y HlBlH^ ■ Ci 1/ I I Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/completearithmetOOwhitrich WHITE'S GRADED-8CHOOL SE7iJjS3'^ COISIPLETE AEITHMETIC, MENTAL AND WRITTEN EXERCISES NATURAL SYSTEM OF INSTRUCTION. By E. E. AVHITE, M. A. CINCINNATI: WILSON, H INKLE & CO NEW YORK: CLARK & MAYNARD. Entered according to Act of Congress, in the year 1870, by WILSON, HINKLE & CO., In the Clerk's Office of the District Court of the United States for the Southern District of Ohio. aPUCATIOS^^ OEFT, EtECTnOTVPED AT THE FRA.NKMN TYPE FOUN'DUY, CINCINNATI. CUL PREFACE. This work is called a Complete Arithmetic, because it em- braces all the subjects which properly belong to a school arith- metic, and because it treats these subjects both analytically and inductively. It is designed to be a complete text-book for pupils who have mastered the elements of numbers. The work is characterized by the same features as the lower books of the series, viz. : 1. It combines 3Iental and Written Arithmetic in a practical and philosophical manner. This is done by making the mental exer- cises preparatory to the written ; and thus these two classes of exercises, which have been so long and so unnaturally divorced, are united as the essential complements of each other. 2. It faithfally embodies the inductive method of teachinff. The written methods are preceded by the analysis of mental problems, and both the written methods and the principles which they in- volve, are derived inductively from the analytic processes. The successive steps of each process are mastered by the pupil through the solution of problems, and he is required to deduce and state the rules before he is confronted with the author's generalization. All definitions which are deducible from the processes, and, with few exceptions, all principles and rules, are placed after the j^roblems— a feature peculiar to this Series. 3. It is specially adapted, both in matter and method, to the grade of pupils for ivhich it is designed. The greater portion of the work is devoted to a progressive and thorough treatment of subjects not embraced in the lower books — an arrangement which spe- 54t>{49 (iii) IV • ' ' ' PREFACE. ^ciall'y mte6fe' the ivataits of Graded Schools. But sixty pages cover the same ground as the Intermediate Arithmetic, and of these not more than twenty pages are in any sense a repetition. The repeated matter consists of definitions, principles, and rules, all the problems being new. The subjects before treated are not only concisely reviewed, but from a higher stand-point. Of the twenty-four pages devoted to the fundamental rules, eight present new abbreviated methods; and of twenty-eight pages devoted to Denominate Numbers, simple and compound, more than sixteen discuss new topics. A similar difference is observable in the treat- ment of Common Fractions and United States Money. Among the added articles worthy of special mention are those on Denom- inate Fractions, the Metric System, and Longitude and Time. In the number of problems, the author has aimed to hit the golden mean between a paucity and an excess, and the greatest pains have been taken to make them sufiiciently progressive, varied, and difficult, to afford the requisite drill and practice. Instead of rehashing old problems, with their incorrect data and obsolete terms, the author has gone to science and history for statistical information of practical value, and he has aimed to present the current values, terms, forms, and usages of American business. The mental problems will be found as difficult and comprehensive as those which constitute the latter half of the standard Mental Arithmetics, and are sufficiently numerous to afford thorough drills in analysis. The explanations of the written processes are not designed to serve as models for the pupil to memorize and repeat. They are intended to supplement the analysis. In some cases, a formal analysis is given ; in others, a principle is deduced or demon- strated; and in others, the process is described or its principles stated. Neither teacher nor pupil is denied the privilege of de- termining his own explanations. Another characteristic feature of this work is the prominence PREFACE. V given to Principles. A clear comprehension of the principles of arithmetic is essential to its thorough mastery, and their in- duction, proof, and illustration are mental exercises of great value. Until the pupil can step inductively from processes to principles, he has not a thorough knowledge of numbers. In this work the principles are concisely and formally stated in connec- tion with the rules which are based upon them. The author invites special attention to the sections treating of Percentage, Ratio and Proportion, and Involution and Evo- lution. Over eighty pages are devoted to Percentage and its applications, and it is believed that the treatment will be found not only full and thorough, but of great practical value. The student who masters these pages will certainly have a fair knowledge of the nature, laws, and usages of the business of the country. The introduction of Formulas, it is hoped, Avill prove a useful feature. The thorough treatment of Ratio before Proportion, and of the latter before its application to the solution of problems, will make the mastery of this subject easy. The treatment of Involution and Evolution will not escape notice. The geometrical explana- tions of Square Root and Cube Root are the reverse of those usually given, and are believed to be new. They will be found both simple and natural. All useless and obsolete subjects have been omitted; and those of interest only to more advanced students and teachers, are presented in an appendix. The typography and illustrations are beautiful and appropriate. The Complete Arithmetic is submitted to American teach- ers in the hope that it may not only be found new in its general plans and in many of its methods and details, but that it may prove eminently adapted to the present wants and condition of Graded Systems of Instruction. Columbus, Ohio, July, 1870. SUGGESTIONS TO TEACHERS. 1. The Mental Problems should be made a thorough drill in analysis; but, since the reasoning faculty is not trained by mere logical verbiage, the solution should be concise and simple. They should also be made introductory to the written processes of which they are often a com- plete elucidation. The corresponding problems in the two classes of exercises should be recited in connection, as well as separately. 2. All Written Problems should be solved by the pupils on slate or paper, and the solutions should be brought to the recitation for the teacher's inspection and criticism. From three to five minutes at the beginning of the recitation will suffice to compare the answers of the class, and ascertain the accuracy and neatness of each pupil's work. Time thus taken from the class-drill is more than made good by the increased interest, self-reliance, and accuracy, which the absence of answers secures. The explanations of the written processes, given by the pupil, should be both analytic and inductive. 3. The Definitions should be deduced and stated by the pupils under the guidance of the teacher, and this can usually be done in connection with the solution of the problems. See Int. Arith., p. 5, Sug. 3. When the definitions are placed before the problems, as in the applications of Percentage, they should be studied by the pupils, but their recitation may be deferred until the problems are solved, and the processes mastered. 4. The Principles should be taught inductively, when this is pos- sible, and each should be proven or illustrated, or both, by the pupil. A thorough mastery of every principle should be made an essential condition of the pupil's progress. The recitation should secure a constant application of known principles, and a clear comprehension of all new ones. 5. The Rules should also be deduced and stated by the pupils. The true order is this: 1. A mastery of the process. 2. Recognition of the successive steps in order, and a statement of each. 3. Combi- nation of these several statements into a general statement. 4. Com- parison of this generalization with the author's rule. 5. Memorizing of the rule approved. See Int. Arith., p. 6. 6. When two or more methods or solutions are given, the one pre- ferred should be thoroughly taught. It is well for pupils to understantl different processes and explanations, but they should be made familiar with one of them. 7. Before a subject is left, the pupils should be required to make a topical analysis of the definitions, principles, and rules, and the same should be recited with accuracy and dispatch. Their knowl- edge of the subject should finally be tested by a series of questions and problems. See General Review, pp. 268-284. (vi) CONTENTS. SECTION I -VI.— The Fundamental Rules. Notation and Numeration Addition .... The Addition of Two Columns Subtraction . . PAGE 10 13 15 17 Multiplication . Abbreviated Processes Division Abbi-cviated Processes PAGE . 20 . 22 . 25 . 28 SECTION VII.— Properties of Numbers. Divisors and Factors Cancellation 32 Greatest Common Divisor . . 37 35 I Least Common Multiple • . .40 Notation and Numeration . 43 Reduction of Fractions . . 46 Addition of Fractions . . 53 Subtraction of Fractions . . 55 Multiplication of Fractions . 57 SECTION VIII.— Fractions. Division of Fractions Complex Fractions Numbers Parts of Other Num- bers 66 Review of Fractions . . 68 SECTION IX.— Decimal Fractions. Numeration and Notation . 73 Reduction of Decimals . . 79 Addition of Decimals . . 82 Subtraction of Decimals . . 83 Multiplication of Decimals . 84 Division of Decimals . . 85 SECTION X.- Notation and Reduction . Addition and Subtraction . Multiplication and Division ■United States Money. Abbreviated Methods Aliquot Parts Bill. Surfaces SECTION XI.— Mensuration. . 100 I Solids . 104 SECTION XII.— Denominate Numbers. Reduction 107 Denominate Integers and Mixed Numbers . . . .107 Denominate Fractioi Tlio Metric System Metric Tables 110 119 120 (Vii) Vlll CONTENTS. SECTION XIII.— Compound Numbers. Addition and Subtraction Multiplication and Division PAGE 125 128 Longitude and Time SECTION XIV.— Percentage. The Four Cases of Percentage 137 Review of the Four Cases . 144 Applications of Percentage . 146 Profit and Loss . . 146 Commission and Brokerage 149 Capital and Stock . . 154 Insurance .... 159 Life Insurance . . . 163 Taxes . . . .164 Customs or Duties . . 168 Bankruptcy . . . 170 Interest . . . . . 171 General Method . . 172 Six Per Cent. Method Method by Days Partial Payments The Problems in Interest Review of Problems . Present Worth and Discount Bank Discount Promissory Notes and Drafts Bonds .... Annual Interest Compound Interest . Equation of Payments . Equation of Accounts SECTION XV.— Ratio and Proportion. Ratio . . . , . 220 Proportion .... 224 Simple Proportion . . . 225 Compound Proportion . . 230 Partnership Simple Partnership . Compound Partnership Problems for Analysis SECTION XVI. — Involution and Evolution. Involution .... 246 i Geometrical Explanation Another Method of Involution 248 Cube Root Evolution .... 249 ' Geometrical Explanation Square Root .... 251 I Applications SECTION XVII.— General Review. Test Problems . . . . 268 [ Test Questions . APPENDIX. Notation 287 Proofs by Excess of 9's . . 287 Circulating Decimals . . 289 Tables of Denominate Numbers 291 Legal Rates of Intcrrst . . 295 Life Insurance . . . 295 Equation of Payments . . 296 Arithmetical Progression . 297 Geometrical Progression . . 300 Alligation Duodecimals Permutations . Annuities . Rules of Mensuration Henkle's Method of Writing Decimals .... Schuyler's Contracted Method of Multiplying Decimals page 130 175 179 181 185 190 192 194 198 204 205 208 211 215 234 235 237 239 255 257 262 264 281 303 306 308 309 309 311 312 COMPLETE ARITHMETIC. SECTION I. PRELIMINARY DEFINITIONS. Art. 1. Arithmetic is the science of numbers, and the art of numerical computation. As a science, Arithmetic treats of the relations, properties, and prin- ciples of numbers ; and, as an art, it applies the science of numbers to their computation. 2. A Unit is one thing of any kind. 3. A dumber is a unit or a collection of units. 4. An Iwtegev is a number composed of whole or integral units ; as, 5, 12, 20. It is also called a Whole Number. 5. Numbers are either Concrete or Abstract A Concrete I^timher is applied to a particular thing or quantity ; as, 4 stars, 6 hours. When a concrete number expresses the denominate units of cur- rency, weiglit, or measure, it is called a Denominate Number. (Art. 174.) An Abstract Wumber is not applied to a particular thing or quantity ; as, 4, 6, 20. A concrete number is composed of concrete units, and an abstract number of abstract units. (9) 10 COMPLETE ARITHMETIC. ;;R. A JProMem^is a question proposed for solution. 7. An lExanipl6 is a problem used to illustrate a process or.a.principi»^j ; ;- 8. A Itiile is a general description of a process. 9. An Arithmetical Sign is a character denoting an operation to be performed upon numbers, or a relation between them. 10. In the Mental Solution of a problem, the suc- cessive steps are determined mentally, and the results are held in the mind. In the Written Solution of a problem, the results are written on a slate, paper, or other substance. SECTION II. NOTATION AND NUMERATION. MENTAL EXERCISES. 1. How many hundreds, tens, and units in 368? 427? 549? 608? 724? 806? 870? 2. How many hundred-thousands, ten-thousands, and thousands in 456048? 607803? 680435? 700450? 3. Read the thousands' period in 3045; 40607; 150482; 405360; 920400; 600060. 4. Read first the thousands' period and then the units' period in 65671 ; 120408 ; 400750 ; 650400 ; 80008. 5. Read 45037406; 520600480; 138405050. 6. Read 50008140 ; 600650508; 805000030. 7. Read 5308008450; 35006060600; 120500408080. 8. Read 7008360004; 302000860060; 500080800008. NUMERATION AND NOTATION. 11 WRITTEN EXERCISES. 9. Express in figures the number composed of 5 thou- sands, 7 tens, and 3 units; 4 ten-thousands, 6 hundreds, and 5 units. 10. Express in figures 50 thousands and 40 units; 406 thousands and 30 units; 700 thousands and 7 units. Express the folloAving numbers in figures: 11. Five million five thousand five hundred. 12. Sixty million sixty thousand and sixty. 13. Seven hundred million seven hundred thousand seven hundred. 14. Five hundred and sixty million sixty-eight thousand. 15. Four billion fi)urteen million fi)rty-five thousand. 16. Sixty-five billion six thousand and fifty. 17. Three hundred and fifty billion forty-nine million. 18. Seventeen trillion seventy billion seven hundred thousand four hundred. 19. Fifty-six trillion sixteen million and ninety. 20. Seven quadrillion eighty-five billion two hundred and four. DEFINITIONS AND TRINCIPLES. 11. There are three methods of expressing numbers : 1. By words; as, five, fifty, etc. 2. By letters, called the Roman method. 3. By figures, called the Arabic method. 12. Notation is the art of expressing numbers by fig- ures or letters. 13. Nmneration is the art of reading numbers ex- pressed by figures or letters. Note. — Notation may be defined to be the art of writing numbers, and Numeration, the art oi reading numbers. In Arithmetic, the term notation is used to denote tlie Arabic method. 14. In the Roman Notation, numbers are expressed by means of seven capital letters, viz: I, V, X, L, C, D, M. 12 COMPLETE ARITHMETIC. I stands for one; V for five; X for ten; L for fifty; C for one hundred; D for five hundred; M for one thousand, All other numbers are expressed by repeating or combining these letters. 15. In the Arabic Notation, numbers are expressed by means of ten characters, called figures, viz: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The first of these characters, 0, is called Naught, or Cipher. It denotes nothing, or the absence of number. The other nine characters are called Significant Figures, or Numeral Figures. They each express one or more units. They are also called Digits. 16. The successive figures which express a number, denote successive Orders of Units. A figure in units' place denotes units of the first order; in tens' place, units of the second order; in hundreds' place, units of the third order, and so on — the term units being used to express ones of any order. 17. Figures have two values, called Simple and Local. The Simple Value of a figure is its value when stand- ing alone. It is also called its Absolute value. The Local Value of a figure is its value arising from the order in which it stands. The local value of a figure is tenfold greater in hundreds' order than in tens' order. 18. The local value of each of the successive figures Avhich express a number, is called a Term. The terms of 325 are 3 hundreds, 2 tens, and 5 units. 19. The figures denoting the successive orders of units, are divided into groups of three figures each, called Periods. The first or right-hand period is called Units; the second. Thousands; the third. Millions; the fourth. Billions; the fifth, Trillions; the sixth. Quadrillions; the seventh, Quintillions ; the eighth, Sextillions; the ninth, Septillions; the tenth. Oc- tillions; the eleventh, Nonillions ; the twelfth, Decillions, etc. Note. — The division of orders into periods of three figures each is the French method. In the English method, the period contains six ADDITION. 13 orders, the name of the first period being Units, the second Millions; the third Billions, etc. 20. The three orders of any period, counting from the right, denote respectively Units, Tens, and Hundreds of that period. They may be briefly read by calling the first order by the name of the period, and uniting the words ten and hundred in each period after the first with the period's name. Thus, the orders of thousands' period may be read thousands, ten-thousands y hundred-thousands; the orders of millions' pe- riod, millions, ten-millions, hundred-millions, etc. 21. Principles. — 1. Ten units of the first order make one unit of the second order, ten units of the second order make one unit of the third, and, generally, ten units of any order make one unit of the next higher order. Hence, 2. The value of the successive orders of figures increases ten- fold from right to left. 3. The value of a figure is increased tenfold by each removal of it one order to the left, and is decreased tenfold by each re- moval of it one order to the right. SECTION III. ADDITION. MENTAL PROBLEMS. 1. Add by 6's from 1 to 73, thus: 1, 7, 13, 19, 25, etc. 2. Add by 7's from 3 to 73; from 6 to 90. 3. Add by 8's from 5 to 77; from 7 to 95. 4. Add by 9's from 4 to 76; from 8 to 98. 5. The ages of five boys are respectively 12, 10, 9, 8, and 7 years: what is the sum of their ages? 6. A rode 45 miles the first day, 42 miles the second day, and 38 miles the third day : how far did he ride in all ? Suggestion. — Add the tens and then the units of each couplet, thus: 45 H- 40 = 85, 85 + 2 -= 87 ; 87 + 30 = 117, 117 + 8 = 125. Or name - only results, thus: 45, 85, 87; 117, 125. (Art. 22.) 14 COMPLETE ARITHMETIC. 7. A drover bought 37 sheep of one farmer, 44 sheep of another, 48 sheep of another, and 27 sheep of another: how many sheep did he buy? 8. The Senior class of a college contains 27 students, the Junior class 34, the Sophomore class 38, and the Freshman class 46: how many students in the college? 9. A grocer sold 18 sacks of flour on Monday, 23 on Tuesday, 27 on Wednesday, 24 on Thursday, 35 on Friday, and 37 on Saturday: how many sacks did he sell during the week? 10. A lady paid $36 for a carpet, $34 for a bureau, $16 for awashstand, $28 for a bedstead, and $42 for chairs: how much did she pay for all? 11. A man paid $85 for a horse, and $17 for his keeping; and then sold him so as to gain $15: for how much did he sell the horse? 12. Two men start from the same point, and travel in opposite directions, the one at the rate of 54 miles a day, and the other at the rate of 48 miles a day: how far will they be apart at the close of the second day? "WRITTEN PROBLEMS. 13. Add 347, 4086, 7080, 29408, and 67736. 14. 667 + 3804 + 45608 -f 304867 + 87609 = what? 15. Add four thousand and fifty-six ; sixty-three thousand seven h\indred ; seven million nine thousand and ninety-nine ; and fifty-six million nine hundred and seventy-eight. 16. Add eight million eighty thousand eight hundred ; seven hundred thousand and seventy; five million eighty-six thousand seven hundred and eight; and sixty million six hundred thousand and seventy. 17. A grain dealer bought wheat as follows: Monday, 2480 bushels; Tuesday, 788 bushels; Wednesday, 1565 bushels; Thursday, 2684 bushels; Friday, 985 bushels; Sat- urday, 3867 bushels. How many bushels did he buy during the week? ADDITION. 15 18. Ohio contains 39964 square miles; Indiana, 83809; Illinois, 55409 ; Michigan, 56243 ; Wisconsin, 53924 ; Min- nesota, 83531 ; Iowa, 55045 ; and Missouri, 65350. What is the total area of these eight States ? 19. The population of these States in 1860 was as follows : Ohio, 2339511 ; Indiana, 1350428 ; Illinois, 1711951 ; Michi- gan, 756890 ; Wisconsin, 778714 ; Minnesota, 189923 ; Iowa, 674913; Missouri, 1182012. What was their total popu- lation ? 20. The territory of the United States has been acquired as follows : Square miles. Territory ceded by England, 1783, .... 815615 Louisiana, as acquired from France, 1803, . . . 930928 Florida, as acquired from Spain, 1821, .... 59268 Texas, as admitted to the Union, 1845, .... 237504 Oregon, as settled by treaty, 1846, .... 280425 California, etc., as conquered from Mexico, 1847, . . 049762 Arizona, as acquired from Mexico by treaty, 1854, . 27500 Alaska, as acquired from Russia by purchase, 1867, . 577390 What is the total area of the United States ? ADDITION OF TWO COLUMNS. 22. There is a practical advantage in adding two columns at one operation. Some accountants add three or more col- umns in this manner. 21. Add 67, 58, 43, 36, and 54. Process. 67 Add thus : 54 + 30 -= 84, + 6 = 90 ; 90 + 40 = 130, + 58 3 = 133 ; 133 + 50 == 183, -f 8 -^ 191 ; 191 + 60 = 251, 43 + 7 = 258. 36 Or thus, naming only results: 54, 84, 90; 130, 133, 183; _54 191 ; 251, 258. 258 Note. — The process consists in first adding the tens of each couplet, and then the units. If preferred, the units may first be added, and then the tens. Suflicient practice will enable the accountant to add two columns without separating the numbers into tens and units. 16 COMPLETE ARITHMETIC. 22. Add 37, 40, 63, 84, 67, 22, and 70. 23. Add 95, 46, 77, 66, 88, 63, 33, and 44. 24. Add 67, 76, 45, 54, 38, 83, 27, and 72. 25. Add 68, 86, 97, 79, 86, 68,78, and 87. 26. Add 45, 60, 57, 86, 83, 76, 49, 58, and 84. 27. Add 56, 75, 83, 96, 69, 73, 37, 38, and 205. 28. Add 27, 72, 33, 38, 69, 96, 75, 57, and 336. 29. Add 235, 88, 77, 66, 55, 44, 33, 22, and 11. 30. Add 405, 56, 43, 47, 74, 36, 63, 75, and 6Q. 31. Add 46, 67, 72, 38, 99, 87, 65, 74, and 88. 32. Add 73, 86, 47, 56, 69, 65, 58, 33, 52, and 94. DEFINITIONS AND PRINCIPLES. 23. Addition is the process of finding the sum of two or more numbers. The Sum of two or more numbers is a number contain- ing as many units as all of them, taken together. It is also called the Amount 24. The Sign of Addition is a short vertical line bi- secting an equal horizontal line, thus : -|-. It is called plus. 25. The Sign of Equality is two short horizontal parallel lines, thus: =. It is read equals or is equal to. Thus, 7 -[- 8 = 15 is read 7 plus 8 equals 15. 26. Lihe Numbers are composed of units of the same kind. Thus, 4 balls and 8 balls, or 4 dimes and 8 dimes, or 4 and 8, are like numbers. 27. Principles. — 1. Only like numbers can be added. 2. Only like orders of figures can be added. 3. Tlie sum is of the same hind or order as the numbers added. 4. The sum is the same whatever be the order in which the numbers are added. Note. — See appendix for method of proof by "casting out theO's." ADDITION. 17 SECTION IV. SUBTRACTION. MENTAL PKOBLEMS. 1. Count by 4's from 61 back to 1, thus: 61, 57, 53, etc. 2. Count by 6's from 53 back to 5 ; from 74 back to 2. 3. Count by 7's from 66 back to 3 ; from 85 back to 1. 4. Count by 8's from 75 back to 3 ; from 94 back to 6. 5. Count by 9's from 73 back to 1 ; from 96 back to 6. 6. A grocer having a certain number of sacks of flour, bought 48 sacks, and sold 33 sacks, and then had 34 sacks on hand : how many sacks had he at first ? 7. A man sold a horse for $95, which was $28 more than the horse cost him : what was the cost of the horse ? 8. Two men start at once from the same point, and travel in the same direction, one traveling 52 miles a day, and the other but 39 miles : how far will they be apart at the close of the second day? 9. A man earns $85 a month, and pays $18 for house rent, and $35 for other expenses : how much does he save each month? 10. A gentleman being asked his age said, that if he should live 27 years longer, he should then be three score and ten : what was his age ? 11. From a piece of carpeting containing 68 yards, a mer- chant sold 27 yards to one man and 18 yards to another: how many yards of the piece were left? 12. A man bought a carriage for $135, paid $21 for re- pairing it, and then sold it for $170: how much did he gain? 13. A boy earned 65 cents, and his father gave him 33 cents; he paid 45 cents for an arithmetic and 18 cents for a slate: how much money had he left? 14. There are 85 sheep in three fields; there are 36 sheep C.Ar.-2 18 COMPLETE ARITHMETIC. in the first field, and 28 sheep in the second : how many sheep in the third field? 15. John had 33 chestnuts, and Charles 25; John gave Charles 14 chestnuts, and Charles gave his sister as many as he then had more than John: how many chestnuts did the sister receive? WRITTEN PROBLEMS. 16. A builder contracted to build a school-house for $25460, and the job cost him $21385: what were his profits? 17. The earth's mean distance from the sun (old value) is 95274000 miles, and that of Mars is 145168136: how much farther is Mars from the sun than the earth? 18. The population of Illinois in 1860 was 1711951, and in 1865 its population was 2141510 : what was the increase in five years? 19. The population of Massachusetts in 1860 was 1231066, and in 1865 it was 1267031 : what was the increase in five years ? 20. The area of the Chinese Empire is 4695334 square miles, and the area of the United States is 3578392 square miles : how much greater is the Chinese Empire than the United States? 21. The area of Europe is 3781280 square miles: how much greater is Europe than the United States? The Chi- nese Empire than Europe? 22. In 1866, Ohio produced 99766822 bushels of corn, and Illinois 155844350 bushels : how many bushels did Illi- nois produce more than Ohio? 23. A man bought a farm for $5867, and built upon it a house at a cost of $1850, and then sold the farm for $7250: how much did he lose? 24. An estate of $13450 was divided between a widow and two children; the widow's share was $6340, the son's $1560 less than the widow's, and the rest fell to the daughter: what was the daughter's share? 25. A man deposited in a bank at one time $850, at an- SUBTRACTION. 19 other, $367, and at another, $670; he then drew out $480, and $375 : how much money had he still in bank ? 26. A man bought a farm for $6450, giving in exchange a house worth $4500, a note for $1150, and paying the dif- ference in money: how much money did he pay? 27. A grain dealer bought 15640 bushels of wheat, and sold at one time 3465 bushels, at another, 4205, and at an- other, 1080: how many bushels remained? 28. A has 320 acres of land ; B has Q5 acres more than A ; C has 124 acres less than both A and B ; and D has as many acres as both A and C less the number of acres owned by B. How many acres have B, C, and D respectively? How many have all? 29. From the sum of 45003 and 13478, take their differ- ence. DEFINITIONS AND PRINCIPLES. 28. Subtraction is the process of finding the difference between two numbers. The Difference is the number found by taking one number from another. It is also called the Remainder. The Winuend is the number diminished. The Subtrahend is the number subtracted. 29. The Sign of Subtraction is a short horizontal line, made thus — . It is called minus. Thus 12 — 5 is read 12 minus 5. 30. Principles. — 1. The 7ninue7idj subtrajiend, and differ- ence are like numbers. 2. The minuend is the sum of the subtrahend and difference. 3. If the minuend and subtrahend be equally increased, the dfference will not be changed. 4. The adding of 10 to a term of the minuend and 1 to the next higher term of the subtrahend, increases the minuend and subtrahend equally. 20 COMPLETE ARITHMET[C. SECTION V. MULTIPLICATION. MENTAL PROBLEMS. 1. There are 24 hours in a day : how many hours in 7 days? In 9 days? 11 days? 2. There are 60 minutes in an hour: how many minutes in 8 hours? In 12 hours? 15 hours? 3. If a man earn $63 a month, and spend 848, how much will he save in 12 months? 4. If 12 men can do a piece of work in 15 days, how long will it take one man to do it ? 5. If 35 bushels of oats will feed 8 horses 25 days, how long will they feed one horse ? 6. Two men start from the same place and travel in op- posite directions, one at the rate of 28 miles a day, and the other at the rate of 32 miles a day : how far will they be apart at the end of five days ? 7. Two men are 450 miles apart: if they approach each other, one traveling 30 miles a day and the other 35 miles a day, how far apart will they be at the end of 6 days ? 8. A cask has two pipes, one discharging into it 90 gallons of water an hour, and the other drawing from it 75 gallons an hour: how many gallons of water will there be in the cask at the end of 12 hours? 9. A had $24, B four times as much as A less $16, and C twice as much as A and B together plus $17 : how much money had B and C? 10. A farmer sold to a grocer 15 pounds of butter, at 30 cents a pound, and bought 8 pounds of sugar, at 15 cents a pound, and 9 pounds of coffee, at 20 cents a pound : how much was still due him? MULTIPLICATION. 21 WRITTEN PROBLEMS. 11. Multiply 624 by 45 ; by 405 ; by 4005. 12. Multiply 38400 by 27 ; by 607 ; by 6007. 13. Multiply 7863 by 69 ; by 6900 ; by 64000. 14. Multiply 48000 by 760 ; by 7600000. 15. There are 5280 feet in a mile : how many feet in 608 miles ? In 3300 miles ? 16. The earth moves 1092 miles in a minute : how far does it move in 1440 minutes, or one day? 17. A square mile contains 640 acres, and the state of Ohio contains, in round numbers, 40000 square miles : how many acres in the state ? 18. If a garrison of 380 soldiers consume 56 barrels of flour in 75 days, how many soldiers will the same amount of flour supply one day? 19. A man bought a farm, containing 472 acres, at $24 an acre, and after investing $3450 in buildings, he sold the farm, at $33 an acre : did he gain or lose, and how much? DEFINITIONS AND PRINCIPLES. 