fwood, &c., found? 150 COMPOUND [SECT. VIL 70. How many cubic feet in a room, 16 fe^i long, 15 teet wide, and 9 feet high ? 71. How many cubic feet in a load of wood, 8 feet long., 4 feet wide, and 3- feet high ? 72. How many cubic feet in a pile of wood, 16 feet long, 6 feet wide, and 5 feet high ? How many cords ? 73. How many cords of wood in a pile, 140 feet long 4 feet wide, and 6 feet high ? WINE MEASURE. (ART. 155.) 74. In 4624 gills, how many gallons ? 75. In 24260 quarts, how many hogsheads ? 76. How many pints in 1 5 hogsheads, and 20 gallons ? 77. How many gills in 40 barrels ? BEER MEASURE. (ART. 156.) 78. How many barrels of beer in 5000 pints ? 79. How many hogsheads in 7800 quarts ? 80. How many quarts in 25 hogsheads, and 7 gallons 1 81. How many pints in 110 gallons, 3 qts. and I pt. 1 DRY MEASURE. (ART. 157.) 82. Reduce 536 bushels, and 3 pecks to quarts. 83. Reduce 821 chaldrons to pints. 84. Reduce 1728 pints to pecks. 85. Reduce 85600 quarts to bushels. TIME. (ART. 153.) 86. In 1 5 days, 6 hours, and 9 min., how many seconds V 87. In 365 days and 6 hours, how many minutes ? 88. How many seconds in a solar year ? 89. Allowing 365d. 6h. to a year, how many mmutet has a person lived who is 21 years old ? 90. How many hours in 568240 seconds ? 91. How many weeks in 8568456 minutes? 92. How many lunar months in 6925600 hours ? 93. How many years in 56857200 hours? 94. How many years in 1 000000000 seconds ? A.RT. 165.] NUMBERS. 157 CIRCULAR MEASURE. (ART. 159.) 95. In 75 degrees, how many seconds ? 96. In 8 signs, and 15 degrees, how many minutes? 97. In 12 signs, how many seconds? 98. In 86860 seconds, how many degrees ? 99. In 567800 minutes, how many signs ? 100. In 25000000 seconds, how many signs? COMPOUND NUMBERS REDUCED TO FRACTIONS. Ex. 1. Change 7s. 6d. to the fraction of a pound. 7s. 6d. 1 or 20s. 7 , 12 (Art. 161, I,) we Numerator 90d. Denominator 240d. have 90d. which Ans. 2 a 3 Q o=i2 a r, or f. j s me numerator of the fraction. Then reducing 1 to the same denomi- nation as the numerator, we have 240d., which is the denominator. Consequently & is the fraction required. But -5-W may be reduced to lower terms. Thus &=* ft,oii. (Art. 120.) Hence, 165* To reduce a compound number to a common fraction of a higher denomination. First reduce the given compound number to the lowest denomination mentioned for the numerator; then reduce a TOUT of the denomination of the required fraction to the same denomination as the numerator, and the result will be the denominator. (Art. 161.) OBS. When the given number contains but one denomination, it of course requires no reduction. 2. Reduce 3s. 7d. 2 far. to the fraction of 1. Ans. fr. or -&. 3. Reduce 9d. 3 far. to the fraction of Is. 4. What part of a bushel is 3 pecks and 5 qts. ? QUEST. 165* How is a compound number reduced to a common 158 COMPOUND [SECT. VII 5. What part of a peck is 5 qts. and 1 pt. ? 6. What part of a gallon is 3 qts. 1 pt. and 3 gills 1 7. What part of 1 gallon is 1 pt and 1 gill? 8. What part of 1 hogshead is 15 gals, and 3 qts.? 9. What part of 1 ton is 5 cwt. and 2 qrs. ? 10. What part of 1 hundred weight is 2 qrs. and 7 Ibs/ 11. What part of 1 quarter is 1 Ib. and 5 oz. ? 12. What part of 1 mile is 45 rods? 13. What part of 1 mile is 10 fur. and 35 rods ? 14. What part of 1 league is 1 m. 1 fur. and 1 r.? 15. What part of 1 yard is 2 qrs. and 3 nails? 16. What part of 1 is 1 penny? Ans. frfrr- 17. What part of 1 is -f of a penny? Note. The lowest denomination mentioned in this example, is thirds of a penny. Hence, 1 must be reduced to thirds of a penny for the denominator, and 2, the given number of thirds will be the numerator. Ans. ~T%~5i or ^TaT- 18. What part of 1 is 5$ shillings? Ans. ff. 19. What part of 1 day is 2 hours ? 20. What part of 1 day is 4 h. and 8-J- min. ? 21. What part of 1 hour is 3 rhin. and 40 sec. ? 22. What part of 1 hour is 15f sec.? 23. What part of 1 pound is -f- of an ounce ? 24. What part of 1 ton is of a pound ? 25. What part of 1 hogshead is f of a gallon ? 26. What part of 1 gallon is -f of a gill ? FRACTIONAL COMPOUND NUMBERS REDUCED TO WHOLE NUMBERS OF LOWER DENOMINATIONS. Ex. 1. Reduce f of 1 to shillings and pence. Operation. Multiply the numerator by 20, to 3 eighths . reduce it to shillings, as in reduc 20 tion. (Art 161, I.) ix20s.=^s. g\50 or 7s. and 4 remainder. Again, mul- tiplying- the remainder 4 by 12, we dul. 7,and4rem. J/ e | g . an(J 48 ^ 8=6 d The quotients, 7s. and 6d. are the an- 8)48 swer required. Hence. Pence 6. Ans. 7s. 6d. S. 166, 167.] NUMBERS. 159 166. To reduce fractional compound numbers to whole numbers. First reduce the given numerator to the next lower denom- ination; (Art. 161, I;) then divide the product by the de- nominator, and the quotient will be an integer of the next lower denomination. Proceed in like manner with the re- mainder, and the several quotients will be the whole numbers required. 2. Reduce of 1 to shillings. Ans. 12s. 3. How many shillings and pence in ^ ? 4. How many shillings, &c., in f ? 5. In of 1 week, how many days, hours, 6. In -f^ of 1 day, how many hours, minutes, 7. Change f of 1 league to miles, &c. 8. Change \ of 1 mile to furlongs, &c. 9. Reduce -fa of 1 hundred weight to quarters, 10. In f of 1 ton, how many hundred weight, &c. ? 11. In -f of 1 bushel, how many pecks, quarts, &c. ? 12. In -^g- of 1 peck, how many quarts, &c. ? 13. Reduce -fr of 1 to shillings. Suggestion. Since the numerator, when reduced to the denomination required, cannot be divided by the denomi- nator, the division must be represented. Note. This, in effect, is reducing -^ of 1 to the fraction of a shilling. 14. Reduce yfj of 1 to pence. Ans. -fffd. 167. From the last two examples it is manifest, that a fraction of a higher denomination may be changed to a fraction of a lower denomination, by reducing the given numerator to the denomination of the required fraction, and 'placing the result over the given denominator. QUEST. 166. How are fractional compound numbers reduced to whole ones ? 167. How is a fraction of a higher denomination clanged wo a fraction of a lower denomination t 160 COMPOUND [SECT. VII 15. Reduce Tar f 1 to tne fraction of a shilling. Ans. y^ys. 16. Reduce rlir f 1 week to the fraction of a day. 17. Change TT*ST f 1 m ^ e to tne fraction of a rod. 18. Change TOT of 1 rod to the fraction of a foot. 19. Change \Yli of 1. yard to the fraction of a nail. 20. Change 1 1 of 1 ton to the fraction of a pound. ADDITION OF COMPOUND NUMBERS. lat is the sum of 4, 9s. 6d. 2 far. : 3, 12s. 8d 1 . What is the sum of 4, ' 3 far. of iron, how much will 50 miles require ? 30. How much wheat will it require to make 1000 barrels of flour, allowing 4 bu. 2 pks. 6 qts. to a barrel ? AHT. 173.] DIVISION. 169 DIVISION OF COMPOUND NUMBERS. Ex. 1. Divide 17, 6s. 9d. by 4. Operation. Beginning with the pounds 5. d. far. we find 4 is contained in 17, 4)17 " 6 " 9 " 4 times and 1 over. Set the 4 " 6 " 8 " 1 4 under the pounds, and re- duce the remainder 1 to shil- lings, which added to the 6s., make 26s. 4 in 26s. ; 6 times and 2s. over. Set the 6 under the shillings, and reduce the remainder 2s. to pence, which added to the 9d. make -33d. 4 in 33d., 8 times and Id. over. Set the 8 under the pence, reduce the Id. to farthings, and divide as before. Ans. 4, 6s. 8d. 1 far. 173* Hence, we deduce the following general RULE FOR DIVIDING COMPOUND NUMBERS. Begin with the highest denomination, and divide each separately. Reduce the remainder, if any, to the next lower denomination, to which add tJie number of that de- nomination contained in the given example, and divide the sum as before. Proceed in this manner through all the de- nominations. OBS. 1. Each partial quotient will be of the same denomination, as that part of the dividend from which it arose. 2. When the divisor exceeds 12, and is a composite number, it is idvisable to divide first by one factor and that quotient by the other. (Art. 78.) If the divisor exceeds 12, but is not a composite number, long division may be employed. (Art. 77.) 3. The process of dividing different denominations, is called Com- pound Division. QUEST. 173. Where do you begin to divide a compound number ? What is tione with the remainder ? Obs. Of what denomination is each partial quotient ? When the divisor is a composite number, how pro- ceed ? What is the process of dividing different denominations, called ? 170 DECIMAL [SECT. VIL 2. Divide 274, 4s. 6d. by 21. Operation. 5. d. 3)274 " 4 " 6 Divide by the factors of 21, 7)91 " 8"^~2 13 " 1 " 2 Ans. 3. Divide 635, 17s. by 31. Operation. . s. . s. The remainder 15, is reduced 31)635, 17 (20. 10-^p to shillings, to which we add the 620 given shillings, making 31.7, and 15 rem. divide as before. The remain 20 der 7s. may be reduced to pence ~o7y and divided again if necessary. 310 7 rem. 4. Divide 7, 8s. 2d. by 3. 5. Divide 35, 10s. 8d. 3 far. by 6. 6. Divide 42, 17s. 3d. 2 far. by 8. 7. A man bought 5 cows for 23, 16s. 8d. : how much did they cost apiece 1 8. A merchant sold 10 rolls of carpeting for 62, 12s. 9d. : how much was that per roll ? 9. Paid 25, 10s. 6^d. for 12 yards of broadcloth: what was that per yard ? 10. A silver-smith melted up 2 Ibs. 8 oz. 10 pwts. of silver, which he made into 6 spoons: what was the weight of each spoon ? 11. The weight of 8 silver tankards is 10 Ibs. 5 oz. 7 pwts. 6 grs. : what is the weight of each ? 12. If 8 persons consume 85 Ibs. 12 oz. of meat in a month, how much is that apiece ? 13. A dairy- wo man packed 95 Ibs. 8 oz. of butter in IG boxes : how much did eah box contain ? 16. A tailor had 76 yds. 2 qrs. 3 na. of cloth, out of which he made 8 cloaks : how much did each cloak con tain? ARTS. 175, 176.] FRACTIONS. 171 17. A man traveled 50 m. and 32 r. in 11 hours: at what rate did he travel per hour ? 18. A man had 285 bu. 3 pks. 6 qts. of grain, which he wished to carry to market in 15 equal loads : how much must he carry at a load ? 19. A man had 80 A. 45 r. of land, which he laid out into 36 equal lots : how much did each lot contain ? SECTION VIII. DECIMAL FRACTIONS. ART. 175. When a number or thing is divided into equal parts, those parts we have seen are called Fractions. (Art. 105.) We have also seen that these equal parts take their name or denomination from the number of parts into which the integer or thing is divided. (Art 103.) Thus, if a unit is divided into 10 equal parts, the parts are called tenths; if divided into 100 equal parts, the parts are called hundredths; if divided into 1000 equal parts, the parts are called thousandths, &c. Now it is manifest that if a tenth is divided into 10 equal parts, 1 of those parts will be a hundredth ; for, -rV^-lO^Tuir- (Art. 138.) If a hundredth is divided into 10 equal parts, 1 of -the parts will be a thousandth; for, Tifu-^lO^TTMnrj &c. Thus a new class of fractions is ob tained, which regularly decreases in value in a tenfold ratio ; that is, a class which expresses simply tenths, hun- dredths, thousandths, &c., without the intervening parts, as m common fractions, and whose denominators are always 10, 100, 1000, &c. 176. Fractions which decrease in a tenfold ratio, or >rhich express simply tenths, hundredths, thousandths, &c., are called DECIMAL FRACTIONS. QUEST. 175. What are fractions? From what do the parts take their name ? 176. What are decimal fractions * From what do they arise 1 Why called decimals ? 17'xJ DECIMAL [SECT. VIII OBS. Decimal fractions obviously arise from dividing a unit intc ten equal parts, then subdividing each of those parts into ten other equal parts, and so on. They are called decimals, because they de- crease in a tenfold ratio. (Art. 10. Obs. 2.) 177. Each order of integers or whole numbers, it has been shown, increases in value from units towards the left in a ten-fold ratio ; (Art. 9 ;) and, conversely, each order must decrease from left to right in the same ratio, till we come to units' place again. 178. By extending this scale of notation below units towards the right hand, it is manifest that the first place on the right of units, will be ten times less in value than units' place ; that the second will be ten times less than the first ; the third ten times less than the second^ &c. Thus we have a series of orders below units, which de- crease in a tenfold ratio, and exactly correspond in value with tejiths. hundredths, thousandths, &c., in com. fractions. 179. Decimal Fractions are commonly expressed by writing the numerator with a point ( . ) before it. OBS. If the numerator does not contain so many figures as there are ciphers in the denominator, the deficiency must be supplied by prefixing ciphers to it. For example, -fa is written thus . 1 ; -fc thus .2 ; T 3 u- thus .3 ; &c. -j-J-u- is written thus .01, putting the 1 in hundredths place ; ifa thus .05 ; &c. That is, tenths are written in the first place on the right of units ; hundredths in the second place ; thousandths in the third place. &c. 180. The denominator of a decimal fraction is always 1 with as many ciphers annexed to it as there are decimal figures in the- given numerator. (Art. 175.) OBS. The point placed before decimals, is called the Decimal Poinf, or Separatrix. Its object is to distinguish the fractional parts fir am whole numbers. QUEST. 177. In what manner do whole numbers increase and decrease ? 178. By extending this scale below units, what -would be the value of the first place on the right of units ? The second I The third ? With what do these orders correspond ? 179. How are deci- mal fractions expressed. 180. What is the denominator of a decimal fraction I Obs. What is the point placed before decimals called ? ARTS. 177-183.] FRACTIONS. 173 1 8 1 The names of the different orders of decimals or places below units, may be easily learned from the following DECIMAL TABLE. 423 .267145986274 182. It will be seen from this table that the value of each figure in decimals, as well as in whole numbers, depends upon the place it occupies, reckoning from units. Thus, if a figure stands in the first place on the right of units, it expresses tenths ; if in the second, kundr&UM, which is the same result as before. 2. 3. 4. Multiply 8.45 96.071 456.03 By -25 .0032 4.5 4225 192142 228015 1690 288213 182412 Ans. 2.1125 .3074272 2052.135 191. From the preceding illustrations we deduce I'he following general RULE FOR MULTIPLICATION OF DECIMALS. Multiply as in whole number 's, ami point off as many figures from the right of the product for decimals, as there are decimal places both in the multiplier and multiplicand. If the product does not contain so many figures as then are decimals in both factors, supply the deficiency by prefixing ciphers. QUEST. 191. How are decimals multiplied together ? How do you point off the product? When the product does not contain so many figures as there are decimals in both factors, what is to be done 1 180 MULTIPLICATION OP [SECT. VIIL PROOF. Multiplication of Decimals is proved in the sam manner as Simple Multiplication. (Arts. 53, 74.) OBS. The reason for pointing off as many decimal places in tae product as there are decimals in both factors, may be illustrated thus : Suppose it is required to multiply .25 by .5. Supplying the denom- inators .25=1% and .5=T 5 o. (Art. 180.) Now (Art. 135.) But Tffl,= 125; (Art. 179;) that is, the product o! .25X-5, contains just as many decimals as the factors themselves. In like manner it may be shown that the product of any two or more de- cimal numbers, must contain as many decimal figures as there are places of decimals in the given factors. EXAMPLES. Ex. 1. In 1 piece of cloth there are 31.7 yards: how many yards are there in 7.3 pieces ? 2. In 1 barrel there are 31.5 gallons: how many gal- lons are there in 8.25 barrels ? 3. In one rod there are 16.5 feet: how many feet are there in 35.75 rods ? 4. How many cords of wood are there in 45 loads, al- lowing 8.25 of a cord to a load ? 5. How many rods are there in a piece of land 25.35 rods long, and 20.5 rods wide ? 6. If a man can travel 38.75 miles per day, how far can he travel in 12.25 days? 7. How many pounds of coffee are there in 68 sacks, allowing 961.25 pounds to a sack? 8. If a family consume .85 of a barrel of flour in a week, how much will they consume in 52.23 week?? 9- What is the product of 10.001 into .05? 10. What is the product of 50.0065 into 1.003 ? 192. When the multiplier is 10, 100, 1000, &c., the multiplication may be performed by simply removing the decimal point as many places towards the right, as there are ciphers in the multiplier. (Arts. 59, 191.) QUEST. How is multiplication of decimals proved ? 192. How proceed when the multiplier is 10 100, 1000, &c. A.RTS. 192, 193.] DECIMALS. 181 11. Multiply 4.6051 by 100. Ans 460.51. 12. Multiply 2.6501 by 1000. 13. Multiply .5678 by 10000. 14. Multiply .000781 by 2.40001. 15. Multiply 1.002003 by .0024. 16. Multiply .58001 by .0001003. 17. Multiply 8.00 1502 by .00005. 18. Multiply 85689.31 by .000001. 19. Multiply .0000045 by 69.5. 20. Multiply .0340006 by .000067. 21. Multiply .5 by 5 millionths. 22. Multiply .15 by 28 ten thousandths. 23. Multiply 25 hundredth thousandths by 7.3 24. Multiply 225 millionths by 2.85. 25. Multiply 2367 ten millionths by 3.0002. DIVISION OP DECIMAL FRACTIONS. 193. Ex. 1. Divide .75 by .5. Operation. 5). 75 We divide as in whole numbers, and point ~L5 Ans. ff 1 decimal figure in the quotient. OBS. We have seen in the multiplication of decimals, that the pro- duct has as many decimal figures, as the multiplier and multiplicand. (Art. 191.) Now since the dividend is equal to the product of the jivisor and quotient, (Art. G5,) it follows that the dividend must have as many decimals as the divisor and quotient together; consequently, as the dividend has two decimals, and the divisor but one, we must point off one in the quotient ; that is, we must point off as many de- cimals in the quotient, as the decimal places in the dividend exceed those in the divisor. 2. Divide .289 by 2.4. Operation. 2.4).289(. 12+ Ans. 24 Since the divisor contains two figures, 49" we substitute long division for short, 48 and point off the quotient as before 1 rem. 182 DIVISION OF [SECT. V1IL y te. When there is a remainder, the sign + should be annexed t ie quotient, to show that it is not complete. 3. Divide 1.345 by .5. Ans. 2.69. 4. Divide .063 by 9. Operation. In this example the dividend has thre% 9). 063 more places of decimals than the divisor , ~007 Ans. nence tne quotient must have three places of decimals. We must, therefore, prefix two ciphers to the quotient. 194. From these illustrations we deduce the follow- ing general RULE FOR DIVISION OF DECIMALS. Divide as in whole numbers, and point off as many fig- ures for decimals in the quotient, as the decimal places in the dividend exceed those in the divisor. If the quotient does not contain figures enough, supply the deficiency by prefixing ciphers. PROOF. Division of Decimals is proved in the same man- ner as Simple Division. (Art. 73.) OBS. 1. When the number of decimals in thedivisor is the same aa that in the dividend, the quotient will be a whole number. 2. When there are more decimals in the divisor than in the divi- dend, annex as many ciphers to the dividend as are necessary to make its decimal places equal to those in the divisor. The quotient thence arising will be a whole number. (Obs. 1.) 3. After all the figures of the dividend are divided, if there is a re- mainder, ciphers may be annexed to it and the division continued at pleasure. The ciphers annexed must be regarded as decimal places belonging to the dividend. Note. For ordinary purposes, it will be sufficiently exact to carry the quotient to three or four places of decimals ; but when great accu- racy is required, it must be carried farther. QUEST. 194. How are decimals divided ? How point off the quo- tient ? How is division of decimals proved ? Obs. When the number of decimal places in the divisor is equal to that in the dividend, what is the quotient ? When there are more decimals in the divisor than in the dividend, how proceed I When there is a remainder, what may be done ? A.RTS. 194, 195.] DECIMALS. 183 EXAMPLES. 1. If 1.7 of a yard of cloth will make a coat, how many coats will 10.2 yards make? 2. In 6.75 cords of wood, how many loads are there, Allowing .75 of a cord to a load? 3. If a man mows &2 acres of grass per day, how long will it take him to mow 39.36 acres? 4. If 23.25 bushels of barley grow on an acre, how many acres will 556 bushels require ? 5. In 74.25 feet, how many rods ? 6. In 99.225 gallons of wine, how many barrels? 7. If a man chops 3.75 cords of wood per day, how many days will it take him to chop 91.476 cords? 8. If a man can travel 35.4 miles per day, how long will it take him to travel 244.26 miles ? 9. A dairy-man has 187.5 pounds of butter, which he wishes to pack in boxes containing 12.5 pounds apiece : how many boxes will it require ? 10. In 3.575, how many times .25? 195. Whrn the divisor is 10, 100, 1000, &c., the di vision may be performed by simply removing the decimal point in the dividend as many places towards the left, as there are ciphers in the divisor, and it will be the quotient required. (Arts. 80, 194.) 11. Divide 756.4 by 100. Ans. 7.564. 12. Divide 1268.2 by 1000. Ans. 1.2682. 13. Divide 1 by 1.25. 14. Divide 1 by 562.5. 15. Divide .012 by .005. 16. Divide 2 by .0002. 17. Divide 5 by .000001. 18. Divide 13.2 by .75. 19. Divide .0248 by .04. 20. Divide 2071.31 by 65.3. QUEST. 195. When the divisor is 10, 100, 1000, &c., how may the division be performed ? 184 REDUCTION OP [SECT. VIIL REDUCTION OF DECIMALS. CASE I. Ex. 1. Change the decimal .25 to a common fraction. Suggestion. Supplying the denominator, (Art. 180.) Now -j 2 ^- is expressed in the form of a com mon fraction, and as such may be reduced to lower terms, and be treated in the same manner as any other common fraction. Thus -?fc=-u, or -J-. Hence, 196. To reduce a Decimal to a Common Fraction. Erase the decimal point ; then write the decimal denomina- tor under the numerator ', and it will form a common fraction^ which may be treated in tJie same manner as other common fractions. 2. Change .125 to a common fraction, and reduce it to '.he lowest terms. Ans. -J-. 3. Reduce .66 to a common fraction, &c. 4. Reduce .75 to a common fraction, &c. 5. Reduce .375 to a common fraction, &c. 6. Reduce .525 to a common fraction, &c. 7. Reduce .025 to a common fraction, &c. 8. Reduce .875 to a common fraction, &c. 9. Reduce .0625 to a common fraction, &c. 10. Reduce .000005 to a common fraction, &c. CASE II. Ex. 1. Change % to a decimal. Suggestion. Multiplying both terms by 10 the fraction becomes f-ft. As^ain dividing both terms by 5, it becomes -tV (Art. 116.) "But ^ -.6, (Art. 179,) which is the decimal required. QUEST.196. How are Decimals reduced to Common Fractions ? ARTS. 196, 197.] DECIMALS. 185 Now since we make no use of the denominator 10 after it is obtained, we may omit the process of getting it ; for if we annex a cipher to the numerator and divide it by 5, we shall obtain the same result. Operation. 5)3.0 A decimal point is prefixed to the quo- .6 tient, to distinguish it from a whole number. PROOF. .6 reduced to a common fraction is ^ ; (Art 196;) andTV-f. (Art. 120.) 2. Reduce 4- to a decimal. 8)1.000 Annex ciphers to the numerator and 125 proceed as before. Hence, 197. To reduce a Common Fraction to a Decimal. Annex ciphers to the numerator and divide it by the de- nominator. Point off as many decimal figures in the quo- tient, as you have annexed ciphers to the numerator. OBS. 1. If there are not as many figures in the quotient as you have annexed ciphers to the numerator, supply the deficiency by pre- fixing ciphers to the quotient. 2. The reason of this process may be illustrated thus. Annexing a cipher to the numerator multiplies the fraction by 10. (Arts. 59, 133.) If, therefore, the numerator with a cipher annexed to it, is divided by the denominator, the quotient will obviously be ten times too large. Hence, in order to obtain the true quotient, or a decimal equal to the given fraction, the quotient thus obtained must be divided by 10, which is done by pointing off one figure. (Art. 80.) Annexing 2 ciphers to the numerator multiplies the fraction by 100; annexing 3 cipners by 1000, &c., consequently, when 2 ciphers are annexed, the quotient will be 100 times too large, and must therefore be divided by 100; when three ciphers are annexed, the quotient will be 1000 times too large, and must be divided by 1000; &c. (Art. 80.) QUEST. 197. How are Common Fractions reduced to Decimals? Obs. When there are not so many figures in the quotient as you have fcnnexed ciphers, what is to be done I 186 REDUCTION 07 [SECT. VIII 3. Reduce f to decimals. Ans. 1.5. 4. Reduce -, and % to decimals. 5. Reduce -^-, and -fe to decimals. 6. Reduce -f. -|, and f to decimals. 7. Reduce -f-, , and -^ to decimals. 8. Reduce -^V, ^ and ^ to decimals. 9. Reduce -f, -f, and -fa to decimals. 10. Reduce 4^, and ^ 7 5 8 5 to decimals. 1 1. Reduce -g^-. and -nfW to decimals. 12. Reduce i to a decimal. .Arcs. .333333-f-. 13. Reduce iff to a decimal. Ans. .128128128+. 198. It will be seen that the last two examples can not be exactly reduced to decimals ; for there will continue to be a remainder after each division, as long as we con- tinue the operation. In the 12th, the remainder is always 1 ; in the 13th, after obtaining three figures in the quotient, the remainder is the same as the given numerator, and the next three figures in the quotient are the same as the first three, when the same remainder will recur again. The same remainders, and consequently the same fig- ures in the quotient, will thus continue to recur, as long as the operation is continued. 199. Decimals which consist of the same figure or set of figures continually repeated, as in the last two ex- amples, are called Periodical or Circulating Decimals; also, Repeating Decimals^ or Repetends. CASE III. Ex. 1. Reduce 7s. 6d. to the decimal of a pound. Suggestion. First, reduce 7s. 6d. to pence for the nu- merator, and 1 to pence for the denominator of a com QUEST. 199. What are Periodical 01 Repeating Decimals ? ARTS. 198-200.] DECIMALS. 137 mon fraction, and we have fo. (Art. 165.) Now -fft reduced to a decimal is .375. Ans. Hence, 2OO. To reduce a compound number to the decimal of a higher denomination. First reduce the given compound number to a common frac- tion ; (Art. 165 ;) then reduce tJie common fraction to a de- cimal. (Art. 197.) 2. Reduce 5s. 4d. to the decimal of 1. Ans. .2666+. 3. Reduce 15s. 6d. to the decimal of 1. 4. Reduce 12s. 6d. to the decimal of 1. 5. Reduce 9d. to the decimal of 1 shilling. 6. Reduce 7d. 2 far. to the decimal of a shilling. 7. Reduce 1 pt. to the decimal of a quart. 8. Reduce 18 hours to the decimal of a day. 9. Reduce 9 in. to the decimal of a yard. 10. Reduce 2 ft. 6 in. to the decimal of a yard. 11. Reduce 6 furlongs to the decimal of a mile. 12. Reduce 13 oz. 8 dr. to the decimal of a pound. CASE IV. Ex. 1. Reduce .123 to shillings, pence, and farthings. Operation. Multiply the given decimal by 20, as P -, 9 o if it were a whole pound, because 20s. ~20 ma ke 1> an d point off as many figures for decimals, as there are decimal places shil. 2.460 i n the multiplier and multiplicand. 12 (Art. 191.) The product is in shillings pence 5.520 arj d a decimal of a shilling. Then 4 multiply the decimal of a shilling by far. 2.080 12, and point off as before, &c. The numbers on the left of the decimals, Ans. 2s. 5d. 2 f. viz : 2s. 5d. 2 far. form the answer. Hence, QUEST. 200. How is a compound number reduced to the decimal of a higner denomination ? 188 FEDERAL [SECT. VIIL 20 1. To reduce a decimal compound number to whole numbers of lower denominations. Multiply the given decimal by that number which it takes of the next lower denomination to make ONE of this higher, as in reduction, (Art. 161, 1,) and point off the product, as in multiplication of decimal fractions. (Art. 191.) Proceed in this manner with the decimal figures of each succeeding pro- duct, and the numbers on the left of the decimal point in th( several products, will constitute the whole, number required. 2. Reduce .125 to shillings and pence. Ans. 2s. 6d. 3. Reduce .625s. to pence and farthings. 4. Reduce .4625 to shillings and pence. 5. Reduce .756 gallons to quarts and pints. 6. Reduce .6254 days to hours, minutes, and seconds, 7. Reduce .856 cwt. to quarters, &c. 8. Reduce .6945 of a ton to hundreds, &c. 9. Reduce .7582 of a bushel to pecks, &c. 10. Reduce .8237 of a mile to furlongs, &c. 11. Reduce .45683 of an acre to roods and rods. 12. Reduce .75631 of a yard to quarters and nails, FEDERAL MONEY. 202. FEDERAL MONEY is the currency of the United States. The denominations are, Eagles, Dollars, Dimes ] Cents, and Mills. TABLE. 10 mills (m.) make 1 cent, marked ct. 10 cents " 1 dime, " d. 10 dimes " 1 dollar, " doll, or $. 10 dollars " 1 eagle, " E. OBS. Federal Money was established by Congress, Aug. 8th, 1786. Previous to this, English or sterling money was the principal curren^ cy of the country. QUEST. 201. How are decimal compound numbers reduced to whole ones? 202. What is Federal Money ? Recite the Table. Obs. When and by whom was it established ? ARTS. 201-204.] MONEY. 189 Note. Many foreign coins are still in circulation. Indeed some of the rates of postage established by the government, were, until re- cently, adapted to foreign coins. To the 28th Congress belongs the honor of abolishing these anti-national rates, and of establishing others in Federal Money. 203. The national coins of. the United States are of three kinds, viz : gold, silver, and copper. 1. The gold coins are the eagle, the double eagle * half tagle, quarter eagle, and gold dollar * The eagle contains 258 grains of standard gold ; the dou- ole eagle, half eagle, and quarter eagle, like proportions. 2. The silver coins are the dollar, half dollar, quarter dollar, the dime, and half dime. The dollar contains 412 grains of standard silver; the others, like proportions. 3. The copper coins are the cent, and half cent. The cent contains 168 grains of pure copper ; the half cent, a like proportion. Mills are not coined. OBS. 1. The fineness of gold used for coin, jewelry, and other pur- poses, also the gold of commerce, is estimated by the number of parts of gold which it contains. Pure gold is commonly supposed to be divided into 24 equal parts, called carats. Hence, if it contains 10 parts of alloy, or some baser metal, it is said to be 14 carats fine ; if 5 parts of alloy, 19 carats fine ; and when absolutely pure, it is 24 car- ats fine.-}- 2. The present standard for both gold and silver coins of the United States, by Act of Congress, 1837, is 900 parts of pure metal by weight to 100 parts of alloy. The alloy of gold coin is composed of silver and copper, the silver not to exceed the copper in weight. The alloy of silver coin is pure copper. 20 4, All accounts in the United States are kept in QUEST. 203. Of how many kinds are the coins of the United States 1 What are they ? What are the gold coins ? The silver coins ? The topper ? Obs. How is the fineness of gold estimated ? Into how many carats is pure gold supposed to be divided ? When it contains 10 parts of alloy, how fine is it said to be I 5 parts of alloy ? 2 parts ? 4 parts ? What is the standard for the gold and silver coins of the United States * What is the alloy of gold coins ? What of silver coins ? 204. In what re accounts kept ? How would you express 5 eagles ? 7 E. and 5 lolls. ? 10 E. ? How express 6 dimes ? 8 dimes ? 10 dimes ? * By Act of Congress, Feb. 20th. 1849. * SillimHn's Chemistry. 190 FEDERAL [SECT. VIIL dollars, cents, and mills. Eagles are expressed in dollars, and dimes in cents. Thus, instead of 5 eagles, we say, 50 dollars ; instead of 7 eagles and 5 dollars, we say, 75 dollars, &c. So, instead of 6 dimes, we say, 60 cents ; instead of 8 dimes and 7 cents, we say, 87 cents, &c. 2O 5. It will be seen from the Table that Federal Money is based upon the Decimal system of Notation ; that its denominations increase and decrease from right to left and left to right in a tenfold ratio, like whole num- bers and decimals. 2OG. The Dollar is regarded as the unit; cents and mills are fractional parts of the dollar, and are distin- guished from it by a decimal point or separatrix (.) in the same manner as common decimals. (Art. 179.) Dollars therefore occupy units' place of simple numbers ; eagles, or tens of dollars, tens' 1 place, &c. Dimes, or tenths of L dollar, occupy the place of tenths in decimals ; cents or hundredths of a dollar, the place of hundredths ; mills, or thousandths of a dollar, the place of thousandths ; tenths of a mill, or ten thousandths 'of a dollar, the place of ten thousandths, &c. OBS. 1. Since dimes in business transactions are expressed in cents, two places of decimals are assigned to cents. If therefore the number of cents is less than 10, a cipher must always be placed on the left hand of them; for cents are hundredths of a dollar, and hundredths occupy the second decimal place. (Art. 181.) For example, 4 cents are written thus .04; 7 cents thus .07; 9 cents thus .09, &c. 2. Mills occupy the third place of decimals; for they are thou- sandths of a dollar. Consequently, when there are no cents in the given sum, two ciphers must be placed before the mills. Hence, 2O 7 . To read any sum of Federal Money. Call all the figures on the left of the decimal point fol lars ; the first two figures after the point, are cents , tkt QUEST. 205, How do the. denominations of Federal Money increase and decrease 7 Upon what is it based ? 206. What ; s regarded as the unit in Federal Money ? What are cents and mills ? How are they distinguished from dollars ? 207. How do you read Federal Money ? Obs. What other mode of reading Federal Money i ARTS. 205-207.] MONEY. 191 third figure denotes mills ; the other places on the. right are decimals of a mill. Thus, $3.25232 is read, 3 dollars, 25 cents, 2 mills, and 32 hundredths of a mill. OBS. Sometimes all the figures after the point are read as decimals of a dollar. Thus, $5.356 i s rea d, " 5 and 356 thousandths dollars." Read the following sums of Federal Money : 1. 2. 3. $250.56 $44.081 $3.7542 105.863 60.05 0.6054 200.057 75.003 4.0151 506.507 20.501 6.0057 850.071 30.065 8.0106 Write the following, sums in Federal Money : 4. 63 dollars, and 85 cents. Ans. $63.85. 5. 150 dollars, and 73 cents. 6. 201 dollars, and 9 cents. 7. 300 dollars, 5 cents, and 3 mills. 8. 4 dollars, 6 cents, and 8 mills. 9. 100 dollars, 7 cents, 5 mills, and 3 tenths of a mill. 10. 1000 dollars, 6 mills, and 36 hundredths of a mill. Note. In business transactions, when dollars and cents are ex- pressed together, the cents are frequently written in the form of a common fraction. Thus, $"76.45 are written 76-A-5.- dollars. REDUCTION OF FEDERAL MONEY. -CASE I. Ex. 1. How many cents are there in 75 dollars ? Suggestion. L.'nce in 1 dollar there are 100 cents, in 75 dollars there are 75 times as many. And 75x100- 7500. Ans. 7500 cents. 2. In 9 cents, how many mills 1 Ans. 90 mills. 3. In 25 dollars, how many mills ? Ans, 25000 mills. 192 FEDERAL [SECT. VIII Note. To multiply by 10, 100, &c., is simply annexing as many ciphers to the multiplicand, as there are ciphers in the multiplier. (Art. 59.) Hence, 208. To reduce dollars to cents, annex two ciphers. To reduce dollars to mitts, annex three ciphers. To reduce cents to mills, annex one cipher. OBS. To reduce dollars and cents to cents, erase the sign of dottart and the separatrix. Thus, $25.36 reduced to cents, becomes 2536 cents. 4. In $5, how many cents ? 5. How many mills in $364 ? 6. How many mills in $621 ? 7. How many cents in $6245 ? 8. Reduce $75.26 to cents. Q Reduce $625.48 to cents. CASE 1 1 ." 10. In 4500 cents, how many dollars? Suggestion. Since 100 cents make 1 dollar, 4500 cents will make as many dollars as 100 is contained times in 4500. And 4500+ 100=45. Ans. $45. 11. In 150 mills, how many cents'? Ans. 15 cents. 12. In 25000 mills, how many dollars ? Ans. $25. Note. To divide by 10, 100, &c., is simply cutting off as man/ figures from the right of the dividend as there are ciphers in the Di- visor. (Art. 80.) Hence, 209. To reduce cents to dollars, cut off two figures on the right. To reduce mills to dollars, cut off three figures on tht right. To reduce mills to cents, cut off one figure on the right. OBS. The figures cut off are cents and mills. QUEST. 208. How are dollars reduced to cents ? Dollars to mills ? Cents to mills ? Obs. Dollars and cents to cents ? 209. How are cents reduced to dollars ? Mills to dollars ? Mills to cents ? Obs. What are the figures cut. off? ARTS. 208-2 i 1 .] MONEY. 193 13. In 325 cents, how many dollars ? Ans. $3.25. 14. In 423 mills, how many cents ? Ans. 42c. 3m. 15. In 4320 mills, how many dollars ? 16 How many dollars in 63500 cents ? 17. How many cents in 4890 mills ? 2 1 0. Since Federal Money is expressed according to the decimal system of notation, it is evident that it may be subjected to the same operations and treated in the same manner as decimal fractions. ADDITION OF FEDERAL MONEY. Ex. 1. A man bought a cow for $15.75, a calf for $2.375, a sheep for $3.875, and a load of hay for $8.68 how much did he pay for all ? Operation ^ e wr * te tne dollars under dol- lars, cents under cents, &c. Then $15.75^ a ^ each co l umn separately, and 2.375 point off as many figures for cents 3-87^, and mills, in the amount, as there are places of cents and mills in either of $30.680 Ans. the given numbers. 211* Hence, we derive the following general RULE FOR ADDING FEDERAL MONEY. Write, dollars under dollars, cents under cents, fyc., 50 that the same orders or denominations may stand under each other. Add each column separately, and point of the amount as in addition of decimal fracti'MS. (Art 187.) OBS. If either of the given numbers have no cents expressed, it i customary to supply their place by ciphers. 2. A farmer sold a firkin of butter for $9.28, a cheese for $1.17, a quarter of veal for 56 cents, and a bushel of wheat for $1. 12 : how much did he receive for the whole ? QUEST. 211. How is Federal Money added? How point off the amount ? Obs. When any of the given numbers hare no cents ex- 1, how is their place supplied ? 7 i94 FEDERAL [SECT. VIII 3. A man bought a hat for $5.375, a cloak for $35.68, and a pair of boots for $4.75 : how much did he pay fof all? 4. What is the sum of $37.565, $85:20, $90.03, and $150.638? 5. What is the sum of $10.385, $46.238, $190.62 and $23.036? 6. What is the sum of $23.005, $16.03, $110.738, and $131.26? 7. What is the sum of 63 dolls, and 4 cts., 86 dolls, and 10 cts., and 47 dolls, and 37 cts. ? 8. What is the sum of $608.05, $365.205, $2.268, and $47.006? 9. What is the amount of 1 1 dolls. 3 cts. and 5 mills, 16 dolls, and 8 mills, 49 dolls. 7 cts. and 8 mills? 10. What is the amount of 100 dolls, and 61 cts., 51 dolls, and 3 cts., 65 dolls. 8 cts. and 3 mills ? 1 1. What is the amount of 95 dolls. 67 cts. and 8 mills, 1 20 dolls. 45 cts., 101 dolls. 7 cts. and 9 mills? 12. A lady bought a bonnet for $6.67, a pair of gloves for $0.625, a pair of shell combs for $0.75, and a cap for $2.50 : what was the amount of her bill ? SUBTRACTION OP FEDERAL MONEY. Ex. 1. A man bought a horse for $56.50, and a cow for $23.38 : how much more did he pay for his horse than his cow ? Operation ^ e wr ^ te tne ^ ess number under ' the greater, placing dollars under dollars, &c., then subtract, and point off the answer as in subtraction of $33.12 Ans. decimals. 212* Henee, we derive the following general RULE FOR SUBTRACTING FEDERAL MONEY. Write the less number under the greater, with dollars undci dollars, cents under cents, fyc., then subtract, and point ojf the remainder as in subtraction of decimal fractions. (Art 189.) ARTS. 2 1 2, 2 1 3.] MONET. 195 OBS. If either of the given numbers have no cents expressed, it b customary to supply their place by ciphers. 2. A man owing $57.35, paid $17.93 : how much does he still owe ? Ans. $39.42. . 3. A grocer bought two hogsheads of molasses for $68.90, and sold it for $79.26 : how much did he gain by the bargain ? 4. A man owed a debt of $105, and paid all but $23. 67 : how many dollars did he pay ? 5. A merchant bought a quantity of silks for $237.63. and sold it for $196.03 : how much did he lose? 6. A drover bought a flock of sheep for $357, and sold them for $17.33 less than he paid for them: how much did he sell them for ? 7. What is the difference between 365 dolls. 7 cts. and 208 dolls. 20 cts. ? 8. From 1 cent subtract 6 mills. 9. From 1 dollar, 6 cts. and 7 mills, take 89 cts. and 3 mills. 10. From 96 dollars, 6 cents, take 41 dolls., 63 cents, and 8 mills. 11. From 100 dollars, 10 cents, and 3 mills, take 1 cent and 5 mills. 12. From 1000 dollars, 6 cents, take 100 dolls, and 5 mills. MULTIPLICATION OF FEDERAL MONET. 213, In Multiplication of Federal Money, as well as in simple numbers, the multiplier must always be consid- ered an abstract number. (Art. 45. Obs. 2.) Ex 1. How much will 5 yards of cloth cost, at $1.75 per yard ? QUEST. 212. How is Federal Money 8 ubtracted ? How point off the remainder t Obs. When either of the given numbers have no cent* expressed, how is their place supplied ? 213. Jn Multiplication of Ffe. dral Money, what must one of the given factors be considered ? 196 FEDERAL [SECT. VIIL Operation If 1 yard cost $1.75, 5 yards will obvious. $1.75 ly cost 5 times as much. Hence, we multi 5 ply the price of 1 yard by the number ol $8.75 Ans. .7 ar ^ s ? an ^ point off two figures for decimals 'in the product. (Art. 191.) 2. How much will 15.8 yards of fringe cost, at 12 cents per yard ? Operation. Reasoning as before, 15.8 yards will cost 15.8 15.8 times 12 cents. But in performing the mul- .1 2 tiplication, it is more convenient to take $1 89l5 ^ e ^ f r tne mu ltiplie r 5 an d tf 16 result will be the same as if it was placed for the multiplicand. (Art. 47.) Point off the product as before. 214. Hence, when the price of one article, one pound, one yard, &c., is given to find the cost of any number ol articles, pounds, yards, &c. Multiply the price of one article and the number of articles together, and point off the product as in multiplication of decimals. (Art. 191.) 3. Multiply $45.035 by 6.2. Ans. $279.217. 215* From the preceding illustrations we derive the following general RULE FOR MULTIPLYING FEDERAL MONEY. Multiply as in simple numbers, and point off the product as in multiplication of decimal fractions. (Art. 191.) OBS. 1. When the price or the quantity contains a common frac- tion, the fraction should be changed to a common decimal. (Art. 197.) 2. In business operations, when the mills in the answer are 5, 01 over, it is customary to call them a cent; when under 5, they ar disregarded. QUEST. 214. When the price of 1 article, 1 pound, &c., is given, how is the cost of any number of articles found ? 215. What is the rul for Multiplication of Federal Money ? Obs. When the price or quan- ty contains a common fraction, what should b don with it! ARTS. 214, 215.] MONEY. 197 4. What will 10 Ibs. of beef cost, at 6 cents a pound? Solution. 6% cts.=.065, and .065x1 0=.65. Ans. 65 cents. 5. What cost 14 Ibs. of starch, at 10-J- cts. per pound? 6. W T hat cost 15^ pounds of sugar, at 9 cts. a pound? 7. What cost 25 gals, of molasses, at 18-f- cts. a gallon? 8. What cost 23-J- Ibs. of raisins, at 8 cts. per pound ? 9. What cost 33 Ibs. of candles, at 12^ cts. per pound? 10. What cost 16f Ibs. of hyson tea, at 56^ cts. a pound? 11. What will 83 Ibs. of beef cost, at $4.62i per hund. ? Analysis. 83 pounds are i 3 ^- of 100 pounds; there fore 83 pounds will cost -ffo of $4.625; and ffa of Operation. We multiply the price of 100 $4.625 ($4.625) by 83, the given num- 83 ber of pounds, and the product - 13875 $383.875, is the cost of 83 Ibs. at 3 70 00 $4.625 ^QI pound. But the price --- is $4.625 per hundred; conse $3.83 875 Ans. que ntly, the product $383.875 is 100 tin:es too large, and must therefore be divided by 100, to give the true answer. But to divide by 100, we simply remove the decimal point two places toward the left. (Art. 195.) 12. What will 825 feet of boards cost, at $6.75 per 1000? Reasoning as before, 825 feet will cost ja^ of $6.75. We multiply the price of 1000 feet by the given number of feet, and divide the product by 1000. To divide by 1000, we remove the de- cimal point three places towards the 56875 left " ( Art 195 '> Hence > 198 FEDERAL [SECT. VIIL 216. To find the cost of articles bought and sold by the 100, or 1000. Multiply the given price by the given number of articles , then if the price is for 100, divide the product by 100 ; but ij ike price is for 1000, divide it by 1000. (Art. 195.) 13. At $4.50 per 1000, what will 1250 bricks cost? 14. A farmer sold a quarter of beef, weighing- 256.5 pounds, at $5.37 per 100 : how much did he receive for it? 15. At $4.62 per hundred, what will 1675 pounds of pork cost ? 16. What cost 2129 feet of spruce boards, at $18.25 per 1000? 17. How much will 456f yards of shirting cost, at 12-J- cts. per yard ? 18. What cost 156 Ibs. of chocolate, at 15 cents a pound ? 19. What cost 235 pounds of cheese, at 6-J- cents a pound ? 20. What cost 175 dozens of eggs, at 10 cents per dozen? 21. At 47 cents per bushel, what will be the cost oi 300 bushels of corn ? 22. What will 153 Ibs. of sugar cost, at 8 cents per pound ? 23. What will 1500 pounds of butter cost, at $8.50 per hundred? 24. What cost 28500 feet of timber, at $3.76 per 100? 25. What cost 8230 feet of mahogany, at $70.20 per 1000? 26. What cost 7630 hemlock shingles, at $3.50 per 1 000 ? 27. What cost 15024 pine shingles, at $8.37 per 1000 ? 28. At 16i cts. a pound, what will 219^ pounds of honey cost ? QUEST. 216. How do you find the cost of articles bought and W by the 100, or 1000? ARTS, 216, 217.] MONEY. 199 29. At $2.67-f per yard, what will 400 yards of cloth eost? 30. At $5f per barrel, what will 1560 barrels of flour EOSt? DIVISION OF FEDERAL MONEY. Ex. 1. A man bought 6 hats for $25.68: how much did they cost apiece ? Operation. If 6 hats cost $25.68, 1 hat will cost 6)25.68 one sixth of $25.68. Divide as in sim- $4.28 Ans. P^ e num bers, and point off two decimal figures in the quotient. (Art. 194.) Proof. $4.28 If 1 hat costs $4.28, 6 hats will cost 6 times 6 as much ; and $4.28x6=$25.68, which is the $25^68 iven cost - Hence, 217* When the number of articles, pounds, yards, &c., and the cost of the whole are given, to find the price of one article, one pound, &c. Divide the whole cost by the whole number of articles, and point off" the quotient as in division of decimal fractions. (Art. 194.) 2. How marly yards of cloth, at $3.13 per yard, can be bought for $20.345 ? Operation. Since $3.13 will buy 1 yard 3 loxonq/is/A s A $20.345 will buy as many yards } ?878 ( as 3 ' 13 is contained times in $20.345. Divide as in simple numbers, and point off one decimal figure in the quotient. (Art. 194.) Proo/ $3.13x6.5=$20.345. Hence, QUEST. 217. When the number of articles, pounds, &o., and the ot of the whole are given, how is the cost of one article found ? 200 FEDERAL [SECT. 218. When the price of one article, pound, yard, find the cost of the whole are given, to find the number oi articles, &c. Divide the whole cost by the price of one, and point offtht quotient as in Art. 217. 3. Divide $149.625 by $2.375. Ans. 63. 4. If $75 are divided equally among 18 men, how much will each receive ? Operation. 18)75($4.166 Ans. After dividing the $75 by 18, 72 there is a remainder of 3 dollars, 3QQQ which must be reduced to cents and 18 mills, (Art. 208,) and then be di- vided as before. The ciphers thus annexed must be regarded as deci- mals; consequently there will be 120 three decimal figures in the quo- 108 tient. 12 rem. 219* From the preceding illustrations we derive the following general RULE FOR DIVIDING FEDERAL MONEY. Divide as in simple numbers, and point oj[jhe quotient as in division of decimal fractions. (Art. 194jp OBS. After all the figures of the dividend are divided, if there is a remainder, ciphers may be annexed to it, and the operation may be continued as in division of decimals. (Art. 194. Obs. 3.) The ciphers thus annexed must be regarded as decimal places of the dividend. 5. How many pounds of cheese, at 7 cts. a pound, can you buy for $1.47 ? QUEST. 218. When the price of 1 article, 1 pound, &c., and the cost of the whole are given, how is the number of articles found ? 219. What is the rule for Division of Federal Money ? Obs. When there i a remainder after all the figures of the dividend are divided, how proceed '? ARTS. 218, 219.] MONEY. 201 6. A man paid $0.75 for the use of a horse and buggy to go 8 miles : how much was that per mile ? 7. How many quarts of cherries, at 7 cents a quart, can you buy for $1.12? 8. How many pounds of figs, at 14 cents a pound, can you buy for $3.57 ? 9. How many watermelons, at 12-J- cts. apiece, can be bought for $3 ? 10. How many pen-knives, at 20 cts. apiece, can be oought for $7.20? 11. At 1 7- cts. a quart, how many quarts of molasses can be bought for $4.40? 12. A man bought 50 pair of thick boots for $175' how much did he give a pair ? 13. A man paid $485.50 for 260 sheep: how much did he give per head ? 14. At $2.50 a cord, how many cords of wood can T buy for $165? 15. At $4.75 per barrel, how many barrels of flour can I buy for $8.50 ? 16. If a man's income is $1.68 per day, how much is it per hour ? 17. If a man pays $3. 62- per week for board, how long can he board for $188.50 ? 1 8. Suppose a man's income is $500 a year, how much is that per day ? 19. Suppose a man's interest money is $28.80 per day how much is it per minute ? 20. A mason received $94.375 for doing a job, which took him 75 days : how much did he receive per day ? 21. At $1.1 2^- per bushel, how many bushels of wheat can be bought for $523.75 ? 22. If $1285.20 were divided equally among 125 men, what would each receive ? 23. If $1637.10 were divided equally among 150 men, what would each receive ? 24. The salary of the President of the United States js $25000 a year : how much does he receive per day? 202 BILLSL [SECT. VIII APPLICATIONS OF FEDERAL MONEY. BILLS, ACCOUNTS, &C. 2 2O. A Bill, in mercantile operations, is a paper containing a written statement of the items, and the pric* or amount of goods sold. Ex. 1. What is the cost of the several articles, and what the amount, of the following bill ? BOSTON, May 25th, 1845. James Brown, Esq. Bought of Fair/kid $ Lincoln, 5 yds. Broadcloth, at $3.25 3 yds. Cambric, " .12 3 doz. Buttons, " .15 6 skeins Sewing Silk, " .06| 4 yds. Wadding, .08 Amount, $17.77. Received Pay't, Fairfield tSf Lincoln. (2.) NEW HAVEN, Sept. 2d, 1845. Hon. R. S. Baldwin. Bought of Durrie fy Peck t 4 Lo veil's Young Speaker, at $ .62 5 Olmsted's Rudiments, " .58 6 Morse's Geography. " .50 8 Webster's Spelling Book, " .10 3 Day's Algebra, " 1.25 What was the cost of the several articles, and what tin amourt of his bill ? ART. 220.] BILLS. 203 (3.) NEW YORK, Aug. 18th, 1845. John Jacob Aslor, Esq. Bought of G. W Lewis < Co 25 Ibs. Sugar, 50 Ibs. Coffee, at $.09 - .11 - 12 Ibs. Tea, u .75 - 14 Ibs. Raisins, u .14 - 9 doz. Eggs, a .10 - 15 Ibs. Butter, u .12*- What was the cost of the several articles, and what the amount of his bill ? f (4.) PHILADELPHIA, June 3d, 1845. W. A. Sanford, Esq. To James Conrad, Dr. For 28 yds. Silk, at SI. 25 22 yds. Muslin, .56 " 16 pair Cotton Hose, .37-i- - 35 " Silk "1.10 25 Shoes, 1.25 Wha was the cost of the several articles, and haw much is due on his account 1 (5.) CINCINNATI, July 1st, 1845. Messrs. Holmes fy Homer . To H. W. Morgan 4* Co., Dr. For 100 bbls. Flour, at $4.50 50 Pork, 8.25 " 25 " Be"ef, 9.75 112 kegs Xard, 3.25 25 bush. 'Corn, .34' What was the cost of the several articles, and how ir.uch is due on his account ? 204 PERCENTAGE. [SECT. IX, (6.) NEW ORLEANS, Aug. 12th, 1845. F. C. Emerson, Esq. To W. H. Arnold $ Co., Dr. For 35 hhds. Molasses, at $12.60 2100 Ibs. Sugar, .05 - " 14000 Ibs. Cotton, .07* " 1350 Ibs. Coffee, " .06} - 31200 Ibs. Rice, .08 rt 150 boxes Oranges, 4.12 - CREDIT. By 500 Clocks, at $5.00 " Note to balance account, What was the amount of charges, and what the amoun of the note ? SECTION IX. PERCENTAGE. ART. 222. The terms Percentage and Per Cent, signi- fy a certain allowance on a hundred ; that is, a certain parl of a hundred, or simply hundredths. Thus the expres- sions 2 per cent., 4 per cent, 6 per cent., &c.. of any number or sum of money, signify 2 hundredths (TOT:) 4 hundredths (TOT?) 6 hundredths (TOT?) & c - f that num- ber or sum. For example, 1 per cent, of $100, is ^fa of that sum, which is 1 dollar; 2 per cent, of $100, is - T -2_ of that sum, which is 2 dollars; 4 per cent, of $100, is -p^- of that sum, which is 4 dollars; 6 per cent, of $100, is -^fo of that sum, which is 6 dollars, &c. Hence, universally, QUEST 222. What do the terms percentage and per cent, signify I What is meant by 2 per cent. , 4 per cent. , &c. , of any sum ? What then does any given percentage of any number or sum of money imply 1 Obs, From what are the terms percentage and per cent, derived * ARTS. 222, 222. a.] PERCENTAGE. 205 222. ft. Any given percentage of any number, or sum of money, implies so many units for every 100 units; so many dollars for every 100 dollars; so many cents for every 100 cents; so many pounds for every 100 pounds, s. Wha js meant b> the term per annum ? 234. What is meant by the princi pal ? The rate ? The amount ? 235. How is the rate usually detea mined ? Is it the same everywhere ? w ARTS. 233-236.] INTEREST. 215 235* The rate of interest is usually established by aw. It varies in different countries and in different parts of our own country. OBS. 1. The legal rate of interest in New England, New Jersey, Pennsylvania, Delaware, Maryland, Virginia, North Carolina, Ten- nessee, 'Kentucky, Ohio, Indiana, Illinois, Missouri, and Arkansas, is 6 per cent. In New York, South Carolina, Michigan, Wisconsin, and Iowa, it is 7 per cent. In Georgia, Alabama, Mississippi, and Florida, it is 8 per cent. ; and in Louisiana but 5 per cent. On debts and judgments in favour of the United States, interest is computed at 6 per cent. 2. In England and France the legal rate is 5 per cent. ; in Ireland^ 6 per cent. In Italy about the commencement of the 13th century, Lt varied from 20 to 30 per cent. 236. Any rate of interest higher than the legal rate, is called usury, and the person exacting it is liable to a neavy penalty. Any rate less than the legal rate may be taken, if the parties concerned so agree. OBS. 1. When no rate is mentioned, the rate established by the laws of the State in which the transaction takes place, is always un- derstood to be the one intended by the parties. 2. The term per annum, is seldom expressed in connexion with the rate per cent., but it is always understood ; for the rate is the per cent. paid per annum. (Art. 234.) Ex. 1. What is the interest of $15 for 1 year, at 4 per cent. ? Suggestion. 4 per cent, is -fta ; that is, $4 for $100, 4 cents for 100 cents, &c. (Art. 222. a.) Now as the in- terest of $1 (100 cents) for a year, is 4 cents, the interest of $15 for the same time, is 15 times as much. And 15 times 4 cents are 60 cents. A?is. 60 cents. QUEST. Obs. What is the legal rate of interest in New England, New Jersey, &c. ? What is the legal rate of interest in New York, South Carolina, &c. ? In Georgia, Alabama, &c. ? On debts due the United States ? What is the legal rate of interest in England and France ? Ireland ? 236. What is any rate higher than the legal rate called? What is the consequence of exacting usury 7 Is it safe to take le* than legal interest? Obs. When no rate is mentioned, what rat* ia understood ? 216 INTEREST. [SECT. IX, Operation. We multiply the principal by the given $15 Prin. rate per cent, expressed in decimals, as in .04 Rate, percentage; (Art. 225;) and point off aa $.(50 i n t many decimals in the product as there are decimal places in both factors. 2. What is the interest of $45 for 1 year, at 3 per cent? $1.35 Ans. 3. What is the interest of $32.125 for 1 year, at 4 per cent. ? Operation. 4% per cent, expressed in deci $32. 125 Prin. mals is .045. (Art. 223.) Multi- .045 Rate. ply, &c. as above, and point off 6 160625 decimals in the product. (Art. 191.) 128500 '^ ne fractions of a mill may be omit- $L445625 Ans. ted in the answer " Hence ' 237* To find the interest of any sum, at any given rate for 1 year. Multiply the principal by the given rate per cent, expressea in decimals, and point off the. product as in multiplication of decimal fractions. (Art. 191.) The amount is found by adding the principal and interest together. (Art. 234.) OBS. 1. In adding the principal and interest, care must be taken to add dollars to dollars, cents to cents, &c. (Art. 211.) 2. When the rate per cent, is lees than 10, a cipher must always oe prefixed to the figure denoting it. (Art. 223. Obs. 1.) It is highly important that the principal and the rate should both be written cor- rectly, in order to prevent mistakes in pointing off the product. 4. What is the interest of $75.21 for 1 year, at 6 pe; cent? $4.5126. Ans. 5. What is the interest of $100 for 1 year, at 5 pel cent. ? at 6 per cent. ? at 4 per cent. ? at 7 per cent. ? QUEST. 237. How do you compute interest for 1 year? How find the amount ? Obs. What precaution is necessary in adding the princi pal and interest together ? When the rate is less than 10 per cent, now is it written ? ARTS. 237, 238.1 INTEREST. 217 6. What is the interest of $35.31 for 1 year, at 6 per cent. ? 7. What is the interest of $50.10 for 1 year, at 7 per cent. ? 8. What is the interest of $63 for 1 year, at 5 per cent. ? 9. What is the interest of $136.75 for 1 year, at 4% per cent. ? 10. What is the interest of $260.61 for 1 year, at 6 per cent. ? What is the amount ? Ans. $15.636 int. $276.246 amount. 11. What is the interest of $140.25 for 1 year, at 7 per cent. ? What is the amount ? 12. What is the interest of $163.40 for 1 year, at 8 per cent. ? What is the amount ? 13. What is the interest of $400 for 1 year, at 6 per cent. ? What is the amount ? 14. What is the amount of $500 for 1 year, at 7 per cent. ? 15. What is the amount of $1000 for 1 year, at 8 per cent. ? 16. What is the interest of $100 for 3 years, at 6 pei cent, per annum ? Operation. , T1 J e interest for 3 Y^rs is ^ . plainly 3 times as much as for 1 year. We therefore first find the interest for 1 year as above, $6.00 Int. 1 y. which is $6 ; then multiplying 3 No. of y. this by 3, gives the interest for $18JOO Int. for 3 y. 3 Y ears - Hence, 238. To compute the interest of any sum for a given number of years. First find the interest of the given sum for I year, at the given rate; (Art. 237 ;) then multiply the interest of 1 yea/r by the given number of years. ^ 238. How is interest computed for any number of years? 218 INTEREST [SECT. IX. 17. At 5 per cent, per annum, what is the interest ol $45 for 4 years? Ans. $9. 18. At 6 per cent., what is the interest of $200 for 5 years ? What is the amount ? 19. At 7 per cent., what is the interest of $250 for 10 years? What is the amount? 20. At 8 per cent., what is the interest of $340.50 for 3 years ? What is the amount? 21. At 6 per cent, per annum, what is the interest of $100 for 1 month? Operation. * month is -^ of 12 months ... or a year, therefore the inter- ' est for l month wil1 be as nr much as the interest for 1 12)6.00 Int. for 1 y. year. Now the interest of $^50 Int. for 1m. $ 100 for l 7 ear is $ 6 > and -& of $6, is 50 cts. In like manner any number of months may be considered a frac- tional part of a year, and the interest for them may be computed in the same way. Hence, 239. To compute the interest of any sum for a given number of months. First find the interest for 1 year as above ; then take such a fractional 'part of 1 year's interest^ as is denoted by the given number of months. Thus, for 1 month take -fa of 1 year's interest ; for 2 months, -& or ; for 3 months, -^ or -J- ; for 4 months, -fa or ; for 6 months, -fa or % ; &c. 22. At 5 per cent., what is the interest of $600 for 6 months. Ans. $15. 23. A*. 7 per cent., what is the interest of $250 for 4 months? QUEST. 239. How is interest computed for months ? For 2 months, what part would you take? For 3 months? 4 months? 5 months I 6 months? 7 months? S months? 9 months? 10 months? 11 months? A.Rrs. 239, 240.] INTEREST. 219 24. What is the interest of $375.31 for 3 months, at 6 per cent. ? 25. What is the interest of $60 for 7 months, at 8 per cent. ? What is the amount ? 26. What is the interest of $96 for 10 months, at 6 per cent. ? What is the amount ? 27. At 6 per cent., what is the interest of $600 for 1 day? Operation. 1 day is -gV of 30 days, or $600 Prin. ? m < on 1 th J he e th ? int ,. er f ' 1j 06 Rate. * or 1 " a W1 ^ " Q * interest for 1 month. If, I In. for 1 y. therefore, we find the inter- 30)3.00 In. for 1 m. est for j monthj and take ^ Ans. $0.10 In. for 1 d. of this, it will evidently be the interest for 1 day. In like manner, any number of days may be considered a fractional part of a month, and the interest for them may be found in the same way. Hence, 24O. To compute the interest of any sum for a given number of days. First find the interest fo Y \ month as above, then take such a fractional part of 1 montn's interest as is denoted by the. given number of days. Thus for 1 day take $ of 1 months interest ; for 2 days, -g'V, or tV ; for 3 days. -^5-, or -rV ; for 10 days, i ] for 20 days, -f ; 4* c - 28. At 4 per cent., what is the interest of $470 for 10 days? Ans. $0.522. 