31. JKultijilicatioil is the process of taking one num- ber as many times as there are units in another. The 31ultlplicand is the number taken or multiplied. The 3Iultipliev is the number denoting how many times the multiplicand is taken. The JProdlict is the number obtained by multiplying. The multiplicand and multiplier are Factors of the product, and the product is a Multiple of each of its factors. 32. The Sif/n of Hiiltij^lication is an oblique cross, made thus: X- It is read multipled hy, or times. When placed between two numbers, it shows that they are to be multiplied together ; and, since the order of the factors does not affect the product, either number may be made the multiplier. The mul- tiplier is usually written after the sign. 22 COMPLETE ARITHMETIC. 33. The product may be obtained by adding the multipli- cand to itself as many times less one as there are units in the multiplier. Hence, Multiplication is a short 7nethod of finding the sum of several equal numbers. 34. Principles. — 1. The ^fultiplicand may be either con- crete or abstract. 2. The 7mdtiplier must always be regarded as abstract. 3. The product and multiplicand are like numbers. 4. The product is not affected by changing the order of the factors. 5. The midtiplicand equals the product divided by the multi- plier. 6. The midtiplier equals the product divided by the midtipli- cand. 7. The division of either the multiplicand or the multiplier by any number divides the product by that number. ABBREVIATED PROCESSES. Case I. The Mlialtiplier lO, lOO, lOOO, etc. 1. There are 7 days in a week: how many days in 10 weeks? In 100 weeks? 2. There are 24 hours in a day : how many hours in 10 days? 100 days? 3. If a railway train run 30 miles an hour, how far will it run in 10 hours? 1000 hours? 4. If a freight car will carry 18 head of cattle, how many cattle will 10 cars carry? 100 cars? 1000 cars? 5. There are 12 months in a year: how many months in 100 years? 1000 years? WKITTEN PEOBLEMS. 6. Multiply 648 by 100. Process : 648 X 100 — 64800. The annexing of a cipher to a num- ber removes the significant figures one place to the left, and hence increases their value 10 times ; the annexing of two ciphers removes MULTIPLICATION. 23 the significant figures two places to the left, and increases their value 100 times. Hence, the annexing of two ciphers to 648 multiplies it by 100. 7. Multiply 456 by 10 ; by 10000. 8. Multiply 3050 by 100 ; 100000. 9. Multiply 347000 by 1000 ; by 1000000. 10. Multiply 889000 by 10000 ; by 100. 35. Principle. — The removal of a figure one order to the left increases its value tenfold. 36. Rule.— To multiply by 10, 100, 1000, etc., Annex to the multiplicand as many ciphers as there are ciphers in the mul- tiplier. Case 11. The Mlultiplier a convenient part of lO, lOO, lOOO, etc. Note. — If the class is not sufficiently familiar with the subject of fractions, this case may be omitted. 11. There are 24 sheets of paper in a quire: how many sheets in 2^ quires ? In 3^ quires ? 12. There are 60 minutes in an hour : how many minutes in 3i hours? In 12^ hours? 13. If a workman earn $40 a month, how much w411 he earn in 2^ months? In 12|^ months? 14. At 36 cents a yard, what will 25 yards of cloth cost? 33i yards? 15. At 24 cents a dozen, what will 12|^ dozens of eggs cost? 16 J dozens? WHITTEW PEOBLEMS. 16. Multiply 459 by 33^. Process. Since 33i is i of 100, 33^ times 459 = i of 100 3 ) 45900 times 459 = -^ of 45900. Or, multiply the multi- 15300 Prod. plicand by 100, and divide the product by 3. 17. Multiply 486 by 3^; by 33^. 18. Multiply 1688 by 12^; by 25; by 50. 24 COMPLETE ARITHMETIC. 19. Multiply 40648 by 16|; by 33i; by 333^. 20. Multiply 3468 by 25 ; by 125 ; by 250. 21. Multiply 4086 by 16| ; by 166|; by 333|. 22. Multiply 10366 by 50 ; by33|; by 66^. 37. Principle. — Jj the multiplier be multiplied by a given number, and the resulting product be divided by the same num- ber, the quotient ivill be the true product. 38. KuLE. — To multiply by a convenient part of 10, 100, 1000, etc., 3Mti2}ly by 10, 100, 1000, etc., and divide the product by the number of times the multiplier has been increased. Case III. The MlTaltipliex- a little less tlxan lO, lOO, lOOO, etc. 23. Multiply 467 by 98. Process. Since 98 = 100 — 2, the product of 467 by 98 = 467 46700 X 100 — 467 X 2, or 46700 — 934. In multiplying ^^4 \yy 100 the multiplicand is taken two times more than 45766, Prod, it should be. 24. Multiply 5672 by 99 ; by 999. 25. Multiply 40863 by 97 ; by 997. 26. Multiply 8679 by 998 ; by 9998. 27. Multiply 618734 by 95 ; by 99995. 39. Rule. — To multiply by a number a little less than 10, 100, 1000, etc., Multiply by 10, 100, 1000, etc., and subtract from the product tJie multiplicand midtiplied by the difference be- tween the multiplier and 10, 100, 1000, etc., as the case may be. Case IV. The Mlxaltiplier 14, 15, lO, etc., or 31, 51, 61, etc. 28. Multiply 7856 by 14 ; by 41. 1st Process. 2d Process. 7856 X 14 7856 X 41 31414 31414 109974, Product. 321996, Pivduct. Note. — An inspection of each process will suggest its explanation. The second partial product need not be written, as the successive terms can be added mentally to the proper terms of the first partial product. DIVISION. 25 29. Multiply 38407 by 13 ; by 15 ; by 17. 30. Multiply 4960 by 16 ; by 18 ; by 19. 31. Multiply 360978 by 31 ; by 51 ; by 71. 32. Multiply 48706 by 61 ; by 81 ; by 91. 33. Multiply 34087 by 17 ; by 71 ; by 18. 40. KuLES. — 1. To multiply by 13, 14, 15, etc., Multiply by the units' term, and add the successive products after the first, ivhich is units, to the successive terms of the midtiylicand. 2. To multiply by 31, 41, 51, etc.. Multiply by the tens' tsrm, and add the successive products to the successive terms of the mul- tiplicand beginning with tens. SECTION VI. DIVISION. MENTAL PROBLEMS. 1. There are 7 days in a week: how many weeks in 63 days? 98 days? 126 days? 2. There are eight quarts in a peck: how many pecks in 72 quarts? 120 quarts? 144 quarts? 3. There are 60 minutes in an hour : how many hours in 480 minutes? 720 minutes? 1440 minutes? 4. A man paid $3600 for a farm, paying at the rate of $40 an acre : how many acres in the farm ? 5. A grocer bought 12 barrels of flour for $90, and sold them so as to gain $18: how much did he receive per barrel ? 6. Two men are 120 miles apart, and are traveling toward each other, one at the rate of 7 miles an hour, and the other at the rate of 8 miles an hour : in how many hours will they meet? 7. If a man can build a wall in 84 days, how long will it take 7 men to build it? C.Ar.— ;^ 26 COMPLETE ARITHMETIC. 8. If 8 men can do a piece of work in 15 days, how long will it take 12 men to do it? 9. If a quantity of provisions will supply a ship's crew of 20 men 15 weeks, how large a crew will it supply 25 weeks ? 10. If a man can do a piece of work in 40 days, by work- ing 8 hours a day, how long would it take him if he should work 10 hours a day? 11. A man earns $16 while a boy earns $9: how many dollars will the man earn while the boy is earning $12 ? 12. The fore wheels of a carriage are each 9 feet in cir- cumference, and the hind wheels are each 12 feet: if the fore wheels each rotate 400 times in going a certain distance, how many rotations will each hind wheel make ? 13. Five times Harry's age plus 4 times his age, minus 6 times his age, plus 7 times his age, minus 5 times his age, equals 60 years : how old is Harry ? 14. A number multiplied by 6, divided by 3, multi- plied by 8, and divided by 4, equals 96 : what is the num- ber? WRITTEN PROBLEMS. 15. Divide 486 by 6 ; by 8 ; by 9. 16. Divide 8408 by 12; by 24; by 36. 17. Divide 84600 by 900 ; ^by 12000. 18. Divide 412304 by 3600;' by 303000. 19. The dividend is 1059984 and the divisor is 306 : what is the quotient? 20. The dividend is 2185750 and the quotient is 250 : what is the divisor? 21. The product is 1123482 and the multiplier is 246: what is the multiplicand? 22. How many passenger cars, costing $2450 each, can be bought for $100450? 23. There are 5280 feet in a mile, and the height of Mount Everest, in Asia, is 29100 feet : what is its height in miles? DIVISION. 27 24. There are 3600 seconds in an hour : how many hours in 738000 seconds ? 25. Divide the product of 480 and 256 by their difference. DEFINITIONS AND PRINCIPLES. 41. Division is the process of finding how many times one number is contained in another ; or, it is the process of finding one of the equal parts of a number. The JJivulend is the number divided. The Divisor is the number by which the dividend is divided. The Quotient is the number of times the divisor is con- tained, in the dividend. The Jletnainder is the part of the dividend which is left undivided. 42. The Sign of Division is a short horizontal line between two dots, thus : -^. It is read divided by. Thus, 16 -^ 4 is read 16 divided by 4. Division is also expressed by writing the dividend above and the divisor below a short horizontal Hne. Thus, \^ is read 18 divided by 3. 43. There are two methods of division, called Short Di- vision and Long Division. In Short Division^ the partial products and partial dividends are not written, but are formed mentally. In Long Division^ the partial products and partial dividends are WTitten. 44. — 1. One number is contained in another as many times as it must be taken to produce it. Hence, Division is the reverse of midtiplication. 2. One number is contained in another as many times as it can be taken from it. Hence, Division is a brief method of finding how many times one number can be subtracted from another. 28 COMPLETE ARITHMETIC. 45. Principles. — 1. The divisor and quotient are factors of the dividend. 2. When division finds how many times one number is contained in another, the divisor and dividend are like NUMBERS, and the quotient is an abstract number. 3. When division finds one of the equal parts of a num- ber, the divisor is an abstract number, and the dividend and quo- tient are like numbers. 4. The m^dtiplying of both divisor and dividend by the same number does not change the value of the quotient. 5. The dividing of both dividend and divisor by the same num- ber does not change the value of the quotient. ABBREVIATED PROCESSES. Case L The IDivisor lO, lOO, lOOO, etc. 1. There are 10 cents in a dime : how many dimes in 80 cents? 120 cents? 240 cents? 2. There are 10 dimes in a dollar : how many dollars in 70 dimes? 250 dimes? 2500 dimes? 3. There are 100 cents in a dollar: how many dollars in 800 cents? 2400 cents? 7500 cents? 4. At $10 a barrel, how many barrels of flour can be bought for $90? For $150? 5. At $100 apiece, how many horses can be bought for $1200 ? For $2500 ? For $45000 ? WRITTEN PROBLEMS. 6. Divide 450 by 10. 7. Divide 3852 by 100. Process. Process. 45K) 38152 45, Quotient. 38, Quotient. 52, Remainder. The explanation of these processes is obvious. The cutting off of the right-hand figure removes all the other figures one place to the right, and thus decreases their value ten times. The cutting off of two figures removes the other figures two places to the right, and de- DIVISION. 29 creases their value one hundred times. Tlie figures cut off denote the remainder. 8. Divide 356000 by 100 ; by 1000. 9. Divide 46035 by 100 ; by"^1000. 10. Divide 384602 ; by 1000 ; by 10000. 11. Divide 95000000 by 10000; by 1000000. 46. Principle. — The removal of a figure one order to the right decreases its value tenfold. 47. Rule.— To divide by 10, 100, 1000, etc., Cut off as many figures from the right of the dividend as there are ciphers in the divisor. The figures cut off denote the remainder. Case II. T'lxe IDivisor ending \vitli one or nriore Cipliefs. 12. There are 20 quires of paper in a ream : how many reams in 80 quires? 160 quires? 13. There are fifty cents in a half-dollar : how many half- dollars in 150 cents ? 350 cents ? 14. There are 60 minutes in an hour : how many hours in 240 minutes? 720 minutes? 15. A barrel of beef contains 200 pounds : how many barrels will 1200 pounds make ? 3600 pounds ? \^7■RITTEN PROBLEMS. 16. Divide 71400 by 3400. Process. First divide both divisor and dividend by 34100)714100(21 100, which is done by cutting off the two "^ right-hand figures. Then divide 714, the ^ , new dividend, by 34, the new divisor. 17. Divide 58864 by 4500. Process. First divide both dividend and divisor by 45100 ) 588i64 ( 13 100, which, in the case of the dividend, leaves 45 a remainder of 64. Next divide 588 by 45, 1^^ leaving a remainder of 3, which is 3 hundreds -^ since the dividend (588) is hundreds. The first p • ^ QRA remainder is 64 units which, annexed to the ' 3 hundreds, gives 364, the true remainder. 30 COMPLETE ARITHMETIC. 18. Divide 63200 by 7900 ; by 7000. 19. Divide 116000 by 2500; by 4800. 20. Divide 172800 by 14400; by 18000. 21. Divide 129600 by 4800; by 64000. 48. Principle. — The dividing of both divisor and dividend by the same number does not change the value of the quotient. 49. Rule. — To divide by a number ending in one or more ciphers, 1. Cut off the ciphers from the right of the divisor, and an equal number of figures from the right of tlie dividend. 2. Divide the new dividend thus formed by the new divisor, and tJie residt will be tJie quotient. 3. Annex the figures cut off from tlie dividend to tJie remainder, if tJiere be one, and the residt will be the true remainder. Case III. The [Divisor a convenient part of lO, lOO, etc. 22. At 3|- cents apiece, how many lemons can be bought for 90 cents ? For 240 cents ? Suggestion. — Since 10 is 3 times Z\, multiply the dividend by 3 and divide the product by 10. 23. At 12^ cents a yard, how many yards of cloth can be bought for 75 cents? For 225 cents? 24. At 16f cents a bushel, how many bushels of coal can be bought for 150 cents ? For 550 cents ? 25. At $33| a head, how many cows can be bought for $200 ? For $1200 ? ^WKITTEN EXERCISES. 26. Divide 4375 by 125. 27. Divide 13600 by 333^. Process. Process. 4375 13600 351000 401800 35, Quotient. 40, Quotient. 800, Remainder. 28. Divide 6250 by 33^ ; by 50. DIVISION. 31 29. Divide 4365 by 250; by 166|. 30. Divide 15300 by 16f ; by 333 J. 50. Principle. — The multi'plying of both divisor and divi- dend by the same number does not change the value of the quotient. 51. Rule. — To divide by a convenient part of 10, 100, 1000, etc., 3fidtiply the dividend by tJie number denoting how many times the divisor w contained in 10, or 100, or 1000, etc., and divide the product by 10, or 100, or 1000, etc. Case IV. The Divisor a CoiTiposite IN'-uixi.'bei'. 31. Divide 18315 by 45. Process. Illustrative Process. 45 = 5X9 5 ) 18315 -r- 45 =^ 3663 -~- 9 5 ) 18315 9 ) 3663^ 9 :=^ 407 -f- 1 9 ) 3603 407 -f- 1 = 407 407, Quotient. Since 45 -- 5X9, the quotient obtained by dividing 18315 by 5, is 9 times too large, and hence this quotient (3663) divided by 9, is the true quotient. The process of dividing by the factors of the divisor successively is the same in principle as the division of both dividend and divisor by these factors successively, which (Art. 48) does not change the value of the quotient. See " Illustrative Process." 32. Divide 58636 by 28 ; by 77. 33. Divide 13328 by 49 ; by 56 ; by 70. 34. Divide 31360 by 64; by 70; by 81. 35. Divide 3687 by 64. Process. 2 )_3687 64 = 2 X 8 X 4 8 ) 1843 .... 1 (1st Kern.) = 1 4 )^ .... 3 (2d " ) = 3 X 2 = . . . . 6 57 2 (3d " ) = 2 X 8 X 2 = 32 True Kemainder = 39 A unit of the first quotient equals 2 units of the dividend, and hence the second remainder (3) equals 3X2 units of the dividend. 32 COMPLETE ARITHMETIC. A unit of the second quotient equals 8 units of the first quotient, and hence the third remainder (2) equals 2X8 units of the second quo- tient = 2X8X2 units of the dividend. Hence the first remainder is 1 ; the second 6 ; the third 32 ; and the total, or true remainder, 39. Note. — The teacher can illustrate this process by considering the dividend (3687) 'pints. The first quotient will be quarts, the second pecks, and third bushels, and the first remainders will be 1 pt., the second, 3 qt., and the third, 2 pk. 1 pt. + 3 qt. -\- 2 pk. = 39 pt. 36. Divide 34567 by 63 ; by 72. 37. Divide 120473 by 56; by 81. 38. Divide 400671 by 64 ; by 77. 39. Divide 346000 by bb ; by 96. 40. Divide 47633 by 90; by 110. 52. Pkinciple. — The division of both divisor and dividend by tJw same number does not change the value of the quotient. 53. Rule. — To divide by a composite number, 1. Resolve the divisor into convenient factors ; divide the dividend by one of these factors, the quotient thus obtained by another, and so on until all the factors are used as divisors. The last quotient will be the true quotient. 2. Multiply each remahider, except the first, by all the divisors preceding its own. TJie sum of these products and the first re- mainder will be the true remainder. SECTION VII. PROPERTIES OF NUMBERS. PRIME AND COMPOSITE NUMBERS AND FACTORS. Note. — The terms number, divisor, and factor, used in this section, denote integral numbers. 1. What two numbers besides itself and 1 will exactly divide 10? 21? 35? 63? 77? 2. What numbers besides itself and 1 will exactly divide 7? 11? 17? 23? 37? 41? PROPERTIES OF NUMBERS. 33 3. What numbers will exactly divide 15? 13? 28? 29? 42? 43? Note. — Since every integer is exactly divisible by itself and 1, these divisors need not be given. 4. What numbers will 'exactly divide 30? 31? 45? 53? 56? 67? 65? 5. Name all the prime numbers between and 20 ; 30 and 50. 6. Name all the composite numbers between 20 and 30 ; 50 and 70. 7. What are the prime divisors of 6 ? 15? 18? 21? 30? 45? 50? 54? 8. What are the prime factors of 12? 24? 35? 39? 42? 9. What are the prime factors of 27? 36? 49? 56? 63? 66? 72? 84? 10. Of what numbers are 2 and 5 prime factors? 2, 3, and 5 ? 2, 5, and 7 ? 3, 5, and 7 ? 11. Of what numbers are 2, 2, and 3 prime factors? 2, 3, 3, and 5 ? 2, 3, 5, and 7 ? 12. What prime factor is common to 9 and 12? 15 and 25? 18 and 30? 21 and 28? 13. What prime factor is common to 24 and 27 ? 35 and 42? 44 and 77? 35 and 50? 63 and 70? WRITTEN EXERCISES. 14. What are the prime factors of 126? Process. Divide 126 by 2, a prime divisor ; next 2 ) 126 divide the quotient 63 by 3, a prime 3 -j g3 divisor, and then divide the quotient 21 3 )'2l by 3, a prime divisor. The prime factors 7 are 2, 3, 3, and 7. 126 = 2 X 3 X 3 X 7. What are the prime factors of 15. 160? 18. 325? 21. 462? 24. 748? 16. 175? 19. 330? 22. 490? 25. 693? 17.256? 20.420? 23. .594? 26.1155? 34 COMPLETE ARITHMETIC. What prime factors are common to 27. 45 and 63? 30. 200 and 250? 28. 50 and 80? 31. 175 and 325? 29. 96 and 256? 32. 144 and 180? DEFINITIONS, PRINCIPLES, AND RULES. 54. The Divisor of a number is any number that will exactly divide it. 55. Numbers are either Prime or Composite. A Prime Number has no divisor except itself and one. A Composite Wuinber has other divisors besides itself and one. Every composite number is the product of two or more numbers, called factors. 56. Two or more numbers are prime to each other, or rela- tively prijne, when they have no common divisor except 1. Thus, 9 and 16 are prime to each other. All prime numbers are prime to each other. Composite numbers may be relatively prime, as 9 and 10 ; 16 and 25. 57. A Factor of a number is its divisor. A Prime Factor of a number is its prime divisor. The terms divisor and factor differ only in their use, the former implying division and the latter multiplication. A divisor or factor of a number is also called its measure. 58. When a number is a factor of each of two or more numbers, it is called their Comm^on Factor, Thus, 5 is a common factor of 15 and 20. 59. Whole numbers are either Even or Odd. An Even N'Um^ber is exactly divisible by 2 ; as, 2, 4, 6, 8, 10, 12, etc. An Odd Number is not exactly divisible by 2 ; as, 1 , 3, 5, 7, 9, 11, 13, etc. CANCELLATION. 35 All the even numbers except 2 are composite. Some of the odd numbers are composite and others are prime. 60. Principles. — 1. A factor of a number is a factor of any number of times that number. 2. A common factor of two or more numbers is a factor of their sum. 3. A composite number is the product of all its prime factors. 4. If a composite number composed of two factors be divided by one fador, the quotient will be the other factor. 5. If any composite number be divided by a factor, or by the product of any number of its factors, the quotient will be the product of the remaining factors. 61. KuLES. — 1. To resolve a composite number into its prime factors, Divide it by any prime divisor, and the quo- tient by any prime divisor, and so continue until a quotient is obtained which is a prime number. The several divisors and tJie last quotient are the prime factors. 2. To find the common factors of two or more numbers, Resolve Uie given numbers into their prime factors and select the factors which are found in all the numbers. CANCELLATION. 33. Divide the product of 4, 7, 9, and 12 by the product of 4, 7, and 9. Process. Instead of forming the prod- Dividend, ^ X y^ X X 12 "^^^' indicate the multiplica- . — VZTirZT^ ^12. tion by the proper sign, and Divisor, /i X ^ X V *u J- • J ^1 J f /\ i> /\ t write the divisor underneath the dividend. Since dividing both dividend and divisor by the same number does not affect the value of the quotient (Art. 48), divide each by 4, 7, and 9. This may be done by canceling, as indicated in the process. The quotient is 12. 34. Multiply 4 X 7 by 12, and divide the product by 4 times 12. 35. Divide 6 X 8 X 20 by 4 X 20. 36. Divide 5 X 7 X 11 X 13i by 7 X 13f 36 COMPLETE ARITHMETIC. 37. Divide 12 X 16 X 2$ by 9 X 24 X 21. Process. Since dividing the factor 4 of a number divides the num- J:^ X 1 X ;^^ _ 8 X 4 32 . her, cancel 12 in the divi- 9 X M X M ~ 9 X S~Y7~^^''^ dend and divide 24 in the ;2 3 divisor by 12, giving 2. Can- cel the 2 and divide 16 in the dividend by 2, giving 8. Divide the 28 in the dividend and 21 in the divisor, each by 7, giving 4 and 3. The uncanceled factors of the divisor are 8 and 4, and those of the dividend are 9 and 3. The quotient is 32 -r- 27 =:^ lo^. 38. Divide 24 X 27 X 12^ by 18 X 54 X 50. 39. Divide 28 X 30 X 100 by 21 X 15 X 33^. 40. 40 X 22 X 35 X 16| -^ 20 X 44 X 50 X 49 :^ what? 41. A farmer exchanged 12 barrels of apples, each con- taining 3 bushels, at 75 cts. a bushel, for 25 sacks of pota- toes, each containing 2 bushels : how much did the potatoes cost a bushel? 42. If 9 men can do a piece of work in 15 days, working 10 hours a day, how many men can do it in 20 days, work- ing 8 hours a day ? DEFINITIONS, PRINCIPLES, AND RULE. 62. Cancellatiou is the omission of one or more of the factors of a number. It is used when both dividend and divisor contain one or more equal factors, to abbreviate the process of division. 63. Principles. — 1. The canceling of one of the factors of a number divides the number by the factor canceled. 2. Canceling equal factors of both dividend and divisor divides them by the same number, and hence does not change the value of the quotient. 3. Dividing one of the composite factors of a product divides the product. 64. Rule. — Indicate the multiplications by the proper sign, and write the divisor underneath the dividend. Cancel the fac- COMMON DIVISOR. 37 tors common to both dividend and divisor, and divide the prod- uct of the factors left in the dividend by the product of those left in the divisor. ]Nf OTE. — When all the expressed factors of either dividend or divisor are canceled, 1 remains as a factor. GREATEST COMMON DIVISOR. 1. What are the divisors of 15? 28? 45? 53? 75? 90? 91 ? 108 ? 2. What is a common divisor of 15 and 35? 42 and 56? 63 and 72 ? 64 and 80 ? 3. What is a common divisor of 27 and 36? 18, 30, and 42? 36, 54, and 72? 4. What is the greatest number that will exactly divide 32 and 48? 45 and 90? 60 and 96? 5. What is the greatest common divisor of 36 and 60 ? 45, 60, and 75? 18, 54, and 90? 6. What is the greatest common divisor of 24, 48, and 72? 16, 48, and 80? 20, 31, and 45? 7. Show that every common divisor of 12 and 16 is a divisor of 28, their sum. 8. Show that a common divisor of any two numbers is a divisor of their sum. 9. Show that every common divisor of 16 and 28 is a divisor of 12, their difference. 10. Show that a common divisor of any two numbers is a divisor of their difference. WKITTEN EXERCISES. 11. What is the greatest common divisor of 126 and 210? Process by Factoring. Resolve 126 and 210 into their prime 126 = ^X$X3X/J' factors. Since every divisor of a num- 2]^0 = 0VJX5X'^ ^^^ ^^ ^ prime factor, or the product of 2X3X7=42, CCA *™ 7 7;'f p'™^ ^Tr' ""* ''"^' ' uct oi all the prime lactors common to 126 and 210 will be their greatest common divisor. 38 COMPLETE ARITHMETIC. What is the greatest common divisor of 12. 60 and 84? 15. 112, 140, and 168? 13. 63 and 126? 16. 84, 126, and 210? 14. 144 and 192? 17. 128, 256, and 1280? 18. What is the greatest common divisor of 288 and 528 ? Process by Dividing. Divide 528 by 288, and 288 by the 288 ) 528 ( 1 first remainder 240, and 240 by the see- 288 ond remainder 48 ; and, there being 240 ) 288 ( 1 no remainder, 48 is the greatest com- ^ mon divisor of 288 and 528. 48 ) 240 ( 5 Since 48, which is the greatest divisor of itself, is a divisor of 240, it is the cpxat- 48 = G.aD.of288and528. ,,, common divisor of 48 ™d 240. Since 48 is a divisor of both 48 and 240, it is a divisor of 288, their swm, and since the greatest common divisor of two numbers is a divisor of their difference, 48 is the greatest common divisor of 240 and 288. Since 48 is the greatest common divisor of 240 and 288, it is a divisor of 528, their sum ; and since the greatest common divisor of two num- bers is a divisor of tlieir difference, 48 is the greatest common divisor of 288 and 528. Note. — Let the pupil show, in like manner, that the last divisor, in the solution of each of the following problems, is the greatest com- mon divisor required. What is the greatest common divisor of 27. S260 and $416? 28. $1815 and $3465? 29. 21451b. and 34711b.? 30. 175, 225, and 275? 31. 240, 360, and 480? 32. 144, 216, and 648? 33. 140, 308, and 819 ? 34. 240, 336, and 1768? 35. What is the greatest common divisor of 1065, 1730, and 2845? 36. What is the greatest common divisor of 156, 585, 442, and 1287? 37. What is the greatest common divisor of 2731 and 3120? 19. 196 and 1728? 20. 336 and 576? 21. 407 and 888? 22. 326 and 807 ? 23. 756 and 1764? 24. 1064 and 1274? 25. 768 and 5184? 26. 741 and 1938? LEAST COMMON MULTIPLE. 39 DEFINITIONS, PRINCIPLES, AND RULES. 65. A Divisor of a number is a number that will ex- actly divide it. A Conifnon Divisor of two or more numbers is a number that will exactly divide each of them. The Greatest Conimou Divisor of two or more numbers is the greatest number that will exactly divide each of them. 66. Principles. — 1. Every prime factor, and every product of any two or more prims factors of a number, is a divisor of tliat number. Conversely, 2. Every divisor of a number is a prime factor, or the product of two or more of its prime factors. 3. The product of all tlie prime factors common to two or more numbers is their greatest common divisor. 4. The divisor of a number is a divisor of any- number of times that number. 5. A common divisor of two numbers is a divisor of their sum, or of their difference. 6. Any common divisor of eitJier of two numbers and their difference is a common divisoi' of the tivo numbers. 67. Rules. — ^1. To find the greatest common divisor of two or more numbers by factoring, Resolve the given ^lumbers into tlieir prime factors, and select the factors which are common. The product of the common factors will be the greatest common divisor. 2. To find the greatest common divisor of two numbers by division, Divide the greater number by the less, and the divisor by the remainder, and the second divisor by the second re- mainder, and so on, until there is no remainder. The last divisor will be the greatest common divisor. Note. — When there are three or more numbers, first find the great- est common divisor of two of them, and then the greatest common divisor of this G. C. D. and a third number, and so on. 40 COMPLETE ARITHMETIC. LEAST COMMON MULTIPLE. 1. What number will 16 exactly divide? 25? 30? 45? Note. — A number will exactly divide its multiple. 2. What number is a multiple of 15? 24? 32? 54? 75? 100? 120? 150? 200? 3. How many multiples has every number? 4. What number will 8 and 10 both exactly divide? 9 and 12? 20 and 25? 5. What number is a common multij^le of 5 and 12 ? 15 and 30? 25 and 50? 6. How many common multiples have two or more num- bers? 7. What is the least number that 7 and 8 will both ex- actly divide? 9 and 12? 20 and 30? 25 and 75? 8. What number is the least common multiple of 7 and 10? 12 and 18? 8, 12, and 16? 9. How many least common multiples have two or more numbers ? 10. Show that all the prime factors of a number are factors of its multiple, and, conversely, that a number con- taining all the prime factors of another number is its mul- tiple. WKITTEN EXERCISES. 11. What is the least common multiple of 12, 18, and 30? Process by Factoring. Resolve tlie numbers into 12 = ^ X ^ X 3 their prime factors, and select .r, 9 V ^ V S ^^^ *^^ different factors, re- peating each as many times as /\ /\ 5(/ j^. jg found in any number. 2X2X3X3X5 = 180, L. C. M. The factor 2 is found twice in 12; the factor 3, twice in 18; and the factor 5, once in 30. The product of 2X2X3X3X5 is the least common multiple required, since it is the least number which contains all the prime factors of 12, 18, and 30. Process by 2)12 15 Division. 42 70 3)6 15 21 35 5)2 5 7 35 7)2 1 7 7 2 1 2X3X5X7X 1 2 = 1 420, L. a M. LEAST COMMON MULTIPLE. 41 What is the least common multiple of 12. 8, 12, 20? 16. 18, 24, 72, 48? 13. 9, 21, 42? 17. 15, 35, 70, 105? 14. 32, 48, 80? 18. 25, 75, 100, 150? 15. 27, 54, 108? 19. $16, $40, $60, $72? 20. What is the least common multiple of 12, 15, 42, 70? Find all the prime factors by dividing the given num- bers by any prime number that will exactly divide two or more of them, thus : Dividing by 2, it is found to be a prime factor of 12, 42, and 70. Write the quotients with the 15 un- derneath. Dividing by 3, it is found to be a prime factor of 6, 15, and 21, and hence it is a prime factor of 12, 15, and 42. Dividing by 5, it is found to be a prime factor of 5 and 35, and hence of 15 and 70. Dividing by 7, it is found to be a prime factor of 7 and 7, and hence of 42 and 70. The remaining quotient 2 is a prime factor of 12. Hence, all the prime factors of 12, 15, 42, and 70 are 2, 3, 5, 7, and 2, and since the product of these several prime factors (2 X 3 X 5 X 7X2 = 420) is the least number that contains each of them, it is the least common multiple of 12, 15, 42, and 70. What is the least common multiple of 21. 12, 18, 30? 26. 30, 45, 48, 80, 120? -22. 8, 28, 70? 27. 16, 30, 40, 50, 75? 23. 9, 20, 15, 36? 28. 15, 27, 35, 42, 70? 24. 15, 24, 25, 30? 29. 8, 28, 20, 24, 32, 48? 25. 18, 21, 27, 36? 30. 2, 3, 4, 5, 6, 7, 8, 9? DEFINITIONS, PRINCIPLES, AND RULES. 