29. What is the interest of $1000 for 1 y. 1m. and 1 d., at 6 per cent. ? 30. What is the interest of $42.50 for 2 years and 6 months, at 7 per cent. ? 31. What is the interest of $69.46 for 1 year and 8 months, at 8 per cent. ? QUEST. 240. How is interest computed for days ? For 2 days, tthat part would you take ? Far 5 day ? 7 days ? 12 days ? 25 days * 220 INTEREST. [SECT. IX, 241. From the foregoing principles we may deduce the following general RULE FOR COMPUTING INTEREST. I. FOR ONE YEAR. Multiply the principal by the given "ate, and from the product point off as many figures for deci> mals, as there are decimal places in both factors. (Art. 237.) II. FOR TWO OR MORE YEARS. Multiply the interest oj 1 year by the given number of years. (Art. 238.) III. FOR MONTHS. Take such a fractional part of 1 year's interest, as is denoted by the given number of months. (Art. 239.) IV. FOR DAYS. Take such a fractional part of 1 months as is denoted by the given number of days. OBS. 1. In calculating interest, a month, whether it contains 30 or 31 days, or even but 28 or 29, as in the case of February, is usually assumed to be one twelfth of a year. 2. In calculating interest 30 days are considered a month ; conse- quently the interest for 1 day, or any number of days under 30, is so many thirtieths of a month's interest. (Art. 170. Obs. 2.) This practice seems to have been originally adopted on account of its convenience. Though not strictly accurate, it is sanctioned by custom, and is everywhere allowed by law. 32. What is the interest of $45.23 for 1 year and 2 months, at 5 per cent. 1 33. What is the interest of $43.01 for 2 years, at 7 per cent. ? 34. What is the interest of $215.135 for 2 years and 3 months, at 6 per cent. ? 35. At 8 per cent., what is the interest of $75.98 for 3 years ? 36. At 5 per cent., what is the interest of $939 for 4 years ? 37. At 6 per cent., what is the interest of $137.50 for 6 months ? QUEST. 241. Wha : s the general rule for computing interest ? In reckoning interest, y f r > tne i nt - f SI is equal to half as many cents as there $4.93 Ans. ' are months in the given time. 247. From these illustrations we may derive a SECOND RULE FOR COMPUTING INTEREST. Multiply the principal by the interest of $1 for the given time, and point off the product as before. (Art. 241.) Or, multiply the principal by half the number of months, ^ind point off two more decimals in the product than there are decimal jigures in the multiplicand. OBS. 1. In the latter method, the years must be reduced to months, and the days to the fraction of a month, then take half of them. The interest at any other rate, greater, or less than 6 per cent, may be found by adding to, or subtracting from the interest at G per cent., such a fractional part of itself, as the required rate exceeds or falls short of 6 per cent. Thus, if the required rate is 7 per cent., first find the interest at 6 per cent., then add of it to itself; if 5 per cent., subtract - of it from itself, &c. QUEST. 247. What is the second method of computing interest! 0/8. When the rate is greater or less than 6 per cent., how proceed ? \ 224 INTEREST. [SECT. IX. 2. When it is required to compute the interest on a note, we must first find the time for which the note has been on interest, by sub- tracting the earlier from the later date; (Art. 170;) then cast the in- terest on the face of the note for the time, by either of the preceding methods. (Arts. 241, 247.) 13. What is the interest of $300 for 4 months, and 18 days, at 7 per cent. ? Operation. $300 Prin. .023 int. of $1 for? The required rate is 1 the time. 5 , QQQ per cent, more than 6 per gOO cent - j we therefore find the interest at 6 per cent, and 6)$6.900=Int. at 6 per ct. add i rf it (0 ^ 1 150= of 6 per cent. Ans. $&050~ Int. at 7 per ct. 14. At 5 per cent., what is the interest of $256.25 for 9 months and 15 days? 15. What is the interest of $450 from Jan. 1st, 1844. to March 13th, 1845, at 6 per cent. ? Operation. $450 Principal. Yr mo -072 Int. of $1 for the time. 1845 " 3 " 13 "900 1844 " 1 " I IIJ^L Time 1 " 2 " 12 $32.400 Ans. EXAMPLES FOR PRACTICE. 1. What is the interest of $45.25 for 8 months, at 6 per cent. ? 2. What is the interest of $167.375 for 6 months, at 6 per cent ? 3. What is the interest of $93.86 for 3 months and 15 days, at 6 per cent. ? 4. What is the interest of $110 for 1 month and 20 days, at 6 per cent. ? 5. At 7 per cent., what is the interest of $158.91 for I year and 3 months ? QUEST. 347. How compute the interest on a nete? ART. 247.] INTEREST. 225 6. At 7 per cent, what is the amount of $217 for 1 year and 8 months ? 7. At 6 per cent., what is the amount of $348.10 for 2 years and 1 month ? 8. At 7 per cent, what is the interest of $400 for 1 year and 6 months ? 9. At 7 per cent, what is the amount of $213.01 for 9 months? 10. At 5 percent, what is the amount of $603 for 2 years and 5 months ? 11. What is the amount of $861 for 8 months and 24 days, at 6 per cent. ? 12. What is the amount of $1236 for 3 months and 14 days, at 7 per cent ? 13. What is the interest of $1400 for 1 year, 1 month and 9 days, at 7 per cent. ? 14. What is the interest of $469.20 for 27 days, at 8 per cent. ? 15. What is the amount of $705 for 5 years, at 9 per cent. ? 16. What is the amount of $1000 for 10 years, at 5 per cent ? 17. What is the amount of $1650.06 for 20 years, at 7 per cent. ? 18. What is the amount of $2500 for 7 years, at 15 per cent. ? 19. At 4 per cent., what is the interest of $17000 for 1-J- years? 20. At 7i per cent, what is the interest of $1625.81 for 45 days ? 21. At 121 per cent, what is the amount of $165.13 for 33 days ? 22. At 7 per cent., what is the amount of $8531 for 63 days? 23. At 6 per cent, what is the amount of $16021 foi 93 days ? 24. What is the interest on a note of $65, dated Jan. 10th, 1844, to May 16th, 1845, at 6 percent.? 25. What is the interest of $170 from June 19th, 1840, to July 1st, 1841, at 7 per cent? 226 INTEREST. [SECT. IZ. 26. What is the interest of $105.63 from Feb. 22d 1839, to Aug. 10th, 1840, at 5 per cent. ? 27. What is the interest of $234 from April I Oth, 1834, to Oct. 1st, 1835, at 6 per cent.? 28. What is the interest of $195.22 from June 25th 1838, to March 31st, 1840, at 6 per cent.? 29. What is the interest of $391 from Sept. 1st, 1840 to Nov. 30th, 1841, at 8 per cent.? 30. What is the interest of $510.83 from March 21st, 1842, to Dec. 30th, 1842, at 7 per cent? 31. At 6 per cent, what is the interest of $469.65 from August 10th, 1843, to Feb. 6th, 1844 ? 32. At 7 per cent., what is the amount due on a note of $285, dated March 15th, 1844, and payable Sept. 18th 1845? 33. At 6 per cent., what is the amount due on a note of $S9l, dated Oct. 9th, 1844, and payable March 1st, 1845? 34. At 5 per cent., what is the amount of $623 from Feb. 19th, 1844, to Aug. 10th, 1844? 35. At 4 per cent., what is the amount of $589.20 from January 10th, 1844, to January 13th, 1845? 36. At 4 per cent, what is the amount of $731.27 from July 1st, 1844, to April 4th, 1845? 37. What is the interest of $849 from July 4th, 1841, to July 7th, 1845, at 6 per cent. ? 38. What is the interest of $966 from Jan. 1st, 1842, to March 20th, 1844, at 7 per cent. ? 39. What is the interest of $1539 from May 21st, 1842 to Aug. 19th, 1843, at 6 per cent ? 40. What is the amount of $ 1 100 from June 1 5th,l 840, to Aug. 3d, 1845, at 5 per cent. ? 41. What is the amount of $1 for 50 years, at 6 per ct. ? At 7 per cent. ? 42. What is the amount of one cent for 500 years,al 7 per cent 1 A.RT. 248.] INTEREST. 227 PARTIAL PAYMENTS. 248* W T hen partial payments are made and endorsed upon Notes and Bonds, the rule for computing the inter- est adopted by the Supreme Court of the United States, is the following. 1 " The rule for casting interest, when partial payments have been made, is to apply the payment, in the first place, to the discharge of the interest then due. II. ^ 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 principal remaining dm. III. "If the payment be less than the interest, the surplus of interest must not be tdken to augment the principal ; but interest continues on the former principal until the period when the payments, taken together, exceed the interest due, and then the surplus is to be applied tc wards discharging the principal ; and interest is to be computed on the balance as aforesaid." Note. The above rule is adopted by Massachusetts, New York, and the other States of the Union, with but few exceptions. It is given in the language of the distinguished Chancellor Kent. Johnson's Chancery Reports, Vol. I. p. 17. $850. NEW HAVEN, Jan. 1st, 1841. 43. For value received, I promise to pay George How- land, or order, eight hundred and fifty dollars, on demand, with interest at 6 per cent. JOHN HAMILTON. The following payments were endorsed on this note July 1st, 1841, received $100.62. Dec. 1st, 1841, received $15.28. Aug. 13th, 1842, received $175.75. What was due on taking up the note, Jan. 1st, 1843 1 QUEST. 248. What is the general method of casting interest oa Motes laid Bonds, when partial payments have been made 1 228 INTEREST. [SECT. Operation. Principal, Interest to first payment, July 1st, (6 months,) Amount due on note July 1st, - 1st payment, (to be deducted from amount,) Balance due July 1st, Int. on Bal. to 2d pay't Dec. 1st, (5 mo.,) $19.37 2d pay't (which is less than the inter- est then due,) Surplus interest unpaid Dec. 1st, Int. continued on Bal. from Dec. 1st, 1842, to Aug. 13th, (8 mo., 12 d.,) Amount due Aug. 13th, 1842. 3d payment (being greater than the interest now due) is to be deducted from the am't. Balance due Aug. 13th, Int. on Bal. to Jan. 1st, (4 mo., 18d.,) Bal. due on taking up the note, Jan. 1st, 1843, $650.38 15.28 $409" 32.54 $850.00 25.50 $875.50 100.62 $774.88 36.63 $811.51 175.75 $635.76 14.62 $500. NEW YORK, May 10th, 1842. 44. For value received, I promise to pay James Mon- roe, or order, five hundred dollars on demand, with in- terest at 7 per cent HENRY SMITH. The following sums were endorsed upon it : Received, Nov. 10th, 1842, $75. Received, March 22d, 1843, $100. What was due on taking up the note, Sept. 28th, 1843 $692.35. BOSTON, Aug. 15th, 1843. 45. Three months after date, I promise to pay John Warren, or order, six hundred and ninety-two dollars and thirty-five cents, with interest at 6 per cent., value re- ceived, SAMUEL JOHNSON. A.RTS. 249, 249. a.\ INTEREST. 229 Endorsed, Nov. 15th, 1843, $250.375. " March 1st, 1844, $65.625. How much was due July 4th, 1845 ? $1000. PHILADELPHIA, June 20th, 1841. 46. Six months after date, I promise to pay Messrs. Carey, Hart & Co., or order, one thousand dollars, with interest, at 5 per cent., value received. HORACE PRESTON. Endorsed, Jan. 10th, 1844, $125. " June 16th, 1844, $93. Feb. 20th, 1845, $200. What was the balance due Aug. 1st, 1845 ? CONNECTICUT RULE. xJ49 "Compute the interest on the principal to the time of me first payment; if that be one year or more from the time the in- terest commenced, add it to the principal, and deduct the payment 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 in like manner, from one payment to an- other, till all the payments are absorbed ; provided the time between one payment and another be one year or more. But if any payments be made before one year's interest hath 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 arisen at the time of such payment, no interest is to be computed, but only on the principal sum for any period." Kirby's Reports. THIRD RULE. 24:9. a. First find the amount of the given principal for the whole time ; then find the amount of each of the several payments from the time it was endorsed to the time of settlement. Finally, subtract the amount of the several payments from the amount of the principal, and the remainder will be the sum due. * If a year does not extend beyond the time of payment; but if it does, then find the amount of the principal remaining unpaid, up to the time of settlement, likewise the amount of the endorsements from the time they were paid \ the lime of settlement, and deduct the sum of these several amounts from the mount of the principal. 230 INTEREST. [SECT. IX, Note. It will be an excellent exercise for the pupil to cast the in- terest on each of the preceding notes by each of the above rules. 47. What is the interest of 175, 10s. 6d. for 1 year, al 5 per cent. ? Operation. We first reduce the 10s. 6d. 175.525 Prin. to the decimal of a pound. .05 Rate. (Art. 200,) then multiply the 877625 Int. for 1 yr. Principal by the rate and point 2Q on the product as in Art. 241. The fiure 8 on the left of the decimal point is pounds, and those on the right are decimals d. 6.30000 of a pound, and must be re- 4 duced to shillings, pence, and far. 1.20000 farthings. (Art. 201.) Ans. 8, 15s. 6|d. Hence, 25O To compute the interest on pounds, shillings, &c. Reduce, the given shillings, pence, and farthings to the de- cimal of a pound; (Art. 200 ;) then find the interest as cm dollars and cents ; finally, reduce the decimal figures in the answer to shillings, pence, and farthings. (Art. 201.) 48. What is the interest of 56, 15s. for one year and 5 months, at 6 per cent. ? Ans. 5, 2s. If d. 49. What is the interest of 75, 12s. 6d. for 1 year and 3 months, at 7 per cent. ? 50. What is the interest of 96, 18s. for 2 years and 6 months, at 4- per cent. ? 51. What is the amount of 100 for 2 years and 4 months, at 5 per cent. 'I 52. What is the amount of 430, 16s. lOd. for 1 year and 5 months, at 6 per cent. ? QUEST. 250. How is interest computed on pounds, shillings, &t ? ARTS. 250-252.] INTEREST. 231 PROBLEMS IN INTEREST. 251, It will be observed that there are four 'parts or terms connected with each of the preceding operations, viz : the principal, the rate per cent., the time, and the inter- est, or the amount. These parts or terms have such a re- lation to each other, that if any three of them are given, the other may be fonnd. The questions, therefore, which may arise in interest, are numerous ; but they may be reduced to a few general principles, or Problems. OBS. I. The term Problem, in its common acceptation, means a question proposed, which requires a solution. 2. A number or quantity is said to he given, when its value is stat- ed, or may be easily infened from the conditions of the question under consideration. Thus, when the principal and interest are known, the amount may be said to be given, because it is merely the sum of the principal and interest. So, if the principal and the amount are known, the interest may be said to be givsn, because it is the differ- ence between the amount and the principal. 252* To find the interest on any given sum, as in the "oregoing examples, the principal, the rate per cent., and the time are always given. This is the First and most important Problem in interest. The other Probkms will now be illustrated. PROBLEM II.* To find the RATE PER CENT., the principal, the interest, and the time being given. 1. A man loaned $75 to one of his neighbors for 4 years, and received $24 interest : what was the rate per cent. 1 QUEST. 251. How many terms are connected with each of :he pre- ceding examples ? What are they ? Are they all given ? When three are given, can the fourth be found ? Obs. What is a problem ? When is a number or quantity said to be given ? 252. What terms are given when it is required to find the interest ? * Should this and the following Problems bo deemed too difficult for beginners they can be omitted till review. 232 INTEREST. [SECT. IX. Analysis. The interest of $75 at 1 per cent, for 1 year, is $.75, and for 4 years it is $.75x4=$3. (Art. 238.) Now since $3 is 1 per cent, interest on the principal for the given time, $24 must be -\ 4 - of 1 per cent., which is equal to 8 per cent. (Art. 121.) Or, we may reason thus : If $3 is 1 per cent, on tho principal for the given time, $24 must be as many per cent, as $3 is contained times in $24 ; and $24~$3^8. Ans. 8 per cent. PROOF. $75X-08=$6.00, the interest for 1 year at 8 per cent., and $6x4=$24, the interest of $75 for 4 years at 8 per cent. Hence, 253. To find the rate per cent, when the principal, interest, and time are given. First find the interest of the principal at 1 per cent for the given time ; then make the interest thus found the denom- inator and the given interest the numerator of a common fraction, which being reduced, to a whole or mixed number, will give the required per cent. (Art. 121.) Or, simply divide the given interest by the interest of the principal at 1 per cent, for the given time, and the quotient will be the per cent. 2. If I borrow $300 for 2 years, and pay $42 interest, what rate per cent, do I pay ? Operation. The interest of $300 for 2 yrs. $6)$42 at 1 per cent, is $6. (Art. 238.) 7 Ans. 7 per ct. PROOF. $300x.07x2=$42. 3. If I borrow $460 for 3 years, and pay $82.80 in- terest, what is the rate per cent. ? 4. A man loaned $500 for 8 months, and received $40 interest : what was the rate per cent. ? 5. At what rate per cent, must $450 be loaned, to gain $56.50 interest in 1 year and 6 months? QUEST. 253. When the principal, interest, and time are giTen, hew is the rate per cent, found ? V. ART. 253.] INTEREST. 233 6. At what per cent, must $750 be loaned, to gain $225 in 4 years ? 7. A man has $8000 which he wishes to loan for $60C per annum for his support : at what per cent, must he loan it? 8. A gentleman deposited $1250 in a savings bank, for which he received $31.25 every 6 months ; what per cent, interest did he receive on his money ? 9. A capitalist invested $9260 in Railroad stock, and drew a semi-annual dividend of $416.70 : what rate per cent, interest did he receive on his money ? 10. A man built a hotel at an expense of $175000, and rented it for $8750 per annum : what per cent, inter- est did his money yield him ? PROBLEM III. To find the PRINCIPAL, the interest, the rate per cent., and the. time being given. 11. What sum must be put at interest, at 6 per cent, to gain $30 in two years? Analysis. The interest of $1 for 2 years at 6 per cent., (the given time and rate,) is 12 cents. Now 1 2 cents interest is -iW of its principal $1 ; consequently, $30 the given interest, must be -^ of the principal required. The question therefore resolves itself into this : $30 is -^ of what number of dollars? If $30 is -fW, liir is iV of $30. which is $2i; and -HH}=$2ixlOO, which is $250, the principal required. Or, we may reason thus: Since 12 cents is the interest of 1 dollar for the given time and rate, 30 dollars must be the interest of as many dollars for the same time and rate, as 12 cents is contained times in 30 dollars. And $30-H.12=250. Am. $250. PROOF. $250x06=$! 5.00, the interest for 1 year at the given per cent., and $!5x2=$30, the given interest, Hence. 234 INTEREST. [SECT. IX, 254. To find the principal, when the interest, rate per cent, and time are given. Make the interest of $1 for the given time and rate, the nu- merator, and 100 the denominator of a common fraction ; then divide the given interest by this fraction ; and the quotient will be tJie principal required. (Art. 141.) Or, simply divide the given interest by the interest of $ 1 for the given time and rate, expressed in decimals ; and the quotient will be the principal 12. What sum put at interest will produce $13.30 in 6 months, at 7 per cent. 1 Operation. The int. of $1 for 6 $.035)$13.300 mo. at 7 per cent, is $.035 "380. Ans. $380. ( Art 239 ') 13. A father bequeaths his son $500 a year: what sum must be invested, at 5 per cent, interest, to produce it? 14. What sum must be put at 6 per cent, interest, to gain $350 interest semi-annually ? 15. A gentleman retiring from business, loaned his money at 7 per cent., and received $1200 interest a year how much was he worth ? PROBLEM IV. To find the TIME, the principal, the interest, and the rate per cent, being given. 16. A man loaned $80 at 5 per cent., and received $10 interest : how long was it loaned ? Analysis. The interest of $80 at 5 per cent, for 1 year is $4. (Art. 237.) Now, since $4 interest requires the principal 1 year at the given per cent, $10 interest will require the same principal -^of 1 year, which is equal to 2i years. (Art 121.) QUEST. 254. When the interest, rate per cent., and time are given, how is the principal found ? ARTS. 254, 255.] INTEREST. 235 Or, we may reason thus : If $4 interest requires the use of the given principal 1 year, $10 interest will re- quire the same principal as many years as $4 is contained times in $10. And $10-=-$4=2.5. Ans. 2.5 years. Hence, 255* To find the time when the principal, interest, and rate per cent, are given. Make the given interest the numerator, and the interest of the principal for 1 year at the given rate the denominator of a common fraction, which being reduced to a whole or mixed number, will give the time required. Or, simply divide the given interest by the interest of the principal at the given rate for 1 year, and the quotient will be the time. OBS. If the quotient contains a decimal of a year, it should be re- duced to months and days. (Art. 201.) 17. How long will it take at 5 per cent, to double itself; that is, to gain $100 interest? Operation. The interest of $100 for 1 year, at 5 pel cent., is $5. (Art. 237.) 20 Ans. 20 years. PROOF. $100x.05x20=$100. (Art. 238.) TABLE. Showing in. what time any given principal will double itself at any rate, * from 1 to 20 per cent. Simple Interest. Percent. Years. Per cent. Years. Per cent. Years. Per cent. Years. 1 100 6 16-f 11 9-rV 16 6i 2 50 7 14f 12 8- 17 H4 3 33-3 8 12"2 13 7^ 18 54 4 25 9 IH 14 7-? 19 5 20 10 10 15 6* 20 ) 5 QUEST 255. When the principal, interest, and rate per cent, are given, how is the time found? Obs. When the quotient contains a 'de- cimal of a yeai , what should be done with it ? 236 COMPOUND. [SECT. IX, 18. In what time will $500, at 6 per cent., produce $100 interest? 19. How long will it take $100, at 6 per cent., to double itself? 20. How long will it take $100, at 7 per cent., to double itself? 21. How long will it take $7250, at 10 per cent., to double itself? COMPOUND INTEREST. S56. Compound Interest is the interest arising not only from the principal, but also from the interest itself. after it becomes due. OBS. 1. Compound Interest is often called interest upon interest. 2. When the interest is paid on the principal only, it is called Sim- vie Interest. Ex. 1. What is the compound interest of $500 for 3 years, at 6 per cent. ? Operation. $500 principal. $500x.06=$ 30 Int. for 1st year. 530 Amt. for 1 year. $530x.06= 31.80 Int. for 2d year. 561.80 Amt. for 2 years. $561.80x06= 33.70 Int. for 3d year. $595.50 Amt. for 3 years. 500.00 Prin. deducted. Ans. $95.50 compound Int. for 3 years. 'QuEST. 256. From what does compound interest arise ? O6. What is compound interest often called ? What is Simple Interest I ARTS. 256, 257.] INTEREST. 237 257* Hence, to calculate compound interest. Cast the interest on tlie given principal for 1 year, or the specified time, and add it to the principal ; then cast the inter- est on this amount for the next year, or specified time, and add it to the principal as before. Proceed in this manner with each successive year of tlie, proposed time. Finally, subtract the given principal from the last amount, and the remainder will be the compound interest. 2. What is the compound interest of $350 for 4 years, at 6 per cent. ? 3. What is the compound interest of $865 for 5 years, at 7 per cent. ? 4. What is the amount of $250 for 6 years, at 5 per cent, compound interest ? 5. What is the amount of $1000 for 3 years, at 4 per cent, compound interest, payable semi-annually ? 6. What is the amount of $1200 for 2 years, at 6 per cent, compound interest, payable quarterly ? 7. What is the amount of $800 for 3 years, at 5 pel cent, compound interest, payable semi-annually ? 8. What is the amount of $1500 for 5 years, at 7 per cent, compound interest ? 9. What is the amount of $2000 for 2 years, at 3 per cent, compound interest, payable quarterly? 10. What is the amount of $3500 for 6 years, at 6 pel cent, compound interest? Note. This and the next two examples may be solved either by the rule, or by the Table below. 11. What is the amount of $1860 for 8 years, at 7 per cent, compound interest ? 12. What is the amount of $20000 for 10 years, at 3 per cent, compound interest ? QUEST. 257. How is compound interest calculated t 238 INTEREST. [SECT. IX. TABLE, Showing the amount of SI, or 1, at 3, 4, 5, C, and 7 per cent*, mn pound interest, for any number of years, from I to 35. Yrs. | 3 per cent. \ 4 per cent. \ 5 per cent, \ 6 per cent. \ 7 percent.\ 1. 1.030,000 1.040,000 1.050,000 1.060,000 1.07,000 I 2 - 1.060,900 1.081,600 1.102.500 1.123,600 1.14.490 3. 1.092,727 1.124,864 1.157,625 1.191,016 1.22J504 4. 1.125,509 1.169,859 1.215,506 1.262,477 1.31,079 5. 1.159,274 1.216,653 1.276,282 1.338,226 1.40,255 6. 1.194,052 1.265,319 1.340,096 1.418,519 1.50,073 7. 1.229,874 1.315,932 1.407,100 1.503,630 1.60,578 8. 1.266,770 1.368,569 1.477,455 1.593,848 1.71,818 9. 1.304,773 1.423,312 1.551,328 1.689,479 1.83,845 10. 1.343,916 1.480,244 1.628,895 1.790,848 1.96,715 11. 1.384,234 1.539,454 1.710,339 1.898,299 2.10,485 12. 1.425,761 1.601,032 1.795,856 2.012,196 2.25,219 13. 1.468,534 1.665,074 1.885,649 2.132,928 2.40,984 14. 1.512,590 1.731,676 1.979.932 2.260,904 2.57,853 15. 1.557,967 1.800,944 2.078^28 2.396,558 2.75,903 16. 1.604,706 1.872,981 2.182,875 2.540,352 2.95,216 17. 1.662,848 1.947,900 2.292,018 2.692,773 3.15,881 18. 1.702,433 2.025,817 2.406,619 2.854,339 3.37,293 19. 1.753.506 2.108,849 2.526,950 3.025,600 3.61,652 20. 1.806,111 2.191,123 2.653,298 3.207,135 3.86,968 21. 1.860,295 2.278,768 2.785,963 3.399,564 4.14,056 i 22. 1.916,103 2.369,919 2.925,261 3.603,537 4.43,040 23. 1.973.587 2.464,716 3.071,524 3.819,750 4.74,052 24. 2.032J94 2.563,304 3.225,100 4.048,935 5.07,236 25. 2.093,778 2.665,836 3.386,355 4.291,871 5.42,743 26. 2.156,592 2.772,470 3.555,673 4.549,383 5.80,735 27. 2.221,289 2.883,369 3.733,456 4.822,346 6.21 ,386 28. 2.287,928 2.998,703 3.920,129 5.111,687 6.64,883 29. 2.356,566 3.118,651 4.116,136 5.418,388 7.11,425 30. 2.427,262 3.243,398 4.321,942 5.743,491 7.61,225 31. 2.500,080 3.373,133 4.538,039 6.088,101 8.14,571 32. 2.575,083 3.508,059 4.764,941 6.453,386 8.71,527 33. 2.652,335 3.648,381 5.003,189 6.840.590 9.32,533 34. 2.731,905 3.794,316 5.253,348 7.251 ;025 997,811 35. 2.813,862 3.946,089 5.516,015 7.686,087 10.6,765 ARTS. 258, 259.] DISCOUNT. 239 258. To calculate compound interest by the preced- ing Table. Find the amount of $1, or 1 for the given number of years by the table, multiply it by the given principal, and the product will be the amount required. Subtract the principal from the amount thus found, and the remainder will be the compound interest. 1 3. What is the compound interest of $200 for 1 years, At 6 per cent? What is the amount ? Operation. $1.790848 Amt. of $1 for 10 years by table. 200 the given principal. $358.169600 amount required. $200 principal to be subtracted. Ans. $158.1696 interest required. 14. What is the amount of $350 for 12 years, at 4 per cent. ? 15. What is the amount of $469 for 15 years, at 3 per cent. ? What the interest ? 16. What is the interest of $500 for 24 years, at 6 per cent. ? 17. What is the interest of $650 for 30 years, at 7 per cent. ? DISCOUNT. 259. DISCOUNT is the abatement or deduction made for the payment of money before it is due. For example, if I owe a man $100, payable in one year without interest, the present worth of the note is less than $100 ; for, if $100 were put at interest for 1 year, at 6 per cent., it woul-l amount to $106 ; at 7 per cent., to SI 07; &c. In con- sideration, there fore, of the present 'payment of the note, jus- tice requires that he should make some abatement from it This abatement is called Discount. QUEST. 258. How is compound interest computed by the Table ? 259. What is discount 1 What is the present worth of a debt, payable 8,1 some future time, without interest? 240 DISCOUNT. [SECT. IX The present worth of a debt payable at some future time without interest, is that sum which, being put at legal interest, mil amount to the debt, at the time it becomes due. Ex. 1. What is the present worth of $545, payable in 1 year and 6 months without interest, when money is worth 6 per cent, per annum ? Analysis. The amount, we have seen, is the sum o 1 the principal and interest. (Art. 234.) Now the amount of $1 for 1 year and 6 months, at 6 per cent, is $1.09 ; (Art. 237 ;) that is, the amount is -ffl-g- of the principal $1. The question then resolves itself into this : $545 is -}-$ of what principal ? If $545 is -H-f, -^ is 545-*- 109, 01 $5; and -HHM&5xlOO, which is $500. Or, we may reason thus: Since $1.09 (amount) requires $1 principal for the given time, $545 (amount) will re- quire as many dollars as $1.09 is contained times in $545; and $545-*-$1.09=$500. That is, the present worth of $545, payable in 1 year and 6 months, is $500, which is the answer required. PIIOOF. $500x.09=$45, the interest for 1 year and 6 months; and $500+$45=$545 the given amount. (Art. 247.) Hence, 26 O. To find the present worth of any sum, payable at a future time without interest. First find the amount of $ 1 for the time, at the given rate, as in simple interest ; (Art. 247 ;) then divide the, given sum by this amount, and the quotient will be the pre- sent worth. The present worth subtracted from the debt, will give thi true discount. OBS. This process is often classed among the Problems of Interest in which the amount, (which answers to the given sum or debt,) the rate per cent., and the time are given, to find the principal, which answers to the present worth. QUEST. 260. How do you find the present worth of a debt ? Hov find the discount \ ART. 260.] DISCOUNT. 241 2. What is the present worth of $250.38, payable in 8 months, when money is worth 6 per cent, per annum ? What is the discount ? Operation. 1.04)250.38(240.75 The amount of $1 for the 208 given time and rate, is $1.04. ^23 (Art. 247.) Dividing the given 4 I Q sum by this amount, the quotient $240.75, is the present worth. * And $250.38240.75=89.63, ___ the discount. C $240.75 the present 520 Ans. < worth ; ( $9.63 the discount. 