68. A 3Iultiple of a number is any number which it will exactly divide. Note. — Every number is an exact divisor of its product by an integer. C.Ar.— 4 42 COMPLETE ARITHMETIC. A Common Multiple of two or more numbers is any number which each of them will exactly divide. The Least Common Multiple of two or more numbers is the least number which each of them will ex- actly divide. Note. — The following definitions may be preferred by some teachers : A Muhiple of a number is the product arising from taking it one or more times. A Common Multiple of two or more numbers is a number which is a multiple of each of them. The Least Common Multiple of two or more numbers is the least number which is a multiple of each of them, 69. Principles. — 1. Every multiple of a number contains all its 'prime factors. 2. A common multiple of two or more numbers contains all their prime factors. 3. The least common multiple of two or more numbers con- tains all their prime factors, and no other factors. 4. The least common multiple of two or more numbers con- tains each of their prime factors the greatest number of times it occurs in either number. 70. Rules. — 1. To find the least common multiple of two or more numbers by factoring, Resolve each of the num- bers into its prime factors, and then select all the different factors, taking each the greatest number of times it is found in any number. The product of the different factors, thus selected, will he the least common multiple. 2. To find the least common multiple of two or more numbers by division, Write the numbers in a line, and divide by any prime divisor of two or more of them, tvriting the quotients and the undivided numbers underneath. Divide these resulting numbers by any prime divisor of two or more of them, and so proceed until no two of the resulting numbers have a common prime divisor. The product of the divisors and tJie last result- ing numbers will he the least common multiple required. js^OTE.— If no two of the given numbers have a common divisor, their product will be the least common multiple FRACTIONS. 43 SECTION VIII. FRACTIONS FOURTHS HALVES SIXTHS NUMERATION AND NOTATION. 1. If a single thing be divided into two equal pieces, what part of the whole will one piece be ? 2. If a single thing be divided into four equal pieces, what part of the whole will one piece be? Two pieces? Three pieces? 3. How many halves in a single thing or unit? How many fourths ? 4. Which is the greater, one half or one fourth of a unit? How many fourths in one half? 5. What is meant by one third of a unit? Two thirds? One sixth? Three sixths? Two fifths? Four fifths? 71. Such parts of a unit as one half, two thirds, three fourths, etc., are called Fractions. A fraction may be ex- pressed in figures by writing the figure denoting the number of equal parts, into which the unit is divided, below a short 44 COMPLETE ARITHMETIC. horizontal line (^), and the figure denoting the number of equal parts taken, above the same line (^). Thus, f ex- presses five sixths of a unit. 6. What does f express? What does the figure 7, below the line, denote? The figure 5, above the line? Bead the following fractions, and tell, in each case, what each figure denotes : 7. f 10. 1 13. J, 16. A 8. i U. f 14. A 17. U 9. f 12. i 15. A 18. U Write the following fractions in figures : (19) (20) (21) Two fifths ; Seven twelfths ; Twenty-four fortieths ; Seven ninths; Ten thirteenths ; Thirty-five fiftieths; Ten ninths. Twenty seventeenths. Forty fifty-fifths. 22. Is the fraction f greater or less than 1? Why? 23. Is I greater or less than a unit? Why? Compare the value of each of the following fractions with a unit or 1 : 24. f 26. Y 28. I 30. {-} 25. I 27. j\ 29. H 31. U 32. Deduce from the above examples a general statement of the value of fractions as compared with a unit or 1. DEFINITIONS AND PRINCIPLES. 72. A Fraction is one or more of the equal parts of a unit. The unit divided is called the Unit of the Fraction ; and one of the equal parts, into which it is divided, is called a Fractional Unit. An integer is composed of integral units, and a fraction of fractional units. 73. A Common Fraction is expressed in figures by two numbers, one written over the other^ with a line between them. Note. — Decimal fractions are a varietv of common fractions. (Art. 112.) FRACTIONS. 45 The number above the line is called the Numerator ; and the one below the line, the Denominator. The Denominator of a fraction denotes the number of equal parts into which the unit is divided. The Numerator of a fraction denotes the number of equal parts taken. The numerator and denominator are called the Terms of the fraction. 74. Principle. — The value of a fraction is less than 1 ivhen its numerator is less than its denominator ; equal to 1 when its numerator equals its denominator ; and more Uian 1 when its numerator is greater than its denominator. 75. Common Fractions are Proper or Improper. A JProper Fraction is one whose numerator is less than its denominator ; as, |, |. An Improx^er Fraction is one whose numerator is equal to or greater than its denominator. The Talue of a proper fraction is less than one ; and the value of an improper fraction is equal to or greater than one, and hence it is regarded as not properly the fraction of a unit. 76. Fractions are Simple^ Compound , or Complex. A Simple Fraction is a fraction not united with an- other, and both of whose terms are integral ; as, |. A Compound Fraction is a fraction of a fraction ; as, I off; |of 3f A Complex Fraction is one having a fraction in A Mixed Wumber is an integer and a fraction united ; as, 5 J, 16|. 77. The fraction | may be considered as expressing 3 fifths of 1 unit, or 1 fifth of 3 units ; and hence the numer- 2 one or both of its terms ; as, h 3' f' qi 46 COMPLETE ARITHMETIC. ator of a fraction may denote the number of units to be divided, and the denominator the number of parts into which the numerator is to be divided. Thus, f may be read 5 sixths, or 1 sixth of 5, or 5 divided by 6. Hence, A frojction may express an unexecuted division, the numer- ator denoting the dividend, tJie denominator the divisor, and the fraction itself the quotient. EEDUCTION OF FRACTIONS. Case I. "Wh-ole or IVTixed ^N'mnbers reduced to Improper Fractions. 1. How many thirds in an apple? In 4 apples? 7 ap- ples? 10 apples? 20 apples? 2. How many fifths in 3 melons ? In five melons ? 8 melons? 12 melons? 15 melons? 3. How many sixths in 1 ? 5? 8? 12? 20? 4. How many fourths of an inch in 2 and 1 fourth inches? In 3| inches ? 6f inches ? 30^ inches ? 5. How many fifths in 3|? 4|? 12f? 16f? 6. How many tenths in 5xV? 8fV? 12 ,\? 15^? WRITTEN PROBLEMS. 7. Reduce 225 to sevenths. 225 1 to sevenths. Process. Process. 225 225^ 7 7 1575, Ans, l^P, Ans, 7 7 8. Reduce 324 to ninths. 324|^ to ninths. 9. Reduce 48fi to 15ths. 65|| to 15ths. 10. Reduce 54^V to 20ths. 135|J to 30ths. 11. Reduce 63 j\ to an improper fraction. 12. Reduce 74 {i to an improper fraction. 13. Reduce 206/^ to an improper fraction. REDUCTION OF FRACTIONS. 47 14. Reduce 145/^ to an improper fraction. Reduce to an improper fraction, 15. 137^ 17. eOO^f 19. 208^ 16. 408^0 18. 365^ 20. 607^% 78. Rules. — 1. To reduce an integer to a fraction, 3ful' tiply the integer by the given denominator, and write the denomi- nator under the prodmit. 2. To reduce a mixed number to a fraction, Multiply the integer by the denominator of the fraction, to the product add tJie numerator y and write the denominator under the residt. Case II. IiTiproper jFractions reclncecl to "Whole or IVEixed. 21. How many dollars in 8 half-dollars? 16 half-dollars? 30 half-dollars ? 22. How many pints in 9 thirds of a pint? 15 thirds of a pint ? 33 thirds of a pint ? 23. How many days in 20 fifths of a day? 35 fifths of a day? 42 fifths of a day? 24. How many units in 36 ninths? 63 ninths? 75 ninths? 25. How many units in -2/-? A^? -%^-? V? 26. How many units in fl? f^? V^? %^-'i WKITTEN PKOBLEMS. 27. Reduce ^^~ to a whole number. Process : ^6^^.= 256 h- 16 = 16, Am. 28. Reduce -\^/ to a mixed number. Reduce to a whole or a mixed number, 29. W 32. -«# 35. m^ 88. -V^- 30. -W- 33. -V/ 36. ^-^^A 39. ^ffs 31. %5. 34 1128 37, 426_a 40. i^j. 79. Rule. — To reduce an improper fraction to an integer or mixed number, Divide the numerator of the fraction by the denominator. 48 COMPLETE ARITHMETIC. Case III. Siinple Frac^tions redxiced to Lowest Terms. 41. How many fourths of an inch in 2 eighths of an inch ? In 4 eighths ? 6 eighths ? 42. How many sixths in 2 twelfths ? In 4 twelfths ? 8 twelfths? 10 twelfths? 43. How many sevenths in 4 fourteenths? In 6 four- teenths? 8 fourteenths? 12 fourteenths? 44. How many eighths in 3% ? ig? i|? f-g? 45. How many tenths in 3«o? M ? IV W 46. Reduce f-^, ^f , ^ each to fifths. 47. Reduce J^, f f, Jf, and -g-J each to sixths. 48. Divide both terms of -f^ by 3, and show that the value of the fraction is not changed. 49. Show that the division of both terms of any fraction by the same number does not change its value. WRITTEN PROBLEMS. 50. Reduce -f Jf to its lowest terms. Pkocess. Divide both terms of |£^- by 5, re- 105 -^ ^ 21 21jj-7^3 ducing it to fi; then divide both 140-^5 "28 28 -r- 7 4 terms of f| by 7, reducing it to f. Since f can not be reduced to smaller Or : — — ^ - ' Ans. or lower terms, it is in its lowest terms. 140 ^35 4 Qj.^ ^j^.^g ^^^1^ ^^j,^^ ^f ^05 by 35, their greatest common divisor, reducing the fraction to |. Reduce to lowest terms, 51. t'^ 54. IIJ 57. m 60. ^VA 52. m 55. m 58. ,V^ 61. T%V 53. Ill 56. m 59. tWf 62. |^^ 63. Express the quotient of 195 divided by 105 in its simplest form. Ans. -^-^. 64. Express the quotient of 462 divided by 441 in its simplest form. 128 -- 256. 360 ^ 288. REDUCTION OF FRACTIONS. 49 65. Express the quotient of 576 ^- 432 in its simplest form. 216 ^ 324. 828 -- 506. DEFINITIONS, PRINCIPLE, AND RULES. 80. A fraction is reduced to lower terms when it is changed to an equivalent fraction with smaller terms. A fraction is in its lowest terms when its terms are prime to each other. 81. Principle. — The division of both terms of a fraction by the same nwnher does not cha7ige its value. 82. Rules. — To reduce a fraction to its lowest terms, 1. Divide both temns of the fraction by any common divisor; then divide both terms of the resulting fraction by any common divisor ; and so on, until the terms of the resulting fraction have no common divisor except 1. Or, 2. Divide both terms of the fraction by ihdr greatest common divisor. Case IV. Fractioias reduced to Higlier Tex-ms, and to a Coin- ixioii IDenoininator. 66. How many eighths of a foot in 1 fourth of a foot? In 2 fourths ? 3 fourths ? 67. How many twelfths in 3 sixths ? 4 sixths ? 5 sixths '! 68. How many fifteenths in |? |? |? 69. Change |, f , and ^ each to twelfths. 70. Change f , f , and /^ to fortieths. 71. Change f, y%, and -j^^ to sixtieths. 72. Change f , -f , and j\- to thirtieths. 73. Change f, f, and ^^ to fortieths. 74. Multiply both terms of | by 4, and show that the value of the fraction is not changed. 75. Show that the multiplication of both terms of anv fraction by the same number does not change its value. C.Ar.— 5 50 COiMPLETE ARITHMETIC. WKITTEN PKOBLEMS. 76. Reduce |-, |-, and ||- to equivalent fractions having a common denominator. 7 1 1* Reduce the fractions to twenty-fourths, thus : f = ^ t T^ 10.7 21.11 22 10 .2 1 2 2 24 » 8' 1¥) TI 2 4* 2f 2i 14 Reduce to a common denominator, 77. I, I, f 80. -I, t, tV 83. I, f, ^, IT. ih -]* 91. 1, H. i5> !!' « 92. 1, tt. H, 1S"^J Hi 9ti II tf ft nominator. Change the fractions to 96ths. 5 — 60. 7 42. 11 44. 21. .63 ?■ — ^5 > T^ — 96 > 1¥ ^^ » -^-^ M- Reduce to the least common denominator, 88. I, I, I, i kQ _7 8 1 1 1 H ^^- l¥» yS"' TTT' a^ DEFINlTIOi\S, PRINCIPLE, AND RULES. 83. A fraction is reduced to higher terms when it is changed to an equivalent fraction with greater terms. 84. Several fractions are reduced to a Common Denomi- nator when they are changed to equivalent fractions, with the same denominator. When the common denominator of several fractions is the smallest denominator which they can have in common, it is called their Least Common Denominator. 85. Principle. — TJie multiplication of both terms of a frac- tion by the. same number does not change its value. REDUCTION OF FRACTIONS. 51 86. EuLEs. — 1. To reduce a fraction to higher terms, Divide the given denominator by the denominator of the fraction, and multiply both terms by the quotient. 2. To reduce fractions to the least common denominator, Divide the least common midtiple of the denominators by the de- nominator of each fraction, and multiply both terms by the quotient. Case V. CoiTipovind lETractions I'ecliaced to Siixiple Fractions. 93. How much is 1 half of 1 third of a pear ? 1 half of 1 fourth of a pear? 94. A father divided |^ of a pine-apple equally between 3 boys : what part of the pine-apple did each boy receive ? 95. What is ^ of i? \ of |? \ of | ? 96. What is ^ of i? i of f ? 97. What is i of ^? i of f ? I of |? 98. What is I of ^? j of I? f of |? 99. What is I of I? I of f ? I of |? 100. What is I of 4? ^ of f ? I of ^? 101. What is I of 12? \ of 12^? \ of 13^? SoLUTiox.— I of 13|^ = ^ of 12, which is 4, + ^ of 1^ or |, wliich is f or i. 4 + I == 4|. Hence, \ of 131 is ^, 102. What is \ of 17i? i of 21|? \ of 33^? 103. What is I of 12? I of 12^? | of 16^? 104. What is f of 22^? f of 25 J? J of 37^? 105. What is g of 33^? | of 42^? j\ of 62^? WKITTEN PROBLEMS. 106. Reduce % of y^ of 7| to a simple fraction. Process. f of ^ of 7i = ^^X-AXl5L225_5 ^,,,, ^ '" ^ 9 X'15X 2 270 6 '' ^ 0Xi:0X 2 6 3 52 COMPLETE ARITHMETIC. Reduce to a simple fraction, 107. I of I of 4 111. f of f of f of 3J 108. f of ^ of 2^ 112. I of y\ of f of 4f 109. j- of If of 1| 113. i of ^ of 2f of 2^ 110. i of f of 21 114. I of 1% of tV of 3i 87. Rules. — To reduce a compound fraction to a simple fraction, 1. Multiply the numerators together for a numerator, and the denominators together for a denominator. Or, 2. Indicate the continued multiplication of the numerators, and also of the denominators, and reduce the resulting fraction to its lowest terms by cancellation. REVIEW PROBLEMS. 115. Reduce 16 to a fraction having 8 for a denominator. 116. Change 35-1-21 to a fraction in its lowest terms. 117. How many 15ths of a gallon in 33^ gallons? 118. Reduce ^/^ to a mixed number with the fraction in its lowest terms. 119. Reduce $f g-, $W, ^W' and $^Y each to whole or mixed numbers. 120. Reduce |, /^, and \l each to 30ths. 121. Reduce 12^, 18|, and 33^ each to 12ths. 122. Reduce y\ of ^^ of 2-^ of 24 to a simple fraction. 123. Reduce |, f , and ^^ to a common denominator ; to the least common denominator. 124. Reduce f, 5^, and f of -f to their least coTnmon denominator. Suggestion. — First reduce 5| and f of f to simple fractions. 125. Reduce ^, J of f , and f of 6| to their least common denominator. 126. Reduce ^ of 2^, f of 3, and ^ of ^ of 6^ to their least common denominator. 127. Reduce J of J, | of 2|, and -f of l^ to their least common denominator. 128. Reduce J, f of 5^, 2^ of 3J, and 2y»7 to their least common denominator. ADDITION OF FRACTIONS. 53 ADDITION OF FRACTIONS. 1. A clerk spends f of his salary for board, | of it for clothing, and ^ for other expenses: what part of his salary does he spend? 2. How many ninths in -f, f, and ^? 3. A man traveled ^ of his journey the first day, and J of it the second day : what part of the journey did he travel in the two days? 4. How many twelfths in ^ and i? | and ^? 5. A owns I of a vessel, and B |- of it : what part of the vessel do both own ? 6. What is the sum of | and | ? | and f ? 7. fandyV? f and ^? | and |? fV and I? 8. 4 and ^? ^ and f? f and ^^^^ ? | and J? 9. i and 51? 2 J and 6^? 5^ and 6^? 8' and 9|? Suggestion. — First add the fractions and then the integers. 10. Show that fractions having a common denominator, express like fractional units, and that only like fractional units can be added. WRITTEN PROBLEMS. 11. What is the sum of ^% ^|, and ^? Process: ± + il + iP — 9 + 16 + 10 _ 3_,5 _ 11.2, Am. 2 3 ' 8 3 ' 23— 23 ' 2 3 2 3' ^"^' 12. What is the sum of \i, jf, -]|, and yV? 13. What is the sum of ^, |?, ||, and f J? 14. What is the sum of J-fJ, j\\, and J-gl? 15. What is the sum of f, j\, and ii? Process. " Since unlike fractional units can not be added, f + 1^2.+ xi = reduce the fractions f, f^, and \^ to a common f f + 4 f + If = denominator, and then add the resulting frac- fi-=r 1||, Am. tions. 54 COMPLETE ARITHMETIC. 16. Add f and If 21. ^%, ^% |3, and U- 17. I, ,V, and j\. 22. |, |, j^, and i^. 18. I, li, and J-f. 23. ||, if f ?, and f. 19. f, 1^0' and If. 24. |, y^* A^ and ^^. 20. A M, and fi 25. ^-L, _i_, ^i^, and ^^,. 26. Add I, f of f , and f of j of 2-^. PJ^ocess. Since -| of | = f , and f of | of 2^ = f , the sum of | -f f of r=lH f+70ff0f2^=f + f + i 27. Add I of I, I of fV of 2^, and |. 28. Add I of 2i 'l of I, and f of ^ of 6. 29. Add ^\, } of 5, and f of f of |. 30. Add j\ of 2, I- of f , i of i of T%, and 3f 31. Add 33^, 37^, 55f, and 66|. Process. ooi^ 4 The sum of 33|, 37|, 55f, and 66f, equals the sum 37 1 Y of i + ^ +1 + f added to the sum of 33 + 37+ 55 + 66. 553 V ^ i + ^ + f + t==2x\or2f Write the i under the frac- n(^2 8 tions and add the 2 with the integers. The sum is 193^, ^ns. 32. Add 39i, 56f, 88|-, and 104^^^. 33. Add 45, 87f, 66|, and 75i. 34. Add 121, 16|, 18f, 30f 33i, and 62^. 35. Add f , I of f , 16|, and 48^. 36. Add $5.12^, $3.18|, $8.25, and $3.81j. 37. Add f, j% } of 5i, and 65/x. 38. Add f , I l^\, and ,\ of 2|. PRINCIPLES AND RULES. 88. Principles. — 1. Only like fractional units can be added. Hence, 2. Fractions must have a common denominator before they can be added. SUBTRACTION OF FRACTIONS. 00 89. Rules. — 1. To add fractions, Reduce the fractions to a common denominator, add the numerators of the nmu fractions, and under Hie sum write tJie common denominator. 2. To add mixed numbers, Add the fractions and the in- tegers separately, and combine the results. Notes. — 1. Compound fractions must be reduced to simple fractions before they can be added. 2. When mixed numbers are small they may be reduced to im- proper fractions and then added. SUBTRACTION OF FRACTIONS. 1. A boy spent | of his money for a slate: what part of his money has he left? 2. How much is f less f ? f less f? f less f? 3. How much is \^ less 3^? ]-^ less /j? |^ less j\? 4. A bought f of a bushel of clover seed and sold ^ of a bushel to B : what part of a bushel has A left ? Suggestion.— Change f and ^ to twelfths. 5. How much is f less | ? J less | ? | less J? 6. f less f ? f less f ? | less |? | less J ? 7. ^ less f ? I less ^? f less | ? f less |? 8. f less I-? I less f ? | less |? ^ less |? 9. 1^^ less I? fi less |? | less jV • I less | ? 10. 51 less 3|? 6| less 4^? 9 J less 7^? 12^ less GJ? 11. Why can not | be subtracted from | without first re- ducing the fractions to a common denominator? WRITTEN PROBLEMS. 12. Subtract i| from ||. Pkocess: ||-i| = 27-i9_^^^^^ 13. Subtract |i from {§ ; ^ from |f; f| from ff 14. Subtract ^2^ from jf Process : \i - /^ = M - M = M, ^ns. 56 COMPLETE ARITHMETIC. How much is 15. a- f? 18. il-U? 21. M-A? 16. H-A? 19. M-ii? 22. 2 3 17 ? 3« 54 • 17. ^-vV? 20. A-A? 23. It-i*? 24. From | of 4 take |^ of J of |. Process : \ of * - | f of | of | — | f — -^ = ^%, ^?is. 25. From | of f take | of J of |. 26. From | of 7 take i of | of 7. 27. From ^ of | of | take f of | of f . 28. From | of ^ of 2^ take j\. 29. From 340| take 247f . Process. First subtract the fractions and then the integers. 0-Q2 8 Since h% is greater than -2%, add | g- to ^^0, making j^, ^^ and then subtract ^f from y, writing the diiference, 4' 2 i.|^ under the fractions, and adding 1 (|^) to the 7 92|-|, -4ws. units before subtracting the integers. 30. 93| — 46^ = ? 31. 561 — 37^:-=? 32. 108f — 901=:? 36. What fraction added to 37. What number added to 6f will make 16 f ? 38. From the sum of § and | take their difference. 39. From f-f | take | — |. 40. From| + |+T^o take ^ of If 41. From -J + I take -^-^ — ^ ^f 5^ 42. From a cask containing 45^ gallons of sirup, a grocer sold one customer 16| gallons and another 21f gallons: how many gallons remained unsold ? 43. A man bequeathed yu ^^ ^^^ property to his wife, j\ of it to his children, and the remainder to a college for its better endowment. What part of his property did the col- lege receive ? 44. A man owning f of a factory, sold ? of his share : what part of the factory did he still y I^ractions. 20. How many times is | of a cent contained in 4 cents ? Solution. — In four cents there are 20 fifths of a cent, and 2 fifths of a cent are contained in 20 fifths of a cent 10 times. 21. If a fruit jar hold f of a gallon, how many jars will hold 6 gallons? 12 gallons? 18 gallons? 22. If f of a yard of silk will make a vest, how many vests will 5 yards make? 7 yards? 10 yards? 23. If a yard of cloth cost $f, how many yards can be bought for $10 ? For $15 ? For $20 ? 24. How many times is | contained in 8? f in 12? f in 9 ? I in 15 ? I in 9 ? f in 12 ? 25. How many times is f contained in 12?^ in 15? 26. Show that 8 -- f == ^^. o WRITTEN PROBLEMS. 27. What is the quotient of 25 -f-|? Process : 25 -:- | =- ^^^ ^ 28f , Ans. Note. — It will be noticed that the integer is muhiplied by the denominator of the fraction and the product divided by its nu- merator. What is the quotient of 28. 21 --3^? 31. 100--!^? 34. 75 - -6i? 29. 42 --14? 32. 96 --tt? 35. 120- -3i? 30. 72 ^|4? 33. 125 ^li? 36. 225- -5i? 100. Rules. — To divide an integer by a fraction, 1. Mul- tiply the integer by the denominator of the fraction, and divide the product by the numerator. Or, 2. Divide the integer by the numerator, and multiply the quo- tient by the denominator. 64 COMPLETE ARITHMETIC. Case III. Fractions clivitled by Fractions. 37. How many times is f of an inch contained in i of an inch ? -I of an inch in 4 of an inch ? 38. How many times f in f ? | in f ? f in Y? 39. How many times | in |? | in V? f in Y-? f in 15? _6_ ;„ _9_9 _6_ in 13 9 T • 1 1 "M 1 • 1 1 "* 1 1 • 40. How many times is ^ contained in f ? -^ in |? Suggestion. — Change the fractions to tAvelfths. 41. How many times | in |? f in |- ? i in |? f i^i A? 49 3 in 4 ? 3 in 7 ? 3 in 1 1 *? 2 in 3 9 2 in 5 9 43. Show that the quotient of two fractions having a com- mon denominator, equals the quotient of their numerators. -WKITTEN PKOBLEMS. 44. Divide | by |. Peocess : |. - f = |^-| = H = IH, Ans. Since | = ^><^, and f=.^><-8, |_^|^7X_5 ^3X8^7^6^ ^ 40 ' ^ 40 ' ^ ^ 40 40 3X8 It is thus seen that inverting the terms of the divisor, and taking the product of the numerators for the numerator, and the product of the denominators for the denominator, is the same as reducing the frac- tions to a common denominator, and dividing the numerator of the dividend by the numerator of the divisor. 7 V 5 Note.— That | -^ f = ^ mav also be thus explained : ^ = 3 times i and since 1 ~ 4 =r. 1-XA t -- ^ = \ of ^^^ = ^^. What is the quotient of 45. A^ii? 49. 3i-2i? 53. A-^4off|? 46. i|-^-TV? 50. 5i-3i? 54. |ofA-|of4? 47. A 48. H-^A- 52. 16^ il? 51. 6-1-121-? 55. -jV of 31- 1 of 2i? o o ra ..,. . 31^9 56_ ||^^of|of3-|? COMMON DIVISOR. 65 57. If a family use ^ of a barrel of flour in a month, how long will 2^ barrels last ? 58. If a bushel of corn cost $|, how many bushels can be bought for $^? For $9 J? 59. If 13 yards of silk cost $171 how many yards can be bought for $48f? For $62^? 60. If a man walk 3^ miles an hour, in how many hours will he walk 20J miles ? 61. At $33^ an acre, how many acres of land can be bought for $841|? 62. By what must f be multiplied that the product may be 26|? 63. Divide the product of 6^ multiplied by 3^ by the quotient of 4^ -4- 5^ ? PRINCIPLES AND RULES. 101. Principles. — 1. The quotient of two fractions having a common denominator, equals the quotient of their numerators. 2. The multiplying of both dividend and divisor by the same number does not change the value of the quotient. 102. Rules. — To divide a fraction by a fraction, 1. Re- duce the fractions to a common denominator, and divide the numerator of the dividend by the numerator of the divisor. Or, 2. Invert the terms of the divisor, and then midtiply the numerators together and also the denominators. Or, 3. Midtiply both dividend and divisor by tJie least common multiple of the denominators of the fractions, and divide the resulting dividend by the resulting divisor. Notes. — 1. The third rule depends on the second principle; and, since multiplying two fractions by their least common multiple changes them to integers, the new dividend and divisor are always integral. Thus, multiplying both fractions by 24, tlie /. c. m., f-f- y'j -^ 15 -^- 14= lj\f ; multiplying by 6, the /. c. m., 6-| -f- 5^ = 40 -T- 33 — 1^^. Compound fractions should first be reduced to simple fractions. 2. It is Hot necessary that the pupil be made equally familiar with these three methods of dividing one fraction by another. He should thoroughly master one of them. C.Ar.—C) 66 COMPLETE ARITHMETIC. COMPLEX FEACTIONS. 4 64. Reduce the complex fraction -I, to its simplest form. Process ■ 1 ^i-f = -4X7^ 5X6 H, Ans. 3 to the simplest form 1- 69. 16| 25 73. -foff foff 77. f-A 1 70. 24 25 16| 74. ?of2i 5* 78. f + f L^ 71. * 12* 1 75. 79. s - 1 12* 76. toftt 80. 1 . 1 1 ■ 1 65. 66. 67. 68. 103. A complex fraction expresses an unexecuted divis- ion, the numerator being the dividend and the denomi- nator the divisor. It is reduced to its simplest form by performing the division as expressed. Notes. — 1. A complex fraction may be changed to a fraction with integral terms, by multiplying both of its terms by the least common mul- tiple of the denominators of its fractions. (Art. 102, Note 1.) Compound fractions must first be reduced to simple fractions. 2. Let the above problems also be solved by this method. NUMBERS PARTS OF OTHER NUMBERS. MENTAL PKOBLEMS. 1. If ^ of a barrel of flour cost $3, what will a barrel cost ? 2. If ^ of a ream of note paper cost 75 cents, what will a ream cost ? 3. Charles gave Henry 7 marbles, which were ^ of all he had : how many marbles had Charles ? DIVISION. 67 4. 15 is I of what number? 5. 16 is -^ of what number? 6. 12J is -|- of what number? 7. 16f is ^ of what number? 8. 22| is i of what number? 9. 24 is f of what number? Solution. — ^^If 24 is f of a number, A is i of 24, which is 12. If 12 is ^ of a number, | is 5 times 12, or 60. Hence, 24 is | of 60. 10. 27 is f of what number? 11. 45 is 1^ of what number? 12. 64 is -jSj- of what number? 13. 27^ is f of what number? 14. 46f is -j^ of what number? 15. 37^ is f of what number? 16. 87^ is -^j of what number? 17. 45 is ^ of how many times 9 ? 18. 63 is -J of how many times 12? 19. 81 is ^ of how many times 20? 20. 108 is H of how many times 15 ? 21. What part of 4 is 1 ? What part of 4 is 3? 22. What part of 6 is 5 ? 9 is 8 ? 12 is 6 ? 23. 11 is 7? 16 is 12? 20 is 15? 18 is 12? 30 is 15? 24. 7 is what part of 21? 8 of 32? 9 of 27? 25. 13 of 39 ? 16 of 72 ? 15 of 25? 60 of 90? 26. i is what part of f ? i of |? ^ of f ? f of J? 27. iof i? I of I? f of I? I of A? |of li? 28. I of 11? f of 4? I of 10? 4 of 8? j- of 10? 29. 5i of 16|? 6J of 33|? m of 37^? 33| of 16|? 30. 3| of 6^? 5^ of 2|? 21 of 3^? 6\ of 12^? PRINCIPLE AND RULE. 104. Principle. — Only like numbers can be compared. 105. Rule. — To find what part one number is of another, Divide the number denoting the part by the number denoting the whole. bg COMPLETE ARITHMETIC. KEVIEW OF FRACTIONS. MENTAL PROBLEMS. 1. A boy having $J gave S| for a knife : how much money had he left? 2. If f be added to a certain fraction, the sum will be ^ : what is the fraction? 3. A laborer spends f of his wages for board and ^ for clothing: what part has he left? 4. A man did -^ of a piece of work the first day, ^ of it the second day, ^ of it the third day, and the remainder the fourth day: what part of the work did he do the fourth day? 5. A man bought a farm, paying f of the price down, ^ of it the first year, ^ the second year, and the remainder the third year: what part did he pay the third year? 6. A man is 42 years of age, and ^ of his age equals the age of his son : how old is the son ? 7. A man bought a cow for $33^ and sold her for | of what she cost : how much did he lose ? 8. If a yard of velvet cost $8^, what will f of a yard cost? 9. Jane's age is 16| years, and Mary's age is f of Jane's : how old is Mary? 10. A man owning -f of a mill sells f of his share: what part of the mill does he still own? 11. Charles bought f of a pound of candy and gave his sister f of a pound, and his playmate f of what remained: what part of a pound had he left? 12. A wife is 35 years of age, and her age is 4 of the age of her husband: how old is her husband? 13. The diflference between f and f of a certain number is 14: what is the number? 14. A farmer sold 50 sheep, which were f of his flock: how many sheep had he before the sale? 15. When Charles is -f older than he now is, he will be 21 years of age: how old is he? REVIEW PROBLEMS. 69 16. A farmer sold f of his farm for $1645: at this rate, ■what was the value of the farm ? 17. A man sold f of his farm and had 64 acres left: how many acres had he at first ? 18. A man sold a horse for $90, which was } more than it cost him : what' was the cost of the horse ? 19. A lady paid $30 for a cloak, which was f more than she paid for a dress : what was the cost of the dress ? 20. f of 42 is ^ of what number ? 21. A man is 45 years old, and | of his age is f of the age of his wife: how old is his wife? 22. Samuel is | as old as Harry, and James, who is 9 years old, is f as old as Charles: how old are Charles and Samuel ? 23. A man gave $150 for a watch and chain, and the chain cost f as much as the watch : what did each cost ? 24. If to A's age there be added | and f of his age, the sum will be 48 years : what is A's age ? 25. A farmer's sheep are in 4 fields; the first contains f of all, the second ^, the third |, and the fourth 46 sheep : how many sheep in the 4 fields? 26. A saddle cost $35, and -f of the cost of the saddle was f of the cost of a bridle : what was the cost of the bridle? 27. If to f of a man's age 15 years be added, the sum will be f of his age : how old is he ? 28. The distance from Cleveland to Columbus is 138 miles, f of which is |^ of the distance from Columbus to Cincinnati : what is the distance from Columbus to Cin- cinnati ? 29. f is f of what number? 30. If I of the value of a house equals f of the value of a lot, and the value of both is $4400, what is the value of each? 31. If -| of A's money equals f of B's, and both together have $340, how much has each? 70 COMPLETE ARITHMETIC. 32. If I of A's age is f of B's, and | of B's is 20 years : what is the age of each ? 33. If ^ of a yard of velvet cost $2|, what will | of a yard cost? 34. How many pounds of honey, at $| a pound, can be bought for $3 ? ^ • 35. How many bushels of apples, at $| a bushel, can be bought for $16|? 36. If a barrel hold 2| bushels, how many barrels will be required to pack 55 bushels of apples? 