3. What is the present worth of $475, payable in 1 year, when money is worth 7 per cent, per annum? 4. What is the present worth of' $175, payable in 2 years, when money is worth 7 per cent, per annum ? 5. What is the present worth of $1000, payable in 4 months, when the rate of interest is 6 per cent. ? 6. What is the discount on $750, due 6 months hence, when interest is 5 per cent, per annum ? 7. A man sold a farm for $ 1800, payable in 15 months : what is the present worth of the debt, allowing the rate to be 6 per cent. ? 8. I have a note of $1150.33, payable in 9 months: what is its present worth at 7 per cent, interest per an- num? 9. A merchant sold goods amounting to $840.75, pay- able in 6 months : how much discount should he make for cash down, when money is worth 7 per cent. ? 10. What is the discount on a draft of $2500, payable in 3 months, at 4 per cent, per annum ? 1 1. What is the present worth of $5000, payable in 2 months, at 6 per cent, per annum ? 12. VVhat is the difference between the discount on $500 for 1 year, and the interest of $500 for 1 year, at 6 per cent. ? 242 DISCOUNT. [SECT. IX BANK DISCOUNT. 261. It is customary for Banks in discounting a no.a or draft, to deduct in advance the legal interest on the given sum from the time it is discounted to the time when it becomes due. Bank discount, therefore, is the same as simple interest paid in advance. Thus, the bank discount on a note of $106, payable in 1 year at 6 per cent., is $6.36, while the true discount is but $6. (Art. 260.) OBS. 1. The difference between bank discount and true discount, is the interest of the true discount for the given time. On small sums for a short period this difference is trifling, but when the sum is large, and the time for which it is discounted is long, the difference is con- siderable. 2. Taking legal interest in advance, according to the general rule of law, is usury. An exception is generally allowed, however, in favor of notes, drafts, &c., which are payable in less than a year. The Safety Fund Banks of the State of New York, though the legal rate of interest is 7 per cent., are not allowed by their charters to take over 6 per cent, discount in advance on notes and drafts which mature within 63 days from the time they are discounted.* 262* According to custom, a note or draft is not pre- sented for collection until three days after the time speci- fied for its payment. These three days are called days of grace. It is customary to charge interest for them. Banks, therefore, always calculate the interest for three days more than the time stated in the note. 13. What is the bank discount on a note of 8500, pay- able in 1 year, at 6 per cent. ? What is the present worth ? QUEST. 261. How do banks usually reckon discount? What thr is bank discount ? Obs. What is the difference between bank discos and true discount? Is this difference worth noticing ? How is interest in advance generally regarded in law ? What exceptica this rule is allowed ? 262. When is it customary to present r months, at 4 per cent. ? 19. What is the bank discount of $ 1 825.60, payable in 4 months and 15 days, at 6 per cent? 20. What is the present worth of a draft of $1292, payable in 60 days, at 7 per cent, discount? 21. What is the present worth of a draft of $5000, payable in 15 days, at 6 per cent, discount? 22. What is the present worth of a draft of $15000, payable in 3 days, at 6 per cent, discount? 23. What is the present worth of $1326, payable in 10 months, at 5% per cent, discount? 24. What is the bank discount, at 7 per cent., on a note of $836.81, payable in 90 days? 25. What is the bank discount, at 8 per cent., on a draft of $1261.38, payable in 60 days? 26. What is the bank discount, at 6 per cent., on a draft of $10000, payable in 30 days? 27. What is the difference between the true discount and bank discount on $1000, payable in 5 years, at 6 per cent ? 28. What is the difference between the true discount and bank discount on $100000, payable in 1 year, at 7 per cent. ? 244 INSURANCE. [SECT. INSURANCE. 263. INSURANCE is security against loss or damage of property by fire, storms at sea, and other casualties. This security is usually effected by contract with Insurance Companies, who, for a stipulated sum, agree to restore to the owners the amount insured on their houses, ships, and other property, if destroyed or injured during the specified time of insurance. 264. The written instrument or contract is called the Policy. The sum paid for insurance is called the Premium. The premium paid is a certain per cent, on the amount of property insured for 1 year, or during a voyage at sea, or other specified time of risk. Hence, 265* To compute Insurance for 1 year, or the speci fied time. Multiply the sum insured by the given rate per cent., as in interest. (Art. 237.) OBS. 1. Insurance on ships and other property at sea is sometimes effected by contract with individuals. It is then called out-door in- surance. 2. The insurers, whether an incorporated company or individuals! are often termed Underwriters. Ex. 1. How much premium must a mechanic pay an nually for the insurance of his shop and tools worth $350 at !- per cent. ? Solution. $350x.015=$5.25. Ans. 2. What amount of premium must be paid annually for insuring a house worth $875, at -f per cent. ? 3. Shipped a box of books valued at $1000, from New QUEST. 263. What is Insurance? 264. What is meant by the policy? The premium? 265. How is insurance computed? Obs. When insurance is effected with individuals, what is it called ? What are tlxe insurers sometimes called ? ARTS. 263-265.] INSURANCE, 245 York to New Orleans, and paid !} per cent, insurance . tvhat was the amount of premium ? 4. A powder mill worth $925, was insured at 15^ per cent. : what was the annual amount of premium ? 5. A merchant shipped a lot of goods worth $1560, from Boston to Natchez, and paid 1-f- per cent, insurance ^ what amount of premium did he pay ? 6. A gentleman obtained a policy of insurance on his house and furniture to the amount of $2500, at 3-J- per cent, per annum : what premium did he pay a year 1 7. A man owning a sixteenth of a whale ship, which cost him $2750, got it insured,at 7-J- per cent, for the voy- age : how much did he pay ? 8. A man owning a schooner worth $3800, obtained insurance upon it, at 5^ per cent, for the season : what amount of premium did he pay ? 9. A crockery merchant having a stock of goods valued at $7500, paid 2 per cent, for insurance : how much pre- mium did he pay a year 1 10. A merchant shipped 83765 worth of flour, from Cincinnati to New York, and paid 1 per cent, insurance : how much premium did he pay ? 11. What is the "i^mai premium for insuring a store worth $7350, at f per cent. ? 12. An importer effected insurance on a cargo of tea worth $65000, from Canton to Philadelphia, at 3 per cent. how much did his insurance cost him? 13. A manufacturer obtained insurance to the amount of $76500 on his stock and buildings, at -f- per cent. : how much premium did he pay annually ? 14. A policy was obtained on a cargo of goods valued at $95600, shipped from Liverpool to New York, at 2-J per cent. : what was the amount of premium ? 15. The owners of the whale ship George Washing- ton obtained a policy of $58000 on the ship and cargo, at 7\ per cent, for the voyage : what was the amount of premium ? 16. A gentleman paid $60 annually for insurance on his house and furniture, which was 2 per cent, on its value what amount of property was covered by the policy? 246 INSURANCE. [SECT. IX , Note. Tliis example is similar to those of Problem III, in interest (Art. 254.) Solution. Since the rate of insurance is 2 per cent. 01 .02, it is plain that $60 is -rjhr of the amount insured. Now if $60 is -rihr, ifa is half as much, or $30 ; and H-4 is $30X100. or $3000. Or thus: 60^.02=3000 Ans. $3000. PROOF. $3000x.02=$60, which was the annual pre- mium paid. 17. If I pay $250 premium on silks, from Havre to New York, at !-- per cent., what amount of property does my policy cover ? 18. A merchant paid $1200 premium, at 2-^ per cent, on a ship and cargo from London to Baltimore, which was lost on the voyage : what amount should he recover from the Insurar.ee Company ? 19. If a man pays $60 premium annually for the in- surance of his house, which is worth $3000, what rate per cent, does he pay ? Note. This example is similar to those of Problem II, in interest (Art. 253.) Solution. $60-*-$3000=.02. Ans. 2 per cent. PROOF. $3000x.02=$60, which is the premium paid. 20. A merchant paid $40 premium for insuring $5000 on his stock : what rate per cent, did he pay? 21. If a man pays $75 for insuring $15000, what rate per cent, does he pay 1 22. If the owner pays $2800 for insuring a ship worth $40000, what rate per cent, does he pay ? 23. A blacksmith owns a shop worth $720: what amount must he get insured annually, at 10 per cent, so that in case of loss, both the value of the shop and the. premium may be repaid ? Analysis. Since the rate of insurance is 10 per cent., on a policy of $100, the owner would actually receive but $90 ; for he pays $ 1 for insurance. The question then resolves itself into this: $720 is -rVg of what sum? ART. 266.] PROFIT AND LOSS. 247 If 720 is-flfo -rh- is 720+90=8, ai 1 ^ is 8x100= 800. Ans. $800. PROOF. $800X-10=$80, the premium he would pay, and $800 $80=$720, which is the value of his shop. 24. If I send an adventure to China worth $6250, what amount of insurance, at 8 per cent., must I obtain, that in case of a total wreck I may sustain no loss by the operation 1 25. What amount of insurance must be effected on $1 1250, at 5 per cent., in order to cover both the premium and property insured ? PROFIT AND LOSS. 266* PROFIT and Loss in commerce, signify tb3 sum gained or lost in ordinary business transactions. They are reckoned at a certain per cent, on the purchase price, or sum paid for the articles under consideration. MENTAL EXERCISES. 1. A merchant bought a barrel of flour for $6, and sold it at a profit of 10 per cent. : how much did he sell it for ? Suggestion. Since he made 10 per cent, profit, if we add 10 per cent, to the purchase price, it will give the selling price. Now 10 per cent, of $6 is 60 cents, (Art 225,) which added to $6, make $6.60. Ans. He sold it for $6.60. 2. A grocer bought a box of oranges for $5, and sold it, at 12 per cent, profit : how much did he receive for his oranges ? 3. A farmer bought a ton of hay for $9, and sold it QUEST 266. What is meant by profit and loss ? How are they reckoned ? 248 PROFIT AND LOSS. [SECT. IX for 10 per cent, more than he gave : how much did he sell it for? 4. Bought a sleigh for $12, and sold it at a loss of 8 per cent. : how much did I receive for the sleigh ? Solution. 8 per cent, of $12, is 96 cents; and $12 96 cents leaves $11.04. Ans. 5. Bought a box of honey for $5, and having lost a portion of it, sold the remainder, at 1 1 per cent, loss : how much did I receive for it ? 6. A shop-keeper bought a piece of calico for $7, and sold it, at 12 per cent, profit : how much did he sell it for ? 7. A lad bought a sheep for $3, and on his way home was offered 15 per cent, for his bargain: how much was he offered for his sheep ? 8. A farmer bought a colt for $20, and offered to sell it for 5 per cent, less than he gave: how much did he ask for it? 9. A gentleman bought a horse for $100 ; after using it awhile, he sold it, at 7 per cent, loss : how much did he get for his horse ? 10. A man bought a building lot for $150, and in con sequence of the rise of property, sold it for 10 per cent, advance : how much did he get for it? 11. A hack-man bought a carriage for $200, and after using it for one season, sold it for 15 per cent, less than he gave for it : how much did he sell it for ? 12. A man bought a house for $800, and sold it the next day for 10 per cent, advance : how much did he sell it for ? EXERCISES FOR THE- SLATE. CASE I. i. A merchant bought a quantity of grain for $75, and sold it for 8 per cent, profit : how much did he gain bj the bargain ? Solution. $75X.08=$6.00. (Art. 225.) Hence, ART. 267.] PROFIT AND LOSS. 249 267* To find the amount of profit or loss, when the purchase price arid rate per cent, are given. Multiply the. purchase price by the given per cent, as in percentage ; and the product will be the amount gained or lost by the transaction. (Art. 225.) 2. A man bought a sleigh for $60, and afterwards sold it for 10 per cent, less than cost : how much did he lose ? 3. A grocer bought a cask of oil for $96.50, and re- tailed it, at a profit of 6 per cent. : how much did he make on his oil ? 4. A pedlar bought a lot of goods for $215, and retail- ed them, at 20 per cent, advance : how much was his profit? 5. A merchant bought a cargo of coal for $450, which he afterwards sold for 12- per cent, less than cost: what was the amount of his loss'? 6. A manufacturer purchased $1000 worth of wool, and after making it up, sold the cloth for 25 per cent, more than the cost of the materials : how much did he receive for his labor 1 CASE II. 7. A man bought a span of horses for $350, and wished to dispose of them for 12 per cent, profit: how much must he sell them for? Operation. Reasoning as before, he $350 purchase price, must sell them for the jmr- .12 per cent, profit, chase price, together with $42.00 gained. 12 per cent, of that price. ^w~toQQsplliTi0-iirirp Having found 12 per cent. [mg price ' of $350, (Art. 225,) add ft to the cost, and the sum $392, is manifestly the soling vrice. 8. A stage proprietor bought a coach for $480 ; find- QUEST. 267. How is the amount of profit or loss found, when th tost and rate per cent, are given ? PROFIT AND LOSS. [SECT. IX, ing it damaged, he was willing to sell it, at 5 per cent loss : at wha* price would he sell it ? Operation. Having found the sum $480 purchase price, lost, (Art. 225,) subtract it 05 per cent. loss, from the cost, and the T e- $24.00 sum lost mainder is obviously *h* sellin rice. sellin S P rice ' flence > 268. To find ho~v ary article must be sold, in orde, to gain or lose a given rate per cent. First find the amount of profit or loss on the purcha**. price at the given rate, as in the last Case ; then the amou*v thus found added to, or subtracted from the 'purchase price as the case may be, will give the selling price required. 9. A merchant bought a firkin of butter for $22.75- how much must he sell it for in order to gain 15 per cent by his bargain ? 10. Bought a chest of tea for $37.50: for how mucb must I sell it, in order to make 18 per cent, by the opera tion? 11. Bought a quantity of produce for $89.33, which 3 propose to sell, at 20 per cent, loss : how much must } receive for it ? 12. A drover bought a flock of sheep for $275, am taking them to market, sold them, at 25 per cent, ad vance : how much did he sell them for ? 13. A merchant had a quantity of groceries on hand which cost him $367.13; for the sake of closing up hh business he sold them, at 15 per cent, less than cost : how much did he get for them ? 14. A man bought a farm for $875, and was offered 33 per cent, advance for his bargain : how much was he offered? 15. A merchant bought a cargo of cotton for $30000; QUEST. 268. What is the method of finding how an article must b* sold, in order to p-iin or lose a given per cent. ? ARTS. 268, 269. J PROFIT AND LOSS 251 the price declining, he sold it at 2 per cent, less than cost : for how much did he sell it ? CASE III. 16. A man bought a cow for $25, which he afterwards 252 PROFIT AND LOSS. [SECT. IX the article, and not on the selling price, or sum received, as it is often supposed. 17. A merchant bought a piece of cloth for $2.75 pei yard, and sold it for $3.25 : what per cent, did he gain 'I Solution. Since he gained 50 cents on a yard, his gain was -gfe of the cost. And &=. 1 S-ft. Ans. 18-fV per cent. 18. A boy purchased a book for 20 cents, and sold jt for 30 cents : what per cent, did he make ? 19. A merchant bought a box of sugar, at 6 cents a pound, and sold it for 7-J- cents a pound : what per cent, was his profit ? 20. A grocer bought eggs at 9 cents, and sold them for 12 cents per dozen : what per cent, was his profit? 21. A man bought a hat for $4.50, and sold it for $6 : what per cent, did he gain ? 22. A jockey bought a horse for $73, and sold him for $68 : what per cent, did he lose? 23. A merchant bought a quantity of goods for $155.63 and sold them for $148.28 : what per cent, did he lose? 24. A gentleman bought a house for $3500, and sold it for $150 more than he gave : what per cent, was his profit? 25. A speculator laid out $7500 in land, and afterwards sold it for $10000 : what per cent, did he make ? 26. A drover bought a herd of cattle for $1175, and sold them for $1365: what per cent, did he gain; and how much did he make by the operation ? 27. A merchant bought $10000 worth of wool, and sold it for $12362: what per cent. ; and how much was his profit? CASE IV. 28. A jockey sold a horse for $250, which was 25 per cent, more than it cost him : how much did he pay for tho horse ? Analysis. It will be observed that the selling price ($250) is equal to the cost and the amount gained added ART. 270. j PROFIT AND LOSS. 253 together. Now considering the cost a unit or 1, the gain which is a certain per cent, of the cost, (Art. 266,) is T^-, consequently l+^^-Hnh (Art. 127,) will denote the sum of the cost and the gain. The question therefore resolves itself into this : 250 is +*% of what number ? If 250 is -HHh -rb- is 2 ; and iffr is 100 times 2, or 200. Or ; we may simply divide 250 by the fraction -ftf-ft. (Art. 141.) The quotient 200 is the cost required. PROOF. $200x.25=$50 ; and $200+$50=$250, the selling price ? 29. A merchant sold a quantity of goods for $180, which was 1 per cent, less than cost : how much did the goods cost him ? Analysis. It will be observed that the selling price ($180) is equal to the cost diminished by the sum lost. Now reasoning as in the last example, 1 rVu^iW w iM denote the cost diminished by the loss. The question now is this : 180 is -^ of what number ? If 180 is T 2 ^-, rb is 2, and -HH} is 200. Or thus: S180^ 1 3 D %-=$200. Ans. PROOF. $200x. 10=820, and $200 $20-$180, the selling price. Hence, 27O. To find the cost when the selling price and the per cent, gained or lost are given. Make the given per cent, added to or subtracted from 100, as the case may be. the numerator, and 100 the. denominator of a common fraction; then divide the selling price by this fraction ; aiid the quotient will be the cost required. OBS. 1. It is not unfrequently supposed that if we find the per- centage on the selling price at the given rate, and add the percentage thus found to, or subtract it from the selling price, as the case may be, the sum or remainder will be the cost. This is a mistake, and leads QUEST. 270. How is the cost found, when the selling price and the rate per cent, gained or lost, are given ? Obs. What mistake is iome made in finding the cost ? How may it be avoided ? s 254 PROFIT AND LOSS. [SECT. IX. to serious errors in the /esult. It will easily be avoided by remem- bering, that the basis on which profit and loss are calculated, i* always the purchase price, or sum paid for the articles under con- sideration. (Art. 2b'9. Obs. 2.) 30. A grocer sold a hogshead of molasses for $24, and gained 20 per cent, on the cost : what was the cost of the molasses ? 31. A merchant sold a piece of broadcloth for $85, which was 10 per cent, less than the cost: what was the cost of it ? 32. A butcher sold a yoke of oxen for $125, and there- by made 15 per cent. : how much did they cost him 1 33. A bookseller sold a lot of books for $200, which was 12 per cent, more than the cost: what was the cost? 34. A wholesale druggist sold a quantity of medicines for $560, and made 50^ per cent, profit on them : what was the cost of them ? 35. A merchant sold a cargo of rice for $1500, which was 12 per cent, less than cost: what was the cost? EXAMPLES FOR. PRACTICE. 1. A merchant bought 25 boxes of raisins for $45 : af what price per box must he retail them to gain 10 per cent, by his bargain ? Suggestion. He must sell the whole for 10 per cent more than the cost. Hence, if we add 10 per cent, to the cost, and divide the sum by the number of boxes, it will give the retail price per box. (Art. 217.) 2. A shopkeeper bought a piece of cotton containing 40 yards, at 6 cents a yard, and sold it for 7 cents a yard : what per cent, profit did he gain ; and how much did he make by the bargain ? 3. A merchant bought 60 yards of domestic flannel at 25 cents per yard, and sold it at 30 cents per yard : what per cent, was his profit ; and how much did he clear by the operation ? 4. A bookseller bought 100 Arithmetics at Slfrcentf ART. 270.] PROFIT AND LOSS. 255 apiece, and retailed them at 37 cents apiece : what per cent. ; and how much did he make by the operation. 5. A drover bought 175 sheep for $350', and sold them ?o as to gain 15 per cent. : how much did he sell them for per head 1 6. A baker paid $2500 for 480 barrels of flour, and finding it damaged, sold it at a loss of 8 per cent. : how much did he sell it for per barrel 1 7. A merchant bought 10 pieces of broadcloth, each piece containg 30 yards, for $1400, and retailed the whole at a profit of 20 per cent. : at what price did he sell it per yard? 8. A grocer bought 500 Ibs. of butter for $75, and sold it at a loss of 7 per cent. : how much did he get per pound 1 9. A merchant bought 12 hogsheads of molasses at 25 cents per gallon : how must he sell it by the gallon in order to gain 20 per cent. ; and how much was his profit ? 10. A farmer raises 750 bushels of wheat at an ex perise of $675 : how must he sell it per bushel, in order to make 18 per cent. ? 1 1. A provision merchant bought 1500 barrels of pork at $10.25 per barrel, and sold it at a loss of 9 per cent. : how much did he lose ; and what did he get per barrel ? 12. An inn-keeper bought 150 bushels of oats, at 25 cents a bushel, and retailed them at the rate of 12 cents a peck : what per cent. ; and how much did he make on the oats ? 13. A miller bought 500 bushels of wheat, at 75 cents per bushel : how much must he sell the whole for in order to gain 20 per cent. ? 14. A grocer bought 1630 pounds of tea, at 62-^- cents per pound, and sold it at 10 percent, loss : how much did he sell it at per pound ? 15. A merchant bought a bale of calico prints contain- ing 750 yards and paid $75 : how must he retail it per yard, in order to gain 20 per cent. ; and how much would he make on a yard ? 16. A bookseller purchased 1000 geographies, at 84 256 DUTIES. [SECT. IX. cents apiece : how must he retail them to gain 20 pei cent. ? 17. A milliner bought 1200 yards of ribbon, at 30 cents per yard : how must she sell it per yard to gain 50 per cent. ? 1 8 A grocer bought 5000 Ibs. of sugar for $350, and retailed it, at 6 cents per pound : what per cent, loss did he sustain ? 19. A man purchased goods amounting to $1635: what per cent, profit must he gain, in order to make $350 ? 20. A speculator bought" 10000 acres of land for $12500, and afterwards sold* it, at 25 per cent, loss: for how much per acre did he sell it ; and how much did he lose by the operation ? DUTIES. 271. DUTIES, in commerce, signify a sum of required by Government to be paid on imported goods. Duties are of two kinds, specific and ad valorem. A specific duty is a certain sum imposed on a ton, hundred weight, hogshead, gallon, square yard, foot, &c. without regard to the value of the article. Ad valorem duties are those which are imposed on goods, at a certain per cent, on their value or purchase price. Note. The term ad valorem is a Latin phrase, signifying according to, or upon the value. 272. Before specific duties are imposed, it is custo- mary to make certain deductions called tare, draft, or tret, leakage, &c, Tare, in commerce, is an allowance of a certain QUEST. 271. What are duties in commerce ? Of how many kind* are they ? What are specific duties \ Ad valorem duties ? Note. What is the meaning of the term ad valorem ? 272. What deductions are made before specific duties are imposed ? What is tare ? Draft or tret ? Leakage ? 271-273.] DUTIES. 257 number of pounds made for the box, cask, &c., which contains the article under consideration. Draft or Tret is an allowance of a certain per cent, (usually 4 per cent.) on the weight of goods for waste, or refuse matter. Leakage is an allowance of a certain per cent, (usually 2 per cent.) for the waste of liquors contained in casks, &c. OBS. 1 . All duties, both specific and ad valorem, are regulated by the Government, and have been different at different times and in different countries. 2. The allowance or deductions for draft, tare, leakage, &c., are also different on different articles, and are regulated by law. 3. In buying and selling groceries in large quantities, allowances are sometimes made for draft, tare, leakage, &c., similar to those in reckoning duties. CASE I. Ex. 1. What is the specific duty on 10 pipes of wine, at 15 cents per gallon, reckoning the leakage at 2 per cent. 1 Suggestion. First deduct the leakage. In 1 pipe there are 2 hogsheads or 126 gallons ; in 10 pipes there are 10 times 126, or 1260 gallons. But 2 per cent, of 1260 gallons, is 1260x.02-25.20 gallons; (Art. 225 ;) and 25.2 gallons subtracted from 1260 gallons leaves 1234.8 for the number of net gallons. Now if the duty on 1 gal- lon is 15 cents, on 1234.8 gallons it is 1234.8x.l5= $185.22, the duty required. Hence, 273. To fin' the specific duty on any given merchan- dise. First deduct the legal draft, tare, leakage, &c. from the given quantity of goods ; then multiply the remainder by the given duty per gallon, pound, yard, <$-., and the product will be the duty required, QUEST. Obs. How are duties regulated ? Ar \ And 2 per ct. on 10560 Ibs. is $211.20. 3. At 3 cents a pound, what is the duty on 25 casks of nails, each weighing 125 Ibs. allowing 8 pounds on a cask for tare ? 4. At 5 cents a pound, what is the specific duty on 75 boxes of raisins, weighing 60 Ibs. apiece, allowing 6 pounds a box for draft ? 5. At 4 cents per pound, what is the specific duty on 110 chests of cinnamon, each weighing 230 Ibs. allowing 16 Ibs. per chest for draft? 6. At 15 cents a pound, what is the specific duty on 300 bags of indigo, each weighing 200 Ibs., allowing 4 per cent, for tret ? CASE II. 7. What is the ad valorem duty, at 15 per cent, on an invoice of calico prints, which cost $150 in Liverpool? Suggestion. When duties are imposed upon the actual cost of merchandise, there are of course no deductions to be made ; consequently we have only to find 15 per cent, of $150, the amount of the given invoice, or cost of the goods, and it will be the duty required. Solution. $150x-15-$22.50. Ans. Hence, 274. To find the ad valorem duty on any given mer- chandise. Multiply the given invoice by the given or legal per cent, and the product will be the duty required. (Art. 225.) OBS. 1. An invoice is a written statement of merchandise, with the value or prices of the articles annexed. QUEST. 274. How are ad valorem duties calculated ? Obs. What i> an invoice? What does the law require respecting the invoiced aaported goods ? ART. 274.] DUTIES. 259 2. The law requires that the invoice shall be verified by the owner, or one of the owners of the goods, wares, or merchandise, certifying that the invoice annexed contains a true and faithful account of tht actual costs thereof, and of all charges thereon, and no othei differ- ent discount, bounty, or drawback, but such as has been actually al- lowed on the same; which oath shall be administered by a consul, or commercial agent of the United States, or by some public officer duly authorized to administer oaths in the country where the goods were purchased, and the same shall be duly certified by the said consul, &c. Fraud on the part of the owners, or the consul, &c. who administers the oath, is visited with a heavy penalty. Laws of Hie United States. 8. What is the ad valorem duty, at 30 per cent, on a box of books invoiced at $250 ? 9. What is the ad valorem duty, at 20 per cent., on a quantity of Java coffee, which cost $356.12? 10. What is the amount of ad valorem duty, at 25 per cent, on a quantity of Turkey carpeting, which cost $526.61. 11. What is the duty on a quantity of bombazines, in- voiced at $310, at 30 per cent. ? 12. What is the duty on a quantity of beeswax, the in voice of which is $460.25, at 15 per cent, ? 13. At 25 per cent, what is the duty on an invoice oi bleached linens, amounting to $745.85. 14. At 20 per cent, what is the duty on an invoice of jewelry, amounting to $4250 ? 15. W'hat is the duty on a bale of goods, invoiced at $2500, at 40 per cent. ? 16. What is the duty on an invoice of silks, amounting to $5650, at 30 per cent. ? 17. What is the duty on a quantity of cutlery, invoiced at $4560, at 33 per cent ? 18. What is the duty on an invoice of broadcloths, which amounts to $8280, at 35 per cent. ? 19. What is the duty on an invoice of wines, amount- ing to $10265, at 35 per cent? 20. What is the duty on a quantity of cotton fabrics, invoiced at S13637.50, at 33 per cent? 21. What is the duty on a quantity of ready-made clothing, amounting to $5638.25, at 50 per cent ? \ 260 ASSESSMENT [SECT. IX ASSESSMENT OF TAXES. 275. A TAX is a sum imposed or levied on indi riduals for the support or benefit of the Government, a corporation, parish, district, &c. Taxes levied by the Government, are assessed either on the person or property of the citizens. When assessed on the person, they are called poll taxes, and are usually a specific sum. Those assessed on the property are usually apportioned at a cer- tain per cent, on the amount of real estate and personal property of each citizen or taxable individual. OBS. Property is divided into two kinds, viz : real estate, and per- sonal property. The former denotes possessions that are fixed ; as houses, lands, &c. The latter comprehends all other property; as money, stocks, notes, mortgages, ships, furniture, carriages, cattle, tools, &c. 276* When a tax of any given amount is to be as- sessed, the first thing to be done is to obtain an inventory 01' the amount of taxable property, both personal and real, in the State, County, Corporation, or District, by which the tax is to be paid ; also the amount of property of every citizen who is to be taxed, together with the number of Polls. OBS. 1. By the number of polls is meant the number of taxable individuals, which usually includes every native or naturalized free- man over the age of 21, and under 70 years. In some States it also includes the young men over the age of eighteen years, who are sub- ject to military duty. 2. When any part or the whole of a tax is assessed upon the polls, each citizen is taxed a specific sum, without regard to the amount of property he possesses. Ex. 1. A certain town is taxed $325. The town con- tains 200 polls, which are assessed 25 cents apiece ; and QUEST. 275. What are taxes t Upon what are they assessed ? When assessed upon the person, what are they called ? When assessed upon the property, how are they apportioned ? Obs. How is pro^ j "y ui\kled? W Ii.it does real estate denote ? What is personal property ? $76. When a tax is to be assensed, what is the first step ? Oft*. Wl.at is meant by the number of polls ? ARTS. 275-278.] OF TAXES. 261 the whole amount of property both real and personal, is valued at $13750. How much is the tax on a dollar; that is, what per cent, is the tax, and how much is a man's tax who pays for 1 poll, and whose property is valued at $850 ? Suggestion. The tax on the polls is 200x.25=$50. And $50 subtracted from $325 leaves $275, which is to be assessed equally on the amount of property possessed by the citizens of the town. The next step is to find how much must be paid on a dollar. Now if $13750 pay $275, $1 must pay ia } 50 part of $275. And $275-*- $13750=$.02, the tax on $1, which is 2 per cent. Fi- nally, at 2 per cent., or 2 cents on $1, the tax on $850, the amount of the man's property, is $850x.02=$ 17.00. And $17+.25 (the poll)=$ 17.25, the man's tax. Hence, 27 7 To assess a State, County, or ether tax. 1. First find the amount of tax on all the polls, if any, at the given rate, and subtract this sum from the whole Lax to be assessed. Then dividing the remainder by the whole amount of taxable property in the Stale, County, fyc., the quotient will be the per cent, or tax on 1 dollar. II. Multiply the amount of each man's property by the per cent, or tax on one dollar, and the product will be the tax vn his property. III. Add each marts poll tax to the tax he pays on his property, and the amount will be his whole tax. 2 7 8 . PROOF. When a tax bill is made out, add together the taxes of all the individuals in the town, district, Ans. is also 8. PROOF OF DIVISION BY CASTING OUT THE NINES. S83. First cast the 9.5 out of the divisor and quotient^ and multiply the remainders together ; to the product add the. remainder, if any, after division ] cast the 9s out of this sum. and set down the excess ; finally cast theQs out of the dividend, and if the excess is the same as that obtained from the divisor and quotient, the work may be considered right. AXIOMS. 284 In mathematics, there are certain propositions whose truth is so evident at sight, that no process of rea- soning- can make it plainer. These propositions are called axioms. An axiom* therefore, is a self-evident proposition. 1. Quantities which are equal to the same quantity, are equal to each other. 2. If the same or equal quantities are added to equai quantities, the sums will be equal. 3. If the same or equal quantities are subtracted from equals, the remainders will be equal. 4. If the same or equal quantities are added to unequal^, the sums will be unequal. QUEST. 283. How is division proved by casting out the 9s? . 283-286.] OF NUMBERS. 369 5. If the same or equal quantities are subtracted from U'/iequals, the remainders will be unequal. 6. If equal quantities are multiplied by the same or equal quantities, the products will be equal. 7. If equal quantities are divided by the same or equal quantities, the quotients will be equal. 8. If the same quantity is both added to and subtracted from another, the value of the latter will not be altered 9. If a quantity is both multiplied and divided by the same or an equal quantity, its value will not be altered. 10. The whole of a quantity is greater than a part. 11. The 'whole of a quantity is equal to the sum of all its parts. OBS. The term q uantity signifies any thing which can be multiplied, divided, or measured. Thus, numbers, yards, busJids, weight, time, &c., are called quantities. 285. The following principles will at once be recog- nized by the pupil as deductions from the four Fundamen- tal Rules of Arithmetic, viz : Addition, Subtraction, Mul- tiplication, and Division. 286. When the sum of two numbers and one of the numbers are given, to find the other number. From the given sum subtract the given number, and the remainder will be the. other number. Ex. 1. The sum of two numbers is 25, and one 01 them is 10 ; what is the other number? Solution. 25 10=15, the other number. (Art. 40.) PROOF. 15-1-10=25, the given sum. (Art. 284. Ax. 1 1.) 2. A and B together own 36 cows, 9 of which belong to A : how many does B own ? 3. Two farmers bought 300 acres of land together, and one of 'them took 115 acres: how many acres did the other have ? QUEST. 284. What is an axiom ? What is the first axiom ? The second? Third? Fourth? Fifth? Sixth? Seventh? Eighth? Ninth! Tenth ? Eleventh ? Obs. What is meant by quantity ? 286. When the mm of two numbers and cne of them r riven, how ia the other found f 270 PROPERTIES [SECT. 287* When the difference and the greater of two numbers are given, to find the less. Subtract the difference from the greater , and the remainder icill be the less number. 4. The greater of two numbers is 37, and the difference between them is 10: what is the less number? Solution. 37 10=27, the less number. (Art. 40.) PROOF. 274-10=37, the greater number. (Art. 39. Obs.) 5. A had 48 dollars in his pocket, which was 12 dollars more than B had : how many dollars had B ? 6. D had 450 sheep, which was 63 more than E had : kow many had E ? 289. When the difference and the less of two num- bers are given, to find the greater. Add the difference and less number together, and the sum icill be the greater number. (Art. 39.) 7. The difference between two numbers is 5, and the less number is 15: what is the greater number? Solution. 154-5=20, the greater number. PROOF. 20 15=5, the given difference. (Art. 40.) 8. A is 16 years old, and B is 8 years older: how old isB? 9. The number of male inhabitants in a certain town, is 935 ; and the number of females exceeds the number of males by 115: how many females does the town con- tain? QUEST. 287. When the difference and the greater of two numbeia are given, how is the less found ? 289. When the difference and *fcfl less of two numbers are given, how is the greater found ? AJB/S. 287-291.] OF NUMBERS, 27 1 29O. When the sum and difference of two numbers ij-e given, to find the two numbers. From the sum subtract the difference, and half the remainder till be the smaller number. To the smaller number thus found, add the given difference, and the sum will be the larger number. 10. The sum of two numbers is 35, and their difference is 1 1 : what are the numbers ? Solution. 35 1 1=24 ; and of 24=12, the smaller number. And 12+11=23, the greater number. PROOF. 23+12=35, the given sum. (Art. 284, Ax. 11.) 1 1. The sum of the ages of 2 boys is 25 years, and the difference between them is 5 years : what are their ages ? 12. A man bought a chest of tea and a hogshead of molasses for $63 ; the tea cost $9 more than the molasses : what was the price of each ? When the product of two numbers and one o the numbers are given, to find the oilier number. Divide the given product by the given number, and the quotient will be the number required. (Art. 74.) . 13. The product of two numbers is 84, and one of the numbers is 7 : what is the other number ? Solution. 84-7-7=12, the required number. (Art. 72.) PROOF. 12x7=84, the given product. (Art. 54.) 14. The product of A and B's ages is 120 years, and A's age is 12 years : how old is B ? 15. A certain field contains 160 square rods, and the length of the field is 20 rods: what is its breadth ? QUEST. 290. When the sum and difference of two numbers nre given, how are the numbers found? 291. When the product of two numbers and one of them are given, how is the other found ? 272 PROPERTIES [SECT. jfl Note. The area of a field is found by multiplying its length and breadth together. (Art. 163.) Hence the area of a field may be con- idered as a product 292. When the divisor and quotient are given to find the dividend. Multiply the given divisor and quotient together, and tht oduct will be the dividend. (Art. 73.) 16. If a certain divisor is 9, and the quotient is 12 what is the dividend ? Solution. 12x9=108, the dividend required. PROOF. 108-^-9=12, the given quotient. (Art. 72.) 17. A man having 1 1 children, gave them $75 apiece how many dollars did he give them all ? 18. A farmer divided a quantity of apples among 90 ooys, giving each boy 15 apples : how many did he give them all ? When the dividend and quotient are given, to find the divisor. Divide the given diviflend by the given quotient, and the. quotient thus obtained will be the number required. (Art. 73. Obs. 2.) 19. A certain dividend is 130, and the quotient is 10: what is the divisor ? Solution. 130-^-10=13, the divisor required. (Art. 72.) PROOF. 13x10==- 130, the given dividend. (Art. 73.) 20. A gentleman divided $120 equally among a com- pany of sailors, giving them $10 apiece : how many sail- ors were there in the company ? QUEST. 292. When the divisor and qxiotient are given, how is tha dividend found ? 293. When the dividend and quotient are given hov HJ the divitor found ! ARTS. 292-295.] oy NUMBERS, 273 21. A farmer having 600 sheep, divided them into flocks of 75 each : how many flocks had he ? 294* When the product of three numbers and iico of the numbers are given, to find the other number. Divide the given product by the product of the two given numbers, and the quotient witt be tJie other number. 22. There are three numbers whose product is 60 ; one of them is 3, and another 5 : it is required to find the other number? Solution. 5x3=15; and 60-1-15=4, the number re- quired. PROOF. 5x3x4=60, the given product. 23. The product of A, B, and C's ages, is 210 years ; the age of A is 5 years, and that of B is 6 years : what is the age of C ? 24. The product of three boys' marbles, is 1728 ; two of them have a dozen apiece : how many has the other? SECTION XI. ANALYSIS. ART. 29 5 Business men have a method of solving practical questions, which is frequently shorter and more expeditious, than that of arithmeticians fresh from the schools. If asked, by what rule they perform them, their reply is, " they do them in their head" or by the " no rule method" Their method consists in Analysis, and may, with propriety, be called the COMMON SENSE RULE. QUEST 294. When the product of three numbers and two of them are given, how is the other found ? 295. What is said of the method by which business men solve practical questions ? In what does their method consist ? What may it with propriety be called ? 274 ANALYSIS. [SECT. The term analysis, in physical science, signifies the re> tolving of a compound body into its elements or compo* nent parts. ANALYSIS, in Arithmetic, signifies the resolving of num- bers into the factors of which they are composed, and the tracing of the relations which they bear to each other. OBS. In the preceding sections the student has become ac- quainted with the method of analyzing particular examples and com" binations of numbers, and thence deducing general principles and rides. But analysis may be applied with advantage not only to the development of mathematical truths, but also to the solution of a great variety of problems both in arithmetic and practical life. MENTAL EXERCISES. Ex. 1. If 8 barrels of flour cost $40, how much will 5 barrels cost 1 Analysis. 1 is 1 eighth of 8 : therefore 1 barrel will cost 1 eighth as much as 8 barrels ; and I eighth of $40 is $5. Now it is obvious that 5 barrels will cost 5 times as much as 1 barrel ; and 5 times $5 are $25, the answer required. Or, we may reason thus ; 5 barrels are -f of 8 barrels ; 5 barrels will therefore cost -f as much as 8 barrels. Now 1 eighth of $40 is $5, and 5 eighths is 5 times $5, which is $25. Ans. 2. If 7 Ibs. of tea cost 42 shillings, what will 10 Ibs cost? 3. If 9 sheep are worth $27, how much are 15 sheep worth ? 4. If 10 barrels of flour cost $60, what will 12 barrels cost? 5. Suppose 30 gallons of molasses cost $15, how many dollars will 7 gallons cost ? QUEST. 295. a. What is meant by analysis in physical science 1 What in arithmetic ? Obs. To what may analysis be advantageously applied ? . 295. a.\ ANALYSIS. 275 6. If a man earns 54 shilling? m 6 days, how much can he earn in 15 days 1 7. If 12 men can build 48 rods of wall in a day> how juany rods can 20 men build in the same time 1 8. A gentleman divided 90 shillings equally among 15 beggars: how many shillings did 7 of them receive? 9. Suppose 75 pounds of butter last a family of board- ers 25 days, how many pounds will supply them for 12 days? *10. If 7 yards of cloth cost $30, how much will 9 yards cost? 11. If 10 barrels of beef cost $72, how much will 8 barrels cost ? 12. If 7 acres of land cost $50, what will 12 acres cost? 13. A farmer bought an ox cart, and paid $15 down, which Avas -fo of the price of it : what was the price of the cart ; and how much does he owe for it ? Analysis. The question to be solved is simply this . 15 is -A- of what number ? If 15 is -fV, -^ is of 15, which is 5. Nt>w if 5 is 1 tenth, 10 tenths is 10 times 5, which is 50. . ^ $50 is the price of the cart, and ' } $50 $15-35, the sum unpaid. Note. In solving examples of this kind, the learner is often per- plexed in finding the value of -JL-, &c. This difficulty arises from supposing that if -fc of a certain number is 15, -fo of it must be -j-L- of 15. This mistake will be easily avoided by substituting in his mind the word parts for the given denominator. Thus, if 3 parts cost $15, 1 part will cost .1 o f $15, which is $5. But this part is a tenth. Now if 1 tenth cost $5, then 10 tenths will cost 1 times as much. 14. A man bought a yoke of oxen, and paid $56 cash down, which was of the price of them : what did they cost 2 15. A merchant bought a quantity of wood and paid $45 in goods, which was f of the whole cost : how much iid he pay for the wood ? 16. A whale ship having been out 24 months, the cap 276 ANALYSIS. [SECT. tain found that his crew had consumed -f of his provis- ions : how many months' provision had he when he em- barked ; and how much longer would his provisions last ? 17. How many times 7 in -f- of 35? Analysis. |- of 35 is 7, and is 4 times 7, which is 28. Now 7 is contained in 28, 4 times. Ans. 4 times. 18. How many times 6 in f of 45 ? 19. How many times 10 in f of 60? 20. How many times 12 in -f- of 84 ? 21. -f of 42 are how many times 6 ? 22. of 40 are how many times 5 ? 23. -^ of 80 are how many times 12 ? 24. f of 48 are how many times 4 ? 25. of 64 are how many times 7 ? 26. T^- of 100 are how many times 12 ? 27. -ft of 110 are how many times 8 ? 28. -f of 180 are how many times 10 ? 29. -fa of 84 are how many times 9 ?- 30. How many yards of cloth, at $ 7 per yard, can be bought for i of $54 ? 31. How many barrels of flour, at $5 per barrel, can be bought for -f of $60 ? 32. A man had $64 in his pocket, and paid -f- of it for 10 barrels of flour : how much was that per barrel ? 33. 40 is -f- of how many times 6 ? Analysis. Since 40 is f, i is of 40, or 8 ; and f is 9 times 8, or 72. Now 6 is contained in 72, 12 times. Ans, 12 times 34. 56 is -f of how many times 7 ? 35. 81 is T^T of how many times 30 ? 36. 72 is T 8 !- of how many times 9 ? 37. 96 is -f- of how many times 12 ? 38. 64 is -fo of how many times 20 ? 39. 54 is -f of how many times 24 ? 40. 108 is -ft- of how many times 12 ? . 296.] ANALYSIS. 277 41. Frank sold 10 peaches, which was f ol all he had ; He then divided the remainder equally among 5 com- panions: how many did they receive apiece? 42. Lincoln spent 60 cents for a book, which was if of his money ; the remainder he laid out for oranges, at 4 cents apiece : how many oranges did he buy ? 43. A man paid away $35, which was f of all he had : he then laid out the rest in cloth at $2 per yard : how many yards did he obtain ? 44. A farmer bought a quantity of goods, and paid $20 down, which was -f of the bill : how many cords of wood, at $3 per cord, will it take to pay the balance ? 45. A man bought a horse and paid $60 in cash, whi-;h was f of the price : how many barrels of flour at $6 per barrel, will it take to pay the balance? 46. ij- of 27 is f of what number ? Analysis. f of 27 is 9. And if 9 is of a certain number, -| of that number is 3 ; and f is 4 times 3, which ss 12, the number required, 47. -rV of 30 is f of what number ? 48. -f- of 40 is -f- of what number ? 49. 4- of 35 is -rV of what number? 50. - of 54 is i^s of what number ? EXERCISES FOR THE SLATE. 296. It will be seen from the preceding examples, that no particular rules can be prescribed for solving questions by analysis. None in fact are requisite. The process will be easily suggested by the judgment of the pupil, and the conditions of the question. OBS. The operation of solving a question by analysis, is called an analytic solution. In reciting the following examples, the pupil QITEST. ?%. Can any particular rules be prescribed for solving uestions by analysis ? How then will you know how to proceed! Obs. What is the operatian of solving questions by analysis, called ? 278 ANALYSIS. [SECT. should be required to analyze each question, and give the reason ibr each step, as in the preceding mental exercises. Ex. 1. If 40 barrels of beef cost $320, how much will 52 barrels cost ? Analytic Solution. Since 40 bbls. cost $320, 1 bbl, will cost -fa of $320. And -fa of $320 is $320-^40^$8. Now if 1 bbl. cost $8, 52 bbls. will cost 52 times as much ; and $8x52 $416, which is the answer required. Or thus : 52 bbls. are ^ of 40 bbls. ; therefore 52 bbls will cost -f-2- of $320 ; (the cost of 40 bbls. ;) and -f- o* $320 is $320x-f-tr=$416,thesame result as before. (Arts 132, 133.) OBS. 1. Other solutions of this example might be given; but our present object is to show how this and similar examples may be solved I y analysis. The former method is the simplest and most strictly analytic, though not so short as the latter. It contains two steps: First, we separate the given price of 40 bbls. ($320 ) into 40 equal parts, to find the value of one part, or the cost of 1 bbl., which is $8. Second, we multiply the price of 1 bbl. ($8 ) by 52, the number oi barrels, whose cost is required, and the product is the answer sought. 2. In solving questions analytically, it may be remarked in general, that we reason from the given number to 1 , then from 1 to the re- quired number. 3. This and similar questions are usually placed under Simple Pro- portion, or the " Rule of Three;" but business men almost invariably solve them by analysis. 2. If 30 cows cost $360.90, how much will 47 cows cost, at the same rate? 3. If 25 barrels of apples cost $15, how much will 37 barrels cost ? 4. If 15 hogsheads of molasses cost $450, how much will 21 hogsheads cost? 5. If 31 yards of cloth cost $127, how mucn will 89 yards cost ? 6. If 55 tons of hay cost $660, what will 17 tons come to ? 7. An agent paid $159 for 530 pounds of wool : how much was that per 100? 8. A man bought 30 cords of wood for $76.80 : how much must he pay for 65 cords? A.RT. 296.] ANALYSIS. 279 9. A gentleman bought 85 yards of carpeting for $106.25 : how much would 38 yards cost *? 10. A drover bought 350 sheep for $525 : how much would 65 cost, at the same rate ? 11. If 12-| pounds of coffee cost $1.25, how much will 45 pounds cost ? 12. If 16 bushels of corn are worth $8, how much are 25 bushels worth ? 13. Paid $20 for 60 pounds of tea : how much would 12 pounds cost, at the same rate ? 14. Bought 41 yards of flannel for $16.40: how much would 8f yards cost 1 15. Bought 18 pounds of ginger for 84.50: how much will lOf pounds cost? 16. If a stage goes 84 miles in 12 hours, how far will it go in 15-^ hours? 17. If 8 horses eat 36 bushels of oats in a week, how many bushels will 25 horses eat in the same time ? 18. If a Railroad car runs 120 miles in 5 hours, how far will it run in 1 2-f- hours ? 19. If a steamboat goes 180 miles in 1 2 hours, how far will it go in 5-f hours ? 20. If 4 men can do a job of work in 12 days, how long will it take 6 men to do it ? Solution Since the job requires 4 men 12 days, it will require 1 man 4 times as long; and 4 times 12 days are 48 days. Again, it requires 1 man 48 days, it will re- quire 6 men as long ; and 48 days-^-6=8 days, which is the answer required. 21. If 6 men eat a barrel of flour in 24 days, how long will it last 10 men? 22. If a given quantity of corn lasts 9 horses 96 days, how long will the same quantity last 15 horses ? 23. If 12 men can build a house in 90 days, how long will it take 20 men to build it ? 24. If 100 barrels of pork last a crew of 20 men 45 months, how long will it last a crew of 28 men ? 25. If 4 stacks of hay will keep 60 cattle 120 days, how long will they keep 25 cattle ? 280 ANALYSIS. [SECT. 26. If -f of a bushel of wheat cost 30 cents, what will ^ of a bushel cost ? 27. If -f- of a ton of hay cost $7, what will of a ton cost? 28. If of a pound of imperial tea cost 27 cents, how much will of a pound cost ? 29. If f of a ton of coal cost $2.61, how much will f of a ton cost ? 30. If -f of a yard of silk cost 6 shillings, how much will -f of a yard cost ? Solution. Since f of a yard cost 6s., -J- will cost 3s.. and -f or 1 yard will cost 9s. Again, if 1 yard costs 9s.,' \ yd. will cost 1^-s. ; and yd. will cost 7-J shillings, which is the answer required. -w 31. If f of a cord of wood cost $1.80, how much will f- of a cord cost ? 32. If f of a yard of broadcloth cost 14 shillings, how much will ^ of a yard cost ? 33. A man bought -f- of an acre of land for $56, and afterwards sold -jf- of an acre at cost : how much did he receive for it ? 34. A grocer bought 7 barrels of vinegar for $28, and sold -f- of a barrel at cost : how much did it come to ? 35. A grocer bought a firkin of butter containing 56 Ibs. for $1 1.20, and sold ^ of it at cost : how much did he get a pound? 36. If 6-J- bushels of peas are worth $5. 50, how much are 20^- bushels worth ? 37. If a man pays $47 for building 23 rods of orna- mental fence, how much would it cost him to build 42^ rods? 38. A farmer paid $45.42 for making 36f rods o stone wall : how much will it cost him to make 60-^- rods ? 39. A man paid -ny a : : 20 : Ans. fore. 44. lOhr. 12hr. \ 8x5x12x20=9600; and 5x4x10=200. Now 9600- 200=48. Ans. 48 acres. QUEST. 329. What is compound proportion ? OZ>. To waat is riuefly applied ? What is it, sometimes called ? 304 33O. From the foregoing illustrations we derive the following general RULE FOR COMPOUND PROPORTION. 1. Place that number which is of the same kind as the an- swer required for the third term. II. Then take the other numbers in pairs, or two of a kind, and arrange them as in simple proportion. (Art. 327.) III. Finally, multiply together all the second and third terms,divide the result by the product of tite first terms, and the quotient wUl"be the fourth term or answer" required. PROOF. Multiply the ansiver into all of the first terms or antecedents of the first couplets, and if the product is equal to the continued product of all the, second and third terms multi- plied together, the work is right. (Art. 324.) OBS. 1. Among the given numbers there is but one which is of the *ame kind as the answer. This is sometimes called the odd term, and is always to be placed for the third term. 2. Questions in Compound Proportion may be solved by Analysis ; also by Simple Proportion, by making two or mart separate state- ments. (Art. 302. Obs. 327.) 3. If 8 men can clear 30 acres of land in 63 days, working 10 hours a day, how many acres can 10 men rlear in 72 days, working 12 hours a day ? 8m. 63d. lOhr. 'Statement. 10m. } Acrea - 72d. > : : 30 : to the answer. That is, 12hr. 8x63X10 : 10X72X12 : : 30 : to the answer. But the prod. 10X72X12X30^ Ang (Art Divided by 8x63x10 QUEST. 330. In arranging the numbers in compound proportion, which number do you put for the third term I How arrange the other numbers ? HavSig stated the question, how is the answer found ? How are questions in compound proportion proved ? Obs. Among tha given numbers, how many are of the same kind as the answer ? Can questions in compound proportion lx? solved by simple proportion ? How? j 331.] PROPORTION. 305 Now by canceling equal factors, (Art. 116,) we have t Xtf2xl2x30 360 or 5 If acres. Ans. Hence, x* / 7 331* After stating the question according to the rule above, if the antecedents or first terms hare factors common to the consequents or second terms, or to the third term, they should be CANCELED before performing the. multiplication and, division. Note. Instead of placing points between the first and second terms, that is, between the antecedents and consequents of the left hand couplets of the proportion, it is sometimes more convenient to put a perpendicular line between them, as in division of fractions. (Art. 140.) This will bring all the terms whose product is to be the dividend on the right of the line, and those whose product is to form the divisor, on the left. In this case the third term should be placed below the se- cond terms, with the sign of proportion ( : : ) before it, to show its origin, and its relation to the answer. 4. If a man can walk 192 miles in 4 days, traveling 12 hours a day, how far can he go in 24 days, traveling 8 hours a day? Operation. The product of the antecedents, 4x12, ' '4 d. 2 has the same ratio to the product of the $ hr. 2 consequents, 24x8, as 192 has to the : 192m. answer required. Ans. | 192x2x2x2-768 miles. 5. If 8 men can make 9 rods of wall in 12 days, how many men will it require to make 36 rods in 4 days ? 6. If 5 men make 240 pair of shoes in 24 days, how many men will it require to make 300 pair in 15 days? 7. If 60 Ibs. of meat will supply 8 men 15 days, how orig will 72 Ibs. last 24 men ? 8. If 12 men can reap 80 acres of wheat in 6 clays, how long- will it take 25 men to reap 200 acres? 9. If 18 horses eat 128 bushels of oats in 32 days, how many bushels will 12 horses eat in 64 days? 10. If 8 men can build a wall 20 ft. long, 6 ft. high, QUEST. 331. When the antecedents have factors common to the toneequenls, what should be dona with them? 306 DUODECIMALS. [SECT. and 4 ft. thick, in 12 days, how long- will it take 24 men to build one 200 ft. long, 8 ft. high, and 6 ft. thick ? 11. If 8 men reap 36 acres in 9 days, working- 9 hours per day, how many men will it take' to reap 48 acres in 12 days, working 12 hours per day? 12. If $100 gain $6 in 12 months, how long will it take $400 to gain $18? 13. If $200 gain $12 in 12 months, what will $400 gain in 9 months ? 14. If 8 men spend 32 in 13 weeks, how much will 24 men spend in 52 weeks ? 15. If 6 men can dig a drain 20 rods long, 6 feet deep, and 4 feet wide, in 16 days, working 9 hours each day, how many days will it take 24 men to dig a drain 20G rods long, 8 ft. deep, and 6 ft. wide, working 8 hours per day ? SECTION XIII. DUODECIMALS. ART. 332* DUODECIMALS are a species of compound numbers, the denominations of which increase and decrease uniformly in a twelvefold ratio. Its denominations are feet, inches or primes, seconds, thirds, fourths, fifths, fyc. Note. The term duodecimal is derived from the Latin numeral duodecim, which signifies twelve. TABLE. 12 fourths ("") make 1 third, marked '" 12 thirds " 1 second, IS seconds " 1 inch or prime, " in. or 12 inches or primes " I foot, " ft. Hence 1'= fV of 1 foot. i"=-Ar of i in. or iV of iV of i ft.=rlr of 1 ft. \'"-^ of 1", or -1^2 of iV of iV of 1 ft.=T7W of I ft. QUEST. 332. What are duodecimals ! What are its denominations Note. What is the meaning of the term duodecimal ? Repeat the Table, Obs. What are the accents called, which are used to distinguish the different denominations ? 132-335.] DUODECIMALS. 307 OBS. The accents use , A der inches, and carry the 1 ft. to the ' next product. 2 times 8 ft. are 16 ft. and 1 to carry makes 17 ft. Again, since 6'=^- of a ft. and 9'=A of a ft., 6' into 9' is -fa of a ft.=54", or 4' and 6". Write the 6" one place to the right of inches, and carry the 4' to the next product. Then 6' or -^ of a foot multiplied into 8 ft. ^ of a ft., or 48', and 4' to carry make 52' ; but 52'=4 ft. and 4'. Now adding the partial products, the sum is 21 ft. 10' 61". OBS. It will be seen from this operation, that feet multiplied into feet, produce feet; feet into inches, produce inches; inches into inches, produce seconds, &c. Hence, 335. To find the denomination of the product of any two factors in duodecimals. Add the indices of the two factors together, and the sum will be the index of their product. Thus, feet into feet, produce feet ; feet into inches, pro- duce inches ; feet into seconds, produce seconds ; feet into thirds, produce thirds, &c. QUEST. 333. How are duodecimos added and subtracted ? 334. To what are duodecimals chiefly applied ? 335. How find the denomina- tion of the product in duodecim Is ? What do feet into feet produce ! Feet into inches ? Feet into seconds ?" 308 DUODECIMALS. [SECT. Inches into inches, produce seconds ; inches into seo onds, produce thirds; inches into fourths, produce fifths, &c. Seconds into seconds, produce fourths ; seconds into thirds, produce fifths ; seconds into sixths, piodtrce eighths, &c. Thirds into thirds, produce sixths ; thirds into fifths, pro duce eighths ; thirds into sevenths, produce tenths, &c. Fourths into fourths, produce eighths ; fourths intc eighths, produce twelfths, &c. Note. The foot is considered the unit, and has no index. 336* From these illustrations we have the following RULE FOR MULTIPLICATION OF DUODECIMALS. 1. Place the several terms of the multiplier under the cor- responding terms of the multiplicand. II. Multiply eack term of the multiplicand by each term of the multiplier separately r , beginning with the lowest de- nomination in the multiplicand, and the highest in the mul' tiplier, and write the first figure of each partial product one or more places to tJie right, under its corresponding denomi- nation. (Art. 335.) III. Finally, add the several partial products togethw, carrying 1 for every 12 both in multiplying and adding, and -the sum will be the answer required. OBS. It is sometimes asked whether the inches in duodecimals are linear, square, or cubic. The answer is, they are neither. An inch is 1 twelfth of a foot. Hence, in measuring surfaces an inch is iV of a square foot ; that is, a surface 1 foot long and 1 inch wide. In measuring solids, an inch denotes iV of a cubic foot. In measuring lumber, these inches are commonly called carpentei* a inches. 