37. If 5| lb. of sugar cost $1, how much will 49 1 lb. cost? 38. If f of a yard of silk cost H}, how many yards can be bought for $10|? 39. If 3f yards of cloth cost $5^, what will G^- yards cost? 40. If a train of cars run f of a mile in If minutes, how many miles will it run in 15 minutes? 41. If 4 pounds of coffee cost $f, what will 7-|- pounds cost ? 42. If 12^ tons of hay will feed 5 horses a year, how many tons will feed 8 horses a year? 43. If a rod 5 feet long casts a shadow 8^ feet long, what is the length of a pole whose shadow, at the same time of day, is 17| feet? 44. If 3 men can do a piece of work in 10 J days, how long will it take 8 men to do it? 45. If a barrel of flour will supply 12 persons 4f weeks, how long will it supply 7 persons ? 46. A can do a job of work in 12 days, and B in 10 days : how long will it take both to do it? 47. A and B can do a certain work in 8 days, and A can do it in 12 days : in what time can B do it? 48. A and B can mow a field in 10 days, and A can mow only f as much as B : how long would it take each to mow the field? 49. How is the value of a proper fraction affected by REVIEW PROBLEMS. 7! adding the same number to both of its terms? By sub- tracting the same number? (Illustrate, taking f.) 50. How is the value of an improper fraction, greater than 1, aifected by adding the same number to both of its terms? By subtracting the same number? (Illustrate.) ^VRITTEN PROBLEMS. 51. Add I, i, A of yV, and 3i 52. From -| of 1^ take 4 of J-|. 53. From the sum of 27f and 20f take their difference. 54. Multiply If by 35; 35 by if; |f by f?; 3^ by 21. 55. Divide fj by 32 ; 32 by if ; if by |; 4i by 31. 9 56. H + T^-what? ii-A? HXtV? ii-A 57. Multiply 2045f by 35 ; 806 by 84| ; 301 by 16f . ^ 58. Divide 347f by 15; 692 by 21|; 19| by 16f. 59. A farm is divided into five fields, containing respect- ively 21f A., 34| A., 45 J A., 56f A., and 29^ A.: how many acres in the farm ? 60. There are 30^ sq. yd. in a square rod : how many square rods in 786^ sq. yd.? 61. A man travels 5J miles an hour: how long will it take him to make a journey of 75f miles ? 62. At $8f a ton, how many tons of hay can be bought for $1081? 63. If 3*3- of an acre of land cost $68, what will 12^ acres cost? 64. If f of a yard of velvet cost $84, how many yards can be bought for $1964? 65. If a number be diminished by 4 of 4-f of itself, the remainder will be 69 : what is the number ? 66. A pedestrian walked -^ of his journey the first day, f of it the second day, and then had 24 miles to travel : how long was the journey? 67. A man pays $350 a year for house rent, which is ^4 of his income : what is his income ? Q8. A man bequeathed to his wife $4860, which was if of his estate : what was the value of the estate ? 72 COMPLETE ARITHMETIC. 69. A graded school enrolls 208 boys, and -j^ of the pupils are girls : how many pupils are enrolled in the school ? 70. A man owning f of a ship sells f of his share for $3480 : at this rate, what is the value of the ship ? 71. A owning f of a mill, sold f of his share to B, and ^ of what he then owned to C for $460 : what was the value of the mill at the rate of C's purchase ? 72. A owns j^ of a section of land ; B, j^ of a section ; and C, y^^ as much as both A and B : what part of a sec- tion does C own? 73. A bought I of a factory for $21840, and sold f of his share to B, and -| of it to C : what part of the factory did A then own? 74. A and B together own 396 acres of land, and f of A's farm equals | of B's : how many acres does each own ? 75. A stock of goods is owned by three partners, A own- ing f , B 1^, and C the remainder ; the goods were sold at a profit of $6160: what was each partner's share? 76. I of a stock of goods was destroyed by fire, and f of the remainder was damaged by water, and the uninjured goods were sold at cost for $5280: what was the cost of the entire stock of goods ? 77. A man paid | of his money for a farm, i of what remained for repairs, ^ of what then remained for stock, ^ of what then remained for utensils, and then had left $650 : how much money had he at first? 78. A merchant tailor has 67f yards of cloth, from w^hich he wishes to cut an equal number of coats, pants, and vests : how many of each can he cut if they contain 3f , 2|-, and li yards respectively? 79. An estate was divided between two brothers and a sister ; the elder brother received | of the estate, the younger ■^jf, and the sister the remainder, which was $450 less than the elder brother received: what was the value of the estate? What was each brother's share ? DECIMALS. 73 SECTION IX. DECIMAL FRACTIONS, NUMERATION AND NOTATION. 1. If a unit be divided into ten equal parts, what is one part called? 2. If a tenth of a unit be divided into ten equal parts, what is one part ? What is -^ of -^ ? 3. If a hundredth of a unit be divided into ten equal parts, w^hat is one part ? What is -^ of y^ ? 4. What part of a tenth is a hundredth? What part of a hundredth is a thousandth? 5. How do the fractions ^, yf g^, and y^qq compare with each other in value? y^^, y^, and y^u^ 106. Since the fractional units, tenths, hundredths, thou- sandths, etc., decrease in value tenfold, they are expressed, like the orders of integers, on a scale of ten. This is done by extending the orders to the right of units, and calling the first fractional order tenths, the second hundredths, the third thousandths, etc. A period is placed at the left of the order of tenths. Thus, ^ is written .5; j^ is written .05; y^"^^ is written .005, etc. Copy and read (6) (7) (8) (9) (10) Cll) .4 .03 .002 .06 .07 .005 .7 .05 .004 .006 .004 .4 .6 .08 .006 .08 .8 .07 .9 .09 .007 .5 .09 .009 12. How many tenths and hundredths in .25? In .45? ,63? .78? .84? .69? .39? C.Ar.-7. 74 COMPLETE ARITHMETIC. 13. How many tenths, hundredths, and thousandths in .325? In .246? .307? .405? .056? 14. How many tenths, hundredths, and thousandths in .045? In .407? .008? .065? .607? .325? 15. How many hundredths in yVo? In yfo? -34? .42? 16. How many thousandths in y||^? In y^w? .325? .065? .205? .008? .046? 107. When the right-hand figure of a decimal denotes hundredths, the whole decimal denotes hundredths, and when the right-hand figure denotes thousandths, the whole decimal denotes thousandths. Thus, .25 is read 25 hun- dreths: .325 is read 325 thousandths. Copy and read (17) (18) (19) (20) (21) .15 .016 .245 .8 .007 .42 .024 .354 .63 .038 .36 .045 .403 .086 .462 .50 .083 .587 .369 .507 .06 .007 .067 .504 .45 108. When fractions denoting tenths, hundredths, thou- sandths, etc., are expressed like integers, on the decimal scale, they are said to I >e expressed decimally. Express decimally (22) (23) (24) (25) (26) t\ TT)\o IT to" tV% Tciro i^ 1000 Ah TfoT A«A xh xioTT tV^ T^TT t¥A -ih T¥Tr t\%\ TTnrTT tMtt t\\ t'A tVA tWit A , lOOO 27. What is the name of the third decimal order? The fourth? The fifth? The sixth? 28. What does each significant figure of .0034 denote? Of .00275 ? Of .03405 ? Of .000325 ? Of .030056 ? DECIMAL FRACTIONS. 75 Copy and read (29) .246 .0246 .708 .0708 .3425 (30) .0635 .00635 .3464 .03464 .32875 (31) (32) 00647 .0307 ,000647 .03007 04056 .030007 004056 .034005 ,32453 .450605 109. When a decimal fraction is expressed decimally, the right-hand figure is written in the order indicated by the name of the decimal. Thus, t^Wo¥ is written .00325. Express decimally (33) (34) Tiny 8 28 TOTF-Q 35(5 TITTO' 10000 33 TiyTFTTO" 3 04 2 TTTOlfO' (35) 7 1 ouiroir 1 00000 2_0 8 lOOTTOlT 3 5 6 TTJTrUT)!}" 380 4 5 1 OO^OlT (36) 2 9 Tm^TJinriJ 609 , TCi^oinnj" 4 04 5 T(n)T)(»0Tr 3 3 033 TT)¥U'OirO^ 204056 \-^ f-U^HWi Express decimally (37) 7 tenths ; 24 hundredths ; 29 thousandths ; 405 thousandths; 65 millionths; 5064 millionths ; 40056 millionths. (38) 42 ten-thousandths; 506 ten-thousandths ; 4008 ten-thousandths ; 65 hundred-thousandths ; 6007 hundred-thousandths ; 54008 hundred-thousandths 3004 hundred-thousandths. 39. Eighty-five thousandths. 40. Four hundred and seven thousandths. 41. Ninety-five ten-thousandths. 42. Six hundred and forty-four ten-thousandths. 43. Seven thousand and eighty-two ten-thousandths. 44. Fifty-seven hundred-thousandths. 45. Seven hundred and eight hundred-thousandths. 7G COMPLETE ARITHMETIC. 46. Nine thousand and forty-eight hundred-thousandths. 47. Six hundred and four millionths. 48. Seven thousand six hundred and forty-three mill- ionths. 49. Forty thousand and sixty-three millionths. 110. An integer and a decimal may be written together as one number, as 6y% or 6.5; 25y^ or 25.07. In reading such mixed decimal numbers, the integer and the decimal are connected by and. Thus, 4.5 is read 4 and 5 tenths. 50. Read45.-6; 30.25; 204.045; 84.0307. 51. Read 2005.045; 408.00075; 3040.0046; 50060.00705. 52. Read 400.045; 500.0063; 7000.0084; 60000.00006. Suggestion. — In such cases read the integer as units; as, four hundred units and forty-five thousandths. The omission of the word units changes the mixed number to a pure decimal. 53. Read 5600.0084; 40508.0307; 75000.000605. 54. Read 300000.000003; 35000000.000035. 55. Write decimally 56ji^ ; 604yf |^ ; 400t^%Vo- 56. Write decimally 207^^11^; 2560r^Vo%W 57. Three hundred units and three hundred and forty- eight millionths. DEFINITIONS, PRINCIPLES, AND RULES. 111. A Decimal Fraction is a fraction whose de- nominator is some power of ten. The word decimal is derived from decern, a Latin word meaning ten. It is applied to this class of fractions because the successive fractional units or orders decrease tenfold, or on the scale of ten. Note.— The powers of ten are 10, 100, 1000, etc. (Art. 388.) 112. Decimal fractions may be expressed in three ways : 1. By words; as, three tenths, twelve hundredths. 2. By writing the denominator under the numerator, in the form of a common fraction ; as, f\, j^J^^. DECIMAL FRACTIONS. 77 3. By omitting the denominator and writing the fraction in a decimal form; as, .3, .012. The denominator is un- derstood. Three tenths, j\, and .3, each express the same decimal fraction, but the term decimal is usiuilly applied to decimal fractions when expressed by the third method. Since common fractions may have 10, 100, etc., for a denominator, it follows that decimal fractions are a class of common fractions. 113. The Decimal I^oint is a period placed at the left of the order of tenths, to designate the decimal orders. 114. A Mixed Decimal is a decimal ending at the right with a common fraction; as, .6f, .033^. 115. A Mixed Decimal Nuinher is an integer and a decimal written together as one number. It is called more simply a Mixed Number. The orders on the left of the decimal point are integral, and those on the right are decimal. The decimal orders are called Decimal Places. 116. The foHowing table gives the names of a few in- tegral and decimal orders, and shows the relation between them: 1 S' Ml . -1 .: 2 ^ I II 000000000.00000000 Integral Orders. Decimal Orders. 117. Prlnciples. — 1. The denominator of a decimal frac- tion is 1 with as many ciphers annexed as there are decimol places in the fraction. -^ 78 COMPLETE ARITHMETIC. 2. The value of the successive decimal orders decreases tenfold from left to right, and increases tenfold from right to left. Hence, 3. The removal of a decimal figure one place to the right DECREASES its volue tenfold, and its removal one place to the left INCREASES its valuc tenfold. 4. The name of a decimal is the same as the name of its right-hand order. Hence, 5. A decimal is read precisely as it woidd he were the denom- inator expressed. 118. Rules. — 1. To read a decimal. Read it as though it were an integer, and add the name of the right-hand order. 2. To write a decimal, Write it as an integer, and so place the decimal point that the right-hand figure shall stand in the order denoted by the name of the decimal. Note. — When the number does not fill all the decimal places, supply the deficiency by prefixing decimal ciphers. ^WRITTEN PROBLEMS. Express decimally 58. Two hundred and five ten-thousandths. 59. Forty thousand and thirty-four millionths. 60. Two thousand and four hundred-thousandths. 61. Six hundred and fifteen ten-millionths. 62. Six hundred units and fifteen ten-thousandths. 63. Fifteen and fifteen thousandths. 64. Three hundred thousand three hundred and three hundred-millionths . 65. Five million and eighty-five ten-millionths. 66. Twelve hundred-thousandths. 67. Four hundred units and four hundred and sixty-fiv^e millionths. 68. Twenty-five and twenty-five thousandths. 69. Five thousand units and five thousandths. 70. Three hundred and seventy-five and three hundred and seventy-five billionths. REDUCTION OF DECIMALS. 79 71. Thirty thousand and forty-six hundred-thousandths. 72. One million and forty-five billionths. 73. Eighty thousand and forty and three hundred and six ten-thousandths. 74. Fifteen thousand units and fifteen ten-thousandths. 75. Seventy-five and five thousand and forty-three mill- ionths. 76. One million units and one millionth. REDUCTION OF DECIMALS. Case I. IDeciixials reclTicecl to IL H • • " 540 ft. Pine Lumber, @ 4.50 per 100 . . . 4 M. Shingles, @ 8.33^ $ Cr. By Cash, Oct. 16, . . . ' . . . ^S25 " " " 23, . . . . . 44 '•' " Medical Services to date . . .15 Received Payment, per due-bill, Charles C. Wilhelm. What is the amount of the due-bill ? 4. Henry Smith bought of John Clarke, of Louisville, Ky., as follows: Mch. 10, 1870, 7 pair calf boots @ $5.75; 6 pair ladies' gaiters @ $3.25; 10 pair children's shoes @ $1.75; Apr. 1st, 12 pair coarse boots @ $3.12^; 6 pair calf BILLS. 99 shoes @ S3. 30; Apr. 12, 7 pair ladies' slippers, $1.33|^; 3 pair calf boots, $d.G2^. Make out and receipt the above bill as clerk of John Clarke. 5. Robert Sterns & Co. bought of Dudley & Bro., Detroit, Mich., Dec. 20, 1869, as follows: 5 doz. ink-stands @ $2. 12 J; 9 boxes steel pens @ $.87^; 8 reams note paper @ $3.50; 5 dozen spellers @ $2.33|^; and 2 dozen copy books @ $1.80. They sold Dudley & Bro. 3 sets outline maps @. $8.25, and paid them $15 in money. Make out the above bill and receipt by due-bill. 6. Mrs. C. B. Jones bought of Cole, Steele & Co., of Indianapolis, as follows: Nov. 12, 1869, 23 yds. calico @ 16|c.; 45 yds. sheeting @ 20c.; Dec. 7th, 12 yds. silk @ $1,621; 8 handkerchiefs @ 45c.; 2 pair kid gloves @ $1.87|-. Make out and receipt the above bill. DEFINITIONS. 141. An Account is a record of business transactions between two parties, with specifications of debts and credits. The party owing the debts specified, is called the Debtor, and the party to whom they are due, is called the Creditor. 142. A JSill is a written statement of an account. It is drawn by the creditor against the debtor, and gives the time and place of the transaction, and the names of the parties. AVhen the debtor has made payments on the account, or has charges against the creditor, such payments or charges are called Credits. They are entered as in Bill 3. 143. A bill is receipted by writing the words ^'Received Payment" at the bottom, and afiixing the creditor's name. This may be done by the creditor, or by a clerk, agent, or any other authorized person. If the debtor is not able to pay a bill when presented, it may be accepted by writing the word ''Accepted" across its face, with date and signature. When a bill is paid by a 100 COMPLEX K ARITHMETIC. promissory note or due-bill, the fact may be added to the words "Beceived Payment" as in Bill 3. 144. A Bill of Goods is a written statement of goods sold, with the amount and price of each article, and the entire cost. It is also called an Invoice. When sales are made at different times, the date is written at the left, as in Bill 1. SECTION XL MENSURATION, I. SURFACES.— Definitions. 145. A Line is length. 146. A Straight Line is a line having the same direction throughout its whole extent. Note. — The word line is commonly used to denote a straight line. 147. An Angle is the divergence of two lines meeting at a common point. The point of meeting is called the vertex. A Thus the divergence of the lines A B and A C is the angle ABC, and the point B is its vertex. 148. When a line so meets another line as to make the two adjacent angles equal, each angle is a Right Angle, and the first line is perpendicular to the second. A Thus the two equal adjacent angles ABC and A B D are right angles, and the line A B is perpendicular to the line CD. MENSURATION. 101 149. An Obtuse Angle is greater than a right angle, and an Acute Angle is less than a right angle. Thus the angle A B D is an obtuse angle, and the angle ABC is an acute angle. The line A B is an oblique line. v, 150. A Surface is that which has length and width, but not depth or thickness. 151. A Plane Surface is a surface such that all possible straight lines connecting each two points of it, lie wholly within the surface. It is also called a Plane. Note. — To determine whether the surface of a table is a plane, take a ruler with a straight edge and apply it to the surface in many different directions. If the edge rests uniformly upon the surface, it is a plane. 152. A Mectangle is a plane figure bounded by four straight lines ^ and having four right angles. 153. A Square is a rectangle with its four sides A Square Inch is a square unch. each side of which is an inch in length. The figure represents a square inch of real size. A square foot, square yard, square rod, etc., are squares whose sides are respectively 1 foot, 1 yard, 1 rod, etc., in length. 154. A Triangle is a plane fig- ure bounded by three straight lines and having three angles. equal. 1 inch. 102 COMPLETE ARITHMETIC. 155. A Right-angled Tri- (ingle is a triangle having a right angle. One of the sides including the right angle is called the Base, and the other the Perpendicular or Altitude. 156. A Circle is a portion of a plane bounded by a curved line, all points of which are equally distant from a point within, called the center. ' The curved line which bounds a circle is its Circumference. One-half of a circumference is a Semi- circumference; one-fourth is a Quadrant; and any portion is an A re. 157. The DiafUete?^ of a circle is a straight line passing through the center and terminating on both sides in the circumference. One-half of a diameter is a Radius. All the diameters of a circle are equal, and all the radii are equal. The circumference of a circle is 3.1416 (nearly 3i) times the diameter. 158. The Avea of a plane figure is its extent of surface, or superficial contents. It is expressed by some unit of measure as a square inch, a square foot, etc. 159. The area of a right-angled triangle is one-half the area of a rectangle with the same base and altitude. The triangle A B C is one- c half of the rectangle A B C D. 160. The area of a circle equals the product of the cir- cumference by the one-half of the radius. Note. — This may be illustrated by dividing a circle by diameters into eighths, and considering each a triangle. MENSURATION. ' 103 \ ^ MENTAL PKOBLEMS. 1. How many square inches in a piece of paper 4 inches . long and 1 inch wide ? 4 inches long and 2 inches wide ? 2. How many square feet in a piece of zinc 4 feet long and 3 feet wide ? 4 feet long and 4 feet wide ? 3. How many square inches in a pane of glass 12 inches square ? Then how many square inches in a square foot ? 4. How many square feet in a piece of oil-cloth 7 feet long and 3 feet wide? 8 ft. long and 6 ft. wide? 5. How many square feet in a square yard? 6. How" many square feet in a room 20 by 15 ft. ? 30 by 24 ft. ? 50 by 30 ft. ? Note. —The dhiiensions of a plane figure are usually expressed by writing the word " by," or the sign " X/' between the figures denoting the length and width. 7. How many square yards in a pavement 40 by 5 yd. ? 50 X 4 yd. ? 80 X 5 yd. ? 8. How many square miles in a township 5 miles square? 6 miles square? 9. How many square inches in a right-angled triangle, whose base is 8 inches and whose altitude is 6 inches? 10. The diameter of a circle is 10 feet: what is its cir- cumference ? ^WKITTEN PKOBLEMS. 11. How many square feet in a floor 374- by 23 ft? 12. How many square yards in a walk 124.5 by 3.25 yd.? 13. How many square feet in the walls of a room 24 by 18f ft. and 10^ ft. high? What is the area of the ceiling? 14. How many square chains in a farm 134 chains long and 52.5 chains wide? 15. How many square feet in a city lot 62\ ft. front by 208 ft. deep? 16. A garden containing 3267 square yards is 494- yards wide : hoAV long is it ? 104 COMPLETE ARITHMETIC. 17. A street containing 800 square rods is 33^ rods long: how wide is it? 18. How many yards of carpeting, J of a yard wide, will cover a room 15 by 8 J yd. ? 19. How many square yards in a triangular garden whose base is 54.5 yards, and altitude 33.2 yards? 20. A triangle contains 270 sq. in., and the base is 36 in. : what is its altitude ? 21. The diameter of a circle is 12 inches: how many square inches in its area ? 22. How many square feet in a circle whose diameter is 20 feet ? 161. Rules. — 1. To find the area of a rectangle. Multiply the length by the width. 2. To find either side of a rectangle. Divide the area by tJie oilier side. 3. To find the area of a triangle. Multiply the base by one half the altitude. 4. To find the area of a circle. Multiply the circumference by one fourth of the diameter. Note. — The two dimensions must be expressed in the same tie- nomination. II. SOLIDS. — Definitions. 162. A Solid is that which has length, width, and depth or thickness. It is also called a Volume or Body. A line has only length ; a surface has length and width ; and a solid has length, width, and depth. iiWfW'iii 163. A Bectanf/tilar Solid k a body bounded by six rectangular surfaces. Illllllllllllllllllllllillliili'^i'^i The surfaces bounding a solid are called Faces, and the sides of these faces are called Edges. A rectangular solid has twelve edges. The face on which a solid is supposed to rest is called its Base. MENSURATION. 105 164. A Cube is a body bounded by six equal squares. All its edges are equal. A Cubic Inch is a cube whose edges are each one inch in length. A cubic foot, cubic yard, cubic rod, etc. are each cubes whose edges are respectively 1 foot, 1 yard, 1 rod, etc. 165. A Cylinder is a solid whose two bases are equal and parallel circles. 166. The volume of a body is called its Solid Contents, or Capacity. It is expressed in some unit of measure, as a cubic inch, a cubic foot, etc. MENTAL PROBLEMS. 1. How many cubic inches in a rectangular solid, 4 inches long, 1 inch wide, and 1 inch thick? 2. How many cubic inches in a rectangular solid, 4 inches long, 3 inches wide, and 1 inch thick ? 3. How many cubic inches in a rectangular solid, 4 inches long, 3 inches wide, and 2 inches thick? 4. How many cubic feet in a block of marble 6 ft. long, 3 ft. wide, and 2 ft. thick? 10 ft. long, 5 ft. wide, and 4 ft. thick? 5. How many cubic feet in a cubic yard? 6. How many cubic feet in a bin 6 ft. long, 3 ft. wide, and 3 ft. deep? 8 ft. long, 5 ft. wide, and 2 ft. deep? 7. How many cubic yards in a room 5 yd. long, 4 yd. wide, and 3 yd. high? 106 COMPLETE ARITHMETIC. WKITTEN PROBLEMS. 8. How many cubic feet in a block of granite 16 ft. long, 8 ft. wide, and 5 ft. thick? Process. ^"^ block IG ft. long, 1 it. thick, and 1 ft. wide, con- IP £, tains 16 cu. ft.; and a block 16 ft. long, 1 ft. thick, g ' ' and 8 feet wide, contains 8 times 16 cu. ft., or 128 128 cu. ft. ^'^- ^^' i ^"d a block 1 6 ft. long, 8 feet wide, and 5 ft. 5 thick, contains 5 times 128 cu. ft. = 640 cu. ft. Hence, 640 cu. ft. solid contents = 16 cu. ft. X 8 X 5. 9. How many cubic feet in a pile of wood 45 ft. long, 3^ ft. wide, and 7 ft. high ? 10. How many cubic yards in a cubic rod? 11. How many cubic feet in a cube each of whose edges is 12i ft. in length ? 12. A building, 65 ft. by 44 ft., has a foundation wall 12 ft. deep and 2 ft. thick: how many cubic feet in the foundations ? 13. A pile of wood, containing 840 cu. ft., is 30 ft. long and S^ ft. wide : how high is the pile ? 14. If 27 bricks make a cubic foot, how many bricks will make a wall 45 ft. long, 27 ft. high, and 2^ ft. thick ? 15. How many cans, 6 by 4 by 2 in., can be placed in a box 30 by 18 by 20 in the clear ? 16. The base of a cylinder is 12 inches in diameter, and its altitude is 25 inches : how many cubic inches in its solid contents ? 167. Rules. — 1. To find the solid contents of a rectan- gular solid. Multiply the length, width, and thichiess together. 2. To find the length, width, or thickness of a rectangular solid, Divide Hie solid contents by the product of the other two dimensions. Note. — The three dimensions must be expressed in the same de- nomination. 3. To find the solid contents of a cylinder. Multiply Hie area of tJie base by the altitude. REDUCTION. 107 SECTION XII. DENOMINATE NUMBERS. Wo&S^-^^-^^fi^ REDUCTION. Case I. RecliAction of IDenoininate Intesf^vs and IVLixecl TdiiTi"ber.s. 1. How many mills in 9 cents? In 12^^ cents? 62-^ cents? 100 cents? 2. How many cents in 7 dimes? 25^ dimes? 45.4 dimes? 56.8 dimes? 75.3 dimes? 3. How many dollars in 50 dimes ? 120 dimes ? 145 dimes? 1250 dimes? 1625 dimes? 4. How many dollars in 800 cents ? 2400 cents ? 1365 cents? 2235 cents? 5. How many farthings in 9 pence ? 72 pence ? 90^ pence? 24.5 pence? Note. — For tables see appendix. 6. How many pence in 8| shillings? 10^- s. ? 33 J s. ? 2.5 s. ? 6.5 s. ? 108 COMPLETE ARITHMETIC. 7. How many shillings in 15 £ ? 2.5 £? 16.4 £? 8. How many pence in 22 far. ? 48 far. ? 105 far. ? 201 far.? 9. How many pounds in 120 s.? 360 s.? 720 s.? 10. How many shillings in.72d. ? 144 d.? 25.2 d.? 34.8 d.? 52.82 d.? 73.44 d.? 11. How many drams in 8 oz. avoir.? 20 oz. ? 4.5 oz.? 12. How many ounces in 5 lb. avoir. ? 10| lb. ? 2.5 lb. ? 13. How many pounds in 64 oz. avoir. ? 19.2 oz. ? 4.8 oz. ? 14. How many grains in 5 pwt. ? 10|^ pwt. ? 2.5 pwt. ? 15. How many pwt. in 7 oz. ? 6.5 oz. ? 12-| oz. ? 16. How many ounces of gold in 7 lb. ? 12| lb. ? 1.5 lb. ? 4.5 1b.? 12.5 1b.? 17. How many pounds of gold in 48 oz. ? 14.4 oz.? 2.52 oz.? 4.68 oz.? 62.4 oz.? 18. How many scruples in 12 5 ? 8| 5 ? 14.5 5 ? 19. How many drams in 15 B ? 12J S ? 11.5 3 ? 20. How many ounces in 9 lb ? 5.5 lb ? 10.5 lb ? 21. How many inches in 8 J ft. ? 15^ ft. ? 33 J ft. ? 22. How many yards in 12 rd. ? 1.6 rd. ? 3.2 rd. ? 23. How many rods in 11 yd.? 33 yd.? 6.6 yd.? 24. How many miles in 18 fur. ? 13.6 fur. ? 7.2 fur.? 25. How many sq. ft. in 3^ sq. yd. ? 16| sq. yd. ? 26. How many square yards in 12.6 sq. ft.? 49.5 sq. ft.? 1.71 sq.ft.? 56.7 sq.ft.? 27. How many quarts in 17 pk. ? 12i pk. ? 301 pk. ? 28. How many gallons in 35 qt. ? 14.8 qt. ? 2.56 qt. ? 29. How many weeks in 365 days? 25.2 days? 30. How many years in 192 mo.? 25.2 mo.? 100 mo.? WKITTEN PROBLEMS. 31. Reduce 5£ 6s. 3d. to pence; 1275 d. to pounds. o £ 6 s. 3d. 12 ) mo Process : _20 Process : 20 ) 106 3 d. ■ 106 8- 5£ Gs. 12 1275 d., Ans. 5 £ s. 3 d., An^. HEDUCnOX. 109 32. Reduce 38 lb. 11 oz. 7 dr. to drams. 33. Reduce 12 bu. 5 qt. to pints. 34. Reduce 13 mi. 5 fur. 3 yd. to yards. 35. Reduce 11 A. 3 R. 22 P. to perches. 36. Reduce 503 pt. to bushels. 37. Reduce 324 gi. to gallons. 38. Reduce 10280 ft. to miles. 39. Reduce 12460'' to signs. 40. Reduce 30684 sec. to higher denominations. 41. How many pence in £45? In £237|? 42. How many perches in 95 A. ? 320| A. ? 43. How many hundred-weight in 4085 oz. avoir. ? 44. How many miles in 12840 ft.? 45. Reduce 13 mi. 5^ fur. to inches.^ 46. Reduce 113420 inches to miles. 47. Reduce 3450 cubic feet of wood to cords. 48. Reduce 5124 quarts to bushels. 49. Reduce 16 common years to hours. 50. How many seconds were in the year 1868? 51. Reduce 4 common yr. 45 d. to minutes. 52. Reduce 3.7 bushels to pints. 53. Reduce 4.5 rods to feet. 54. Reduce 3.65 lb. Troy to ounces. 55. Reduce 15° 40' 36" to seconds. 56. Reduce 588487" to degrees. 57. Reduce 12.3 miles to feet. 58. Reduce 365^ days to weeks. 59. Reduce 706.35 perches to acres. 60. How many acres in 12f sq. miles? 168. Rules. — I. To reduce a denominate number from a higher to a lower denomination, 1. Multiply the number of the highest denomination by the number of units of the next lower which equah a unit of the higher, and to the product add the number of the lower denomi- tiation, if any. 2. Proceed in like m^anner witli this and each successive 110 COMPLETE ARITHMETIC. result tJms obtained, until the number is reduced to tJie required denomination. Note. — The successive denominations of the compound number should be written in their proper orders, and the vacant denomina- tions, if any, filled with ciphers. II. To reduce a denominate number from a lower to a higher denomination, 1. Divide the given denominate number by the number of units of its oivn denomination which equals one unit of the next higher, and 'place the remainder, if any, at the right. 2. Proceed in like manner with this and each successive quotient thus obtained, untiL the number is reduced to the re- quired denomination. 3. The last quotient, with the several remainders annexed in proper order, will be the answer required. Note. — The above rules also apply to the reduction of denominate fractions, both common and decimal. (Art. 169.) Case XL IRedvaction. of Denoiniiiate yraotioiis. 1. What part of a peck is j^^ of a bushel ? ^ bu. ? Solution. — xV hu. = xV of 4 pk. = x\ pk. or ;} pk., and y\ bu. = 3 times \ pk. = -J pk. Hence, x\ bu. — | pk. 2. What part of a quart is ^^ of a peck ? f^ pk. ? 3. What part of a day is y\ of a week ? y^- w. ? 4. AVhat part of an hour is -^^ of a day ? -/^ d. ? 5. What part of an inch is -^^^ of a foot ? y^ ft. ? 6. What decimal part of an inch is .03 of a foot? Solution.— .03 ft. = .03 of 12 in,, or 12 times .03 in. = .36 in. 7. What decimal of an hour is .05 of a day? .025 d. ? 8. What decimal of a day is .12 of a week? .012 w. ? 9. What decimal of a quart is .125 of a peck? .35 pk. ? 10. What part of an inch is ^% of a foot? .08 ft. ? 11. What part of a pint is -}q of a gallon ? .06 gal. ? REDUCTION. Ill 12. What part of a foot is | of an inch ? Solution.— f in. = f of xV ft. = ttV ft. 13. What part of a week is ^j of a day? ^ d. ? 14. What part of an hour is y^ ^^ ^ minute ? -\^ min. ? 15. What part of a gallon is 4 of a pint? | pt. ? 16. What part of a pound avoir, is f of an ounce? 17. What decimal of a foot is .48 of an inch? Solution.— .48 in. = .48 of Jj ft. ^ j\ of .48 ft. = .04 ft. 18. What decimal of a bushel is .12 of a peck? 3.6 pk.? 19. What decimal of a week is .49 of a day? 6.3 d. ? 20. What decimal of a pound Troy is .144 of an ounce? 2.52 oz.? 38.4 oz.? .72 oz.? 9.6 oz.? 21. What decimal of a ream is .8 of a quire? 2.8 quires? 22. What part of a dime is f of a cent? .625 ct. ? 23. What part of a shilling is | of a penny? .6 d. ? .18 d.? 2.4 d.? 1.44 d.? 24. What part of a gallon is || of a pint? .64 pt. ? ^WKITTEN PROBIiEMS. 25. Reduce T^J^rir ^^ ^ ^^J to the fraction of a minute. PROCESS : -^ d. = IX 24 ^ _ TX^XAO ^in. ^ l^ „,i,, 18000 18000 18000 25 26. Reduce g^f^ of a pound avoirdupois to the fraction of a dram. 27. Reduce Jf of a yard to inches. 