2. How many square feet are there in a board 1 8 feet 9 inches long, and 2 feet 6 inches wide ? 3. How many square feet are there in a board 14 feet 1 inches long, and 1 1 inches wide ? QUEST, What do inches into inches produce! Inches into thirds \ Inches into fourth* ? Seconds into seconds ? Seconds into thirds ? Seconds into eighths ? Thirds into thirds t Thirds into sixths ? 336. What is the rule for multiplication of duodecimals ? Obs. What kind of inches are those spoken of in measuring surfaces by duodecimals ? In measuring folids ? In measuring lumber what are they called ? LET. 336.] DUODECIMALS. 309 4. How many square feet in a gate 12 feet 5 inches tvide, and 6 feet 8 inches high? 5. How many square feet in a floor 16 feet 6 inches .ong, and 12 feet 9 inches wide? 6. How many square feet in a ceiling 53 feet 6 inches long, and 25 feet 6 inches wide ? 7. How many square feet are there in a stock of 6 boards 17 feet 7 inches long, and 1 foot 5 inches wide? 8. How many feet in a stock of 10 boards 12 feet 8 nches long, and 1 foot 1 inch wide? 9. How many cubic feet in a stick of timber 12 feet 10 inches long, 1 foot 7 inches wide, and 1 foot 9 inches thick ? 10. How many cubic feet in a block of marble 8 feet 4 inches long, 2 feet 6 inches wide, and 1 foot 10 inches thick ? 11. How many cubic feet in a load of wood 6 feet 7 inches long, 3 feet 5 inches high, and 3 feet 8 inches wide ? 12. How many feet in aload of wood 7 feet 2 inches long, 4 feet high, and 3 feet wide ? 13. How many feet in a load of wood 9 feet long, 4 feet 3 inches wide, and 5 feet 6 inches high ? 14. How many feet in a pile of wood 100 feet long, 5 feet high, and 4 feet wide 1 15. How many feet in a pile of wood 150 feet longf, 8- feet high, and 5 feet wide ? 16. How many cubic feet in a wall 40 feet 6 inches long, 5 feet 10 inches high, and 2 feet thick? 17. How many solid feet in a vat 10 feet 8 inches long, 7 feet 2 inches wide, and 6 feet 4 inches deep ? 18. How many bricks 8 inches long, 4 inches wide, and 2 inches thick, are there in a wall 20 feet long, 10 feet high, and \\ feet thick ? 19. How much will the flooring of a room which is 20 feet long, and 18 feet wide come to, at 6-$- cents per square foot ? 20. How much will the plastering of a wall 16 feet square come to, at 12 cents per square yard ? 310 INVOLUTION. [SEC/ . Xff% SECTION XIV. INVOLUTION. MENTAL EXERCISES. ART. 331. Ex. 1. What is the product of 5 mu.ii plied by 5? Ans. 5X5=25. 2. What is the product of 3 multiplied into 3 twice ? Ans. 3X3X3=27. 3. What is the product of 2 into itself three times ? Ans. 2X2X2X2=16. 338* When any number or quantity is multiplied into itself, the product is called a power. Thus, in the ex- amples above, the products 25, 27, and 16 are powers. The original number, that is, the number which being multiplied into itself, produces a power, is called the root of all the powers of that number ; because they are de- rived from it. 339* Powers are divided into different orders; as the first, second, third, fourth, fifth power, &c. They take their name from the number of times the given number is used QS Q. factor, in producing the given power. Note. 1. The first power of a number is said to be the number itself. Strictly speaking, it is not & power, but a root. (Art. 338.) 3 yards. 1. The second power of a number is also called the square ; (Art. 153. Obs. 1 ;) for, if the side of a square is 3 yards, then the r product of 3x3=9 yards, will be the area of ^ the given square. (Art. 163.) But 3X3=9 w is also the second power of 3 ; hence, it is tailed the square. 3X3=9yards. QUEST. 333. What is a power ? 339. How are powers divided ? From what do they take their name ? Note. What is said to be the first power ? What is the second power called ? The third 1 The fourth ? h. 337-340.] INVOLUTION. 3. The third power of a number is also called the cube; (Art. 154. Obs.2;) for, if the side of a cube is 2 feet, then the product of 2X2X2= 8 feet, will be the solidity of the given cube. (Art. 164.) But 2X2X2=8, is also the third power of 2; hence it is called the cube. 4. The fourth power of a number if culled the biquadrate. 2 feet. 2X2X2=8 feet. 4. What is the square of 4? Ans. 16. 5. What is the cube of 3 ? The fourth power of 3 ? 6. The fourth power of 2 ? The fifth power of 2 ? 7. What is the square of 5 ? Of 6 1 Of 7 ? Of 9 ? Ot 8? Of 10? Of 11? Of 12? 8. What is the cube of 3 ? Of 4 ? Of 5 ? Of 6 ? 3 4O. Powers are frequently denoted by a small figure placed above the given number at the right hand. This figure is called the index or exponent. It shows how many times the given number is employed as a fac- tor to produce the required power. Thus, The index of the first power is 1, but this is omitted ; for, (2) J =2. The index of the second power is 2 ; The index of the third power is 3 ; The index of the fourth power is 4 ; The index of thefiflh power is 5 ; &c. That is, 2*=2, the first power of 2 ; 2 2 =2x2, the square, or 2d power of 2 ; 2* =2x2x2, the cube, or 3d power of 2; 2 4 =2x2x2x2, the biquadrate, or 4th power of 2 ; 25=2x2x2x2x2, the fifth power of 2 ; 2 6 =2x2x2x2x2x2, the 6th power of 2; &c. QUEST. 3-10. How are powers denoted ? What is this figure called ? What does it show ? What is tne index of the first power? Of tho econd ? The third ? Fourth ? Fifth ? Sixth ? 812 INVOLUTION EXERCISES FOR THE SLATE. 9. Express the third power of 6 ; the 4th power of 12. 10. Express the square of 16 ; the cube of 20 ; the fourth power of 25 ; the fifth power of 72 ; the sixth power ot 100 ; the tenth power of 500. 341. The process of finding a power of a given number by multiplying it into itself, is called INVOLUTION 34:2* Hence, to involve a number to any required power. Multiply the given number into itself, till it is talten as a factor, as many times as there are units in the index of the, power to which the number is to be raised. (Art. 339.) OBS. 1. The number of multiplications in raising a number to any given power, is one less than the index of the required power. Thus, the square of 3 is written 3 2 , and 3X3=9, the 3 is taken twice as a factor, but there is but one multiplication. ". A fraction is raised to a power by multiplying it into itself. Thus, the square of -| is ^^^ ==x ^ i Mixed numbers should be reduced to improper fractions, or tlu ixOmmon fraction may be reduced to a decimal. 3. All powers of 1 are the same, viz: 1; for 1X1 XI XI, &c.^l. 1 1 . What is the square of 24 ? Common Operation. Analytic Operation. 24 24=2 tens or 20+4 units. 24 24=2 tens or 20-f4 units. 96 80+16" 48 400+80 576. Ans. And 400+160+16=576. It will be seen from this operation that the square o5 20+4, contains the square of the first part, viz : 20x20 =400, added to twice the product of the two parts, viz : 20x4+20x4=160, added to the square of the last part, viz: 4x4=16. Hence, QUEST. 341. What is involution ? 342. How is a number involved to any required power ? Obs. How many rn duplications are there i* raising a number to a given power ? How is a fraction involved ? A mixed number ? Wha* are all powora of 1 ? ARTS, jit 1-343.] EVOLUTION. 313 ^342. a. The square of any number which consist? of o figures, is equal to the square of the tens, added to twice product of the tens into the units, added to the square of the units. OBS. 1. The product of any two factors cannot have more figures than both factors, nor but one less than both. For example, takej), the greatest number which can be expressed by one figure. (Art. 7.) And" (9) 2 , or 9X9=81, has two figures, the same number which both factors have. 99 is the greatest number which can be expressed by two figures; (Art. 7;) and (99)2, or 99x99=9301, has four figures, the same as both factors nave. Again, 1 i the smallest number expressed by one figure, and (I) 2 , or IX 1 = 1) h as Dut one figure less than both factors. 10 is the smallest number which ran be expressed by two figures ; and (10) 2 , or 10X10=100, has one figure less than both factors. Hence, 2. Any square number cannot have more figures than double the number of the root or first power, nor but one less. 3. A cube cannot have more figures than triple the number of the root or first power, nor but two less. 12. What is the square of 45? 50? 75? 100? 540? 13. What is the cube of 5 ? Of 8? 10? 12? 60? 1 4. What is the fourth power of 3 ? Of 4 ? 16 ? 20 ? 15. What is the fifth power of 2? Of 3 ? 4? 5? 6? 16. What is the square of i? Of i? -J-? |? ? f? 1 7. What is the cube of -f ? Of i ? Of $ ? Of H ? 18. What is the square of 2i? Of3-J-? 5f? 10-f? 19. What is the square of 1.5 ? Of 3.25 ? Of 10.25 ? EVOLUTION. 343* If we resolve 25 into two equal factors, viz: 5 and 5, each of these equal factors is called a root of 25. So if we resolve 27 into three equal factors, viz : 3, 3, and 3, each factor is called a root of 27 ; if we resolve 16 into four equal factors, viz : 2, 2, 2, and 2, each factor is called a root of 16. And, universally, when a number is resolv- ed into any number of equal factors, each of those factors is said to be a root of that number. Hence, QUEST. 342. a. What is the square of any number consisting of two figures equal to ? OBS. How many figures are there in the product of any two factors ? How many figures will the square of a number con- tain ? The cube ? 343. When a number is resolved into any mim ber of equal factors, what is each of thoso factors called ? 314 KVOLIJTiON. ; 344:9 A root of a number is a factor, which, beingsS multiplied into itself a certain number of times, will pro- duce that number. (Art. 338.) OBS. When a number is resolved into two equal factors, each of these factors is called the second or square root ; when resolved into three equal factors, each of these factors is called the third or ?ube root ; when resolved into four equal factors, each factor is called the fourth root ; &c. Hence, The name of the root expresses the number of equal factors into which the given number is to be resolved. For example, the second or square root, shows that the number is to be resolved into two equal factors ; the third or cube root, into three equal factors ; the fourth root, into four equal factors, &c. Thus, The square root of 16 is 4; for 4x4 10. The cube root of 27 is 3; for 3X3X3=27. The fourth root of 16 is 2; for 2X2X2X2=16, &*. MENTAL EXERCISES. Ex. 1. Resolve 25 into two equal factors. Solution. 25=5x5. A?is. 5, and 5. 2. Resolve 8 into three equal factors. Solution. 8=2x2x2. Ans. 2, 2, and 2. 345* The process of resolving numbers into equal factors is called EVOLUTION, or the Extraction of Roots. OBS. 1. Evolution is the opposite of involution. (Art. 341.) One is finding a power of a number by multiplying it into itself; the other is finding a root by resolving a number into equal factors. Powers and roots are therefore correlative terms. If one number is a power of another, the latter is a root of the former. Thus, 27 is the cube of 3 ; and 3 is the cube root of 27. 2. The learner will be careful to remember, that In subtraction, a number is resolved into two parts; In division, a number is resolved into two factors; In evolution, a number is resolved into equal factors. 3. What is the square root of 16? Ans. 4. 4. What is the square root of 36 ? Of 49 ? QUEST. 344. What then is a root ? Obs. What does the name of the root express? What does the square root show ? The cube root? The fourth root 1 345. What is evolution ? Obs. Of what is it the opposite ? Into what are numbers resolved in subtraction ? In divi sion ? In evolution ? V 344-3 46. J EVOLUTION. 315 f5TVV hat is the square root of 64 ? Of 8 H Of 1 00 ? Of 121? Of 144? 6. What is the cube or third root of 8 ? Soluli&n. If we resolve 8 into three equal factors, each of these factors is 2 : for 2x2x2=8. The cube root of 8 therefore, is 2. 7.. What is the cube root of 27 ? 8. What is the cube root of 64 1 9. What is the cube root of 125 ? 10. What is the fourth root of 16? 1 1. What is the square root of -fa ? Solution. The square root of the numerator 9, is 3 ; and the square root of the denominator 16, is 4. There- fore -f- is the square root of -fa ; for -f- xf =tV- 12. What is the square root of $? Ans. -. 13. What is the square root of if ? Of ff? 14. What is the square root of f| ? Of -ftfr ? 15. What is the cube root of ? Ans. . 16. What is the cube root of l Of 346* Roots are expressed in two ways] one by the radical sign (v) placed before a number ; the other by a fractional index placed above the number on the right hand. Thus, v4,or 4 5 denotes the square or 2d root of 4 ; 3 JL 4 JL V27, or 27 3 denotes the cube or 3d root of 27 ; Vl6,or 16 4 denotes the 4th root of 16. OBS. 1. The figure placed over the radical sign, denotes the root, or the number of equal factors into which the given number is to be re- solved. The figure for the square root is usually omitted, and simply the radical sign \/ is placed before the given number. Thus, the square root of 25 is written \/ 25. 2. When a root is expressed by ^.fractional index, the denominator like the figure over the radical sign, denotes the root of the given num- ber. Thus. (25)- denotes the square root of 25; (27) denotes the cube root of 27. Quiv-3T. 316. In how many ways are roots expressed ? What are thy ? Obs. What does the figure over the radical sign denote ? What f .he denominator of the fractional indox ? 316 EVOLUTION. EXERCISES FOR THE SLATE. 17. Express the cube root of 45 both ways. 18. Express the cube root of 64 both ways. Of 125, 19. Express the fourth root of 181 both ways. Of 576 20. Express the 5th root of 32 ; the 6th root of 64. 21. Express the 7th root of 84 ; the 8th root of 91 ; the 9th root of 105 ; the 10th root of 256. 22. Express the cube root of 576 ; the fourth root oi 675 ; the fifth root of 1000 ; the twelfth root of 840. 347. A number which can be resolved into equal factors, or whose root can be exactly extracted, is called a perfect power ) and its root is called a rational number, Thus ; 16, 25, 27, &c.,are perfect powers, and their roots 4, 5, 3, are rational numbers. 34:8* A number which cannot be resolved into equal factors, or whose root cannot be exactly extracted, is called an imperfect power ; and its root is called a Surd, or irra- tional number. Thus, 15, 17, 45, &c., are imperfect pow- ers, and their roots 3.8-f- ; 4.1-f-; 6.7+, &c., are surds, for their roots cannot be exactly extracted. OBS. A number may be a perfect power of one degree and an im- perfect power of another degree. Thus, 16 is a perfect power of the second degree, but an imperfect power of the third degree ; that is, it is a perfect square but not a perfect cube. Indeed numbers are sel- dom perfect powers of more than one degree. 16 is a perfect power of the 3d and 4th degrees : 64 is a perfect power of the 2d, 3d and 6th degrees. 349 Every root, as well as every power of 1, is 1. (Art. 342. Obs. 3.) Thus, (1)*, (I) 3 , (I) 6 , and vl, VJ. c Vl, &c., are alt equni. QUEST. 347. What is a perfect power? What is a rational num- ber ? 348. What is an imperfect power ? What is a surd 1 Obs. Ar numbers ever perfect powers of one degree and imperfect pcwers of another degree ? Are they often perfect powers of more than one d gre ? 84V. What are all roots and powers of 1 ? Art- -350.] EVOLUTION. 317 jU/ EXTRACTION OF THE SQUARE ROOT. 35O. To extract the square root ^ is to resolve a given number into two equal factors ; or, to find a number which being multiplied into itself, will produce the given number. (Art. 344. Obs.) Ex. 1. What is the side of a square room which con- tains 16 square yards? Solution. Let the room be re- 4 yards, presented by the adjoining figure. It is divided into 1 6 equal squares, which we will call square yards. Since the room is square, the question is simply this : What is the square root of 16? Now if we resolve 16 into two equal fac- tors, each of those factors will be the square root of 16. But 16=4 4x4= 1G yards. X4. The square root of 16, therefore, is 4. 2. What is the length of one side of a square room which contains 576 square feet ? Operation. 576(24 4 44)176 176 Since we may not see what the root of 576 is at once, as in the last example, we will separate it into periods of two figures each, by putting a point over the 5, and also over the 6 ; that is, over the units' figure and over the hun- dreds. This shows us that the root is to have two fig- ures ; (Art. 342. a. Obs. 2 ;) and thus enables us to find the root of part of the number at a time. Now the greatest square of 5, the left hand period, is 4, tl 3 root of which is 2. We place the 2 on the right hand of the number for the first part of the root ; then subtract its square from 5, the period under consideration, and to the right of the remainder bring down 76, the next period, for a divi- dend. To find the next figure in the root, we double the 2, (he part of the root already found, and placing it on the left of the dividend for a partial divisor, we find how many times What ft ft to extrart the square root of A mimbw ? S18 SQUARE ROOT. it is contained in the dividend, omitting the right ure. Now 4 is contained in 17, 4 times. Placing the on the right of the root, also on the right of the partial^ divisor, we multiply 44, the divisor thus completed, by 4, the last figure in the root, and subtracting the product 176 from the dividend, find there is no remainder. The an- swer therfore is 24. Note. Since the root is to contain two figures, the 2 stands in tens' place ; hence the first part of the root found is properly 20 ; which be- ing doubled, gives 40 for the divisor. For convenience we omit the cipher on the right; and to compensate for this, we omit the right hand figure of the dividend. This is the same as dividing both the divisor and dividend by 10, and therefore does not alter the quotient. (Art. 88.) PFOOF. 24=2 tens, or 20+4 units. 24-2 20+4 " 96 48 80- 400+80 -16 (24)2=576 = 400+160+16. (Art. 342. a.) ILLUSTRATION BY GEOMETRICAL FIGURE. 20ft. II Let the large square ABCD, represent the room in the last exam- ple ; then the square DE FG will be the greatest square of the left hand period, the root of which is 20 ft., and 20x20= 400, the number of feet in its area. (Art. 163.) But this square 400 ft. taken from 576 ft. leaves a remainder of 176 ft. Now* it is plain, if this remaining space is all added to one side of this square, its sides will become unequal ; consequently it will cease to be a square. (Art. 153. Obs. 1.) But if it is equally enlarged on two sides it will obviously continue to be a QUEST. Note. What place does the first figure of the root occupy ir the example above ? Why is ths right hand figure of the dividend omit- ted ? 20ft. G AET. 3JW ] SQUARE ROOT. 319 gtBSr For this reason the root is doubled for a divi- fcr in the operation. The parallelograms AEFH and jFIC will therefore represent the additions made to the two sides, each of which is 4 ft. wide ; consequently the area of each is 20x4=80 ft., and the area of both is 40x4= 160 ft. But having made these additions to two sides of the square, there is a vacancy at the corner. The square BIFH represents this vacancy, the side of which is 4 ft. or the same as the width of the additions ; and its area is 4x4=16 ft. For convenience of finding the area of this vacancy, it is customa^ - ! j?n& Air. i 2:) SQUARE ROOT. 321 hat is the square root of 53824 ? 12. What is the square root of 531441 ? 352. If there are decimals in the given sum, they must be separated into periods like whole numbers, by placing a point over units, then over kundredths, and so on, over every alternate figure towards the right. If there is a remainder after all the periods are brought down, the operation may be continued by annexing pe- riods of ciphers. OBS. 1 . There will always be as many decimal figures in the root, as there are periods of decimals in the given number. 2. The square root of a common fraction is found by extracting '.he root of the numerator and denominator. 3. A mixed number should be reduced to an improper fraction. When either the numerator or denominator of a common fraction is not a. perfect square, the fraction may be reduced to a decimal, and the approximate root be found as above. 13. What is the square root of 6.25? Ans.2.5. 14. What is the square root of 1.96 ? 15. What is the square root of 29.16 ? 16. What is the square root of 234.09 ? 17. What is the square root of .1225? 18. What is the square root of .776161 ? 19. What is the square root of 2 ? 20. What is the square root o f \7? 21. What is the square root of 175 ? 22. What is the square root of 1 16964 ? 23. What is the square root of 10316944 ? 24. What is the square root of 25. What is the square root of 26. What is the square root of 6-J- ? 27. What is the square root of 52-rV ? QUEST. 352. When there are decimals in the given num^?, how are they pointed off? When there is a remainder, how proc.?*A ? Obs. How do you determine how many decimal figures there sho^tl be in the root ? How is the square root of a common fraction found ? G a mixed number? 322 SQUAIIE ROOT. APPLICATIONS OF THE SQUARE 353* The principles of the square root may be a plied to the solution of questions in which two sides a right-angled triangle are given, and it is required to find the other side. 354* A triangle is a figure which has three sides and three angles, as in the adjoining dia- gram. When one of the sides of a triangle is perpe?idicular to an- other side, the angle between them is called a right-a?igk. (Legendre, B. I. Def. 12.) A Base. B 355. A right-angled triangle is a triangle which has a right-angle. (Leg. B. I. Def. 17.) The side opposite the right-angle is called the hypoth- enuse, and the other two sides, the base and perpendicular. The triangle ABC is right-angled at B, and the side AC is the hypothenuse. 3 56. It is an established principle in geometry, that the square described on the hypothenuse of a right-angled triangle, is equal to the sum of the squares described on the other two sides. (Leg. IV. 1 1 ., Euc. I. 47. ) Thus, if the base of the triangle ABC, is 4 feet, and the perpendicular 3 feet ; then the square of 4 added to the square of 3 is equal to the square of the hypothenuse BC ; that is, (4) 2 +(3) 2 , or 16+9=25, the square of the hypothenuse ; therefore the square root of 25, which is 5. must be the hypothenuse itself. Henee, when any two sides of a right-angled triangle are given, the third side may be easily found. QUEST. 354. What i a triangle ? What is a right-angle ? 355. What is a right-angled triangle ? Draw a right-angled triangle upon the black-board. What is the side opposite the right-angle called? What are the other two sides called ? 356. What is the square de- "* VKH! 02 V hvpothenuse equal to ? Draw a right-angled triangle, and tatadhe a ma cm aoh of i IB sides t Air. ""59.] SQUARE ROOT. 323 i &&/!. When the base and perpendicular are given, to d the hypothenuse. Add. the square of the base to the square of the perpendicular^ d'-id the square root of the sum will be the hypothenuse. Thus, in the right-angled triangle ABC, if the base is 4 and the perpendicular is 3, then (4) 2 -f(3) 2 =25, and V25=5, the hypothenuse. 358. When the hypothenuse and base are given, to find the perpendicular. From the square of the hypotlienuse subtract the square of the base, and the square root of the remainder will be the perpen- dicular. Thus, if the hypothenuse is 5. and the base 4, then (5)* (4)2=^ and v9=3, the perpendicular. 359. When the hypothenuse and the perpendicular are given, to find the base. From the square of the hypothenuse subtract the square oj the perpendic-iar, and the square root of the remainder trill be the base. Thus, if the hypothenuse is 5, and the perpendicular 3, then (5) 2 (3) 2 =16, and Vl6=4, the base. 28. What is the length of a ladder which will just reach to the top of a house 32 feet high, when its foot is placed 24 feet from the house ? Operation. Perpendicular (32) 2 =32x32=~1024 Base (24) 2 =24x24=J576^ The square root of their sum 1600=40. Ans. 29. The side of a certain school-room having square corners, is 8 yards, and its width 6 yards : what is the distance between two of its opposite corners ? QUEST. 357. When the base and perpendicular are given, how if the hypothenuse found I 358. When the hypo'henuse and base are given, how is the perpendicular found ? 359. When the hypothemis and perpendicular are given, h(T\v is h bnsefhTJnr? ' 324 CUBE ROOT. XIV 30. Two men start from the same place, one*gS?IH| actly south 40 miles a day, the other goes exactly west |fe miles a day : how far apart will they be at the close of tfn^ first day ? 31. How far apart will the same travelers be at the end of 4 days ? 32. A line 75 feet long fastened to the top of a flag staff reaches the ground 45 feet from its base : what is the height of the flag staff? 33. Suppose a house is 40 feet wide, and the length of the rafters is 32 feet : what is the distance from the beam to the ridge pole ? 34. The side of a square field is 30 rods : how far is it between its opposite corners ? 35. If a square field contains 10 acres, what is the length of its side, and how far apart are its opposite corners I EXTRACTION OF THE CUBE ROOT. 36O. To extract the cube root, is to resolve a given num- ber into three equal factors; or, to find a number which being multiplied into itself twice, will produce the given number. (Art. 345.) 1. What is the side of a cubical block containing 27 solid feet ? Solution. Let the given block be represented by the adjoining cubical figure, each side of which is divided into 9 equal squares, which we will call square feet. Now, since the length of a side is 3 feet, if we multiply 3 into 3 into 3, the | product 27, will be the solid con- " tents of the cube. (Art. 164.) " 3x3X3=27- Hence, if we reverse the process, i. e. if we resolve 27 into three equal factors, one of these factors will be the side of the cube. (Art. 344. Obs.) Ans. 3ft. 2. A man wishes to form a cubical mound containing 15625 solid feet of earth : what is the length of its side 1 . What is it to extract the uba root f ARTS. 60, 361.] CUBE ROOT. 325 nation. 1. We first separate the given num- ber into periods of three figures each, I5b -D(25 by p} ac i n g a p i n t over the units' figure, then over thousands. This shows us that the root must have two figures, (Art. 342. a. Obs. 3,) and thus enables us to find part of it at a time. 2. Beginning with the left hand pe- 1525 7625 r i oc } ? we fl n( j the greatest cube of 15 is 8, the root of which is 2. Placing the 2 on the right of the given number for the first figure in the root, we sub- tract its cube from the period, and to the remainder bring down the next period for a dividend. This shows that we have 7625 solid feet to be addod to the cubical mound already found. 3. We square the root already found, which in reality is 20, for since there is to be another figure annexed to it, the 2 is tens; then multiplying its square 400 v by 3, we write the product on the left of the dividend for a divisor ; and finding it is contained in the dividend 5 times, we place the 5 in the root. 4. We next multiply 20, the root already found, by 5, the last figure placed in the root ; then multiply this product by 3 and place it under the divisor. We also place the square of 5, the last figure placed in the root, under the divisor, and adding these three results together, multiply their sum 1525 by 5, and subtract the product from the dividend. The answer is 25. PROOF. (25) 3 =25x25x25=:"l5625. (Art. 360.) DEMONSTRATION BY CUBICAL BLOCKS. 361. The simplest method of illustrating the process of ex- tracting the cube root to those unacquainted with algebra and geom- etry, is by means of cubical blocks.* 1. Dividing the number into periods of three figures, shows how * A set of these blocks contains 1st, a cube, the side of which is usually about l| in. square ; 2d, three side pieces about a in. thick, the upper and lower bas of which is just the size of a side of the cube ; 3d, three corner pieces, whose ends are in. squai-e, and whose length is the same as that of the side pieces : 4th, a small cube, the side of which is equal to the end of the corner pieces. It Is desirable for every teacher and pupil to have a set. If not conveniently pro* fured at the jthops, any one can easil v make them fin- himself. 326 CUBE HOOT. [&x XIV. many figures the root will contain, and also enables us to find-. part ot it at a time. Now, placing the large cube upon a table or s?rm;i. -let it represent the greatest cube in the left hand period, whichHrMl example above is 8, the root of which is 2. We subtract this cul from the left hand period, and to the remainder biing down the next period, in order to find how many feet remain to be added. In mak- ing this addition, it is plain the cube must be equally increased on three sides ; otherwise its sides will become unequal, and it will then cease to be a cube. (Art. 154. Obs. 2.) 2. The object of squaring the root already found is to find the area of one side of this cube ; (Art. 163 ;) we then multiply its square by 3, because the additions are to be made to three of its sides ; and, divid- ing the dividend by this product shows the thickness of these addi- tions. Now placing one of the side pieces on the top. and the other two on two adjacent sides of the cube, they will represent these additions. 3. But we perceive there is a vacancy at three corners, each of which is of the same length as the root already found, or the side of the cube, viz: 20 ft., and the breadth and thickness of each is 5 ft., the thickness of the side additions. Placing the corner pieces in these vacancies, they will represent the additions necessary to fill them. The object of multiplying the root already found by the figure last placed in it, is to obtain the area of a side of one of these additions ; we then multiply this area by 3, to find the area of a side of each of them. 4. We find also another vacancy at one corner, whose length, breadth, and thickness are each 5 ft., the same as the thickness of the side additions. This vacancy therefore is cubical. It is represented by the small cube, which being placed in it, will render the mound an exact cube again. The object of squaring 5, the figure last placed in the root, is to find the area of a side of this cubical vacancy. We now have the area of one side of each of the side additions, the area of one side of each of the corner additions, and the area of one side of the cubical vacancy, the sum of which is 1525. W r e \iext multiply the sum of these areas by the figure last placed in the root, in order to find the cubical con- tents of the several additions. (Art. 164.) These areas are added together, and their sum multiplied by the last figure placed in the root, for the sake of finding the solidity of all the additions at once. The result would obviously be the same, if we multiplied them separately, and then subtracted" the sum of their products from the dividend. 