28. Reduce j-J- of a pound Troy to pennyweights. 29. Reduce .005 of a pound to the decimal of a penny. 30. Reduce .0065 of a week to the decimal of an hour. 31. Reduce 9.6 pwt. to the decimal of a pound Troy. 32. Reduce 3.96 inches to the decimal of a rod. 33. Reduce 30.8 rods to the decimal of a mile. 34. Reduce .096 of a bushel to the decimal of a pint. 35. Reduce |f of a rod to the fraction of a league. 112 COMPLETE ARITHMETIC. 36. Reduce ^ of a degree to the fraction of a circum- ference. 37. Reduce ^l of a day to the fraction of a minute. 38. Reduce |-f^ of a week to the decimal of an hour. 39. Reduce | of a minute to the fraction of a day. 40. Reduce 11.2 perches to the decimal of an acre. 41. Reduce 13.62 cords to cord feet. 42. Reduce .037 lb. avoirdupois to drams. 43. Reduce 56f lb. Troy to grains. 44. Reduce y^^ of a gallon to the fraction of a pint. 45. Reduce 2.43 miles to feet. 46. Reduce 777.6 pence to pounds. 47. Reduce 1.408 ft. to the decimal of a mile. 48. Reduce y^- of an hour to the fraction of a day. 49. Reduce .012 of a mile to yards. 50. Reduce | of a yard to the decimal of a mile. 169. Rule. — To reduce denominate fractions from a higher to a lower denomination, or from a lower to a higher, Proceed as in tJie reduction of denoioiinate integers. Note. — Denominate fractions are reduced to a lower denomination by multiplying, and to a higher denomination by dividing, the same as denominate integers ; but in reduction descending there are no units of a lower order to add, and in reduction ascending there are no remainders. Case III. Reduction of* Denoininate Fractions to Lowei* Inteyier:^. 1. How many months in | of a year? } of a year? |- of a year? 2. How many hours in | of a day? | of a day? \\ of a day? 3. How many minutes in ^ of an hour? -^ of an hour? -^ of an hour? 4. How many yards in y^y of a rod ? |^ of a rod ? -j^^ of a rod ? 5. How many quarts in .75 of a peck? 1.25 pk. ? REDUCTION. 113 6. How many months in .25 of a year? .331 yr. ? 7. How many days in .35 of a week? 4.5 w.? 7.3 w.? 8. How many pecks and quarts in .85 of a bushel? Solution.— .85 bu. = .85 of 4 pk. = 3.4 pk., and .4 pk. =::- .4 of 8 qt. = 3.2 qt. Hence, .85 bu. = 3 pk. 3.2 qt. 9. How many feet and inches in .75 of a yard? 10. How many quarts and pints in f of a gallon? 11. How many days and hours in -J of a Aveek? 12. How many pecks and quarts in .55 of a bushel? "WRITTEN PROBLEMS. 13. Reduce -f-^ of a day and .415 of an hour each to integers of lower denominations. Process. Process. 7 V 24 -^^^ ^• /^ da. = t\ of 24 h. = ^-^^ h. =: lOi h. 60 ^^ 24.900 min. i h. = I of 60 rain. = ^ min. = 30 min. 60 2 54.000 sec. yV da. = 10 h. 30 min. .415 h. = 24 min. 54 sec. Reduce to integers of lower denomination 14. -J of a mile. 20. .85 of a lb. avoir. 15. 3^ of a week. 21. .325 of a ton. 16. T^^. of a lb. Troy. 22. .081 of a yard. 17. H of a rod. 23. .9375 of a gallon. 18. -f^ of an acre. 24. .5625 of a cwt. 19. i of a cord. 25. .0135 of a cord. 170. Rule. — To reduce a denominate fraction to inte- gers of lower denominations, 1. Multiply the fraction by the number of units of the next lower denomination, which equals a unit of its denomination. 2. Proceed in like manner with the fractional part of the product and of each succeeding product, until the lowest denomination is reaxihed. C.Ar.— 10 114 COMPLETE ARITHMETIC. 3. The integral parts of the several products, written in proper order, will he the lower integers sought. Note. — When the last product contains a fraction, it should be united with the integer of the lowest denomination, forming a mixed number. Case IV. PtecliactiorL of Integers of Lo^wer Denonainations to Fractions of Higlier II)enoin.inations. 1. What part of a dollar is 25 cents? 50 cts. ? 2. What part of a foot is 8 inches ? 10 in. ? 3. What part of a day is 9 hours? 15 h.? 4. What part of a yard is 2 ft. 6 in. ? Solution. — 1 yd. = 36 in., and 2 ft. 6 in. = 30 in. ; 1 in. =^ j^-^ of a yd., and 30 in. = |f yd. = | yd. Hence, 2 ft. 6 in. = f yd. 5. What part of a gallon is 3 qt. 1 pt. ? 6. What part of a bushel is 2 pk. 5 qt. ? 7. What part of a rod is 3 yd. 2 ft. ? 8. What part of a barrel (31 gal.) is 15 gal. 2 qt. ? 9. What part of 3 pecks is 2 pk. 4 qt. ? 10. What part of 5 yards is 2 yd. 2 ft.? Suggestion. — Each of the above answers should be expressed both as a common fraction and as a decimal. WRITTEN PROBLEMS. 11. Reduce 15 vv. 5 da. to the fraction of a common year. Process. 15 w. 5 da. =r 110 da. iH yr- = ft yi-M -i»i«. 12. Reduce 1 yd. 2 ft. 6 in. to the fraction of a rod. 13. Reduce 1 pk. 2 qt. 11 pt. to the fraction of a bushel. 14. Reduce 9 oz. 2^ dr. to the fraction of a pound. 15. Reduce 9 h. 36 min. to the decimal of a year. 16. Reduce 2 pk. 3*qt. 1.2 pt. to the decimal of a bushel. 17. Reduce 13 s. 4 d. to the decimal of a pound Sterling. REDUCTION'. 115 18. Reduce 1 R. 14 P. to the decimal of an acre. 19. Reduce 8 oz. 8 pwt. to the decimal of a pound Troy. 20. Reduce 1 fur. 18 rd. 1 yd. to the decimal of a mile. 21. What part of 1 bu. 3 pk. is 5 pk. 6 qt.? 22. What part of 3 w. 4 da. is 3 da. 8 h. ? 23. What part of 12 A. 2 R. is 1 A. 2 R. 10 P. ? 24. What part of 3 barrels of flour is 110 lb. 4 oz. ? 171. Rule. — To reduce a denominate number, simple or compound, to the fraction of a higher denomination, Reduce the number which is a part and the number which is a whole to the same denomination, and write the former result as a numerator and the latter as a denominator of a fraction. Notes. — 1. The arivswer may be expressed decimally by changing the common fraction to a decimal. 2. When the whole is a unit and the part a compound number, the process may be somewhat shortened by reducing the number of the lowest denomination to a fraction of the next higher, prefixing the higher number, if any, and then reducing this result to a fraction of the next higher denomination, and so on, until the required fraction u reached. Thus, in the 16th problem above, the 1.2 pt. = .6 qt, ; and 3.G qt. =^ .45 pk.; and 2.45 pk.=:.6125 bu. DEFINITIONS. 172. A Denominate Number is a number com- posed of concrete units of one or several denominations. It may be an integer, a mixed number, or a fraction. 173. Denominate numbers are either Simple or Compound. A Simple Denominate Number is composed of units of the same denomination ; as, 7 quarts. A Co^npound Denominate Nttniber is com- posed of units of several denominations ; as, 5 bu. 3 pk. 7 qt. It is also called a Compound Number. Note. — Every compound number is necessarily denominate. 174. Denominate numbers express Currency, Measure, and Weight. Currency is the circulating medium used in trade and commerce as a representative of value. 116 COMPLETE ARITHMETIC. Measure is the representation of extent, capacity, or amount. Weight is a measure of the force called gravity, by which bodies are drawn toward the earth. 175. The following diagram represents the three general classes of denominate numbers, their subdivisions, and the tables included under each : T n f 1. Cc I. Currency, \ ^ ^_ 1. Coin, \ r 1. United States Money, Paper Money. / '-2. English Money. ' 1. Of extension, - II. Measure, 1. Lines and arcs, ( 1. Lo ' 1 2. Ci] 1. Long Measure, Circular Measure. 2. Surfaces : Square Measure. 3. Capacity, 1. Cubic Measure, 2. Wood Measure, 3. Dry Measure, 4. Liquid Measure. . 2. Of duration : Time Measure. III. Weight •W 1. Avoirdupois Weight, 2. Troy Weight, 3. Apothecaries Weight. Note. — For tables see appendix. 176. The Heduction of a denominate number is the process of changing it from one denomination to another without altering its value. 177. Keduction is of two kinds, Reduction Descending and Reduction Ascending. JReduetion Descending is the process of changing a denominate number from a higher to a lower denomi- nation. Heduction Ascending is the process of changing a denominate number from a lower to a higher denomi- nation. REDUCTION. 117 MENTAL PROBLEMS. 1. How many half-pint bottles can be filled with 2h gallons of sweet oil ? 2. A boy bought f of a bushel of chestnuts for $2, and sold them at 10 cents a quart: how much did he gain? 3. If a workman can do a job of work in 120 hours, how many days will it take him if he work 8 hours a day? 4. How much will f of a cwt. of sugar cost, at 16| cents a pound? 5. If a man spend J of each day in sleep, how many hours will he sleep in the last three months of the year? 6. If a man walk 10 hours a day, at the rate of 3.3 miles an hour, how far will he walk in 6 days? 7. How many square inches in the surface of a brick 8 inches long, 4 inches wide, and 2 inches thick? 8. How many square feet in a board 12.6 ft. long and 8 inches wide? 9. How many solid feet in a plank 16 feet long, 1^ feet wide, and 4 inches thick? 10. A man paid $36 for a stack of hay containing 4^ tons, and sold it at 50 cents a hundred : how much did he gain? 'WRITTEN PROBLEMS. 11. How many yards of carpeting, f of a yard wide, will carpet a room 27 feet long and 21^ feet wide? 12. How many acres in a street 2^ miles long and 5 rods wide ? 13. What would be the cost of a township of land 6 miles square, at $10.50 an acre? 14. A rectangular field is 60 rods long and 37|^ rods wide: how many boards, each 12 feet long, will inclose it with a fence 5 boards high? 15. At $5.62|- a cord, what will be the cost of a pile of wood 85 ft. 6 in. long, 6 ft. 4 in. high, and 4 ft. wide? 118 COMPLETE ARITHMETIC. 16. How many bricks, 4 by 8 in., will it take to pave a walk 16 feet wide and 6 J rods long? 17. How many gold rings, each weighing 3.2 pwt., can be made from a bar of gold weighing .75 of a pound? 18. An octavo book contains 480 pages: how many reams of paper will it take to print an edition of 1200 copies, making no allowance for waste ? 19. How many perches of masonry in the wall of a cellar 45 feet long, 34 feet wide, and 2-J- feet thick ? Note. — In measuring walls of cellars and buildings, masons take the distance round the outside of the walls (the girth) for the length, thus measuring each corner twice. 20. How many perches of stone in the walls of a fort 120 feet square, the walls being 33J- feet high and, on an average, 11 feet thick? 21. What will it cost to excavate a cellar 40 ft. long, 21 ft. 6 in. wide, and 4 ft. deep, at $1.75 a cubic yard? 22. A bin is 8 ft. long, 3^ ft. wide, and 4 ft. deep : how- many bushels of grain (2150| cu. in.) will it hold? 23. A circular park is 165 yards in diameter: how many acres does it contain ? 24. How many cubic feet in the capacity of a circular well 3^ ft. in diameter and 20 ft. deep? 25. A cylindrical cistern is 5 ft. in diameter and 6 ft. 4 in. deep : how many gallons of water will it hold ? 26. A congressional township is 6 miles square, and is divided into 36 sections : how many acres in a section ? 27. A tract of land is 4 miles long and 2^ miles wide : how many sections does it contain ? How many acres ? 28. A speculator bought S\ sections of land at $4.50 an acre, and sold them at $6.25 an acre: how much did he gain? 29. A man sold a farm containing a quarter of a section of land, for $3280 : what did he receive per acre ? THE METRIC SYSTEM. 119 THE METRIC SYSTEM. 178. The Metric Syste7n is a system of weights and measures based on the decimal scale. The system was first adopted by France, and it is now in general use in nearly all the countries of Europe. The use of the system in the United States was legalized by Congress in 1866, and it is employed, to some extent, in several departments of the government service. It has long been used by the Coast Survey. The convenience and accuracy of the system have secured its very general adoption in the sciences and arts, but it is not probable that it will soon come into general use in business transactions. 179. The Meter is the primary unit of the system. It is the ten-millionth part of the distance on the earth's surface from the equator to tlie pole. The Liter (le'-ter) is the unit of the measures of capacity. It is the thousandth part of a cubic meter. The Gram is the unit of weights. It is the weight of the thousandth part of a liter of water at its great- est density. 180. The meter, liter, and gram are each multiplied hy 10, 100, 1000, and 10000, giving multiple units, and they are also each divided by 10, 100, 1000, giving the decimal subdivisions of tenths, hundredths, thousandths, etc. 181. The multiples are named by prefixing to the name of the primary unit, or base, the Greek numerals, I)eka (10), Hedo (100), KUo (1000), and Myria (10000); and the subdi- visions are named by prefixing the Latin words, Deci (10th), a}Ui (100th), and Milli (1000th). 120 COMPLETE ARITHMETIC. METRIC TABLES. 182. — I. Measures of Length. i_ i The Unit is a M ETER — 39.37 inches, nearly. Denominations. Values. Equivalents. Myriameter 3rz lOOOO meters = 6.2137 mi. Kilometer = 1000 meters ^ 0.6214 mi. Hectometer ■■^^ 100 meters == 328iV ft. Decameter — 10 meters = 393.7 in. Meter = 1 meter = 39.37 in. Decimeter — .1 meter ^ 3.937 in. Centimeter = .01 meter = 0.3937 in. Millimeter == .001 meter = 0.0394 in. Decimal Scale. 1 i i i a o Decameter. Meter. Decimeter. Centimeter. Millimeter. 0.000 Ten units of any denomination of the above table equal one unit of the next higher de- nomination, and, hence, the successive de- nominations correspond to successive orders of figures in the decimal system : the meter denoting units ; the decameter, tens, etc. The correspondence between the metric denominations and those of United States Money is also noticeable. The millimete)' corresponds to mills; the centimeter to cents; the decimeter to dimes; the metei' to dollars, etc. The above diagram shows that a decimeter is a little less than four inches, and that a centimeter is a little more than f of an inch. Note. — As no abbreviations for the names of the metric units have been agreed upon in this country, the names are given in full in this work. The tables of equivalents need not be memorized by the pnpil. THE METRIC SYSTEM. 121 183. — II. Measures of Surface. The Unit is an Are, or a Square Decameter. Denominations. Values. Equivalents. Hectare = 10000 sq. meters = 2.471 acres. Are (air) -— 100 sq. meters = 119.6 sq. yards. Centiare = 1 sq. meter -- 1.196 sq. yards. Since 100 units of each denomination in the above Decimal Scale. table equal one of the next higher, each occujDies two o £ orders of figures. The centiares correspond, in this | ^ "^ respect, to cents, which occupy two places. K < o , , . -, . .11 0.00 The above table is used in measuring land. The primary unit for the measuring of small surfaces is a square meter. Note. — Centiare is also written Centare. 184. — III. Measures of Capacity. The Unit is a Liter, or a Cubic Decimeter. „ , Equivalents. Denominations. \alues. rv ^r r- -7 tit Dry Measure. Liquid 3[eastire. Kiloliter = 1000 liters = 1.308 cu. yd. = 264.17 gallons. Hectoliter = 100 liters = 2.8375 bu. = 26.417 gallons. Decaliter = 10 liters rz= 9.08 qt. = 2.6417 gallons. Liter = 1 liter = 0.908 qt. -=: 1.0567 quarts. Deciliter = .1 liter = 6.1022 cu. in. = 0.845 gill. Centiliter = .01 liter = 0.6102 cu. in. = 0.338 fluid ounce. Milliliter = .001 liter . r--^ 0.061 cu. in. =0.27 fluid dram. The kiloliter equals a cubic meter, the liter a cubic decimeter, and the milliter a cubic centimeter. The Kiloliter is called a Stere (Stair), and is the principal measure of wood, stone, etc. One-tenth of a stere is a Decistere, and 10 steres are a Decastere. The liter is used in measuring liquids, and the hectoliter in meas- uring grains. Note. — When the tliree dimensions of a regular solid are expressed in decimeters, their product will be the contents in liters. r.Ar.-ll. 122 COMPLETE ARITHMETIC. 185.— IV. Weights. The Unit is a Gram = 15.432 grains. Denominations. Values. iMiuivalonts in Av. Wci^ri,t. Millier, or tonneau I ^^ 1000000 grams =. 2204.G iwiinds Quintal ^=^ 100000 grams = 220.4G pounds Myriagram -_-= 10000 grams =. 22.046 pounds Kilogram, or Kilo — 1000 grams ^ 2.2046 pounds Hectogram r= 100 grams = 0.5274 ounces Decagram = 10 grams — 0.3527 ounce Gram = 1 gram — 15.432 gr. Troy. Decigram = .1 gram — 1.5432 gr. Troy. Centigram = .01 gram ^ 0.1543 gr. Troy. Milligram = .001 gram ^ 0.0154 gr. Trov. A millier equals tlic weight of a cubic meter of water at its greatest density; a kilogram equals a liter of water; the gram, a cubic centi- meter of water;, and the milligram, a cubic millimeter of water. The kilogram, called, for brevity. Kilo, is the ordinary weight of commerce. 186. — V. Metric Equivalents of Common Denominations. Long Measure. An inch = .0254 meter. A foot = .3048 meter. A yard = .9144 meter. A rod = 5.029 meters. A mile -= 1.6094 kilometers. Square Measure. A sq. inch = .000645 sq. meter. A sq. foot = .0929 sq. meter. A sq. yard = .8362 sq. meter. A sq. rod =r .2529 are. An acre ■= .4047 hectare. A sq. mile = 259 hectares. Cubic Measure. A cu. inch ^=r .0164 liter. A cu. foot — .2832 hectoliter. A cu, yard •= .7646 stere. A cord = 3.625 i^tcrcs. Weight. A grain =^ .0648 gram. A pound av. = .4536 kilogram. A pound Troy ■= .373 kilogram. A ton = .907 tonncau. A gallon =^ 3.786 litems. A bushel ^^ .3524 hectoliters. The new nickel 5 cent piece weighs 5 grams, and is 2 centimeters, or A of a meter, in diameter. THE METRIC SYSTEM. 123 MENTAL PROBLEMS. 1. How many meters in a decameter? In a hectometer? A kilometer? A myriaraeter? 2. What part of a meter is a decimeter ? A centimeter ? A millimeter? 3. Name the metric units of length, in order, from the lowest to the highest, 4. How many liters in a hectoliter ? In a decaliter ? A kiloliter ? 5. Name the metric units of capacity from the highest to the lowest? 6. What part oF a gram is a centigram ? A decigram ? A milligram ? 7. How many meters in 5 decameters? 44 decameters? 225 decameters ? 34.6 decameters? 8. How many liters in 3 hectoliters ? 37 hectoliters ? 22.5 hectoliters? 7.45 hectoliters? 9. How many grams in 8 kilograms ? 24 kilograms ? 3.25 kilograms? .456 kilogram? 10. What decimal part of a meter is a centimeter? 15 centimeters ? 72 centimeters ? 11. What decimal part of a gram is a milligram? 24 milligrarns? 245 milligrams? 12. When the metric units are expressed on the decimal scale, which order, from the decimal point, is the decimeter? The millimeter? The centimeter? 13. Which order, from the decimal point, is the deca- liter? The kiloliter? The hectoliter? 14. Read the several orders in 324.56 meters as metric units. Alls. 3 hectometers 2 decameters 4 meters 5 decimeters and 6 centimeters. 15. Read the several orders in 504.046 grams as metric units. 16. Read 4080.57 liters in metric units. 124 COMPLETE ARITHMETIC. -WKITTEN PROBLEMS. 17. Write 5 kilograms 7 hectograms 6 decagrams 5 grams and 6 centigrams on the decimal scale as grams. Ans- 5765.06 grams. 18. Write 6 hectoliters 4 decaliters 3 liters and 5 deci- liters on the decimal scale as liters. 19. How many meters in 6 kilometers 7 decameters and 5 decimeters? 20. How many grams in 6 kilograms 4 decagrams and 8 centigrams ? 21. Reduce 234.56 hectograms to grams. Process : 234.56 X 100 = 23456. Ans. 23456 grams. 22. Reduce 345.8 centigrams to grams. Process : 345.8 -^ 100 = 3.458. Ans. 3.458 grams. 23. Reduce 45.06 Idloliters to liters. 24. Reduce 35.4 hectoliters to liters. 25. Reduce 84.5 ares to square meters. 26. Reduce 132.4 centimeters to meters. 27. Reduce 24000 millimeters to meters. 28. Reduce 434.5 centiliters to liters. 29. Reduce 3.225 quintals to grams. 30. Reduce 746.35 decagrams to kilograms. 31. How many yards in 220 meters? Process : 39.37 in. X 220 -- 12 -h- 3 == 240.59 +. Ans. 240.59 yd. 32. How many miles in 44.5 kilometers? 33. How many inches in 24 centimeters ? 34. How many bushels in 250 hectoliters of wheat? 35. How many gallons in 37| liters of sirup ? 36. How many pounds of butter in 150 kilos? 37. How many liters in 35 cubic feet? 38. How many steres in 20 cords of wood? 39. How many ares in f of ian acre? 40. How many meters in 1760 feet? cwt. lb. oz. dr. 7 44 6 11.5 12 13 7.6 23 56 12 0. 27 00 14 8.4 COMPOUND NUMBERS. 125 SECTION XIII. COMPOUND NUMBERS. ADDITION AND SUBTRACTION. 1. What is the sum of 7 cwt. 44 lb. 6 oz. 11.5 dr.; 12 cwt. 13 lb. 7.6 dr.; 23 cwt. 56 lb. 12 oz. ; and 27 cAvt. 14 oz. 8.4 dr.? Since only like numbers can be added, Process. write the numbers of the same denomina- tion in the same columns. The sum of the drams is 27.5 dr. = 1 oz. 11.5 dr. Write the 11.5 dr. under drams, and add tlie 1 oz. with the ounces. Proceed in like manner 3 T. 10 15 1 11.5 "ntil the numbers of the several denomi- nations are added. 2. Add 16 -mi. 7 fur. 27 rd. 3 yd. 2 ft. 8i in. ; 18 mi. 4 fur. 5 yd. 1 ft. 7f in. ; 27 mi. 35 rd. 4 yd. 5^ in. ; and 6 fur. 24 rd. 3 yd. 2 ft. 3. Add 13 w. 6d. 13 h. 48min.; 8w. 13 h. 51 min. 37 sec; 12 w. 5d. 22 h. 16 min. 44 sec; 1 w. 10 h. 15 min.; and Id. 10 h. 26 sec. 4. Add 24 lb. 10 oz. 17 pwt. 22 gr.; 16 lb. 19 pwt.; 10 oz. 15 pwt. 21 gr.; 45 lb. 9 oz. 18 gr.; and 13 lb. 11 oz. 18 pwt. 23 gr. 5. Add 15 bu. 3 pk. 7 qt. ; 27 bu. 5 qt. 1 pt. ; 8 bu. 2 pk. 1 pt. ; 47 bu. 3 pk. ; 12 bu. 2 pk. 1 qt. 1 pt. ; and 19 bu. 1 pk. 3 qt. 6. Add 16° 32' 43"; 28° 47' 53"; 25° 53"; 4 s. 48' 48"; lis. 16° 36' 59"; and 5s. 18° 7' 8". 7. How many cords of wood in three piles, the first being 23 ft. long, 4 ft. wide, and 7 ft. high ; the second, 28 ft. long, 4 ft. wide, and 6| ft. high; and the third, 17 ft. long, 8 ft. wide, and 7^ ft. high ? Pkocess, Id. 13 9 yd. 3 4 ft. 1 in. 6.4 11.5 3 ^ = 1 6.9 6 126 COMPLETE ARITHMETIC. 8. From 13 rd. 3 yd. 1 ft. 6.4 in. take 9 rd. 4 yd. 11.5 in. Write the subtrahend under the minuend, pLicing the numbers of the several denomina- tions in columns, as in compound addition. Since 11.5 in. is greater than 6.4 in., add 12 in. to 6.4 in. and tlien subtract. To balance the 12 in. added to the minuend, add 1 ft. (12 in.) 3 4 2 .9 to the subtrahend (Art. 30, Pr. 3), or, if pre- ferred, subtract 1 ft. from the minuend. Proceed in like manner until the difference between the numbers of the several denominations is found. Reduce the | yd. to feet and inches, and add the result to the ft. 6.9 in. of the remainder. 9. From 30 mi. 6 fur. 14 rd. 3 yd. 1 ft. 4 in. take 25 mi. 36 rd. 4 yd. 2 ft. 10 in. 10. From 33 rd. 1 yd. 2 ft. 11 in. take 16 rd. 3 yd. 8 in. 11. From 104° 11' 20'^ take 83° 43' 36". 12. Boston is 71° 4' 9" W. longitude, and San Francisco is 122° 26' 15" AV. longitude : what is their difference in longitude ? 13. From the sum of 245 A. 2 R. 27 P. and 187 A. 3 R. 34 P. take their difference. 14. A note was given July 23, 1863, and it was paid Nov. 16, 1868 : how long did it run ? 15. A man was born Sept. 12, 1827, and his eldest son was born Apr. 6, 1855: what is the difference in their ages? 16. Baltimore is situated 76° 37' W., and Vienna 16° 23' E. : what is their difference in longitude ? 17. A ship in latitude 37° 20' north, sails 15° 45' south ; then 12° 36' north; then 18° 40' south: what is her latitude ? DEFINITIONS AND RULES. 187. A Compound dumber is a number composed of units of several denominations. (Art. 173.) 188. Compound numbers are of the same kind when their corresponding terms denote units of the same denomination ; as, 3 bu. 2 pk., and 6 bu. 3 pk. 5 qt. COMPOUND NUMBERS. 127 189. Compound Addition is the process of finding the sum of two or more compound numbers of the same kind. 190. Rule. — To add compound numbers, 1. Write the compound numbers to he added so tJiat units of the same denomination shall stand in tlie same column. 2. Add the colum^n of tlie lowest denomination^ and divide the sum by the number of units of that denomination, ivhich equals a unit of the next higher denomination ; write the re- mainder under the column added, and add the quotient wiHi the next column. 3. Li like manner add the remaining columns, writing the sum of the highest column under it. Note. — In both simple and compound addition, the sum of each column is divided by the number of units of that denomination, ivhick equals one of the next higher denomination. In simple addition this divisor is 10 ; in compound addition it is a varying number, since the several denominations are expressed on a varying scale. 191. Compound Subtraction is the process of finding the diflTerence between two compound numbers of the same kind. 192. KuLE. — To subtract one compound number from another, 1. Write the subtrahend under the minuend, jAacing terms of the same denomination in the same column. 2. Beginning at the right, subtract each successive term of the subtrahend from the corresponding term of the minuend, and write the difference beneath. 3. If any term of the subtrahend be greater than the corre- sponding term of the minuend, add to the term of the minuend as many units of that denomination as equal one of the next higher, and from the sum subtract the term of the subtrahend, writing the difference beneath. 4. Add one to the next term of the subtrahend, and proceed as before. Note.— Instead of adding one to the next term of the subtrahend, one may be subtracted from the next term of the minuend. 128 COMPLETE ARITHMETIC. MULTIPLICATION AND DIVISION. 1. Multiply 15 rd. 3 yd. 1 ft. 7 in. by 11. Process. Since the value of llie units of 1' 'c\ S -1 1 ft 7 in the successive denominations increases * 11 from right to left, begin at the right 4 fur. 11 rd. 5 yd. 2 ft. 5 in. h^n^. 11 times 7 in. r=- 77 in. = 6 ft. 5 in. Write the 5 in. under inches, and reserve the 6 ft. to add witli the product of feet. Proceed in like manner until the numbers of the several denominations are multiplied. 2. If a man can build 7 rd. 11 ft. 6 in. of fence in a day, how much can 15 men build? 3. How many bushels of wheat in 18 bins, each con- taining 124 bu. 3 pk. 5 qt. ? 4. How much hay in 13 stacks, each containing 4 T. 13 cwt. 56 lb. ? 5. What is the weight of 12 silver spoons, each weigh- ing 2 oz. 13 pwt. 14 gr. ? 6. Divide 19 mi. 4 fur. 20 rd. 2 yd. 9 in. by 7. Process. Since the value of the 7 ) 19 mi, 4 fur. 20 rd. 2 yd. ft. 9 in. "nits of the successive de- 2 mi. 6 fur. 14 rd. 1 yd. 2 ft. 81 in. nominations decreases from left to right, begin at the left hand. } of 19 mi. = 2 mi. with 5 mi. remaining. Write the 2 mi. under miles, and reduce the 5 mi. to furlongs, and add tlie 4 fur. which gives 44 fur. } of 44 fur. = 6 fur. with 2 fur. remain- ing. Keduce the 2 fur. to rods, add the 20 rd,, take } of the result, and proceed in like manner until the numbers of all the denomina- tions are divided. 7. Divide 27 mi. 3 fur. 25 rd. 12 ft. 6 in. by 12. 8. A ship sailed 39° 12' 40" in 21 days: how many de- grees did it average each day? 9. If 15 equal bars of silver contain 24 lb. 8 oz. 16 pwt., what is the weight of each bar ? 10. If 12 equal bins hold 430 bu. 2 pk. of wheat, how much wheat is there in each bin ? COMPOUND NUiMBERS. 129 11. From 13 w. 5d. 18 h. 40 min. take 7 w. 23 h. 45 min., and divide the difference by 15. 12. Add 4 fur. 23 rd. 3 yd. 2 ft. and 7 fur. 16 rd. 1 ft., and divide the sum by 22. 13. From the sum of 56 lb. 13 oz. 9 dr. and 47 lb. 15 oz. 15 dr. take their difference, and divide the result by 9. 14. How many rings, each weighing 4 pwt. 15 gr., can be made from a bar of gold weighing 1 lb. 10 oz. ? Suggestion.— Eeduce both divisor and dividend to the same de- nomination. 15. How many kegs, each containing 5 gal. 1 qt., can be filled from a cask holding 63 gal. ? 16. How many rotations will a wheel 12 ft. 6 in. in cir- cumference make in rolling | of a mile? 17. How many lengths of fence, each 11 ft. 6 in., will inclose a square field each side of which is 20 rd. 5 yd. ? 18. How many barrels, each holding 2 bu. 3 pk., will hold 132 bushels of apples? 19. How many axes, each weighing 3 lb. 3 oz., can be made from a ton of iron? 20. How many steps, 2 ft. 6 in. each, will a man take in walking round a field 45 rods square? 21. The length of a solar year is 365 d. 5 h. 48 min. 48 sec. : how much time is ^^ of a solar year ? DEFINITIONS AND RULES. 193. Compound Multiplication is the process of taking a compound number a given number of times. The multiplier is always an abstract number. 194. Rule. — To multiply a compound number, 1. Write the multiplier under the loivest denomination of the midtiplicand. 2. Beginning at tJie right, midtiply each term of the midti- plicand in order, and reduce each product to the next higher 130 COMPLETE ARITHMETIC. denomination, writing the remainder under tJie term multiplied, and adding the quotient to the next jyroduct. Note. — In both simple and compound multiplication, the successive products are each divided by the number of units of their denomination, which equals one of the next hie/her denomination. 195. Conijwiind Division is the process of dividing a compound number into equal parts. 196. Rules. — I. To divide a compound number, 1. Write the divisor at the left of the dividend, as in simple division. 2. Beginning at the left, divide each term of the dividend in order, and write the quotient under the term divided. 3. If the division of any term give a remainder, reduce tJie remainder to the next lower denonmiation, to the result add the number of thai denomination in the divide7id, and then divide as above. Note. — When the divisor is a large number, the successive terms of the quotient may be written at the right of the dividend, as in long division. II. To divide a compound number by another of the same kind. Reduce both compound numbers to tJie same de- nomination, and then divide as in simple division. Note. — This is not properly compound division, since the com- pound numbers are reduced to simple numbers before dividing. LONGITUDE AND TIME. 197. Lofigitiide is distance east or west from a given meridian. It is measured in degrees, minutes, and seconds. Thus, 15° 24' 40" east longitude denotes a position 15° 24' 40" east of the meridian from which longitude is reckoned. vSince every circle is divided into 360 degrees, the length of a degree depends upon the size of the circle of which it is a part. The length of a degree of longitude depends upon the latitude of the parallel on which it is measured. It is greatest at the equator, where it is 09^ miles; and least at the poles, where it is nothing. LONGITUDE AND TIME. 131 198. The earth rotates on its axis from west to east once every twenty- four hours, and the illumi- nated space between any two meridians passes un- der the sun's rays in the same length of time. A degree of surface at the equator passes under the sun's rays in the same time as a degree at any latitude between the equator and the polar circle. 199^ When the vertical rays of the sun are on the meridian of any place, it is noon, or 12 o'clock, at that place; and since the sun's rays pass over the earth's sur- face from east to westy it is after noon at all places cast of this meridian, and before noon at all places west of it. When it is noon at Cincinnati, it is after noon at New York, and before noon at St. Louis. If 24 clocks were placed 15° apart on any parallel of latitude between the polar circles, the difference in time between any two consecutive clocks would be one hour; and the 24 clocks would to- gether represent every hour of the day. The figures in the diagram represent the location of the clocks (15° apart), and also the hour of the day corresponding to noon on the meridian. MENTAL PROBLEMS. 1. The earth rotates on its axis once every 24 hours: what part of a rotation does it make in 1 hour? 2. How many degrees of the earth's surface pass under the sun's rays in 24 hours? In 1 hour? 132 COMPLETE ARITHMETIC. 3. How many degrees of longitude make a difference of 1 hour in time? 4. When it is noon at Washington, what is the hour of day 15° east of Washington? 15° west of Washington? 5. When it is 6 o'clock at Boston, what is the hour of day 30° east of Boston? 45° west of Boston? 6. If 15° of longitude give a difference of 1 hour in time, how much longitude will give a difference of 1 min- ute in time? 7. When it is 4 o'clock at Cincinnati, what is the time 15' east of Cincinnati? 45' west of Cincinnati ? 8. When it is 9 o'clock at Chicago, what is the time 15° 15' east of Chicago? 15° 45' west of Chicago? 9. If 15' difference in longitude gives a difference of 1 minute in time, what difference in longitude will give a difference of 1 second in time ? 10. What difference in longitude gives a difference of 1 hour in time ? 1 minute ? 1 second ? 11. The difference in time between two cities is 2 hours: what is the difference in their longitude? Which has the earlier time ? 12. The difference in time between New York and St. Louis is 1 h. 2^ min. : what is the difference in their longitude ? 13. A gentleman left Boston and traveled until his watch was 1 h. 3 min. too slow : how far had he traveled, and in which direction? 14. Two captains observed an eclipse of the moon, one seeing it at 9 P. M. and the other at 11^ P. M. : what was the difference in their longitude? WRITTEN" PROBLEMS. 15. The difference in longitude between two places is 31° 45' 30" : what is the difference of time ? Process. 15 ) 31° 45^ 30^^ Divide by 15 as in compound division. 2 li. 7 rain. 2 sec. LONGITUDE AND TIME. 133 16. The difference in longitude between two cities is 5° 31': what is the difference in time? 17. The longitude of Cincinnati is 84° 26' W., and San Francisco is 122° 26' 15" W. : when it is noon at Cincin- nati what is the time at San Francisco? 18. Philadelphia is 75° 10' W. : when it is noon at San Francisco what is the time at Philadelphia? 19. Boston is 71° 4' 9" W.: when it is 7 P. M. at Boston what is the time at Cincinnati? At San Francisco? 20. Berlin is 13° 23' 53" E. : when it is noon at Boston what is the time at Berlin ? 21. The difference in time between two cities is 1 h. 35 min. 12 sec: what is their difference in longitude? Pkocess. 1 h. 35 min. 12 sec. Multiply by 15 as in compound multipli- c ^'J cation. 23° 48^ 0^^ Ans, 22. The difference in time in the observations of an eclipse, on two vessels at sea, is 2 h. 15 min. 10 sec. : what is their difference in longitude? 23. An eclipse was observed at New York, 74° W., at 9.30 P. M., and the time of its observation on a vessel in the Atlantic Ocean, was 11.45 P. M. : what was the longi- tude of the vessel? 24. The difference in time between the chronometers of two observatories is 45 min. 30 sec, and the longitude of the observatory having the earlier or faster time is 85° 40' W. : what is the longitude of the other observatory ? 25. The distance from Boston to Chicago is about 890 miles, and a degree of longitude at Boston contains about 52 miles: when it is noon at Boston what is the time at Chicago ? 26. The distance from Washington to St. Louis is about 690 miles, and a degree of longitude at Washington con- tains about 53 miles : when it is 9 o'clock at St. Louis what is the time at Washington ? 134 COMPLETE AKITHMETIC. 27. The difference in the longitude of two vessels, at the time of the observation of an eclipse, Avas 25° 30': what was their difference in time? 28. How much earlier does the sun rise at Baltimore, which is 76° 37' W., than at St. Louis? 29. How much later does the sun set at Chicago, which is 87° 35' ^y., than at Boston? 30. How much later does the sun set at San Francisco than at Cincinnati ? 31. How much earlier does the sun rise at New York than at San Francisco? 32. How much later does the sun set at Boston than at Berlin? TABLE AND RULES. 200. Table : 15° difference in long, gives a difference - of 1 h. in time. 15' difference in long, gives a difference of 1 m. in time. 15" difference in long, gives a difference of 1 sec. in time. 201. EuLES. — 1. To find the difference in time corre- sponding to any difference in longitude. Divide the difference in longitude, expressed in degrees, minutes, and seconds, by 15, and the respective quotients will be hours, mimdes, and seconds of time. 2. To find the difference in time corresponding to any difference in longitude, Multiply the difference in time, ex- pressed in hours, minutes, and seconds, by 15, and the respect- ive products ivill be degrees, minutes, and seconds of longitude. 3. To find the time at one place when the time at another place and their difference of time are known, When the second place is east of the first, add their difference of time; when it is west of the first, subtract their difference of time. PERCENTAGE. 135 SECTION XIV. PERCENTAGE. NOTATION AND DEFINITIONS. 202. One per cent, of a number is one hundredth of it; two per cent, is two hundredths ; and, generally, any per cent, of a number is so many hundredths of it. 1. How many hundredths of a number is 4 per cent, of it? 7 per cent, of it? 15 per cent, of it? 2. How many hundredths of a number is 3J per cent of it? 121 per cent.? 33-J- per cent. ? 3. How many hundredths of a number is \ of one per cent, of it ? f of one per cent. ? f of one per cent. ? 4. How many hundredths of a number is 115 per cent, of it? 135 per cent.? 180 per cent.? 5. What per cent, of a number is .05 of it? .09 of it? .15 of it? .35 of it? 6. What per cent, of a number is .03^^ of it? .33 J of it? .OOfofit? .001 of it? 7. What per cent, of a number is j-J^ of it? HJ of it? 1.25 of it? 1.65 of it? 8. How many hundredths of a number is 45 % of it? 110%? 125%? 170%? 215%? Note. — Tlie character % is often nsed instead of the words " per cent." Thus, 15 % denotes 15 per cent, 9. What fractional part of a number is 10 % of it? 20%? 25%? e50%? Solution.— 50 % of a number is y\% or h of it. 10. What fractional part of a number is 12^ % of it ? 16|%? 331%? (]G|%? 136 COMPLETE ARITHMETIC. 11. What fractional part of 100 % is 25 % ? 50 % ? 75%? 66t%? WRITTEN EXERCISES. 12. Express decimally 1 % ; 3%; 6%; 7%; 8%; 9%. 13. Express decimally 15%; 20%; 25%; 33%; 45%. 14. Express decimally 112%; 125%; 150%; 220%. 15. Express decimally 6i % ; 3^ % ; 12^%; 24|%. Note. — The fractional part of one per cent, may be expressed deci- mally, as thousandths, ten-thousandths, etc. Thus, 6i 4 = .06k or .0625. 16. Express decimally 7f % ; 16|%; 10^%; 30-J-%. 17. Express decimally 1%; |%;'|%; f%; ^%. 18. Express decimally 5 % ; t\ % ; 7 A % ; i%; 20J-%. DEFINITIONS. 203. Any JPer Cent, of a number or quantity is so many hundredths of it. Note. — The term per cent, is a contraction of tlie Latin per centum, wliieh means by the hundred. 204. The Bate JPer Cent, is the fraction denoting the number of hundredths taken. The rate per cent, is expressed numerically either as a common fraction or as a decimal. 205. The Mate is the number of hundredths. Note. — In this work, the terms Per Cent., Rate Per Cent., and Rate, are not used as synonymous. In the statement, "12 is 6 per cent, of 200," 12 is considered the per cent. ; y^^, or .06, as the rate per cent. ; and 6 as the rate. The rate is the numerator of the fraction denoting the rate per cent. 206. The character % is called the Per Cent Sign, and is read per cent. 207. JPercentage embraces all numerical operations in which one hundred is the basis of computation. PERCENTAGE. 137 THE FOUR CASES OF PERCENTAGE. 208. Four numbers are considered in percentage, and such is the relation between them that, if any two of them are given, the other two may be found. These four numbers are : 1. The I^ase^ or the number of which the per cent, is found. 2. The Hate Per Cent,, or the fraction denoting the number of hundredths of the base taken. 3. The Per Cent,, or the part of the base correspond- ing to the rate per cent. It is also called the Percentage. 4. The Amount or Uifferenee, or the number ob- tained by adding the per cent, to, or subtracting it from, the base. Case I. The Base and tlie Rate Pei* Cent, given, to find the Fei- Cent. 1. How much is 5% of 800? Solutions.— 1. Since 5 % =^ yf ^j, 5 % of 800 = y^o "^ 80f>, ^vhich is 40. Or, 2. 1 % of 800 = y^^ of 800, which is 8 ; and 5 % of 800 ------ 5 times 8, which is 40. 2. What is 6% of 1200? 8% of 250? 9% of 4000? 3. What is 5 % of $300? 8 % of $450? 12 % of $500? 4. What is 6% of 500 miles? 10% of 250 miles? \b% of 600 miles ? 20 % of 300 miles ? WKITTEN PROBLEMS. 5. What is 8% of 674.50? 1st Process. 2d Process. $674.50 $6,745 -- 1 % .08 8 $53.96 00 $53,960 = 8 % Note. — Let the pupil use cme method until he is ftimiliar with it, (\Ar.— 12. 138 COMPLETE ARITHMETIC. What is 6. 5% of 245? 16. \% of $540? 7. 9% of 360? 17. tV% of $4000? 8. 15% of 1200? 18. f % of 21700 ft.? 9: 25% of 37.5? 19. 1% of $48.50? 10. 33% of $150? 20. 331% of 965 days? 11. 8% of $37.50? 21. 16|% of $.54? 12. 3% of $180.25? 22. 15% off? Of^^? 13. H % of 1050 lb. ? 23. 66|%of,^? Of2|-? 14. 21% of 60.8 1b.? 24. 12% of .25? Of .45? 15. 121- ^ of 560 days ? 25. 61% of 50? Of .75? 26. What is the difference between 33% and 25 J- % of 480 miles ? 27. If 70 % of a certain ore is iron, how much iron is there in 3740 pounds of ore? 28. If 20 % of air-dried wood is water, how much water is there in 143J tons of wood? 29. A man receives $1650 a year, and his expenses are 87^% of his income: how much has he left? 30. A grain dealer owning 58500 bushels of wheat, shipped 371 % of it by a steamer, 33^ % of it by a schooner, and the rest of it by railroad : how many bushels did he ship by each? FORMI^LAS AND RULES. 209. Formulas. — 1. Per cent. ==^ base X rate per cent. 2. Amount = base -\- per cent. 3. Difference ^= base — percent. 210. Rules. — To find a given per cent, of any number, 1. Multiply the given number by the rate per cent, expressed decimally. Or, 2. Eemx)ve the decimal point two places to the left, and mul- tiply the result by the rate. Note. — When the rate is an aliquot part of 100, tlie per cent, may be found by taking the same aliquot part of the base. Thus, 33] % of $48 — ] of $48. The process in each of the succeeding cases may be shortened by using the fraction denoting the alicpiot part. PERCENTAGE. 139 Case 11. a^h.e Base tiiad the Per Cent, given, to find tlie Rate r*er Cent. 1. What per cent, of 16 is 4? Solutions.— 1. 1 is j\ of 16, and 4 is j% or .25 (Art. 121). Hence, 4 is 25% of 16. Or, 2. 1% of 16 is .16, and 4 is as many per cent, of 16 as .16 is con- tained times in 4.00, which is 25. 2. What per cent, of $50 are $5? $20? 3. What per cent, of $300 are $12? $30? 4. What per cent, of 150 lb. are 75 lb. ? 50 lb.? 5. What per cent, of 250 ft. arc 50 ft.? 100 ft.? WRITTEN PROBLEMS. 6. AVhat per cent, of 62.5 is 16? 1st Process. 2d Process. 16 ^ 62.5 = .24 Ifc of 62.5 --= .625 .24 =: 24%, Ans. 16 ^ .625 = 24, Rate. Note.— The quotient obtained by the first procesn is the rate pa- cent., and the quotient obtained by the second process is the rate. What per cent, of 7. 75 is 4.5? 8. 125 is 25 ? 9. 120 is 40 ? 10. $450 are $90? 11. $192 are $32? 12. $760 are $19? 13. $1000 are $5? 14. $6 are 45 cts. ? 23. A farmer had 320 sheep and sold 48 of them : what per cent, of the flock did he sell ? 24. A gold ring is 22 carats fine : what per cent, of it is gold? 25. What per cent, of $45 is 16|% of $150? 15. 75 lb. are 16.5 lb.? 16. 20 ft. are 1.2 ft.? 17. 371 yd. are 5 yd. ? 18. .75 is .15? 19. .60 is .45? 20. fisi? 21. *isF 22. 2| is f ? 140 COMPLETE ARITHMETIC. 26. A regiment of 750 men lost 160 men in a certain battle: what per cent, of the regiment remained? 27. What per cent, of any number is | of it? | of it? l^ofit? 2^0 of it? 28. AVhat per cent, of a number is | of it ? 2^- of it ? Ilofit? i off of it? FORMULA AND RULES. 211. Formula. — Rate % =z per cent -^ base. 212. KuLES. — To find what per cent, one number is of another, 1. Divide the number which is the per cent, by the base, and the quotient expressed in hundredths ivill be the rate per cent. Or, 2. Divide the number which is the pter cent, by one per cent, of the base, ami the quotient will be the rate. Case III. 3?ei' Certt. tvncl Rate Fer Cent, giveia, to find th.e JBtise. 1. 45 is 15 % of what number? Solutions.— 1. If 15%, or .15, of a number is 45, the number equals 45 -f- .15, which is 300. Or, 2. If 45 is 15% of a number, 1% of it is y^ ^f 45, which is 3, and 100%, or the number, is 100 times 3, which is 300. 2. 320 is 16% of what number? 3. 7.2 pounds are 12% of how many pounds? 4. Charles is 15 years old, and his age is 30% of his father's age : how" old is his father ? 5. A man's expenses are $28 a month, which is 70% of his wages : how much does he earn a month ? 6. In a certain school 56 pupils study arithmetic, which is 28 per cent, of the whole number of pupils in school : how many pupils in the school ? PERCENTAGE. 141 'WRITTEN PROBLEMS. 7. A man owes $4560, which is 30% of his estate: how much is his estate? 1st Process. 2d Process. $4560 -^ .30 = $15200, Ans. $4560 ^ 30 X 100 = $15200, 8. 256 is 35% of what number? 9. 1331 is 162 % of what number? 10. 1071 is 15% of what number? 11. 480 sheep are 36% of how many sheep? 12. 5280 pounds are 66f % of how many pounds ? 13. $189.80 are 104% of what sum of money? 14. $88.66 are 110% of what sum of money? 15. The number of pupils in daily attendance in a cer- tain school is 420, which is 80 % of the number enrolled : how many pupils are enrolled ? 16. The number of youth of school age in a certain city is 5220, Avhich is 36 % of the number of inhabitants : what is the population of the city ? 17. A man spent 60 % of his money for a suit of clothes, 25 % of it for books, and had $7.50 left: how much money had he at first? 18. A man invested $5400 in railroad stock, which was 37^% of his property: what was the value of his property? 19. In a storm, a ship's crew threw overboard 250 bar- rels of flour, which was 40 % of the number of barrels on board : how many barrels of flour were left on board ? 20. A man owning 60 % of a factory, sold 40 % of his share for $9600 : at this rate, what was the value of the factory ? 21. The land surface of the earth is about 50000000 sq. miles, which is 331 % of the water surface : what is the extent of the water surface? 22. The population of a certain city in 1860 was 64000, which is 80% of the population in 1870: what was the population in 1870? 142 COMPLETE ARITHMETIC. FORMULA AND RULES. 213. Formula.— ^ase =rz ^er cent -^- rate per cent 214. KuLES. — To find a number when a certain per cent, of it is given, 1. Divide the number whidi is tJie per cent, by the rate per cent, expressed decimally. Or, 2. Divide tJie number whidh is Uie per cent, by the rate, and multiply the quotient by 100. Case IV. The Ainount or DifFereiice and the Rate I^er Cent, given, to finci the Base. 1. 216 is 8 % more than what number ? Solutions. — 1. If 216 is 8% more than a number, 216 is 108% or 1.08 of it, and hence the number equals 216 -^- 1.08, which is 200. Or, 2. If 216 is 108^ of. a number, 1% of it is xk of 216, which is 2, and 100% is 100 times 2, which is 200. 2. 318 is 6 % more than what number ? 3. $480 is 20 ^ more than what sum of money? 4. 560 pounds are 12% more than how many pounds? 5. 184 is 8 % less than what number ? Suggestion.— If 184 is 8% less than a number, 184 is 100^ — 8%, or 92% of it. 6. 285 is 5 % less than what number? 7. $356 are 11 % less than how many dollars? 8. 425 feet are 15 % less than how many feet? 9. A horse cost $160,* which was 20% less than the cost of a carriage : what was the cost of the carriage ? 10. A school enrolls 230 boys, which is 15 % more than the number of girls enrolled: hoAV many pupils in the school ? WRITTEN PROBLEMS. 11. A farm was sold for $6390, which was 12^% more than it cost : what was the cost of the farm ? PERCENTAGE. 143 1st Process. 2d Process. 100% +12i% =-112.1% =1.125 $6390 -^112.5 = $56.80 = 1% $6390 -^ 1.125 = $5680, Ans. $56.80 X 100 = $5680, Ans. 12. A man's expenses are $400 a year, which is 31-^% less than his income : what is his income ? 13. 276 is 15% more than what number? 14. What number increased by 30 % of itself, equals 162.5? 15. What number diminished by 16|% of itself, equals 2035.8? 16. A's farm contains 306 acres, which is 32 % less than B's : how many acres in B's farm ? 17. When gold was worth 25 % more than currency, what was the gold value of $150 in currency? 18. When gold was worth 50 % more than currency, what was the value in gold of a dollar bill ? 19. The number of pupils in daily attendance at a school is 570, which is 5 % less than the number enrolled : how many pupils are enrolled? 20. The number of jiupils enrolled in a certain town is 920, which is 15 % more than the average number of pupils in daily attendance : what is the average daily attendance ? 21. The population of a certain city in 1870 was 171527, which is 18 % more than its population in 1860 : what was the population in 1860? FORMULAS AND RULES. 215. Formulas.— 1. Base = amount -^ (1 + rate %). 2. Base =^ difference -f- (1 — rate % ). 216. Rules. — To find a number when another number is given, which is a given rate per cent., more or less, 1. Divide the given number by 1 j9^w8 or minus the given rate per cent, expressed decimally. Or, 2. Divide the given number by 100 plus or minus the given rate, and multiply the quotient by 100. 144 COMPLETE ARITHMETIC. REVIEW OF THE FOUR CASES. 217. The formulas of the four preceding cases of per- centage are here presented together for comparison: Case I. — Per cent. = base X rate per cent. Case II. — Rate per cent. = per cent. ^- base. Case III. — Base = per cent. -=- rate per cent. ^ ^^^ _, f Amount -f- (1 4- rate per cent). Case IY.—Base={ ^.^ ). . ^ /^ \ Vifference -^ (1 — rate per cent). Note. — Tlie two formulas, amount = hose + per cent, and difference =^b(ise — per cent., do not involve the operations of percentage, but simply the adding and subtracting of numbers. MENTAL PROBLEMS. 1. What is 121% of 640? 16f % of 860? 2. What is 33i % of 672 ? 66| % of 321 ? 3. 15 is what per cent, of 60? ^Of 90? 4. 16f lb. is what per cent, of 50 lb. ? Of 100 lb. ? 5. 25% of 120 is what per cent, of 90? 6. 33i% of 150 is what per cent, of 250? 7. 80 is 12|^% of what number? 8. 20% of 105 is 25% of what number? 9. 33^ % of 225 is 15 % of what number? 10. 45 is what per cent, of 75% of 120? 11. 360 is 20% more than what number ? 12. 60 is 33 J % more than what number ? 13. 33|-% of 240 is 33^% less than what number? 14. 25% of 280 is 16|% more than what number? 15. A man is 60 years of age, and 20 % of his age is 25 % of the age of his wife : how old is his wife ? WRITTEN PROBLEMS, 16. The population of a certain city in 1860 was 63500, and the census of 1870 shows an increase of 17J % : what was the population in 1870? rERCEN'TAGE. 145 17. A farm contains 480 acres, of which 30 % is meadow, 25|-% pasture, 16f % grain land, and the rest w^oodland: how many acres of each kind of land in the farm ? 18. A merchant failed in business owing $10500 and having $6300 worth of property: what per cent, of his indebtedness can he pay? 19. A clerk, receiving a yearly salary of $950, pays $275 a year for board, $180 for clothing, and $150 for other expenses : what per cent, of his salary is left ? 20. A lady pays $280 a year for board, $175 a year for clothing and other expenses, and lays up 35 % of her in- come: what is her income? 21. A man's expenses are 80% of his income, and 33 J % of his income equals 10 % of his property, which is valued at $27000 : what are his expenses ? 22. A merchant sold a stock of goods for $10811, and gained 13-| % : what was the cost of the goods ? 23. A cargo of damaged corn was sold at auction for $9450, which was 33^ % less than cost : what was the cost of the corn? 24. An orchard contains 1200 trees, of which 45 % are apple, 22 % peach, 12|^ % cherry, and the rest pear : how many trees of each kind in the orchard? 25. A owns 42^% of a factory worth $35000, B owns 37 % of it, and C owns the remainder : what is the value of each of their shares? 26. A man bequeathed $7560 to his wife, which was Q2^% of the sum bequeathed to his children, and the sum bequeathed to his wife and children was 80 % of his estate : what was the value of the estate? 27. The population of a city in 1870 was 41064, which was 16 % more than in 1860, and the population in 1860 was 6^ % less than in 1865 : what was the population in 1865? 28. The number of deaths in a certaiji city in 1869 was 1960, which was equal to 3^ % of the population : what was the population ? ... C.Ar.— 13. 146 COMPLETE ARITHMETIC. APPLICATIONS OF PERCENTAGE. 218. The principal applications of percentage are Profit and Loss, Commission and Brokerage, Capital and Stocks, Insurance, Taxes, Customs, Bankriiptcy, Interest, Discount, Exchange, Equation of Payments, and Equation of Accounts. All the problems are solved by the application of one or more of the four cases of percentage. PROFIT AND LOSS. 219. The Cost of an article is the price paid for it, or the total expense incurred in producing it. 220. The Selling JPrice of an article is the amount asked or received for it by the seller. The selling price of the seller is the cost to the buyer, and vice versa. 221. When an article is sold for more than its cost, it is said to be sold at a p?'q/i^ or gain; when it is sold for less than its cost, it is said to be sold at a loss or discount. Hence, 222. J^rofit or Grain is the amount which the selling price of an article exceeds its cost. 223. Loss or Discount is the amount which the sell- ing price of an article is less than its cost. Note. — The terms gain and loss are not limited to business trans- actions. When any quantity undergoes an increase or decrease, from any cause, there is a gain or loss, and when such gain or loss can be expressed in hundredths, it may be computed by the principles of percentage. MENTAL PROBLEMS. 1. A merchant bought a piece of cloth for $80, and sold it at 25% profit: for how much did he sell it? (Case I.) 2. A dealer bought hats at $5 apiece, and sold them at 20 % profit : what was the selling price ? PROFIT AND LOSS. 147 3. Hats, costing $5 apiece, were sold at a loss of 20 % : what was the selling price ? 4. At what price must flour, costing $6 a barrel, be sold to gain 16| % ? 5. A grocer bought sugar at 12 cts., 16 cts., and 18 ct^. a pound: for how much must each kind be sold to gain 20 % ? To gain 25 % ? 6. A merchant sells broadcloth, costing $4, for $5 a yard: what per cent, does he gain? (Case IL) 7. When broadcloth, costing $5 a yard, is sold for ^4 a yard, what is the loss per cent. ? 8. Teas costing $1.20 and $1.50 a pound, are sold respectively at $1.50 and $1.80 a pound: what is the gain per cent.? 9. A merchant sold velvet at a profit of $2 a yard, and gained 20%: how much did it cost? {Case III.) 10. A dealer sold boots at $1.50 a pair less than cost, and thereby lost 33^% : what did they cost? 11. A grocer sold tea at 30 cents above cost, and gained lQj% : what was the cost of the tea? What was the sell- ing price ? 12. A man sold a horse for $90, and gained 20 % : what was the cost of the horse ? {Case IV.) 13. A man sold a horse for $80, and lost 20 % : what was the cost of the horse? 14. Sold butter at 40 cts. a pound, and gained 25 % : how much did it cost? 15. A watch, costing $80, was sold at a loss of 10 % : for how much was it sold ? 16. How must shoes, costing $2, $2.50, and $3 a pair, be sold respectively to gain 25 % ? How must each kind be sold to gain 30 % ? 17. How must muslin that cost 10 cts., 15 cts., and 18 cts. a yard, be sold to gain 20 % ? 18. Sold tea at 90 cts. a pound, and gained 20 % : what would have been my gain per cent, had I sold it at $1 a pound ? 148 COMPLETE ARITHMETIC. AVRITTEN PKOBIiEMS. 19. A house and lot, which cost $67o0, were sold at a gain of 12-^- % : for how much were they sold ? 20. Carriages, costing $165, are sold at 18% profit: what is the gain on each carriage? 21. A man paid $4500 for a farm, and sold it for $5400 : what was the gain per cent. ? 22. A drover bought cattle at $65 a head, and sold them at $84.50 a head: what was the gain per cent.? 23. Carpeting, costing $1.75 a yard, is sold for $2: what is the gain per cent. ? 24. A drover bought horses at $130 a head, expended $6 each in taking them to market, and then sold them at $153.50 a head: what was the gain per cent.? 25. A cargo of wheat costing $16500, being damaged, is sold for $13750 : what was the loss per cent. ? 26. A merchant sold a lot of goods at 12^% profit, and gained $8160: what was the cost? 27. A grocer sold 21 barrels of apples at 22 % profit, and gained $45.10: what was the cost per barrel? 28. A man sold a watch for $180, and lost 16f % : what was the cost of the watch? 29. A house and lot were sold for $7762.50, at a gain of 15%: what was the cost? 30. A dry goods firm sold $45000 worth of goods in a year, | of the goods being sold at 20 % profit, |- at 25 % profit, and the rest at 33|^ % profit : what was the cost of all the goods? 31. Sold a piece of carpeting for $240, and lost 20 % : what selling price would have given a gain of 20 % ? 32. A merchant sells goods at retail at 30 % above cost, and at wholesale at 12% less than the retail price: what is his gain per cent, on goods sold at wholesale? 33. How must cloth, costing $3.50 a yard, be marked that a merchant may deduct 15 % from the marked price and still make 15% profit? COMMISSION AND BROKERAGE. 149 34. A merchant marked a piece of silk at 25% above cost, and then sold it at 20 % less than the marked price : did he gain or lose, and how much ? FORMULAS AND RULES. 224. Formulas. — 1. Gain or loss = cost X rate %. .2. Rate per cent. = gain or loss -f- cost 3. Cost = gain or loss -f- rate %. 4. Cost = selling price -^ ^ i ■rate%. 225. Rules.— 1. To find the gain or loss when the cost and rate per cent, are given, Multiply the cost by the rate per cent, expressed decimally. (Form. 1.) 2. To find the rate per cent, when the cost and the gain or loss are given. Divide the gain or loss by the cost, and the quotient expressed in hundredths ivill be the rate per cent. (Form. 2.) 3. To find the cost when the gain or loss and the rate per cent, are given, Divide the gain or loss by the rate per cent, expressed decimally. (Form. 3.) 4. To find the cost when the selling price and the rate per cent, of gain or loss are given, Divide the selling price by 1 plus or minus the rate per cent. (Form. 4.) Note.— Let the pupil review the above problems, solving those in wliich tlie rate is an aliquot part of 100, bv using the fraction. (Art. 210, Note.) COMMISSION AND BROKERAGE. 226. An Agent is a person who transacts business for another. 227. A Factor is an agent who buys and sells goods intrusted to his possession and control. A mercantile factor is also called a Commission Merchant. When a factor lives in a different country or part of the country from his employer, he is called a Correspondent or Consignee. The 150 COMPLETE ARITHMETIC. goods shipped or consigned to a Consignee are called a Consignment ; and the sender of the goods is called a Consignor. 228. A JBroUer is a person who buys and sells gold, bills of exchange, stocks, bonds, etc. ; or an agent who buys and sells property in possession of others. 229. A Collector is an agent who collects debts, taxes, duties, etc. (Arts. 263, 267.) 230. Commission is an allowance made to a factor or other agent, for the transaction of business. The commis- sion allowed a broker is called Brokerage. 231. Commission is computed at a certain per cent, of the amount of property bought or sold, or of the amount of business transacted. The rate per cent, is called the Rate of Commission, and the amount of business transacted is the Base. The rate of commission varies with the amount and nature of the business. A broker's commission is usually less than a factor's. 232. The Wet Proceeds of a sale or collection are the proceeds less the commission and other charges. MENTAL PEOBLEMS. 1. An auctioneer sold $300 worth of furniture, and charged a commission of 5 % : how much did he receive ? 2. A peddler bought $500 worth of rags, at a commis- sion of 10% : what was his commission? 3. An agent sold $1200 worth of school furniture, at a commission of 16| % : how much did he receive? 4. An attorney collected bad debts to the amount of $800, and charged 20 % commission : what was his com- mission ? 5. A society paid a lad $6 for collecting membership dues, to the amount of $100: what rate of commission did he receive? 6. A bookseller received $30 for selling $150 worth of maps: what was his rate of commission? COMMISSION AND BROKERAGE. 151 7. A real-estate broker received $40 for selling a house and lot, at 5% commission: for how much was the prop- erty sold? 8. An attorney received $60 for collecting a note, at 10% commission: what was the amount collected? 9. An agent received $108, with which to buy peaches, after deducting his commission at 8 % : how much did he expend for peaches? 10. A factor received $309, with which to buy flour, after deducting his commission, at 3 % : what was the cost of the flour? 11. A lawyer collected a bill at 25 % commission, and remitted $7.50 as net proceeds: what was the amount col- lected? What was the lawyer's commission? 12. A bookseller sold a lot of books on commission, at 20%, and remitted $160 as net proceeds: for how much were the books sold? WRITTEN PROBLEMS. 13. A commission merchant sold 540 barrels of flour, at $6.37^ a barrel: what was his commission at 3 % ? 14. A real-estate broker sold 325 acres of land, at $24.50 an acre, and charged a commission of 2^ % : what was his commission? 15. An auctioneer sold $5160.50 worth of dry goods, and $715.25 worth of furniture: what was his commission, at 16. A lawyer collected 65 % of a note of $950, and charged 6^% commission: what w^as his commission? What was the amount paid over? 17. A factor in New Orleans purchased $75000 worth, of cotton for a Lowell manufacturer, at If % commission : what was his bill for cotton and commission? 18. An architect charged ^ % for plans and specifica- tions, and If % for superintending the erection of a build- ing, costing $120000: how much was his fee? 152 COMPLETE ARITHMETIC. 19. An agent furnished a school-house for $4500, and received $540 commission: what Avas the rate? 20. An attorney charged $75 for collecting rents to the amount of $1125: what was the rate of commission? 