3622. From the preceding illustrations we derive the following general QUEST. 362. What is the first step in extracting the cube root ? The second ? Third ? Fourth ? Fifth ? How is the cube root proved ? Dem. Why separate the given number into periods of three figures <>aeh ? Why subtract the greatest cube from the left hand period ? Why *quare tfie root already found ? Why multiply its square by 3 ? Why di vide the dividend by this product? Why multiply the root already fnnnfl \\v tlif loct CifrTnrp ntapprl in it? "VVhv mllltinlir thimr?il^t Kv 3 ? found by the last figure placed in it ? Why multiply this product by 3* Why square the figure last, placed in the root ? Why multiply the euro of these areas, l>y the last figure placed in the root ? ART. - CUBE ROOT. 327 ,E FOR EXTRACTING THE CUBE ROOT. I. Separate the give,*, number into periods of three figures , placing a point over units, then over every third fig- ure towards the left in ichole numbers, and over every third figure towards the right in decimals. II. Find the greatest cube in t/te first period on the lejt hand ; then placing its root on the right of the number for the first figure of the root, subtract its cube from the period, and to the remainder bring down the next period for a dividend. III. Square the root already found, regarding its local value ; multiply this square by 3, and place the product on the left of the dividend for a divisor ; find how many times it is contained in the dividend, and place the result in the root. IV. Multiply the root previously found, regarding its local value, by this last figure placed in it, then multiply this product by 3, and write the result on the left of the divi- dend under the divisor ; under this result write also tht square of the last figure placed in the root. V. Finally, add these results to the divisor ; multiply the sum by the last figure placed in the root, and subtract the product from the dividend. To the right of the remainder bring down the next period for a new dividend ; find a new divisor, and proceed with the operation as above. PROOF. Multiply the root into itself twice, and if the last product is equal to the given number, the work is right. OBS. 1. When there is a remainder, periods of ciphers may be added, and the operation continued as in square root. 2. If the right hand period of decimals is deficient, this deficiency must be supplied by ciphers. 3. When there are decimals in the given example, find the root as in whole numbers ; then point off as many decimal figures in the answer, as there are periods of decimals in the given number. 4. The cube root of a common fraction is found by extracting the root of its numerator and denominator. A mixed number should be reduced to an improper fraction. 5. W 7 hen there are more than two periods in the given example, it is sufficient to annex mie cipher to the root previously found, be- fore squaring it for the divisor. 3. What is the cube root of 1728 ? 328 EQUATION OF [J$$C 4. What is the cube root of 13824 ? 5. If a box in the form of a cube, solid inches, what is the length of one side ? 6. What is the side of a cubical vat, which 57 1787 solid feet? 7. What is the side of a cubical mound which contains 1953125 solid yards? 8. What is the cube root of 2 ? 9. What is the cube root of 2357947691 ? 10. What is the cube root of 12.167? 1 1. What is the cube root of 91.125 ? 12. What is the cube root of %\ 1 13. What is the cube root of SECTION XV. EQUATION OF PAYMENTS. ART. 363* EQUATION OF PAYMENTS is the process of finding the equalized or average time when two or more payments due at different times, may be made at once, without loss to either party. OBS. The equalized or average time for the payment of several debts, due at different times, is often called the mean, time. 364. From principles already explained, it is mani- fest, when the rate is fixed, the interest depends both upon the principal and the time. (Art. 241.) Thus, if a given principal produces a certain interest in a given time, Double that principal will produce tioice that interest ; HoZf that principal will produce half that interest ; &c, In double that time the same principal will produce twice that interest ; In half that time the same principal will produce haJ that interest ; &c. QUEST. 363. What is Equation of Payments ? Obs. What is the average time for the payment of several debts sometimes called ! 364. When the rate is fixed, upon what does the interest depend I cks66.] PAYMENTS. 829 3 Hence, it is evident that any given principal viTl produce the same interest in any given time, as ^s_jrf/ One half that prin. will produce in double that time ; One third that prin. will " " thrice that time ; Twice that principal will " " half that time ; Thrice that principal will " " a third of that time, &c. For example, at any given per cent., The int. of $2 for 1 year, is the same as the int. of SI for 2 years; The int. of $3 for 1 year, "5.] CURRENCIES. 335 The process of changing money expressed in of one country to its equivalent value the denominations of another country, is called Ex- 'change of Cwre?icies. Ex. 1. Change 20 sterling to Federal money. Suggestion. Since 1 is worth $4.84, 20 are worth 20 times as much ; and $4.84x20=$96.80. Ans. 2. Change 5, 13s. 6d. to Federal money. Operation. Reduce 13s. 6d, 5, 13s. f3d.=5.675. (Art. 200.) to the decimal of a Value of 18 4.84 pound, and multiply Ans. 827.467. (Art. 215.) the sum b Y $ 4 - 84 - 375. Hence to reduce Sterling to Federal money, Set down the pounds as whole, numbers, and reduce ike given shillings, pence, and farthings to the decimal of a pound; then multiply the whole sum by $4.84, (the value of 1,) point off the product as in multiplication of decimals^ mid it will be the answer required. OBS. 1. Guineas, Francs. Doubloons, and all foreign coins, maybe reduced to Federal currency, by multiplying the given number by the value of one expressed in Federal money. 2. The rule usually given for reducing Sterling to Federal money, is to reduce the shillings, pence, and farthings to the decimal of a pound, and placing it on the right of the given pounds, divide tho whole sum by -fo. This rule is based on the law of 1798, which fixed the value of a pound at $4.44 y-, and that of a dollar at 4s. 6d. But $4.44i is 9 per cent, of itself, or 40 cents, less than $4.84, which is the present legal value of a pound ; consequently, the result or an- swer obtained by it, must be 9 per cent, too small. A dollar is now qual to 49.6d. very nearly, instead of 54d. as formerly. 3. What is the value of 100 in Federal money? 4. What is the value of 275, 15s. in Federal money t 5. Change 450 7 7s. 6d. to Federal money. 6. Change $27.467 to Sterling money. Solution. Since there is 1 in $4.84, in $27.467 there QUKST. 374. What is meant by exchange of currencies ! 375. How is Sterling money reduced to Federal ? Obs. How may any foreign e&tas be reduced to Federal money ? 836 EXCHANGE OF are as many pounds, as $4.84 is contained and $27.467-*-4.84=5.675 ; that is, 5.675. the decimal .675 to shillings and pence, (Art. 201,) have 5, 13s. 6d. for the answer. Hence, 376* To reduce Federal to Sterling money. Divide the given sum by $4.84, (the value of 1,) and point off" the quotient as in division of decimals. The figures on the left hand of the decimal point will be pounds ; those on the right, decimals of a pound, which must be reduced tc $hilli?igS) pence^ and farthings. (Art. 201.) 7. Change $486.42 to Sterling money. 8. Change $1452 to Sterling money. 376. a. In buying and selling Bills of Exchange on England, the premium or discount is commonly reckoned at a certain per cent, on the nominal value of a Pound Sterling, which is $4.44^ (Art. 373. Obs.) 9. What is the worth of a bill of exchange of 100 on London, at 9 per cent, premium 1 Solution. 100x$4.44t=$444.44f, the nominal value, Then, $444.44ix.09=$40.00, the premium. And $444.44-f $40-$484.44. Ans. 10. What is the value of 1325, 10s., at 8-J- per cent, premium. 37 7 Previous to the adoption of Federal money in 1786, accounts in the United States Avere kept in pounds, shillings, pence, and farthings. OBS. At the time Federal money was adopted, the colonial currency, or bills of credit issued by the colonies, had more or less depreciated in value : that is, a colonial pound was worth less than a pound Ster- ling; a colonial shilling, than a shilling Sterling, &c. This deprecia- tion being greater in some of the colonies than in others, gave rise ta the different State currencies. Thus, In New England currency, Va., Ky., and Tenn, 6s. or j^ $1. In New York currency, North Carolina, and Ohio, 8s. or -f-=:$l, In Penn. cur., New Jer., Del., and Md., 7s. 6d. (7lb.) or -f =$1, In Georgia cur., and South Carolina, 4s. 8d. (4fs.) or -fa=:'$\< In Canada currency, and Nova Scotia, 5s. or i=$L QUEST. 376. How is Federal money reduced to Sterling ? 377 jpwvkwa to th adoption of Federal money, in what w*a accounts kept/ ART./S70-379.] CURRENCIES. 837 ^ deduce $45 to New England currency. lution. Since there are 6s. in $1, in $45 there are times 6s. And 6s.x45=270s. Now 270s.-s-20=-13, 10s. Ans. Hence, 378. To reduce Federal money to either of the State currencies. Multiply the given sum by the number of shillings which, in the required currency, make $1, and the product will be the answer in shillings, and decimals of a shilling. The shillings should be reduced to pounds, arid the decimals to pence and farthings. (Art. 201.) 12. Reduce $378 to New England currency. 13. Reduce $465.45 to New York Cunency. 14. Reduce $640 to Pennsylvania currency. 15. Reduce $1000 to Canada currency. 16. Reduce 15, 7s. 6d., N. E. cur. to Federal money. Solution. 15, 7s.6d.=307.5s.(Art. 200.) Now since bs. make $1. 307.5s. will make as many dollars, as 6 is contained times in 307.5. And 307.5-^6^851.25. Ans. Hence, 379. To reduce either of the State currencies to Federal money. Reduce the pounds to shillings, and the given pence and farthings to the decimal of a shilling ; then divide the sum by the number of shillings which, in the given currency, make 81, and the quotient will be the answer in dollars and cents. 17. Reduce 48, 15s., N. E. cur., to Federal Money. 18. Reduce 73, 4s., N. E. cur., to Federal Money. 19. Reduce 100, 18s., N. Y. cur., to Federal Money. 20. Reduce 256, 5s., N. Y. cur., to Federal Money. 21. Reduce 296, 12s., Pcnn. cur., to Federal Money. 22. Reduce 430, 8s., Penn. cur., to Federal Money. 23. Reduce 568, 10s., Ga. cur., to Federal Money." 24. Reduce 1000, 15s., Canada cur., to Federal Money. QUEST. 378. How is Federal Money reduced to the State currencies t 379. How are the Krvorol St*t *im0Ticio reduced to Federal Money ? 338 MENSURATION. fqfet XVI. SECTION XVI. MENSURATION". ART. 3 SO* MENSURATION is the art of measuring magnitudes. OBS. The term magnitude, denotes that which has one or more of the three dimensions, length, breadth, and thickness. 381. In measuring surfaces, it is customary to as- sume a square as the measuring unit, as a square inch, a square foot, a square rod, &c. ; that is, a square whose side is a linear unit of the same name. (Thomson's Legendre, IV. 4. Sch. Art. 153. Obs. 1.) Note. For the demonstration of the following principles, seo references. 382. To find the area of a parallelogram, and a square. (Art. 163. Obs.) Multiply the length by the breadth. (Leg. IV. 5.) OBS. When the area and one side of a rectangle are given, the other side is found by dividing the area by the given side. (Art. 291. Note.) 1. How many acres are there in a field 120 rods long, and 90 rods wide 1 Ans. 07$ acres. 2. How many acres in a field 800 rods long, and 128 rods wide 7 3. Find the area of a square field whose sides are 65 rods in length. 4. A man fenced off a rectangular field containing 3750 sq. rods. the length of which was 75 rods: what was its breadth 1 5. One side of a rectangular field is 1 mile in length, and the field contains 1GO acres: what is the length of the other side 1 383. To find the area of a rhombus. (Leg. I. Def. 18.) Multiply tlie length by the altitude. (Leg. IV. 5.) Note. The term altitude, denotes perpendicular height. 6. The length of a rhombus is 17 ft., and its perpendicular height 12 ft. : what is its area 1 An& 204 sq. ft, 7. What is the area of a rhombus whose altitude is 25 rods, aivl its length 28.6 rods'? 384. To find the area of a trapezium. (Leg. IV. 7.) Multiply half the sum of the parallel sides by the altitude. 8. The parallel sides of a trapezium are 15 ft. and 21 ft., and it altitude 12 ft. : what is its area 1 Ans. 216 ft. 9. Find the area of a trapezium whose parallel sides are 25 rods and 37 rods, and its altitude 18 rods. ARTS. j3^j$-3S9.] MENSURATION. 839 f. To find the area of a triangle. (Leg. IV. 6.) the base by half the altitude. s. 1. The base of a triangle is found by dividing the area by iialf the altitude. 2. The altitude of a triangle is found by dividing the area by half the base. 10. What is the area of a triangle whose base is 45 ft., and its altitude 20 ft.? Ans. 450 sq. ft. 11. What is the area of a triangle whose base is 156 ft., and its altitude 63 ft. 1 386. To find the area of a triangle, the three sides being given. From half the sum of the three sides subtract each side respectively; then multiply together half the sum and the three remainders, and extract the square root of the product. 12. What is the area of a triangle whose sides are 10 ft., 12 ft., and 16 ft. ? Ans. 59.92-f-ft. 13. What is the area of a triangle whose sides are each 12 yds. ? 387. To find the circumference of a circle, when the diameter is given. (Leg. V. 11. Sch.) Multiply the given diameter by 3.14159. Note. The circumference of a circle is a curve line, all the points of which are equally distant from a point within, called the centre. The diameter of a circle is a straight line which passes through the centre, and is terminated on both sides by the circumference. The radius or semi-diameter is a straight line drawn from the centre to the circumference. 14. What is the circumference of a circle whose diameter is 15 ft. 1 Ans. 47.123H5 ft. 15. What is the circumference of a circle whose diameter is lOOrods? 388. To find the diameter of a circle, when the circumference is given. Divide the given circumference by 3.14159. OBS. The diameter of a circle may also be found by dividing the area by .7vS54, and extracting the square root of the quotient. 10. What is the diameter of a circle whose circumference is 34.2477 ft. 1 Ans. 30 ft. 17. What is the diameter of a circle whose circumference is 628.318 yards 1 389. To find the area of a circle. (Leg. V. 11.) Multiply half the circumference ly half the diameter or, multiply the circumference by a fourth of the diameter. 840 MENSURATION. [ Note. The area of a circle may also be found by multip square of its diameter by the decimal .7854. 18. What is the area of a circle whose diameter is 100 ft. 7 Ans. 7854 sq. ft. 19. What is the area of a circle whose diameter is 120 rods ? 20. How many square yards in a circle whose circumference is 160 yards'? 21. Required the diameter of a circle containing 50.2656 sq. rods. 22. Required the diameter of a circle containing 201.0624 sq. ft. 39O The side of a square equal in area to any given surface, is found by extracting the square root of the given surface. (Arts. 350, 339. Obs. 2.) OBS. When it is required to find the dimensions of a rectangular field, equal in area to a given surface, and whose length is double, triple, or quadruple, &c., of its breadth, the square root of , , -J, of the given surface, will be the width ; and this being doubled, tripled, or quadrupled, as the case may be, will be the kngth. 23. What is the side of a square, whose area is equal to that of a circle which contains 225 sq. yds. 1 Ans. 15 yds. 24. What is the side of a square, whose area is equal to that of a triangle containing 576 sq. ft, 1 25. The length of a rectangular field containing 80 acres, is twice its breadth : what are its length and breadth 1 39 ! A. mean proportional between two numbers is found by multiplying the given numbers together, and ex- tracting the square root of the product. (Art. 320. Obs. 1.) 26. What is the mean proportional between 9 and 167 27. What is the mean proportional between 49 and 144 ? 28. What is the mean proportional between * and *- ? 392* In measuring solids, it is customary to assume a cube as the measuring unit, whose sides are squares of the same name. Thus, the sides of a cubic inch, are square inches ; of a cubic foot, are square feet, &c. (Art. 154. Obs. 2.) OBS. To find the capacity, solidity, or cubical contents of a body, is to find the number of cubic inches, feet, &c., contained in the body 393* To find the solidity of bodies whose sides are perpendicular to each other. (Art. 164. Leg. VII. 11. Sch.) Multiply the length, breadth, and thickness together. OBS. When the contents of a solid body and two of its sides are given, the other side is found by dividing the contents by the product of the two given sides. (Art. 294.) A.RTS. 3t)0-397.] MENSURATION. 341 2y^ 1 How many cubic feet are there in a stick of timber 60 ft. long, wide, and "2 ft. thick 1 Ans. 400 cu. ft. rfSOTHow many cubic feet in a wall 100 ft. long, 15 ft. high, and Pf ft. thick? 31. A gentleman wishes to construct a cubical bin, which shall contain 19683 solid feet : what must be the length of its side 1 32. If a stick of timber containing 400 cu. ft., is 60 ft. long, and 3 ft. thick, what is its width 1 Ans. 2 ft. 394. To find the solidity of a prism. Multiply the area of the base by the height. (Leg. VII. 12.) OBS. 1. This rule is applicable to all prisms, triangular, quad- rangular, pentagonal, &c. ; also to all parallelopipedvns, whether rectangular or oblique. (Leg. VII. Def. 4, 8, 9.} 2. The height of a prism is the perpendicular distance between the planes of the bases. Hence, in a right prism, the height is equal to the length of one of the sides. 33. What is the solidity of a prism whose base is 5 ft. square, and its height 15 ft. 1 Ans. 375 cu. ft. 34. What is the solidity of a triangular prism whose height is 20 ft., and the area of whose base is 460 sq. ft. 1 395. To find the lateral surface of a right prism. Multiply the length by the perimeter of the base. OBS. If we add the areas of both ends to the lateral surface, the sum will be the whole surface of the prism. 35. Required the lateral surface of a triangular prism whose per- imeter is 4 in., and its length 12 in. Ans. 54 sq. in. 36. Required the lateral surface of a quadrangular prism whose sides are each 2 ft., and its length 19 ft. 396. To find the solidity of a pyramid, or cone. (Leg. VII. 18. VIII. 4.) Multiply the area of the base by -^ of the altitude. 37. Required the solidity of a square pyramid, the side of whose base is 25 ft., and whose height is 60 ft. Ans. 12500 cu. ft. 38. Required the solidity of a cone, the diameter of whose base is 30 ft., and whose height is 90 ft. 397. To find the lateral or convex surface of a regu- lar pyramid, or cone. (Leg. VII. 16. VIII. 3.) Multiply the perimeter of the base by % the slant-height. OBS. The slant-height of a regular pyramid, is the distance from the vertex or summit to the middle of one of the sides of the base. 39. What is the lateral surface of a regular triangular pyramid whose slant-height is 10 ft., and whose sides are each 8 ft. 1 Ans. 120*1. ft. 342 MENSURATION. [3E<. XVI. 40. What is the convex surface of a cone, the perimete base is 500 yds., and whose slant-height is 120 }^ds. 1 398. To find the solidity of a frustum of a pyrami or cone. (Leg. VII. 19. Sch., VIII. 6.) To Ike sum of the areas of the two ends, add the square root of the product of these areas j then multiply this sum fy\f the perpendicular height. 41. The areas .of the two ends of a frustum of a cone are 9 sq. ft., and 4 sq. ft., and its height is 15 ft. : what is its solidity 1 Ans. 95 cu. ft. 42. The two ends of a frustum of a pyramid are 4 ft. and 3 ft. square, and its height is 10 ft. : what is its solidity 1 399* The convex surface of afrustu?n of a pyramid, or cone, is found by multiplying half tlie sum of the circum- ferences of the two ends by the slant-height. (Leg. VII. 17.) 43. The circumferences of the two ends of a frustum of a pyramid are 12 ft. and 8 ft., and its slant-height 7 ft. : what is its convex surface 1 Ans. 70 sq. ft. 44. The circumferences of the two ends of a frustum of a cone are 15 yds. and 9yds., and its slant-height, 7 yds.: what is its convex surfaced 400. To find the solidity of a cylinder. (Leg. VIII. 2.) Multiply tfbe area of the base by the height or length. 45. Required the solidity of a cylinder 6 ft. in diameter, and 20 ft. high. Ans. 5G5.488cu.il. 40. Required the solidity of a cylinder 30 ft. in diameter, and 65 ft. long. 401. To find the convex surface of a cylinder. Multiply the circumference of the base by the height. 47. What is the convex surface of a cylinder 16 inches in circum ference and 40 in. long 1 Ans. 610 sq. in. 48. What is the convex surface of a cylinder, the diameter of whose base is 20 ft., and whose height is 65 ft 1 To find the surface of a sphere or globe. Multiply the circumference by the diameter. (Leg. VIII. 9.) 49. Required the surface of a globe 13 inches in diameter. Ans. 531 sq. in. nearly. 50. Required the surface of the earth, allowing its diameter to b' BOOO miles. ARTS, 98-400.] MENSURATION. 843 To find the solidity of a sphere or globe. the surface by \ of t/ie diameter. 51. What is the solidity of a globe 12 in. in diameter! 52. What is the solidity of the earth, reckoning its diameter at 8000 miles 1 4O4 The solid contents of similar bodies are to each other, as the cubes of their homologous sides, or like di- mensions. (Leg. VII. 20. VIII. 11. Cor.) 53. If a ball 4 inches in diameter weighs 32 Ibs., what is the weight of a ball whose diameter is 5 inches ? Solution.!* : 53 : : 32 Ibs. : to the weight. Ans. 62.5 Ibs. 54. If a ball 3 inches in diameter weighs 4 Ibs., what is the diam- eter of a ball which weighs 32 Ibs. 1 4O5 To find the side of a cube whose solidity shall be double, triple, &c., that of a cube whose side is given. Cube the given side, multiply it by the given proportion, and the cube root of the product will be the side of the cube required. 55. What is the side of a cubical mound, which contains 8 times as many solid feet as one whose side is 3 ft. Ans. 6 ft. 56. Required the side of a cubical vat, which contains 16 times as many solid feet as one whose side is 5 ft. GAUGING OF CASKS. 4O6 To find the contents or capacity of casks. Multiply the square of the mean diameter into the length itn, inches ; then this -product multiplied into .0034 will be the wine gallons required, or multiplied into .0028 will be the beer gallons. OBS. The mean diameter of a cask is found by adding to the heed diameter .7 of the difference between the head and bung diameters when the staves are very much curved ; or by adding .5 when very liUle curved ; and by adding .65 when they are of a medium curve. 57. How many wine gallons does a cask contain whose length is 35 inches, its bung diameter 30 in., and its head diameter 26 in., it being but little curved 1 Ans. 93.296 gals. 58. How many beer gallons in a cask 54 in. long, whose bung diameter is 42 in., and head diameter 36 in., its staves being much curved 1 344 MISCELLANEOUS EXAMPLES. MISCELLANEOUS EXAMPLES. Ex. 1. How much will 500 sheep cost, at $2 apiece 1 2. How much can a man earn in 240 days, at 3?i cts. per day 1 3. What will 690 bushels of apples cost, at 18| cts. per bushel 4. What cost 476 cows, at $12 apiece 1 5. What cost 685 gallons of oil, at 87 cts. per gal. 1 G. What cost 325^ acres of land, at $10} per acre 1 7. How much flour, at $4 per bbl., can be bought for $5257 8. How many yards of cloth, at $5-^- per yard, can be bought for $1230 7 Ans. 240 yds. 9. How many saddles, at $1H, can be bought for $5025? 10. How many horses, at $75f , can be bought for $3780 ? 11. A man bought -| of a ship, and sold -^ of it : how much had he left 1 Ans. -fr. 12. A broker negotiated a bill of exchange of $10360, at 1-| per cent. : what was his commission 1 13. What is the interest of $2345 for 1 year and 6 months, at G per cent. 1 14. What is the int. of $1356.25 for 90 days, at 6 per ct. 7 15. What is the int. of $533.11 for 6 months, at 7 per ct. 1 16. What is the amount of $925 for 1 yr. and 4 mo., at 8 perct.7 17. What is the amount of $4635 for 30 days, at 7 per ct. 1 18. What is the amount of $10360 for 60 days, at 5 per ct. 1 19. What is the present worth of $1365, payable in 6 months, when money is worth 7 per cent, per annum 1 20. At 6 per ct. discount, what is the present worth of $1623.28, due in 1 year 1 21. What is the bank discount on a note of $730, payable in 4 months, at 6 per ct. 1 Ans. C^'16.212. 22. What is the bank discount on a note of $1575, payable in GO days, at 7 per ct. 1 23 What will 35 shares of Railroad stock cost, at 10 per ct advance 7 Ans. $3867.50. 24. What cost 63 shares of bank stock, at 3J per ct. discount] 25. What premium must a man pay annually for insuring $8500 on his store and goods, at 1^ per ct. 1 26. If I obtain insurance on goods, worth $16265, at 2 per ct., and the goods are lost, how much shall I lose 1 27. What is the insurance on $925.68, at 1 % per ct. 1 28. What is the insurance on $63460, at -f per ct. 1 29. What is the insurance on $48256, at 1| per ct. 1 30. A man bought a farm for $5640, and afterwards sold it for 1,1 per ct. more than it cost: how much did he make by his bargain 1 31. A merchant bought a stock of goods for $4390, and retailed them at a profit of 22J^ per ct. : how much did he make 1 MISCELLANEOUS EXAMPLES. 345 32. An oil merchant bought 15000 gallons of oil for $8500, and sold it at 15 per ct. advance: how did he sell it per gal.? ""Saf I buy 1675 yards of flannel for $368.50, how must I retail r yard to gain 25 per ct. 7 Ans. 27 cts. 4. 'A grocer bought 2500 Ibs. of coffee for $250, and sold it at b per ct. loss : what did he get per pound 7 35. A merchant bought 1824 yds. of cloth, at $2.50 per yd., and retailed it at $3 per yd. : what per ct. was his profit, and how much did he make 7 36. A shop-keeper bought 100 pieces of lace, for $250, and sold them for $375 : what per ct. did he make 7 37. If a grocer buys 3680 Ibs of cheese, at 4 cts. per lb., and sells it at 6^ cents, what per ct. is his profit 7 38. What is the ad valorem duty, at 33^ per ct., on a quantity of cloths which cost $104367 39. W 7 hat is the ad valorem duty, at 15 per ct., on a cargo of tea invoiced at $35856 7 40. At 37^ per ct., what is the duty on a quantity of silks which cost $23265 7 41. The sum of two numbers is 856, and their difference is 75: what are the numbers 7 42. The sum of two numbers is 5643, and their difference is 125: what are the numbers 7 43. The difference of two numbers is 63, and the smaller number is 365 : what is the greater number 7 44. The product of two numbers is 3750, and one of the numbers is 75 : what is the other 7 45. What number is that -f- of which is 265 7 Ans. 477. 46. What number is that -f of -f of which is 120 7 47. How long will it take a person to count a billion, if he counts 50 a minute, and works 6 hours per day, for 5 days a week, and 52 weeks a year 7 48. How many dollars, each weighing 412^ grains, can be made from 7 Ibs. 1 oz. 18 pwt. I8grs. of silver 7 49. How many pound^pf silk will it take to spin a thread which will reach round the earth, allowing its circumference to be 25000 miles, and 2^ oz. to make 160 rods of thread? 50. How many times will the hind wheel of a carriage, 7 ft. 6 in. in circumference, turn round in 7 miles, 1 furlong, 30 rods? 51. How many times will the fore wheel of a carriage, 5 ft. 7 in. in circumference, turn round in the same distance 7 52. What cost 645 bushels of salt, at 4s. N. Y. currency per bu. 7 53. What cost 744 yards of muslin, at Is. 4d. N. Y. cur. per yd. 7 54. What cost 241 melons, at 2s. 8d. N. Y. cur. apiece 7 55. What cost 1536 yards of calico, at Is. N. E. cur. per yd. ? 56. What cost 873 baskets of peaches, at 3s. N. E. cur. a basket? 57. What cost 632 bushels of oats, at Is. 6d. N. E.cur. a bushel? 58. What cost 848 lambs, at 5s. sterling apiece ? 59. What cost 258 yards of cloth, at 15s. sterling per yard? 346 ' MISCELLANEOUS EXAMPLES. 60. What cost 912 bushels of rye, at 2s. Gd. sterling r _ _ 61. What cost 657 yards of silk, at (is. 8d. ster. per yard? 62. What co^t 735 bushels of apples, at Is. 8d. ster per busTTefCt^ 63. What cost 3 pieces of cloth, each containing 27 yards, oHj 3s. 4d. per yard? Aits. 13, 10s. * 64. What cost 248 pair of boots, at 12s. 6d. sterling a pair? 65. If 156 Ibs. of butter cost $15.60, what will 730 Ibs. cost? 66. If 48 yards of cloth cost $480, what will 125 yards cost ? 67. If 96 horses eat 192 tons of hay in a winter, how many ton* will ISOJiorses eat? 6*8. If 10 Ibs. of sugar cost 9fs., what will 240 Ibs. cost? 69. If 25 Ibs. of veal cost $|, how much will 872 Ibs. cost? 70 If 50 Ibs. of ginger cost $7f, how much will 460 Ibs. cost? 71. What cost 260 cords of wood, if 45 cords cost $87f ? 72. A man sold a sheep for l^, and a pig for -|s. -|d. : what did he get for both ? 73. A goldsmith melted up $ Ib. 10^ pwts. of gold, at one time, and 3 oz. lOgrs. at another: how much did he melt in all ? 74. A man having 2$ oz. of silver, sold 6| pwts. : how much had he left? 75. A man owing | , 2-^-s., paid 7-J-s. 2^-d. : how much does ha still owe ? 76. If 50 Ibs. of rice cost *-, what will 840 Ibs. cost ? 77. If 13 yards of edging cost $^-9., what will 200 yds. cost ? 78. If -f- of a ton of" iron cost $35, what will 381 tons cost? 79. If I owe a man 6950, and can pay him but 13s. 4d. on a pound, how much will he receive for his debt ? 80. If 385 yards of linen cost 63, how much can be bought for 18? 81. How much brondy can be bought for 396, if 90 gallons cost 18? 82. If 15^ yards silk cost $18|, what will 56| yards cost ? 83. A grocer used a false weight of 13 oz. for a pound: what was the amount of his fraud in weighing 500 pounds ? 84. If f- of a barrel of apples costs $4 , how much will - of a bar- rel cost ? Ans. $2.45. 