21. A commission merchant charged 2|- % for buying produce, and his commission was $750 : how much produce did he purchase? 22. A w^ool agent received 5 % for buying wool, and his commission was $208.50: how much wool did he buy? 23. My agent has bought 3300 barrels of apples, at $1.75 a barrel, and I allow him 3% commission: how much money must I remit to pay both the cost of the apples and the commission? 24. A Boston merchant sent his factor in Cincinnati $3539.20, to be invested in bacon, after deducting his com- mission at 2 % : how much did he expend for bacon, and what was his commission? 25. A cotton broker in Charleston received $11774, with which to purchase cotton, after deducting his commission of 1|- % : how much did he expend for cotton, and what was his commission? 26. A merchant paid a broker f % for a draft of $1280 on New York : how much was the brokerage ? 27. A broker bought $15600 worth of stocks, and charged |- % : what was his fee ? 28. A real estate broker sold a section of land (640 A.) at $7.50 an acre, and invested the proceeds in railroad stock, receiving 1^ % for selling the land and J % for buying the stock : what was his brokerage ? 29. What will be the total cost of 750 yards of carpeting, if a merchant pays 2^ % commission for purchasing, \ % for a draft covering cost and agent's commission, and $12.50 for freight? 30. A grain dealer in Chicago received $5000 with direc- tions to purchase wheat, at $1.10 a bushel, after deducting his commission at 2^ % : how many bushels of wheat did he purchase? COMMISSION AND BROKERAGE. 153 31. An agent sold 45 sewing machines at $75 apiece, and 9 at $125 apiece, and, deducting his commission, re- mitted $3375 to the manufacturer as proceeds: what was his rate of commission ? 32. A factor sold $15000 worth of goods, at 10 % com- mission, and invested the proceeds in cotton, first deducting 5 % commission for buying : how much money did he invest in cotton? 33. Smith & Jones sell for C. Bell & Co. 3040 pounds of butter, at 22 cts. a pound, and 10560 pounds of cheese, at 15 cts. a pound, and invest the proceeds in dry goods, first deducting their commission of 5 % for selling and 3 % for buying : how much did they invest in dry goods ? What was their entire commission ? 34. A commission merchant sold 1300 barrels of flour, at $5.75 a barrel, receiving a commission of 3J^, and in- vested the net proceeds in coffee, at 28 cts. a pound, first deducting 2 % commission : how many pounds of coffee did he purchase? What was his entire commission? FORMULAS AND RULES. 233. Formulas.— 1. Com. or hroh. = base X rate %. 2. Rate % =-z com. or hroh. -r- hase. 3. Bai^e = com. or hroh. -^ rate %. 4. Base = {base -[- com. m- broh.) -r- (1 -{- rate %.) 234. Rules. — 1. To find commission or brokerage, Mtd- tiply the sum of money denoting the amount of b^mness trans- acted, by the rate per cent, expressed decimally. (Form. 1.) 2. To find the sum to be invested Avhen the amount given includes both the sum to be invested and the commission or brokerage. Divide the given amount by 1 plus the rate per cent., and the quotioit luill be the sum. to be invested. (Form. 4.) Note. — These two rules cover all tlie ordinary business transac- tions in commission or brokerage, but the pupil should be required to form rules embodying each of the four formulas. 154 COMPLETE ARITHMETIC. CAPITAL AND STOCK. 235. CajHtal is property invested in trade, manufac- tures, or other business. 2Z6.TneI>ar Value of capital is its original or nominal value. The Market Value of capital is its real value, or the sum for which it will sell. When the market value of capital equals its par value, it is said to be at par ; when the market value is more than the par value, it is above par, or at a premium; when the market value is less than the par value, it is below paj-, ^jte> ^^^ 1-m— ^;s-T-^"- or at a discount. 237. Premium is the amount which the market value of capital exceeds its par value. 238. Discount is the amount which the market value of capital is less than its par value. 239. Premium and discount are computed at a given per cent, of the par value. The rate per cent, is called the Rate of Premium, or the Hate of Dhcount. 240. A Com^Hiny is an association of persons united for the transaction of business. The association of several persons in business as partners, bound by articles of agreement, is called a Partnership, and the company is commonly called a Firm or Hou?e. (Art. 380.) 241. An Incori^orated Company is a company organized and regulated by law. It is called a Corporation, and the law regulating it is called a Charter. CAPITAL AND STOCK. 155 242. The capital of a corporation is called Stock, and is divided into equal parts, usually $100 each, called Shares. The owners of these shares are called Stockholders. 243. Certificates of Stock are official statements of the size and number of shares owned by each stockholder. They are called Scrij), and are bought and sold like other property. Stocks are at par, above par, or below par, according as their mar- ket value equals, exceeds, or is less than their par value or face. The market value of stocks is quoted at a certain per cent, of the par value. Stocks quoted at 108 are worth 108 % of their face, that is, are 8% above par; stocks quoted at 92 are worth 92% of their face, that is, are 8% below par. The business of buying and selling stocks is called Stock Jobbing, and persons engaged in such business are called Stock Jobbers, or Stock Brokers. 244. The Gross JEarninf/s of a company are the total receipts from its business; and the Net Earnings are the net profits, found by deducting all expenses and losses from the gross earnings. 245. A Dividend is the part of the earnings of a company distributed among the stockholders. Dividends are usually declared annually or semi-annually, and they are computed as a per cent, of the piir value of the stock. The rate per cent, is called tlie Rate of Dividend. 246. An Assessment is a sum levied upon the stock- holders to meet the losses or expenses of the business. The business of incorporated companies is usually managed by directors, who are elected by the stockholders, each being entitled to as many votes as he owns shares. Note. — When a business corporation wishes to raise money in ad- dition to that derived from its capital stock, it issues notes or bonds, payable at a specified time with interest, and secured by mortgage on the property of the corporation. These notes are called Mortgage Bonds, and their owners are called Bondholders. These bonds are negotiable and are called Stocks (Art, 330), but they should be care- fully distinguished from Capital Stock. 156 COMPLETE ARITHMETIC. MENTAL PROBLEMS. 1. When stock is 6 % premium, what is the market value of $1? Of $100? 2. When stock is 12 % discount, Avhat is the market value of $1? Of $100? 3. How much will 5 shares of telegraph stock cost, at 4 % premium ? At 4 % discount ? Note. —A share is $100, if no other vakie is named. 4. How is stock quoted when it is 15% premium? When it is 15 % discount ? 5. When stock is quoted at 107|^, what is the value of $1? Of $100? 6. When stock is quoted at 87, what is the value of $1 ? Of $100? 7. How much will 10 shares of mining stock cost when quoted at 104? At 85?" 8. When bank stock is quoted at 105, how many shares can be bought for $525? For $840? 9. A company declares a dividend of 3 % : how much will a stockholder, owning 15 shares, receive? 10. A manufacturing company made an assessment of 5%, to repair damages caused by a freshet: how much must a stockholder, owning 20 shares, pay? "WKITTEN PROBLEMS. 11. A man bought 75 shares of railroad stock at 1\% discount: how much did they cost? 12. Bought 100 shares of Little Miami stock at 109|-, and sold them at 112^: how much did I gain in the trans- action ? 13. A broker bought 70 shares of insurance stock at 6|- % premium, and sold them at f % discount : how much did he lose ? CAPITAL AND STOCK. 157 14. A man bought 52 shares of Illinois Central at 127 ; and sold 36 shares at 135, and the rest at 137|: how much did he gain ? 15. A man exchanged 52 shares of railroad stock at 80, for insurance stock at 104 : how many shares of insurance stock did he receive? 16. A broker bought 84 shares, $50 each, of telegraph stock at 94, and sold them at lOOf: how much did he gain? 17. The Cincinnati Gas Co. declares a dividend of 16J % : how much will a man holding 36 shares receive ? 18. The capital of an insurance company is $500000, and it declares a dividend of 4|% : how much money is distributed among the stockholders? 19. A company with a capital of $125000 declares a div- idend of 4 % , with $3500 surplus : what were the net earn- ings of the company? Note. — The surplus is a part of the net earnings set apart to meet future demands. 20. The entire capital stock of the railroads in Ohio for 1869 was $106686116, and their net earnings for the year were $9051998: what was the average rate of dividend? 21. The net earnings of a gas company are $22425, and the capital stock is $215000: what rate of dividend can be declared, no surplus being reserved ? What will be the div- idend on 45 shares? 22. The capital of a mining company is $450000; the gross receipts are $70680 ; and the expenses are $40325 : what rate of dividend can it declare, reserving a surplus of $6505? 23. How many shares of bank stock at 4 ^ premium, can be bought for $8320 ? 24. How much railroad stock, at 12| discount, can be bought for $8750 ? 25. When N. Y. Central is quoted at 95|, how much stock can be bought for $6894, brokerage j %^ 158 COMPLETE ARITHMETIC. 26. A broker bought 84 shares of coal stock, at 108 J, re- ceived a dividend of 5.J %, and then sold the stock for 106: how much did he gain ? 27. A broker bought stock at 4 % discount, and, selling the same at 5 % premium, gained $450 : how many shares did he purchase ? 28. A man bought Michigan Central at 120, and sold at 124 : what per cent, of the investment did he gain ? FORMULAS AND RULES. 247. Formulas. — 1 . Dividend or assessment = stock X Tate % . 2. B,ate % = divid. or assess, -f- stock. 3. Stock = divid. or assess, -f- rate % . 4. Prem. or dis. =par value X rate%. 5. Rate % ^^=prem. or dis. -=- par value. 6. Par valus=prem. or dis. -~ rate %. 7. Par val. = market val. ^- (1 dz rate % )• 8. Market val. = par val. X (1 ih rate%). 9. Market val. ^=par val. -\-prem. or — dis. Note. — Formulae 4, 7, and 8 cover all ordinary transactions in stock jobbing. Tlie sign dr , used in 7 and 8, is read plus or minus. 248. KuLES. — 1. To find the dividend or assessment when the stock and rate per cent, are given, Midtiphj the stock by the rate per cent, expressed decimally. (Form. 1.) 2. To find the rate per cent, when the dividend or assess- ment and total stock are given. Divide the dividend or assess- ment by the amount of stock, and the quotient, expressed in hundredths, will he the rate per cent. (Form. 2.) 3. To find the stock when the dividend or assessment and the rate per cent, are given. Divide the dividend or assessment by the rate per cent, expressed decimally. (Form. 3.) 4. To find the premium or discount on a given amount of stock, Midtiply the amount of stock by the rate per cent, ex- pressed decimally. (Form. 4.) INSURANCE. Ud 5. To find the cost or market value of a given amount of stock, Multiply the amount of stock (1) by 1 plus or minus the rate per cent. (Form. 8) ; or (2) by the quoted price ex- pressed as hundredUis. Note. -The cost may also be found by adding the premium to, or subtracting the discount from, the par value. (Form. 9.) 6. To find the amount of stock which can be bought for a given amount of money. Divide the amount of money to be invested (1) by 1 plus or minus the rate per cent. (Form. 7) ; or (2) by the quoted price expressed as hundredths. Note. — When brokerage is paid, the rate of brokerage must be subtracted from the quoted price before dividing. INSURANCE. 249. Insurance is a guaranteed indemnity for loss. 250. Fire Insur- ance is a guaranteed in- demnity for loss of prop- erty by fire. 251. Marine In- surance is a guaran- teed indemnity for loss of property while trans- ported by water. Insurance on the property transported is called Cargo In- surance; that on the vessel i-s called Hull Insurance, 252. Life Insurance is a guaranteed indemnity for loss of life. Health Insurance guarantees the insured a certain sum of monev if sick; and Accident Insurance pledges a like indemnity if the in- sured is injured by accident. 160 COMPLETE ARITHMETIC. 253. The I*olicy is the written contract between the insurer and the insured. The insurer is called an Underwriter, and the insured a Policy Holder. 254. The J^reniiuiU is the sum paid by the insured to obtain the insurance. It is a specified per cent, of the amount insured. The act of insuring is called taking a risk. When property is in- sured, the valuation or amount is usually made less than the real value of the property. The insurance business is chiefly carried on by corporations, called Insurance Companies. In Joint Stock Companies the profits and losses are shared by the stockholders, but in Mutual Companies they are divided among the policy holders. MENTAL PKOBLEMS. 1. A house was insured for $2500, at 1 % : what was the premium? 2. A stock of goods was insured for $8000, at f % : what was the premium ? 3. A hotel worth $6000 is insured for f of its value, at ^i%'' what is the premium? 4. What will it cost to get a house insured for $4000, for 10 years, at \% a year ? 5. The premium paid for insuring a library for $500, is $5 : what is the rate of insurance ? 6. An insurance company insures a school house for $10000, and charges $50 premium: what is the rate? 7. The premium for insuring a cargo of goods, at 2%, was $240 : what was the amount of goods insured ? -WRITTEN PROBLEMS. 8. A factory worth $75000 is insured for f of its value, at 1^ % : how much is the premium ? 9. A merchant has his store insured for $7850, slI ^%^^ and his goods for $12400, at |^ % : what premium does he pay ? INSURANCE. 161 10. A house worth $5400 was insured for f of its value, Sit i%, and the cost of the survey and policy was $1.50: what was the cost of the insurance? Note. — When a new risk is taken, a small fee is usually charged for examining the property, called the Survey, and for issuing the policy. 11. The owners of a vessel paid $561 for a hull insurance of $25500 : what was the rate of insurance ? 12., A merchant paid $100 for an insurance of $12500 on a stock of goods: what was the rate of insurance? 13. A school house is insured at | %, and the premium was $93.60: for how much is the house insured? 14. A grain shipper paid $525 for the insurance of a cargo of wheat, at 1^ % : for how much was the wheat in- sured ? 15. A company insured a block of buildings for $150000, at f % , but, the risk being too great, it re-insured $40000 in another company, at f % , and $35000 in another company, at f %. How much premium did it receive more than it paid? 16. A block of buildings worth $135000 is insured for f of its value by three companies, the first taking ^ of the risk at f % ; the second taking f of it at f % ; and the third taking the remainder at ^ %. What was the total premium ? 17. Suppose the above block should be damaged by fire to the amount of $60000, how much of the damage would each company be obliged to pay ? 18. A house which has been insured for $3500 for 10 years, at f ^ a year, was destroyed by fire : how much did the money received from the company exceed the cost of premiums ? 19. A steamer, burned in 1869, had been insured by a single company 20 years, for $40000, at 24- % a year : what was the actual loss to the company, no allowance being made for interest ? 20. A grain dealer had a cargo of wheat, valued at CAr.— 14. 162 COMPLETE ARITHMETIC. $31360, insured at 2 ^, so as to cover both the value of the wheat and the cost of the premium : for how much was the wheat insured? Suggestion.— Since, the premium was 2 % of the amount insured, the vahie of the wheat was 100 % — 2 %, or 98 % of the amount in- sured. Hence, $31360 = x^ir of the amount insured. (Case IV.) 21. For how much must a cargo of lumber, worth $21825, be insured, at 3 % , to cover both the value of the lumber and the cost of the premium ? 22. For how much must property, worth $11859.40, be insured, at 1|- % , to cover both property and premium ? 23. For what must a cargo of goods, valued at $11520, be insured, at 4 % , to cover both goods and premium ? 24. What amount must be insured to cover property worth $2587, and premium at ^ % ? 25. To cover both goods and premium, at 1%, a mer- chant had a cargo of goods insured for $35000 : what was' the value of the goods ? 26. A merchant shipped a cargo of flour from New York to Liverpool, and, to cover both the flour and the premium, he took out a policy for $50400, at Z\%\ what was the value of the flour ? FORMULAS AND RULES. 255. Formulas. — 1. Premium = amount insured X rate % . 2. Bate ^ :=prem. -^ amount insured. 3. Amount insured =prem. -4- rate %. 4. Property and p)remium = propertij -^ (l — rate%). 256. Rules. — 1. To find the premium. Multiply the amount insured by the rate per cent, expressed decimally. (Form. 1.) 2. To find for what sum property must be insured to cover both property and premium. Divide the value of the property insured by 1 less the rate per cent, expressed ded- mally. (Form. 4.) LIFE I^•SURANCE. 163 LIFE INSURANCE. 257. In Life Insurance the insurer agrees to pay- to the heirs of the insured, or to some person named in the policy, a stipulated sum on the death of the insured, or at a specified time, should his death not occur before. 258. When the policy matures at the death of the in- sured, it is called a lAJe Policy; when it matures in a speci- fied number of years, it is called a Term Policy or an Endow- ment Policy, The premium in life insurance may be paid in a single payment ; or it may be paid annually during the term of the policy ; or it may be paid annually for a specified number of years, usually for ten years. A Non-forfeiting Policy guarantees the insured an equitable part of the sum insured in case he should fail to pay his annual premiums after a specified number of payments have been made. 259. The premium is computed at a certain sum or rate per $1000 insured, the rate varying with the age of the in- sured at the time the policy is issued. The basis on which the rate of life insurance is determined is the expectation of life, or the average extension of life beyond the given age, as shoAvn by life statistics. Tables have been formed showing the expectation of life for every year of man's age. (See appendix.) ^^C^EITTEN PEOBLEMS. 27. A man 45 years of age has his life insured for $3000, at $37.30 per $1000: what annual premium does he pay? 28. A man 30 years of age has his life insured for $6000, at $23.60 per $1000: what is his annual premium? 29. A man 38 years of age is insured for $5000 on the ten year plan, at $44.50 per $1000. what will be the sum of his premiums should they all be paid ? 30. A man 35 years of age took out a life policy for $4000, at the rate of $27.50 per $1000; he died at the age 164 • COMPLETE ARITHMETIC. of 60 : how much greater was the amount insured than the sum of the annual paymfents ? 31. A man 27 years of age took out a life policy for $8000, for the benefit of his wife, at the rate of $21.70 per $1000, and his death occurred at the age of 33: how much did the Avidow receive more than had been paid in annual premiums? TAXES. .260. A Tax is a sum of money assessed on the person, property, income, or business of a citizen for the support of government or other public purposes. 261. A tax on the person of a citizen is called a Poll TaXj or Capitation Tax. 262. A tax on property is called a Property Tax. It is assessed either at a given rate per cent, of the valuation, or at the rate of a given number of mills on the dollar. Property is classified as Real Estate and Personal Property ^ the former including all fixed property, as houses and lands; and the latter, all movable property. The taxable value of real estate is appraised by oncers called Appraisers or Assesso7^s, and the value of personal property is fixed by the owner, under oath, or by the assessor. 263. A tax on the annual income of a citizen is called an Income Tax ; and a tax on his business is called an Ex- cise Tax. Income taxes are assessed at a given rate per cent, of a citizen's net income, less specified exemptions and deductions. Excise taxes consist of fees for business licenses, revenue stamps for business papers, taxes on manufactured products, etc. The Internal Revenue of the United States is chiefly derived from income and excise taxes, assessed and collected by United States ofti^ cers, called Assessors and Collectors. Note. — Taxes are classified as direct and indirect. Property and poll taxes are direct; and excise taxes and duties arc indirect, since they are paid indirectly by the consumer. TAXES. 165 WBITTEN PROBLEMS. 1. The valuation of the taxable property of a village was $632000, and a tax of $9480 was assessed to build a school house : what was the rate of tax ? Process. Since the tax was .015 of the property, $9480 -H- $632000 r= .015 the rate was 1.5^, or 15 mills on the Kate = li%, or 15 mills. dollar. 2. The tax levied in a certain city, for all purposes, was $259776, and the taxable property was listed at $21648000 : what w^as the rate of tax in mills? 3. The amount of tax to be assessed in a certain township is $19340.20; the taxable property is $1425400; and the number of polls, assessed at $1.50 each, is 540: w^hat rate of tax must be assessed on property? 4. The cost of the public schools of a certain city for the next school year, is estimated at $36848 : what amount of school tax must be assessed, the cost of collecting being 2^, and allowing 6 % of the assessed tax to be uncol- lectible ? Process. Since 2% of tax collected is paid for QQ X <|!q^o4o collection, $36848 is 98% or .98 of the 'm ) $37600, Ter Cent. 9. The interest on $540 for 8 mo. 18 da. was $27.09: what was the rate per cent. ? Process. ^540 Since the interest on $540 for 8 mo. .043 18 da., at 1%, is $3.87, the rate % which 1620 produced $27.09 interest, was as many 2160 times 1% as $3.87 is contained times in 6 ) $23.220 InLatQfo. $27.09, which is 7. Hence, the interest $3.87 " "1%. accrued at 7%. $27.09 ^$3.87 = 7. Note. — The interest at 1% may be found bv multiplying $540 by ^ of .043, whicli is .0071. 10. The interest of $456 for 3 yr. 5 mo. 18 da., is $79.04: what is the rate per cent.? 11. The interest of $216 for 5 yr. 7 mo. 27 da., is $122.22: what is the rate per cent.? PROBLEMS IN INTEREST. 187 12. The interest of $560 for 2 yr. 4 mo. 15 da., was $106.40: what was the rate per cent.? 13. The interest of $95.40 for 3 yr. 9 mo., is $28.62: what is the rate per cent.? 14. The interest of $240 from Feb. 15, 1868, to Apr. 27, 1869, was $23.04: what was the rate per cent.? 15. The interest of $252 from Aug. 2, 1867, to March 9, 1868, was $12,152: what was the rate per cent.? 16. A note of $345.60, dated Feb. 5, 1863, was paid Aug. 20, 1865, and the amount was $407,088: what was the rate per cent.? FORMULA AND RULK 297. Formula. — Rate % = interest -i- {prin. X 1 % X time). 298. KuLE. — To find the rate per cent.. Divide the given interest by the interest of the principal for the given time, at 1 per cent. Problem III. I'rin.cipal, Interest, and Rate IPer Cent, given, to find the Time. 17. The interest of $300, at 9%, is $60.75: what was the time? Process. Since the interest of $300 for 1 year, ^3QQ at 9%, is $27, $300 must be on interest " ,09 as many years to produce $60.75 inter- ^27.00 Int. for 1 yr. ^^^y ^^ ^^7 are contained times in $60.75, which is 2.25. Hence the time is 2.25 $60.75 -^ $27 = 2.25 years, or 2 yr. 3 mo. (Art. 170). 2.25 yr. = 2 yr. 3 mo. 18. The interest of $908, at 3| %, was $79.45: what was the time? 19. The interest of $56.78 for a certain time, at 10%, was $22.23: what was the time? 20. How long must a note of $300 run to give an amount of $347.25, at 6 % ? 188 COMPLETE ARITHMETIC. 21. In what time will the interest of $150, at 4%, be $9? Sll? 22. In what time will $2040 produce $334.05 interest, at 5%? 23. In what time will any principal double itself at 4 % interest? At 6 % ? At 10 % ? 24. In what time will any principal double itself at 5 % interest? At 7 % ? At 12 % ? 25. In what time will any principal treble itself, at 5 % interest ? At 10 % ? At 6 % ? FORMULA AND RULE. 299. Formula. — Time = interest -=- (^principal X rate %). 300. Rule. — To find the time. Divide the given interest by the interest of the 'principal for 1 year, at the given rate per cent. Note. — Reduce the fraction of a year to months and days. If preferred, the interest may be divided by the interest for 1 month, at the given rate per cent. Problem IV. Interest, Rate 3?er Cent., and Time given, to find the Principal. 26. What principal will produce $49.20 of interest in 1 yr. 4 mo. 12 da., at 6 % ? 1st Process. Since $1 of principal produces $.082=Int. of $1. ^-082 of interest, it will take as many times %1 of principal to pro- $.082 ) $49.20 ( 600 ^^uce $49.20 of interest as $.082 is 492 contained times in $49.20, which is $1 X 600 = $600. Ans. 600. 600 times $1 = $600, the re- quired principal. Or, 2d Process. Since the interest of $1 is .082 .082 ) $49,200 ( $600 of itself, $49.20 is .082 of the re- 492 quired principal. S49.20h- .082 = $600. PROBLEMS IN INTEREST. 189 What principal will produce 27. $15.24 interest in 7 mo. 6 da., at 8 % ? 28. $1000 interest in 5 yr. 6 mo. 20 da., at 5%? 29. $519 interest in 5 mo. 23 da., at 12 %? 30. What sum invested, at 7%, will produce $378 inter- est annually ? 31. What sum invested, at 4|^ %, will yield an annual income of $900? 32. AVhat principal will produce $220 interest, from Oct. 25, 1871, to March 7, 1872, at 8 % ? 33. What principal will produce $17.78 interest, from Jan. 10, 1872, to March 13, 1872, computed by days, at 4%? FORMULA AND RULES. 301. Formula. — Principal = interest -^ (rate % X time). 302. Rules. — To find the principal when the interest, rate per cent., and time are given, 1. Divide the given interest by the interest of $1 for the given time and rate per cent., and multiply $1 by the quotient. Or, 2. Divide the given interest by the decimal corresponding to the interest of $1 for the given time and rate per cent. Problem V. A-iTionnt, R-ate l^er Cent., and Xiine given, to find the Principal. 34. What principal, on interest at 8%, for 1 yr. 6 mo. 18 da., will give an amount of $730.60? The interest of $1 for 1 yr. 6 mo. 18 da. is $.124, and the amount is 1ST FROCEss. ^j^24. If the amount of $1 is $1,124 = amount of $1. $1,124, it will take as many times S1.124 ) $730,600 ( 650. $1 to yield an amount of $730.60 ^ZM_ as $1,124 is contained times in §??? $730.60, which is 650. 650 times $1 is $650, the required principal. Or, 5620 190 COMPLETE ARITHMETIC. 2d Process. 1.124 ) $730,600 ( $650. ^^"^^ ^^^ ^°^^"^* «^ ^^ ^^ 1-124 6744 of itself, $730.60 is 1.124 of tlie re- ~5620 quired principal. $730.60-4-1.124 6620 = $650. 35. What principal on interest, at 5 %, for 1 yr. 10 mo. 12 da., will amount to $70.24? 36. What sum of money put at interest, at 7 %, for 8 mo. 18 da., will amount to $567.09? 37. What sum of money put at interest, at 8 %, for 2 yr. 1 mo. 15 da., will amount to $421.20? 38. What sum of money put at interest March 15, 1870, at 6 %, will amount to $2600.40, Aug. 6, 1871? FORMULA AND RULE. 303. Formula. — Principal = amt.-T-ll -}- (rate % X iime)\. 304. Rules. — 1. To find the principal when the amount, rate per cent., and time are given, 1. Divide Uie given amount by the amount of $1 for the given time and rate per cent.j and multiply $1 by the quotient. Or, 2. Divide the given amount by ike decimal corresponding to the amount of $1 for the given time and rate per cent. Note. — The principal thus found is the present worth of the amount. See Art. 306. REVIEW OF THE FIVE PROBLEMS. 305. The formulas for the five problems in interest are here presented together : Formulas. — 1. Interest =^ principal X fate % X time. 2. Eate% = int. —- (principal X 1% X time) 3. Time = interest -r- (principal X 'rate % ). 4. Principal ::= interest -h- (rate % X time). 5. Principal = amt -i- [I -\-(rat€% Xtinw)']. PROBLEMS IN INTEREST. 191 WRITTEN PROBLEMS. 1. What is the interest of $205 for 2 yr. 5 mo. 24 da., at 7%? 2. What is the amount of $160, from Jan. 12, 1869, to July 3, 1870, at 8 % ? 3. At what rate per cent, will $512.60 yield $25.72 in- terest in 8 mo. 18 da. ? 4. The principal is $126.75, the interest $20.96, and the time 2 yr. 24 da. : what is the rate ? 5. How long will it take $5000 to produce $1125 inter- est, at 8 % ? 6. The principal is $326.50, the interest $2.76, and the rate 8 % : what is the time ? 7. The amount is $1563.75, the interest $63.75, and the rate 7|- % : what is the time ? 8. What principal will yield $1.36 interest in 20 days, at 6%? 9. The interest on a certain principal from Nov. 11, 1857, to Dec. 15, 1859, at 6%, was $4,474: what was the principal ? 10. What principal, at 7 %, will amount to $659.40 in 8 months ? 11. The interest is $12.78, the time 1 yr. 2 mo. 6 da., and the rate 6 % : what is the amount ? 12. What principal will amount to $609.20 in 4 mo. 18 da., at 4%? 13. What principal will amount to $288.85 in 1 yr. 6 mo., at 6 %? 14. What principal will produce $21,757 interest, from Jan. 1. to Oct. 20, 1869, at 7 % ? 15. How long will it take any principal to double itself at 6 % ? At 4 % ? At 10 % ? 16. What is the amount of $420, from June 10, 1869, to Jan. 21, 1870, at lO-^? 17. What sum, bearing interest at 7%, will yield an an- nual income of $1000? 19^ COMPLETE ARITHMETIC. PRESENT WORTH AND DISCOUNT. 306. The most common application of Problem V is In computing Present Worth and Discount. 307. The JPresent Worth of a debt due at a future time, without interest, is the sum or principal which, at Ihe current rate of interest, will amount to the debt when it becomes due. 308. Disco U^lt is the amount deducted from a debt for its payment before it is due. 309. True Discount is the difference between a debt, not bearing interest, and its present worth. • True discount is the interest on the 'present woi^th of a debt, while simple interest is computed on the debt itself. The difference is the . interest on the true discount for the time. WRITTEN PROBLEMS. 1. What is the present worth of a note of S212, due 1 year hence, without interest, the current rate of interest being 6 % ? What is the true discount ? Process. The amount of $1 for 1 year, $1.06 ) $212.00 ( 200 at 6%, is $1.06, and hence the 212 present worth of $1.06, due 1 year $1 X 200 = f2^ Present .oorth. >■;" <•«' '« *\^^[l^' P''^™' ^"'f' $212- 200 = $12, True discount. "{ f-^^ ^ »' "'« P^!