85. If -jZg of a pound of lard costs -[* of a shilling, how much will |-2- af a pound cost ? 86. If -fa of a ton of hay costs -, what will -J-g- of a ton cost ? 87. How much will -fa of a drum of figs come to, at the rate of f- of a dollar for of a drum ? 88. Bought 48 Ibs. of tea for $27| : how much can be bought for SI 25? 89. Paid $35^- for - of an acre of land : how much can be bought for $7500? MISCELLANEOUS EXAMPLES. 347 50. i&Svit yards of camlet make 3 cloaks, how many cloaks can of 7:>74 yards 1 A/is. 75 cloaks. If 57.35 acres of land produce 430.16 bushels of barley, how ny bushels will 172.05 acres produce 7 92. What will 730f yards of cloth cost, if you pay $112 for yards '] 93. If a cane 3 feet in length cast a shadow 5 feet long, how high Is a steeple whose shadow is 175 feet 7 91. Bought a hogshead of molasses for 4 firkins of butter, each containing 06 Ibs., which was worth 10 cents a pound : what did the molasses cost per gallon ] 95. Bought 15 yds. of silk at 7s. per yard, and 12 yds. of muslin at 3s. per yard, and paid the bill in cheese at 9d. per pound : how many pounds did it take to pay the bill ? 96. If a cubic foot of pure water weighs 1000 oz., what will a pail of water weigh which contains 217 cubic inches'? 97. If I pay $8400 for | of a ship, what must I pay for the whole ship 1 98. A farmer sold 174 sheep, which was -|- of all he had; the remainder he divided equally between his two sons : how many did each receive 1 99. A garrison having been besieged 108 days, found that -f of the provisions were consumed : how much longer would they last 1 100. A garrison of 1520 men have 416955 Ibs. of flour: how long will it last them, allowing each man -f- Ib. per day 1 101. How long will 75240 gals, of water last a ship's company of 30 men, allowing each man -^ gal. per day 1 102. If 10 men can dig a cellar in 30 days, how long will it take 25 men to dig it 1 103. If 6 men spend $48 in 7 weeks, how much will 24 men spend in 35 weeks 1 Ans. $960. 104. If 15 horses consume 70 bushels of oats in 27 days, how many bushels will 45 horses consume in 54 days'? 105. If 6 men can build a wall 30 feet long, G feet high, and 3 feet thick, in 15 days, when the days are 12 hours long, how many days will it take 30 men to build a wall 300 feet long, 8 feet high, and 6 feet thick, working 8 hours a day 1 106. A merchant in New York wished to pay 1500 in London: what will a bill of exchange cost him at 9 per ct. premium 1 107. A broker in Boston sold a bill of exchange on Liverpool for 2500, 15s., at 9 per ct. premium: what did he get for it 1 ? 108. What will a bill on England for 3125, 12s. 6d. cost, when exchange is 10 per ct. above par 1 109. A man wishing to remit $2550 to Ireland, bought a draft on London, at 12 per ct. advance : what was the amount of his bill in sterling money 1 110. A farmer wishes to form a square field, which shall contain 1296 squ've rods : what is the length of its side? 348 MISCELLANEOUS EXAMPLES. 111. A man owns a farm which contains 160 acres, and is in the form of a square : what is the length of its side 1 112. What is the length of the side of a square field containing 10 acres ? , f 113. What is the area of a triangle whose hypothenuss is ife yards, and its perpendicular 30 yards '? 114. What is the area of a triangle whose hypothenuse is 100 tods, and its base 60 rods 7 1 15. Required the mean proportional between 49 and 81. 116. Required the mean proportional between 121 and 5.76. 1 17. What is the mean proportional between and -J-f 1 1 18. Required the mean proportional between -f|- and T 8 ^. 119. A regiment containing 6912 soldiers, was so arranged that the number in rank was triple that in file : how many were there in each 1 120. If a board is 8 in. wide, how long must it be to make a sq. ft 1 121. How much silk f yd. wide will it take to make a sq. yd. 1 122. How much cambric yd. wide will it take to line 9 yds. of balzorine 1 yd, wide? 123. How many yds. of unbleached muslin f yd. wide will it take to line 36 yds. of carpeting \\ yds. wide 1 124. If it takes 10 yds. of broadcloth 1 yds. wide to make a cloak, how many yards of camlet f yd. wide will make one? 125. How much will it cost to carpet a parlor 18 ft. square with carpeting f yd. wide, which is worth $1.50 per yard 1 126. A, B, and C, joined in a speculation; A put in $500, B $700, and C put in the balance : they gained $1200, of which C received $480 for his share : how much did A and B receive, and how much did C put in 1 127. A, B, and C, gain $3600, of which A receives $6, as often as B receives $10, and C $14: what was the share of each? 128. The hour and minute hand of a clock are exactly together &t noon : when will they next be together ? 129. A farmer having lost ^ of his sheep, and sold ^ of them, had 500 left: how many had he at first ? 130. If - of a post stands in the mud, \ in the .vater, and 10 feet above the water, what is the length of the post 1 131. Two persons start from the same place, one goes south 4 miles per hour, the other west 5 miles per hour : how far apart are they in 9 hours ? 132. A messenger traveling 8 miles an hour, was sent to Mexico with dispatches for the army ; after he had gone 51 miles, another was sent with countermanding orders, who could go 19 miles at quick as the former could go 16: how long will it take the latter to overtake the former ; and how far must he travel 1 ANSWERS TO EXAMPLES. NOTE. At the urgent request of several distinguished Teach- ers, who have received Thomson's Practical Arithmetic with favor, the publishers have issued an edition of it, containing the answers in the end of the book. It is hoped that pupils, who may use this edition, will have sufficient regard to their own improve- ment, never to consult the answer till they have made a strenu- ous and persevering effort to solve the problem themselves. N. B. The work without the answers is published as here- tofore. ADDITION. EXERCISES FOR THE SLATE. ART. 21. Ex. ANS. Ex. ANS. Ex. Am. Ex. ANS. 1, 2. Given. 21. 16840. 15. 582 a. 34. $512. 3. 8786. 22. 220083. 16. $45. Q , } 611 bu. 4. 8689. 23. 100003. 17. 98 cts. d0 ' ? $513. fi. 57757. 24. 134735. 18. $101. 36. $627. 6. 651465. 25. 104022. 19. $2788. 37. 630 Ibs. 7. 8651761. 20. $102. 38. $3789. 8. 998943483 ART. 29. 21. $846. 39. $1125. 9. 988. 1. $64. C 754 sh. 40. $2385 r. 10. 7673. 2. 46 Ibs. 22. I 365 1. $554 g. 11. 88765. 3. 48 yrs. ( 1119 b. 41. $1582. 12. 85879944. 4. $313. 23. $6821. 42. $1323. 13-15. Given. 5. $31. 24. $2324. 43. 525 m. 6. 40 s. 25. $4900. 44. $4930. ART. 27. 7. $550. 26. $244. 45. 2234822. 16. 23770. 8. $2480. 27. 113 ts. 46. 4604345. 17. 161524. 9. $190. 28. 476 m. 47. 5067843. 18. 131570. 10. $278. 29. 73 yrs. 48. 4984097. 19. 1999990. 11. $58. 30. $1648. 49. 178346. 12. 33 sch. 31. $34950. 50. 17069453, ART. 2. 13. $136. 32. $33700. 30. 1913. 14. 64m. 33. $3147. 350 A N S W E R S . 5., 34-54, SUBTRACTION. Ex. ANS. Ex. ANS. Ex. ANS. Ex. An. ART. 34. 20. 6121. 13. $1291 33. $5250. 1, 2. Given. 21. 2754087. 14. 53 m. 34. $323. 3. $232. 22. 932417. 15. 93 m. 35. 1933 a. 4. 413. 23. 6834501. jg_ 36. 565 men. 5. 353. 24. 8960895. 17. 1706. 37. $773. 6. 418. 25. 31090814 18. 67 yrs. 38. $18053. 7. 3332. 8. 3231. ART. 4O. 19. 20. $72320. 39. $154. 40. $5491. 9. 32352. 1. 13 yds. 21. 427721. 41. $6749. 10. 613134. 2. $221. 22. 214412. 42. $1695. 11. 531141. 3. 189 g. 23. 1056109. 43. $2752. 12. 3151721. 4. 1003 bu. 24. 194099. 44. $1913. 13-15. Given. 5. ,$3791. 25. 11763528. 45. $332. ART. 38. 16. 54182. 17. 124907. 18. 66104149. 6. $1420. 7. $382. 8. $1079. 9. $374 bu. 10. $1989. 26. 100 a. 27. $986. 28. $22. 29. $19. 30. 146 ts. 46. 12520 bu. 47. $1491. 48. $9699. 49. $21422. 50. $8000. ART. 39. 11. $479. 31. $1090. 19. Given. 12. 32. $3838. MULTIPLICATION. ART. 47. 16. 5200 s. ART. 54. 15. $2522. 1 A r^-i \rnri 17. 40030. 16. $2090. 1 "1 VTlVvll* 6QfiO r 18. 608240. 1. $2790. 17. 4935 s. you r * 6 880 m. 19. 76342. 2. $2552. 18. 3071 bu. 7 QflQfi 20. 41479110. 3. $9520. 19. 2944 qts. /. yvjt/o. SOQAQf) 21, 22. Given. 4. 676 s. 20. $22224. OO'ioU* 9*0^05 5. 2511 s. 21. $1482. tlUt/tlUi/. 10. 9036906. nf-rl VATI ART. 53. 6. $13932. 7. $10955. 22. $8991. 23. $10584. VJflVtsil* 8. $3790. 24. $4096. ATTT *^1 23. 2268 s. 9. $153900. 25. 35720 d. /YKr. OJL 24. 3915 bu. 10. $180. 26. 16425 d. 12. $664. 25. 19200 Ibs. 11. $414. 27. 90625 Ibs. 13. 1917 s. 26. $6394. 12. $945. 28. 176175 lb^ 14. $624. 15. $6153. 27-29. Given. |13. $1792. 30. 507166416.114. $1664. 29. 78475 m. 30. $77970. Ams. ANSWERS . $51 JONTRACTIONS IN MULTIPLICATION. ARTS. 55-61. %\. ANS. Ex. ANS. Ex. ANS. 1. Given. 13. 476000. 26. 390677500000. 2. $2295. 14. 534860000. 27. Given. 3. Given. 15. 1204670800000. 28. 11840000. 4. $684. 16. 26900785000000. 29. 373520000. 5. $4950. 17. 890634570000000. 30. 3603200000. 6. 1872 s. 18. 946030506800000. 31. 55447000000. 7. 8610 m. 19. 783120650730000. 32, 33. Given. 8. 25760 bu. 20, 21. Given. 34. 4059360000. 9. 16128 s. 22. 1080 d. 35. 14760000000. 10. $91080. 23. 38400 Ibs. 36. 6204000000. 11. Given. 24. 10940000. 37. 16726^0000000. 12. ?5200 p. 25. 2075994000. 38. 1075635900000. SHORT DIVISION. ARTS. 67-73. 1, 2. Given . 14. 12212. 26. 8111. 39. 71000. 3. 7. 15. 11111. 27. 911. 40. Given. 4. 6. 16. 1243143. 28, 29. Given. 41. $107. 5. 6. 17. Given. 30. 48 Ibs. 42. 2050. 6. 9. 18. 31. 31. 7615. 43. 5070. 7. Given. 19. 61. 32. 6573. 44. 5021. 8. 123 sh. 20. 51. 33. 16334. 45. 80405. 9. 124 a. 21. 312. 34. 3144. 46, 47. Given. 10. 122 tms. 22. 8231. 35. 107 bbls. 48. 151. 11. Given. 23. 711. 36. 6010. 49. 52 y. 12. 321 yds. 24. 7111. 37. 7000. 50. 162|. 13. 21312. 25. 811. 38. 5100. LONG DIVISION. ARTS. 74-76. 1-4. Given. 18. Given. 8. 11 t. 20. 588, & 8 r. 5. 127208. 19. 1080-f. 9. 11 c. 21. 24,&61r. 6. 1342314. 20. 901-Hf. 10. 120 m. 22. 8, & 13 r. 7. 326561. 11. 200 m. 23. 227, & 5 r. 8. 336568. ART. 77. 12. 250 m. 24. 269, & 1 r. 9. 6437612. 1. 24 h. 13. 13^2 mos. 25. 2813, & 9 r. 10. 72225723. 2. 36 yds. 14. 20 hhcls. 26. 34, & 34 r. 11-13. Given. 3. 43 c. 15. 11 m. 27. 173,&25r. 14. 245. 4. 108t 16. 14, & 4 r. 28. 158,&40r. 15. 1326f--. 5. 13 m. 17. 42. 29. 388,&55r. 16. 1212f|-. 6 20 d. 18. 39, & 7 r. 30. 63, & 72 r. 17, 123744- 7. lO^rrn, 19. 72, & 12 r., 352 ANSWERS. 124. CONTRACTIONS IN DIVISION. Ex. ANS. Ex. ANS. Ex. ANS. 'v 4 ARTS. T8-81. 1, 2. Given. 3. 6 p. 4. 35 c. 5. 7-A-. 6. 3f. 7. Given. 8. 10 ; 25 ; 38 d. 9. 65 ; 765 ; $4320. 10. Given. 11. 44, & 360791 r. 12. 8236, & 7180309 r. 13. Given. 14. 8 h. 15. 34 bbls. 16. 210 r. 17. 68-rV- Llu1 -. 21, 22. Given. 23. 76. 24. 75. CANCELLATION. ART. 91. 25,26. Given. 27. 475. 28. 798. 29. 1248. 30, 31. Given. 32. 27. 33. 7. 34. 28. 35. 30. GREATEST COMMON DIVISOR. ARTS. 94-97. 1. Given. 2. 3. 3. 4. 4. 5. 5. 4. 6. 3. 7. Given. 8. 21. 9. 13. 10. 19. 11. 15. 12. 39. 13. Given, 14. 4. 15. 12. LEAST COMMON MULTIPLE. ART. 1O2. 16. Given. 17. 36. 18. 48. 19. 90. 20. 90. 21. 240. 22. 12600. 23. 504. 24. 1134, 25. 144. REDUCTION OF FRACTIONS. ARTS. 120-4. 1, 2. 3. i. 4. I- 5. -f. C. i. 7. f. 8. $ 9. f 10. - 11. - 12. f. 13. u. - Given. i. 16. 18. 21. 22. 23. 24. 25. 26. 27. 28. 29. 4. 5. 2|. 1. 41|. 30. 28^. $28f. 30. 31,32. Given. 33. -^. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45, 46. Given. 47. 48. 49. 50, 51~ Given 52. i-i 53. - 4 \. D '1 ~90" 55. -f. 56. -Jy. 57. -jV- 58. T-fr. I36.J ANSWERS. .REDUCTION OF FRACTIONS CONTINUED. A S5^ V Axs. Ex. ANS. 1-3. Given. 4. -M-; ffi; 5. -iVo- 5 945 . 1440. JULS_jQ.. JjlHi ' T52 0> 2 52 0> 25 2 U> 25 2 7. s. 9. 10. nja_Z-fiJl_ 1837 fi . 1 5J 1262 5 26250) 2626 1 O 2 2.5.0 . 1 1 3 7 B J^' 625000 > 525UOIT- fern. 13-15. Given. IG. if; tt; if. n. if; A; ff. is. fi; ff ; M; 19. A; W; f&; '20. ii; f^; ^; if 21. -i 4 ^-; iVt; T*T. 22. t2 0" > 120 T2D~' 23. fn; 24. *H; 25. YW; ff. ADDITION OF FRACTIONS. ART. 11-13. Given. 14. 2-fr. 15. 1-H-. 16. 17. 18. 19. 20. 2f||. 21. 23. If. 24. 25. 26. 27. 28. 29. 30. SUBTRACTION OF FRACTIONS. ART. 12. 11,13. Given. 14. i. 15. i. 16. f 17. 1. 18. 19. 20. 21. 22. 23. ART. 129. ART. ISO. 24, 25. Given. 26. 5i$. 27. H. 31. 39|-. 32. 1. 33. Hi. 28. I7f. 34. 2. 29, 30. Given. 35. 0. MULTIPLICATION OF FRACTIONS. ARTS. 11,12. Given, 13. 4, 14. 10i. 15. 6, 16. 6. 17. 18. Given. 19. 7*, 20. 13f. 2 1,22, Given, 23, 6, 24, 18, 25, S8fr. 28. 27. 28. 32^j. 29,80, Given, 40. 81. 258. 32, 889, J38, 84. Giv^n, 43-45, 85. 657, 86, 87, 88. Given, 39. -A-. 41. 42, 46, +. 47, i>. 48. -jfr. 49. A- 60. Given, 354 ANSWERS. [ARTS. 137-144 EXAMPLES FOR PRACTICE. ART. 13T. Ex. ANS. Ex. ANS. Ex. AN.-. Ex. ANS. 1. 4cts. 13. 112-ctS. 24. fcf- 35. 273 lets. 2. Gcwt. 14. 235 p. 25. fcfr 36. 61fM. 3. &9. 15. $16|. 26. $f 37. $4 If* 4. 84- bbls. 16. 32 cts. 27. *H- 38. 621-^cts. 5. 10c. 17. 61i. s. 28. 1237|cts. 39. $8i. 6. 8| a. 18. 75 cts. 29. 781-J-cts. 40. 16| s. 7. 2| s. 19. 112icts. 30. 115-1 cts. 41. 391|cts. 8. 5| s. 20. 2 16$ cts. 31. 243^ cts. 42. 652| s. 9. $6}. 21. $56. 32. $3. 43. &65i. 10. $6f. 22. 157icts. 33. $4. 44. $138ff. 11. $12. 23. $16*. 34. 28i s. 45. 743f m. 12. 136 cts. 4 DIVISION OF FRACTIONS. ART. 13-143. 11-1 3. Given. 22, 25. Given. 35. f|. 45. 323-^. 14. -ft. 26. 22i. QA Q 33 OO 4 *5"l; 36* 46-49. Given. 15. W. 27. -f. 37. 7^V. 50. tt; 1-ft ; 16. jj 28. IfB-. 38. 6ff. lif ; 2 irh-; 17. iV*. 29. ||. 39. 3|f. li 2 ? 5 i^f 18. i. 30, 31. Given. 40,41. Given. 52. 13^. 19. -&. 32. 5|. 42. 87i-. \ Q 1 J B. 3. t) . 1 "g g 2 . 20. rh- 33. 1-H-. 43. 75i. 54. -f. 21. 5 a fi 34. -ftV 44. 212-f. 55. lif. EXAMPLES FOR PRACTICE. ART. 144. 9. 5|1 Ibs. 18. $2f|f. 26. 4ffH- 1. 10 bu. 10. 5-i B 5 Ibs. 19. iiif t. 27. l^Vi/W- 2. 24 a. 11. 10^ c. 20. 87-flfr s. 28. -sVV- 3. lli Ibs. 12. St^bbls. 21. 157-ftVb. 29. 17|. 4. 12 bu. J o tf*O B lo. wt> 124. 22. 9^. 30. 1-ft. 5. 4 gals. 14. 7 cts. oq a A. ft *** 475- 31. rh-- 6. 14 A yds. 15 n *4 g 24. 185-Hf. QO ,13 a o ^ . V? o 5 * 7 5~H yds. 16. ilt^r. 25. 182iWr. J V 1-3. Givep. 15. $6.131. 26. 18f bbls. 37. 80 cents?* 4. $100. * 16. Given. 27. $60. 38. 65 we 5. $75. 17. $784. 28. 75 feet. 39. 56, 13s. 4d. 6. $30.75. 18. $216. 29. 100 days. 40. 186, 2s. 4i CUBE ROOT. ART. 362. 1, 2. Given. 3. 12. 4. 24. 5. 72 in. 6. 83 ft. 7. 125ft. 8. 1.25-f. 9. 1331. 10. 2.3. 11 4.5. 12. *. 13. *. ARTS, 367-406. j ANS $i;VsV ^ < - \K V 365 EQUATION OF PAYMENTS. ART? 36%^* *l *} 3. 6 ra >. 4 months. 7. 1 yrs. *> 69.6^ mo. '' q^Effifc 6. 6 months, 8. 3 months. VI * rf^i yrs. ^KJBLTNERSHJP.^ARTS. 369, i F&. I. Given. L $58&4te-i z I a j-, A. $3333.33i,B's. 2. snra'ovAV $112>.8W-M-, B. $4000, C's. $1800, B's. $1670.175^, C. 8. $100, A's. $2000, C's. $2210.526^, D. $120, B's. 8. $120, A's. , 5. $300, A's. $120, C's. $160, B's. $400, B's. 9. $30 apiece. $200, C's. $600, C's. 10. $40.019^, A. $700, D's. $ oo nhh i o T "D 00,4 1 i~Jo5", Of 6. $2666.661, A's. $ 1 1 *7 '7AQ_Z_3_ O 1 1 I . i UOj^ {) 9 ,w. EXCHANGE OF CURRENCIES. ARTS. 3T5-3T9. 1, 2. Given. 8. 300. 14. 240. 20. $640.625. 3. $484. 10. $6391.855. 15. 250. 21. $790.93$. 4. $1334.63. 11. Given. 17. $162.50. 22. $1147.73f 5. $2179.815. 12. 113, 8s. 18. $244. 23. $2436.428. 7 100, 10s. |13. l86,3s.7.2d. 19. $252.25. 24. $4003. MENSURATION. ARTS, 3O-4O6. 1. Given. '18. Given. 36. 152 sq. ft. 2. 640 a. U9. 11309.76 sq. r. 37. Given. 3. 26 a. 65 r. 20. 2037.18496 yds. 38. 21205.8 cu. ft. 4. 50 rods. 21. 8 r. 40. 30000 sq. yds. 5. 80 rods. 22. 16 ft. 42. 123J cu. ft. 6. Given. 23. Given. 44. 84 so. yds. 7. 4 a. 75 r. 24. 24 feet. 45. Given. 8. Given. * 25. 160 rods long; 46. 45945.75+ cu. ft. 9. 558 rods. 80 rods wide. 48. 4084.067 sq. ft. 10. Given. 26. 12. 49. Given. 11. 49 14 sq.ft. 27. 84. 50. 201061760 sq. m. 12. Given. 28. . 51. 904.77792 cu. in. 13. 62.35-|-yds. 30. 5425 cu. ft. 52. 26808234666Gf in. 14. Given. 31. 27 feet. 53. Given. 15. 314.159 rods. 32, 33. Given. 54. 6 in. 16. Given. 34. 9200 cu ft. 56. 12.599-fft* 17, 200 yds. 5, Given. 58. 244,346+ gal* MISCELLANEOUS EXAM Ex. AKS. Ex.^WANs. M Ex. AMS. I. $1250. m# c-f? 2. $90. * 49. 7812.5 IDS.* 50. 5082. /^ 95. |MBl 3. $1 29.374. /L^ 51. 6776. ///y 96. 7 Ibs. l-SrHfcz 4. $H069. /77 52. $322.50. , 97. $13440. 5. $600.031}. 53. $124. $U ** 98. 58 sheep. 6. $3332.531}. 7. 1163 bbl =>- 9. 500 saddles. 54. $80 6t>\\ / 55. $256. f* W 56. $436.50. 99. 72^days. 100. 313ipJaysT 101. 6270 days. 10. 50 horses. 57. $158. 102. 12 days. 12. $142.45. 58. 212. 104. 420 bu. 13. $211.05. 59. 193, 10a 105. 120 days. 14. $20.344. 60. 114. 106. $7266.66f. 15. $18.659. 61. 219. 107. $12170.31f. 16. $1023.667. 62. 61, 5s. 108. $15280.83$. 17. $4662.031. 64. 155. 109. 645, 9s. 44d. 18. $10446.33i. 65. $73. 110. 36 rods. 19. $1318.84^-. 66. $1250. 111. 160 rods. 20. $1531.396 JL - 2 ' 67. 300 tons. 112. 80 rods. 22. $19.294. 68. 11, 5s. 113. 600 sq. yds. 24. $6090. 69. $26.16. 114. 2400 sq. rods. 25. $106.25. 70. $65f. 115. 63. 26. $406.625. 71. $506. 13i- 116. 264. 27. $13.885. 72. 1, 10s. 8}d. 117. ^-. 28. $396.625. 73. lib. loz. 22grs. 118. ff. 29. $579.072. 74. 1 oz. 18} pwts. 119. 144rk.;48fii 30. $846. 75. 11s. 4d. 120. 18 inches. 31. $987.75, 76. 9, 12s. 121. U yds. Q O tft A * r\ 1 O^y. tJpv/.UJg" 77. $14.615. 122. 12 yds. 34. $0.094. 78. $17145. 123. 60 yds. 35. 20 per ct.; $9 12. 79. 4633^. 124. 24 yds. 36. 50 per ct. 80. 110 yds. 125. $72. 37. 44^ per ct. 81. 1980 gals. 126. $300 A's sh. 38. $3478.667. 82. $68.649. $420 B's sh. 39. $5557.68. 83. 78 Ibs. 2 oz. $800 C put in 40. $8724.375. 85. m shil. 127. $720 A's sh. 41. 3904 ; 4654. 86. 4, 8s. 8d. $ 1200 B's sh. 42. 2759 ; 2884. 87. 331 cts. $1680 C's. sh, 43. 428. 88. 220^ Ibs. 128. 1 hr. 5-ft- mia 44. 50. |89. 169-yV a. 1129. 1200 sheep. 91. 1290.48 bu. 130> isfrft. * 48. 100. ; 93. 105ft. <132. 34hrs.;323m fHOS EECOMMEN PRACTICAL ARITHMETIC. j^of Teachers, Superintendents, Trustees and School L is respectfully incited to the following Recommendation* cockers ard. School Committee of New Haven, of Thorn- PRACTICAL ARITHMETIC. From A. D. Stanley, A. M., Professor of Mathematics in Yale College From such an examination as I have been ablo to make of Thomson* ' Practical Arithmetic," I cannot doubt that it will hold a high rank as an el- ementary work in our Academies and Schools. It will commend itself to teachers for the clearness and precision with which its rules and principles are stated, for the number and variety of examples it furnishes as exercises for ths i>upil, and especially for the care which the author has taken to present ap- propriate suggestions and observations wherever they are needed, to clear up any difficulties that are likely to embarrass the learner. In recommending the work a; a class-book for pupils, it is not unimportant to state, that the author has himself had much experience in the business of instruction, and has thus had occasion to know where there was room for improvement rh the elemen- tary treatises in common use. Without such erperionce, no one c;m be quali- fied to prepare a class-book for schools. A. D. STANLEY Vale College, Dec. 4, 1846. We cordially concur in the views expressed by Prof. Stanley, respecting Thomson's Practical Arithmetic. AZAR[AH ELDRIDGE, A. M., Tutor In Nat. Philosophy. JOSEPH EMEUSON, A. M., Tutor in Mathematics. SAMUEL BRACE. A. M., Tutor in Greek. JAMES HADLEY, JR. A. M., Tutor in Latin. EDWARD C. HER RICK, A. M., Librarian. HAWLEY OLMSTEAL), A. M., Principal of Hopkln*' Grammar School. [for Boy A. N. SKINNER, A. M., Princ. of Select Classical School From Stiles French. A. M., Teacher of Mathematics. I have examined Mr. Thomson's new Practical Arithmetic, with careful < tention, and have decided to adopt it for my classes of beginners. To the teachers of our common schools, this Arithmetic may be particularly lecommended, as HI all respects convenient and exctllent for their use. New Haven, Dec. 5, 1845. STILES FRENCH, From the examination which I have been able to make of the Practie.il Arithmetic, by J. B. Thomson, A. M., I coincide fully in the recommendatioa ofstby Mr. French, to whim the department of mathematical instruction in our irstitate is more immediately intrusted. WM. II. RUSSELL, Principal of the Collegiate an# Commercial Institute, New Haven. Yf9 fully concur in the above recommendations. AMOS SMITH, Principal of Slect School fbr Boj*. B. W COLT, The publishers have the satisfaction of announcing that the ~Boardvf Schte* Vititora have unanimously adopted Thomson's " Practrcal-Atithmeti'cV fol the use of the Public Schools in the city of New Havenj\| " At a meeting of the Board of School Visitors fi^^he First School Society SC "certified by ' ALFJRED H. G. LEWIS, Secretary. ^j|f' 7^ 'V From the Hon. Judge Blackman, A M., Chairman of the Boa&toASchool Vis itors of the City of New Haven. 4B James B. Thomson, Esq., Dear Sir, I have examined w"iWsf5me atten- tion your " Practical Arithmetic," and consider it decidedly the best work for inculcating and illustrating the principles and practice of Arithmetic, which I have ever seen. Your illustrations, in the form of problems to be solved, are drawn, in a great measure, from the familiar scenes of early life ; and while the young learner is interested in the solution of problems which he feels are practicable, he is encouraged to persevere in a study which would otherwise be dull and forbidding, and is thus imperceptibly led to acquire and understand the rules of arithmetic, which he now knows to be true. I am glad you have removed " the ancient landmarks" of common school " ciphering," and thus permitted a child to understand what he reac's ; instead of torturing his mind with a jargon of words which he cannot understand, and requiring him to work by a rule which he cannot explain. I need hardly say, that the inductive method which you have adopted, ii decidedly the most philosophical and intelligible mode of acquiring a knowledge of arithmetic ; and as such I shall cheerfully recommend your work for gen eral use in the schools of this city. I ought not to overlook the copious references by which your rules are ex plained, anffthe mind of the student assisted in his labors ; nor the skill with which the publishers have executed their part of the work. I am, dear Sir, very respectfully yours, ALFRED BLACKMAN. Nov. 29th, 1845. From the Principals of the Publit Schools in the City of New Haven. New Haven, Nov. 28th, 1845. I have given Thomson's " Practical Arithmetic" as careful a perusal as my time would permit. I think it a work of very great merit. The plan of it, which has been ably carried out, appears to me, to be natural and philosophi- cal. The definitions and rules are exceedingly clear, and will be easily under- stood by those for whose instruction they are designed. The notes and obser- vations, which frequently occur, are admirably condensed, and afford much valuable aid and information. The examples for both mental and slate exer cises, are appropriate and abundant, and while the former are sufficiently simple to make the principles clear to the tyro's mind, the latter will secure sufficient practice with the pencil, to fix them there. I notice in almost every new rule, suggestion and illustration, that the pupil is pointed, by the means of numbers in brackets, to principles he has already studied ; this is an excellent plan ; it will be found highly useful to him, and very convenient to his in- structor. I will not attempt to make allusion to all the peculiarities and ex- cellencies of the work ; suffice it to say, that I consider it the best of all the excellent works of a similar kind with which I am acquainted. 1 shall, with- out delay, request the sanction of the Board of Visitors, for its adoption in the school under my care. J. E. LOVELL, Principal of the Lancasterian School. We fully concur in Mr. Lovell's views respecting Thomson's " Practica 1 Arithmetic," and are gratified to know that the Board of Visitors have adopt ed it for the Public Schools of this city. PRELATE DEMICK, Principal of Whiting st. School. WM. H. WAY, Principal of Wooster st. School. Ftom R#. J. JBrcwer t A. M., Prln. of Elm st. Female Seminary, New Haven Owur Sixr-After Mai of a number of different work* which ha\% boea brought to my notice, > I have concluded to adopt your Practical Arithmetic hi my Seminary. SBesiael other and higher merits which those more exclusively devoted to. mathematical pursuits will be ready to point out, the following are excel 1< ev&p- experienced teacher will be able to appreciate. 1. I\ umbering bf-ffie' articles, by which one may readily refer to any pro- \\o\i- step. ' . 2. Invariab'lyigiyin!: the important definitions and general rules in Italics. Sr^wwing-jhto smaller type, in the form of Notes and Observations, the ^literature" Qijhe subject, and useful hints for teachers and advanced pupils Neyy Haven, Dec. 5, 1845. JOSIAH BREWER. , A. M., Principal of the Young Ladies' Institute, New Haven, Ct. > Mr. Thomson Sir, In teaching Arithmetic, I have been exceedingly em- barrassed in deciding upon a text book for my pupils, but am now happy to find inks difficulty removed. I can confidently recommend your Practical Arith- metic, as combining excellencies to be found in no other elementary work on this subject. In the lucid and natural arrangement, the analysis of principles, and the full explanation of each step as you proceed, it exhibits many traces of the skill which appears in the other parts of your Mathematical Series al ready published. Yours, truly, VVM. WHITTLESEY Woolsey Hall, New Haven, Nov. 26, 1845. From E. L. Hart, A. M., Principal of English and Classical School for Boys. Messrs. Durrie & Peck, I have carefully examined Thomson's Practical Arithmetic, and fully believe that it is superior to any other Arithmetic now before th public. 1 like it for its excellent arrangement for its very clear il- lustration and exposition of principles for its accuracy in tables of weights and measures, some of which are incorrect in all other Arithmetics with which I am acquainted and for its eminently practical, business-like character. I shall introduce it into my school as soon as it is practicable. Yours, &.c. New Haven, Nov. 27, 1845. EDWARD L. HART. From J. D. Farren, Esq., Principal of Select School for Boys. Mr. J. B. Thomson Dear Sir, I have examined your Arithmetic, and must gay I am very highly pleased with jt. Its merits will, at once, present them- selves to the mind of every one \*ho will examine it. The thorough, syste- matic course pursued, is a grand one. I have introduced it into my school, which is more in its favor than anything I can say. I would say to those teachers who prefer to have their pupils work by the light of the sun rather than that of the moon, use Thomson's Arithmetic. JOSEPH D. FARREN. New Haven, Nov. 28, 1845. From 5. A. Thomas, Esq., Principal of New Haven Practical School for Boys. NEW HAVKN, Dec. 1, 1845. Mr. J. B. Thomson Sir, From the examination which I have been able to give your "Practical Arithmetic," I think it a valuable addition to that class of School Books. It contains many useful improvements, both in arrangement and in matter. The arrangement of subjects is decidedly the best I have seen the examples are judiciously selected and arranged, and the explanations and rules clear and concise. I think the work well calculated to lead the pupil to EU easy and rapid acquisition of the science of Arithmetic. Yours, truly, S. A. THOMAS. From J. H. Rogers, Esq., Principal of Fair Haven Family Boarding School. After a thorough examination of Thomson's Arithmetic, I believe it to be superior to any other extant. It is sufficient to say that I have adopted it in my school. The mathematical labors of Prof. Thomson evince the erudition of a ripe scholar, united with the skill of a practical teacher. I have tested the value of his Algebra and Geometry, in my school, with great satisfaction, and have no doubt his Arithmetic will fully sustain his high reputation as an author. J. H, ROGERS. Fair Raven, Nov. 28, 1845. Trm Wm, B. Oretne, A. B., Principal of Millbrd klt-woRD, Nov. 29, 1,845. J. B. Thomson, , Esq. Dear Sir, I have examined your Practical Arithme Me, and am pleased to observe the clearness and preci-sio'rrwUla vvl>"w.h the sub ject is presented the same that have so highly clwracteTr/ert^yimr Algebra and Geometry,, and so happily. adapted them to the capacities of the young. .Such a work has long been needed in our schools and acadeimtis. It meuts my views so well, that I have introduced it into my schooldMfl Yours, truly, cr^vEENfot From /. O. Hobbs, A. M., Principal of Washington InstLtale, ,J$iy York City. Gentlemen, 1 have carefully examined Mr. 'I'homsoa^j^HEicnl^Arithiiie- tic," and do most heartily add my testimonial to those alreHdy gtven in its fa- vor. It is indeed a work of very great merit, comprising many excellenciesTo, a small compass. Its value as a practical school-book will be more apparenS on a second and thorough examination. While as an elementary work it de- serves the place in our best schools that is occupied by the best, / know of no other so wel! adapted to general use. ISAAC G. HOBBS. New York, Aug. 1, Iri4t5. From the Teachers of the Normal School connected with the Public School* of the City of JVe/o York. Thomson's "Practical Arithmetic" is an exceedingly well arranged bonk. The principles are stated with clearness and precision -the mode of reasoning is analytical and systematic, yet the character of the work is eminently prac- tical, and well deserves the attention of teachers. We think it cannot fail to occupy a prominent place among the best text-books upou this science HOW in use. JOSEPH M. KEESE, President of the New York State Teach- ers' Association, and Principal of Public School No. 5. We heartily concur in the opinion expressed above. DAVID PATTERSON, M.D.. Principal of Public School No. 3. WM. BELPEN, Principal o: Public School No. 2. LEONARD HAZEi/l'lNE, Principal of Public School No. 14. ABM. K. VAN VLECK, Principal of Public School No. 16. From Wm. Belden, Jr., A. M., Principal of Ward School No. 3, New York City. A careful examination of Prof. Thomson's "Practical Arithmetic" has satis- fied me that it is a work of uncommon merit. The plan of presenting examples, in order to introduce the rule by previously analyzing its principles, which I consider the most important distinctive fea- ture of the work, will commend itself to every experienced teacher, as the natural process, both for imparting knowledge of this subject, and giving cor- rect habits of mental discipline. The language of the explanations and rules is peculiarly clear and intelligi- ble and the amount and value of this part of the work much superior to that of any other Arithmetic with which I am acquainted. The number and gradu- ally progressive character of the examples are also worthy of special notice. VVM. BJc