^™' Zf^ ' of $212. IS as many times $1 as $1.06 is contained times in $212, which is 200. Or, since $1.06 is 1.06 of $1, $212 is 1.06 of its present worth. $212 ~ 1.06 =$200. Note. — This is only an application of Problem V., the debt being the amount, the present worth the principal, and the true discount the interest. 2. What is the present worth of a bill of $260 due in 8 months, without interest, the current rate of interest being 6 % ? What is the true discount? PRESENT WORTH AND DISCOUNT. 193 Find the present worth and the true discount of 3. $220 due in 1 yr. 6 rao., without interest, current rate 1%. 4. $145.60 due in 8 mo. 12 da., without interest, cur- rent rate 8 % . 5. $305.75 due in 9 mo. 6 da., without interest, current rate 9 %. 6. $1250 due in 1 yr. 7 mo. 21 da., without interest, current rate 5 % . 7. $1508 due in 90 days, without interest, current rate 7%. 8. $2040.50 due in 36 days, without interest, current rate 10%. 9. $884,125 due in 1 yr. 2 mo. 10 da., without inter- est, current rate 6 % . 10. What is the difference between the true discount and the interest of $216 for 2 years, at 8 % ? What is the difference between the true discount and the interest of 11. $199.80 for 1 yr. 10 mo., at 6% ? At 12 % ? 12. $666.40 for 2 yr. 4 mo. 15 da., at 6 % ? At 8 % ? 13. $534 for 1 yr. 1 mo. 18 da., at 6 % ? At 5 % ? 14. $175.20 for 1 yr. 10 mo. 12 da., at 9 % ? 15. $1250.60 for 1 yr. 7 mo. 24 da., at 5 % ? 16. $884.12 for 96 days, at 7|- % ? At 10 % ? 17. How large a note, due in 1 yr. 6 mo., with interest at 7 %, will cancel a debt of ^$442, due in 1 yr. 6 mo., with- out interest? 18. What is the difference in the present value of a cash payment of $345, and a note of $371 due in 9 months, with- out interest, the use of money being worth 8 % ? FORMULAS AND RULES. 310. Formulas. — 1. Present worth ^^ debt —[1 + (rate % X time)^. 2. True discount = debt — present worth. C.Ar.— 17. 194 COMPLETE ARITHMETIC. 311. Rules. — 1. To find the present worth of a debt due at a future time, without interest, 1. Divide the debt hj tJie amount of $1 for the given time, at the current rate of interest, and multiply $1 by the quotient. Or, 2. Divide the debt by the decimal corresponding to the amount of $1 for Hie given time, and at the current rate of interest. BANK DISCOUNT. 312. ^anh J)iscouut is the interest on a note for the number of days from the time it is discounted to the time it is legally due. 313. TJie Proceeds of a note are its face, or the sum discounted, less the discount. The pro- ceeds are also called Avails and Cash Value. 314. Days of Grace are the three days allowed for the payment of a note after the specified time has expired. Note. — A note is payable, or nominally due, at the expiration of the specified time, but it does not mature, or become legalbj due, until the last day of grace, or the day preceding, when the last day of grace falls on Sunday or a legal holiday. The date of expiration and the date of maturity are usually written with a line between them ; thus, .Fan. 9 / ]2. 315. When a bank loans money, the borrower gives his note payable at a specified time, without interest. This note is then discounted by the bank for the actual number of days in the time plus the three days of grace, and the pro-^ ceeds are paid to the borrower. 316. When a note drawing interest is discounted by a BANK DISCOUNT. 195 bank, the discount is computed on the amount of the note at the time of its maturity. Business men generally discount notes and bills, not drawing in- terest, by deducting the interest for the time, with or without grace as per agreement. This is sometimes called Business Discount. The rate of interest allowed is usually greater than the current rate. Bills due in three, four, or six months are often discounted by de- ducting 5% or more of their face, without regard to time. This is called Per Cent. Off. WRITTEN PROBLEMS. 1. What is the bank discount of a note of S350, payable in 60 days, discounted at 10 ^ ? What are the proceeds? Process. $350. .0105 1750 Time =60 da. + 3 da. = 63 da. 3 50 6 ) $3.6750 $.6125 X 10 = $6,125, Bank discount. $350 — $6,125 = $343,875, Proceeds. 2. What is the bank discount of a note of $250, payable in 90 days, discounted at 8 % ? What are the proceeds ? Find the bank discount and the proceeds of a note of 3. $145, payable in 60 days, discounted at 7%. 4. $80.50, payable in 30 days, discounted at 8 %. 5. $1000, payable in 90 days, discounted at 7^%. 6. $750, payable in 45 days, discounted at 9 %. 7. $1250, payable in 100 days, discounted at 6%. 8. $56, dated Jan. 1, 1870, payable May 1, 1870, dis- count 6%. 9. $120, dated Apr. 3, 1869, payable June 15, 1869, dis. count 8 %. 10. $500, dated Dec. 15, 1870, payable Feb. 18, 1871, discount 9 %. 11. $8.75, dated Nov. 21, 1870, payable Mch. 12, 1871, discount 9 %. 196 COxMPLETE ARITHMETIC. 12. $400, dated Mch. 4, 1871, payable July 24, 1871, discount 5 % ? 13. What is the difference between the bank discount and the true discount of $1319.50, due in 90 days, dis- counted at 6 % ? 14. What is the difference between the bank discount and the true discount of $768, due in 108 days, discount 8%? 15. What is the difference between the bank discount and the true discount of $3330.80, due in 45 days, dis- count 7%? 16. A note, dated Apr. 10, 1871, is payable in 90 days: what is the time of its maturity? Note. — The date of maturity is found by counting forward the number of days phis three days, when the time is expressed in days ; and the number of calendar months plus three days, when the time is expressed in months. 17. A note, dated Feb. 6, 1868, was payable in 60 days: what was the date of its maturity ? 18. A note, dated Aug. 9, 1870, is payable 4 months from date : what is the date of its maturity ? 19. A note of $460, dated Apr. 3, 1870, and payable in 90 days, with interest at 6 %, was discounted May 10, 1870, at 8 % : what were the proceeds ? Suggestion.— Find the amount of $460 for 93 days, at 6%, and then discount this amount for 56 days, at 8%. 20. A note of $125, dated May 21, 1870, and payable in 60 days, with interest at 6 %, was discounted May 25, 1870, at 10 % : what were the proceeds ? 21. A note of $1000, dated Aug. 15, 1869, and payable in 6 months, with interest at 7%, was discounted Nov. 27, 1869, at 9 % : what was the bank discount? Suggestion. — Compute the interest for 6 mo. 3 da., and the dis- count for 83 days. 22. A note of $90, dated Apr. 12, 1870, and payable in BANK DISCOUNT. 197 4 months, with interest at 5 %, was discounted June 1, 1870, at 7 % : what were the proceeds? 23. A note of $650, dated Mch. 2, 1869, payable Apr. 1, 1870, and indorsed $300 Oct. 1, 1869, was discounted Feb. 3, 1870: what were the proceeds? 24. For what sum must a note, payable in 60 days, be drawn to produce $493, when discounted at 8 % ? Since the proceeds of $1 are Process. $.986, it will require as many $1 — $.014 = $.986, Proceeds of $1. times $1 to produce $493 as $493 -^ $.986 =:: 500 $.986 is contained times in $493, $1 X 500 = $500, Face of note. which is 500 ; and 500 times $1 = $500. 25. For what sum must a note, payable in 90 days, be drawn to produce $1969, when discounted ai 6 % ? 26. What must be the face of a note, dated July 5, 1871, and payable in 4 months, to produce $811, when discounted at 9 % ? 27. What must be the face of a note, dated Jan. 10, 1870, and payable in 3 months, to produce $1958, when discounted at 12%? 28. A merchant discounted a bill of $750, payable in 4 months, by deducting the interest for the time without grace, at 10 % : what were the cash proceeds of the bill ? 29. A note of $340, due in 9 months, without interest, was discounted by deducting the interest for the time, at 8 % : what was the cash value of the note ? 30. A merchant having sold a bill of goods amounting to $1030, on three months' time, allowed 5 % off for cash: what were the cash proceeds of the sale? 31. A retail dealer having bought $950 worth of goods, on 6 months' time, cashed the bill for 7^% off: what were the cash proceeds? 32. A merchant bought, March 20, 1870, a bill of goods amounting to $3540, on three months' time, but, being offered 5 % off for cash, he borrowed the money at a bank 198 COMPLETE ARITHMETIC. for the time, at 10%, and cashed the bill. How much did he g^in by the transaction? FORMULAS AND RULES. 317. Formulas. — 1. Bank discount = sum discounted X (int. of $1 for the days -j- 3 days). 2. Business discount = sum discounted y, int. of $1 for the time. 3. Proceeds = sum discounted — discount. 318- EuLES. — 1. To compute bank discount, Find the in- terest on the sum discounted at the given rate per cent, and for the actual number of days in the time plus three days. 2. To compute business discount, Find the interest on the su7n discounted, at the given rate per cent, and for the given time. 3. To find the proceeds, Subtract the discount from the sum discounted. 4. To find the face of a note to yield given proceeds. Divide the given proceeds by $1 minus tfw interest of $1 for the given time ivith grace, and midtiply $1 by the quotient. Notes. — 1. In discounting a note bearing interest, the interest is computed by months or by days, according as the time is expressed in the note, but the discount is usually computed by days. 2. Business discount is computed by months or days, according as the time is expressed in the paper discounted, and with or without grace. In these respects it differs from bank discount. 3. Bank discount is not only interest paid in advance, but the in- terest is computed on both the proceeds and the discount. The bor- rower pays interest on more money than he receives. PROMISSORY NOTES AND DRAFTS. I. PROMISSORY NOTES. 319. A Promissory Note is a written agreement by one party to pay to another a specified sum at a specified time. The sum whose payment is promised, is called the Face of the note. The person wlio signs a note is called its Maker ; the person to whom it is payable is the Payee; and its owner is the Holder. PROMISSORY NOTES. 199 320. A Johif Note is a note signed by two or more persons who are jointly liable for its payment. 321. A Joint and Several JV^ote is a note signed by two or more persons who are both jointly and singly liable for its payment. 322. An Indorser is a person who signs his name on the back of a note as security for its payment. 323. The following are the more common forms of pro- missory notes : FORM I.— Demand Note. $95jVo- Nashville, Tenn., May 1, 1870. For value received, I promise to pay to John Wilson, on demand, Ninety-five yW Dollars. Henry Smith. [ STAMP.] FORM II.— Time Note. $95tVo- St. Louis, Mo., May 1, 1870. Ninety days after date, I promise to pay to John Wilson, or bearer, Ninetyfive -f^ Dollars, with interest, for value re- ceived. Henry Smith. [ STAMP.] FORM III. — Joint and Several Note. $95yVV- Louisville, Ky., March 12, 1870. Four months after date, we jointly and severally promise to pay Henry Cooke, or order, Ninety-five y'^fy- Dollars, with in- terest at S%, for value received. Thomas Hughes, [ STAMP.] Charles G. Knight. 200 COMPLETE ARITHMETIC. FORM IV. — Note Payable at a Bank. $500. Baltimore, Md., Apr. 10, 1870. Sixty days after date, ive -promise to pay to Wilson, HinUe & Co., or order, at the First National Bank, Five Hundred Dollars, for value received. Charles Cooke & Co. [ STAMP.] Kemarks. — 1. A note should contain the words " value received," otherwise the holder may be required to prove that the maker re- ceived its value. 2. When the time for the payment of a note is not specified, it is due on demand. If the place of payment is not mentioned, it is pay- able at the maker's residence or place of business. 3. When a note contains the words " with interest," and no rate is specified (Form II.), interest accrues at the legal rate. If the words " with interest," are omitted (Form IV.), no interest accrues until after maturity, when the note draws interest, at the legal rate, until paid. 324. A JSegotiable Wote is one which may be bought and sold. A note is negotiable when it is made payable " to the bearer," or to the payee " or bearer," or to the payee " or order," or " to the order of" the payee. A note drawn as in Form I. is not negotiable. A note made payable to the bearer is negotiable without indorse- ment. U. S. treasury notes and bank notes, used as money, are pay- able to the bearer, and are transferred by delivery. A note payable to order must be indorsed by the payee before it is negotiable. When the payee indorses a note by simply writing his name on the back, it is called an indorsement in blank, and the note is payable to the holder. When the indorser orders the payment to be made to a particular person, as : " Pay to Charles Williams," it is called a special indorsement. 325. If the maker of a note fails to pay it at maturity, a written notice of the fact, made by a notary public, is served on the indorsers, who are responsible for the payment of the note. Such a notice is called a Protest. BILLS OF EXCHANGE. 201 Note.— A protest must be made out on the day a note matures, and ■ it must be sent on that day or the next, otherwise the indorsers are not responsible. II. DRAFTS, OR BILLS OF EXCHANGE. 326. A Draft is an order made by one person upon another to pay a specified sum to a third person named. It is also called a Bill of Exchange. The person who makes the order, is called the Drawer ; the person to whom it is addressed, is called the Drawee; and the person to whom the money is payable, is the Payee. 327. The following are the common forms of drafts: FORM I.— Sight Draft. $100. Cincinnati, O., Oct. 1, 1870. Pay to the order of Bartlit & Smith, One Hundred Dollars, and place to the account of Charles S. Kelley. To George Brown, Esq., New York. [ STAMP.] FORM II.— Time Draft. $100. Cincinnati, O., Oct. 1, 1870. Thirty days after sight [or date], pay to the order of Bartlit & Smith, One Hundred Dollars, and place to the ac- count of Charles S. Kelley. To George Brown, Esq., New York. [ STAMP.] When the drawee accepts a draft, he writes the word " Accepted," with the date, across the face, and signs his name, thus: "Accepted, Oct. 3, 1870— Charles S. Kelley." The draft is then called an Ac- ceptance, and the acceptor is responsible for its payment. A draft made payable to bearer or order is negotiable, like a prom- issory note, and is subject to protest, in case payment or acceptance is refused. 202 COMPLETE ARITHMETIC. Notes. — 1. In most of the states both time and sight drafts are entitled to three days of grace. In New York no grace is allowed. on sight drafts. 2. When a draft is drawn, ''acceptance waived," it is not subject to protest until maturity ; and when an indorser writes over his name, "demand and notice waived," a protest in his case is not necessary. 3. The liability of an indorser of a note or draft may be avoided by his writing over his indorsement, " without recourse." 328. A Domestic or Inland JBill is a draft which is payable in the country where it is drawn. A Foreign JSill is a draft which is drawn in one country and is payable in another. 329. Exchange is the process of making payments at distant places by the remittance of drafts, instead of money. When a draft can be bought for its face, it is said to be at 'par; when the cost is less than the face, it is below par, or at a discount; and when the cost is more than the face, it is above par, or at a pre- mium. The rate per cent., which the cost of a draft is more or less than its face, is called the Rate of Exchange. Notes. — 1. The rate of exchange between two places depends chiefly on their relative trade. If Cincinnati owes New York, drafts on New York are at a premium in Cincinnati ; if New York owes Cin- cinnati, drafts on New York are at a discount; if the trade of the two cities with each other is equal, exchange is at par. 2. In foreign exchange, drafts are expressed in the currency of the country on which they are drawn. Tlie comparative value of the money of two countries is called the Far of Exchange. AVRITTEN PKOBLEMS. 1. What is the cost of a draft on New York for $800, exchange being | % premium ? Process : $800 X -OOf = $6, Prem. $800 + $6 = $806, Cost. 2. What is the cost of a draft on New Orleans for $1250, at ^ % discount? 3. What is the cost of a draft on Philadelphia for $1050, at J % premium ? 4. A merchant in St. Louis wishes to remit $2500 by draft to New York : what will be the cost of the draft, ex- change being 1^ % premium ? EXCHANGE. 203 5. What will be the cost of a draft for $500, payable in 30 days after sight, exchange being 1 % premium, and interest 6 % ? Process. $500. $500 X -0055 = $2.75 Discount at 6% for 33 days. $497.25 Proceed.^ of Draft (cost at par). $500 X -01 = MO Premium atl%. $502.25 Cost of Draft. Note. — If preferred, the face may be multiplied by the cost of $1, which is $1 — $.0055 -1- $.01, or $1.0045. $500X1.0045 = $502.25. 6. What will be the cost of a draft for $650, payable in 60 days after sight, exchange being | % premium, and in- terest 8 % ? 7. What will be the cost of a draft for $320, payable in 45 days after sight, exchange being f % discount, and in- terest 7 % ? 8. How large a sight draft can be bought for $259.52, exchange being If % premium ? Process : $259.52 -=- 1.01 1 = $256, Face of Draft. (Case IV.) 9. How large a sight draft can be bought for $962.85, exchange being If % discount? 10. A sight draft, bought at ^ % premium, cost $1256.25 : what was its face? 11. How large a draft, payable 30 days after sight, can be bought for $502.25, exchange being 1 %, and interest 6 % ? Process. $1 — $.0055 = $.9945 Proceeds of $1 discounted for 33 days. $.9945 + $.01 =$1.0045 Cost of $1. $502.25 -f- $1 .0045 = $500 Face of Draft. 12. How large a draft, payable 60 days after sight, can be bought for $798.80, exchange being 1^ % premium, and interest 8 % ? 13. A draft, payable in 30 days after sight, was bought for $352.62, exchange being 1^ % discount, and interest 6 % : what was its face? 204 COMPLETE ARITHMETIC. BONDS. 330. The interest-bearing notes issued by nations, states, cities, railroad companies, and other corporations, as a means of borrowing money, are called Bonds. Bonds are issued in denominations of convenient size, with interest usually payable annually or semi-annually, and they are made nego- tiable like certificates of capital stock. (Art. 243.) The Coupons attached to bonds are due-bills for the interest, which, as the interest becomes due, are cut off and presented for payment. 331. The several classes of bonds issued by the United States Government, are called United States Securities^ or Government Securities, the principal of which are known as Sixes of 1881, Five -Twenties, and Ten-Forties. The 6's of '81, bear interest at 6 % in gold, payable semi-annually, and are payable in 1881. The 5-20's bear interest at 6% in gold, payable semi-annually, and are redeemable, at the option of the Government, after five years from date, and are payable in twenty years. The several series were issued in 1862 ( Old 5-20's), 1864, 1865, 1867, and 1868. The 10-40's, issued in 1864, bear interest at 5% in gold, payable semi-annually (except the $50 and $100 bonds bearing annual in- terest), and are redeemable, at the option of the Government, in ten years, and are payable in forty years. 332. The market value of United States bonds is quoted at a certain per cent, of their par value or face. Bonds quoted at 110 are worth in currency 110% of their face; that is, are 10 % above par. The quotation includes ac- crued interest. WRITTEN PROBLEMS. 1. When U. S. 5-20's are quoted at 109^, what will three $500 bonds cost ? 2. When U. S. 6's of '81 are worth 114, Avhat will $1250 in bonds cost? 3. A widow invested $4725 in U. S. 10-40's, at 105 : what amount in bonds did she receive ? ANNUAL INTEREST. 205 4. A broker invested $26250 in 5-20's at 1061 and sold them at 109 : how much did he gain ? 5. When gold is at 115, what amount in currency can be bought for $8500 in gold ? 6. At 112, what amount in gold can be bought for $1400 in treasury notes ? 7. When gold is at 115, and U. S. 10-40's at 105, what is a $500 bond worth in gold ? 8. When gold is at 120 and U. S. 5-20's at 115, what is the gold value of a $1000 bond? 9. When gold is at 110 and U. S. 10-40's at 108, what is the gold value of a $500 bond? 10. When gold is worth 112|^, what is the gold value of a dollar treasury note ? 11. When gold is at 115, what is the semi-annual interest in currency on $9500 in 10-40's ? 12. When gold is at 120, what rate per cent, in currency is the interest on 5-20's? 13. A college invested its endowment fund in U. S. 5-20's, at 106 : what rate of interest in currency will it receive when gold is at 120 ? At 110 ? At par ? ANNUAL INTEREST. 333. When a note reads "with interest payable annually," the interest on the face of the note, due at the close of each year, is called Annual Interest. 334. When annual interest is not paid at the close of the year, when due, it draws simple interest until paid. WRITTEN PROBLEMS. 1. A note of $500, dated May 10, 1870, is due in 4 years, with interest at 6%, payable annually: if both interest and principal remain unpaid, what will be the amount due on the note at maturity? 206 COMPLETE ARITHMETIC. Process. $500 X » 06 = S30 , Interest on principal due annually. $30 X 4 = $120, Total annual interest. $30 X -06 X 3 = 5.40, Simple interest on 1st annual interest for 3 yrs. $30 X .06 X 2 = 3.60, " " " 2d " " " 2 " $30 X .06 X 1 = 1-80, " " " 3c^ " " ^'1 " $130.80, Ihtal interest due at maturity of note. 500.00 $630.80, Amount due at maturity of note. The $30 annual interest, due at the close of the 1st year, being unpaid, draws simple interest until paid, or for 3 years; the $30 annual interest due at the close of the 2d year, draws interest for 2 years ; and the $30 annual interest due at the close of the 3d year, draws interest for 1 year. The fourth annual interest is paid when due. Hence, the total interest due at the maturity of the note, con- sists of (1) the annual interest for 1 year ($30) multiplied by 4, the number of years; and (2) the simple interest on the $30 annual in- terest for 3 years, 2 years, and 1 year, or for 6 years. The amount due is $500 + $30 X 4 + $1.80 X 6. 2. A note of $750, with interest payable annually, at 8%, was paid 3 yr. 3 mo. 18 da. after date, and no interest had been previously paid : what was the amount due ? Process. $750 X .OS = $60. Interest on face due annually. $60 X 3.3 = $198.00, Total annual int. for 3 yr. 3 mo. 18 da. (3.3 yr.) $60 X -312 =r $18.72, Simple interest on %60for 3 ijr. 10 mo. 24 da. $750. ).72, Amount doe. The first annual interest draws simple interest for 2 yr. 3 mo. 18 da. ; the second, for 1 yr. 3 mo. 18 da. ; and the third, for 3 mo. 18 da. ; and hence, the simple interest on the several annual interests equals the interest of $60 for 2 yr. 3 mo. 18 da. + 1 yr- 3 mo. 18 da. + 3 mo. 18 da., or for 3 yr. 10 mo. 24 da. The amount due consists of (1) the principal; (2) the total annual interest; and (3) the simple interest on the annual interest. 3. A note of $1000, with annual interest at 6%, is due 4 yr. 6 mo. after date : no interest having been paid, what will be due at maturity? ANNUAL INTEREST. 207 4. A man bought a farm for $3500, to be paid in 4 years, with interest payable annually, but failed to pay the inter- est: what was due at the close of the 4th year? 5. $650. New York, July 1, 1869. On the first day of January, 1872, for value received, I promise to pay John Black, or order, six hundred and fifty dollars, with annual interest at 7 % . Charles Church. If no interest be paid on the above note, what will be due at maturity? 6. If the above note and interest be not paid until Sept. 12, 1872, what will be the amount due ? 7. A note of $800, dated March 18, 1867, and due in 3 years, with interest at 6%, payable annually, has the fol- lowing indorsements : Oct. 24, 1868, $150 : Nov. 12, 1869, $240. What was the amount due March 18, 1870? Process. $800 X .06 = $4 8, First annual interest. $48 X 2 = $96, Annual interest due Mch. 18, 1869. $48 X '06 ^ = 4.8 0, Interest on \st annual interest. SmSO, Interest due Mch. 18, 1869. 800 §900.80, Amount due Mch. 18, 1869. Payment, $150. | ^ 15360, Amount of $150, Mch. 18, 1869. Int. on same, $3.60. J ^^^^;^^^ New principal. $747.20 X -06 = $_44.832, Interest due Mch. 18, 1870. $792,032, Amount due Mch. 18, 1870. Payment, J?m I ^245.04, ^mown< $240, 3/cA. 18, 1870. Int. on same, vO.04. J ^54(3 992^ Amount due at maturity. Note. — The annual interest and the interest on the same are com- puted to the close of the year in which the first payment is made, and the interest on the payment is computed to the same date. The differ- ence hetween the amount of the face of the note and the amount of the payment is the new principal for the third year. 8. A note of $500, dated Jan. 15, 1865, and due in 2 years, with interest at 10%, payable annually, is indorsed 208 COMPLETE ARITHMETIC. as follows: May 21, 1866, $100; Mch. 9, 1867, $200. What was the amount due July 15, 1867 ? 335. Rules. — 1. To compute unpaid annual interest, Compute the interest on the iwincipal for the entire time it is on interest, and the interest on each year's interest for the time it is unpaid. The sum of the principal, the interest on the principal, and the interest on Hie unpaid interest will be the amount due. Note. — Instead of computing the interest on the several annual interests separately, simple interest may be computed on one year's inter- est for a time equal to the sum of the periods of time the several annual interests are unpaid. 2. To compute annual interest when partial payments have been made, 1. Compute the interest on the principal to the end of the first year in which any payment is made, and also the interest on the unpaid annual interest to the same date, and f(yrm the amount. 2. Compute the interest on the payment or payments to the end of the year, and form tJie amount. 3. Subtract the amount of the payment or payments from the amount of the principal and interest, arid taking the difference for a new principal, proceed as before with succeeding pay- ments, making the date of settlement the last date. Note. — 1. This rule is based on the principle that the payments with added interest should be applied first to the discharge of the accrued interest at the end of a year, and then to the discharge of the principal. When the amount of the payment or payments will not cancel all the interest due, the unpaid interest draws simple interest to the end of the next year in which a payment is made. COMPOUND INTEREST. 336. Compound Interest is interest on the princi- pal and also on the interest which, at regular intervals of time, is added to the principal. It is generally compounded annually, semi-annually, or quarterly. COMPOUND INTEREST. 209 WRITTEN PROBLEMS. 1. A note of S450 is due in 3 years with interest at 6 %, compounded annually. What will be the amount due at maturity? What will be the compound interest due? Process. $450 .06 $27.00 1st year's interest. 450. $477 'Id principal. .06 $28.62 2d yearns interest. 477. $505.62 3d principal. .06 $30.3372 3d year\ interest. 505.62 $535.9572 Amount due at the end of the third year. 450. $85.9572 Compound interest at the end of the third year. 2. What is the amount of $600 for 4 years at 5 %, com- pounded annually? What is the compound interest? 3. What is the compound interest of $1500 for 3 years at 10 % ? At 8 % ? 4. What is the amount of $800 for 2 years at 6 % , com- pounded semi-annually? Suggestion. -Compute the interest at 3 %. 5. What is the compound interest of $650 for 3 yr. 4 mo. 12 d. at 6 % per annum? 337. Rule. — To compute compound interest, Find the amount of tJie given principal for one interval of time; then, taking tJiis amount as a new principal, find the amount for the second interval, and so continue for the entire time. The dif- ference between the UlsI amount and the principal is the com- pound interest for the time. C.Ar.— 18. 210 COMPLEl'E ARITHMETIC. Notes. — 1. When the interest is compounded semi-annually, the rate per cent, is one half the yearly rate, and when compounded quarterly, it is one fourth the yearly rate. 2. When the time contains years, months, and days, the amount is found for the number of whole intervals in the time, and then the interest is computed on this amount for the remaining months and days. 338. Compound interest is usually computed by the aid of a table giving the amount of $1 at several different rates per cent., and for any number of years which may be in- cluded. 339. A Table Slioiving the amount of $1 at compound interest, at 3, 4, 5, 6, 7, or 8 per cent., for any number of years from 1 to 25. YRS. 3 PER CENT. 4 PER CENT. 5 PER CENT. 6 PER CENT. 7 PER CENT. 8 PER CENT. 1 1.03 1.04 1.05 1.06 1.07 1.08 2 1.0609 1.0816 1.1025 1.1236 1.1449 1.1664 3 1.092727 1.124864 1.157625 1.191016 1.225043 1.259712 4 1.125509 1.169859 1.215506 1.262477 1.310796 1.360489 5 1.159274 1.216653 1.276282 1.338226 1.402552 1.469328 6 1.194052 1.265319 1.340096 1.418519 1.500730 ,1.586874 7 1.229874 1.315932 1.407100 1.503630 1.605781 1.718824 8 1.266770 1.368569 1.477455 1.593848 1.718186 1.850930 9 1.304773 1.423312 1.551328 1.689479 1.838459 1.999005 10 1.343916 1.480244 1.628895 1.790848 1.967151 2.158925 11 1.384234 1.539454 1.710339 1.898299 2.104852 2.331639 12 1.425761 1.601032 1.795856 2.012196 2.252192 2.518170 13 1.468584 1.665074 1.885649 2.132928 2.409845 2.719624 14 1.512590 1.731676 1.979932 2.260904 2.578534 2.937194 15 1.557967 1.800944 2.078928 2.396558 2.759032 3.172169 16 1.604706 1.872981 2.182875 2.540352 2.952164 3.425943 17 1.652848 1.947900 2.292018 2.692773 3.158815 3.700018 18 1.702433 2.025817 2.406619 2.854339 3.379932 3.996019 19 1.753506 2.106849 2.526950 3.025600 3.616528 4.315701 20 1.806111 2.191123 2.653298 3.207135 3.869684 4.660957 21 1.860295 2.278768 2.785963 3.399564 4.140562 5.033834 22 1.916103 2.369919 2.925261 3.603537 4.430402 5.436540 23 1.973587 2.464716 3.071524 3.819750 4.740530 5.871464 24 2.032794 2.563304 3.225100 4.048935 5.072367 6.341181 25 2.093778 2.665836 3.386355 4.291871 5.427433 6.848475 INTEREST. 211 340. The amount of $1 for ike given time and rate, multiplied by the given j^rincipal, gives its amount for the same time and rate. Note. — When the interest is compounded semi-annually, it is com- puted from the table by taking one half the rate and twice the number of years. 6. What is the compound interest of $750 for 15 years at 6 % ? What is the amount ? 7. What is the amount of $500 for 6 years at 8 % , com- pounded semi-annually ? 8. What is the amount of $1250 for 10 yr. 4 mo. 15 da. at 5 ^ , compound interest ? EQUATION OF PAYMENTS. 341. Equation of I^ayments is the process of finding an equitable time for the payment of several debts, due at different times, without interest. It is also called the Average of Payments. The equitable time sought is called the Average Time, or the Equated Time. PEOBLEMS. 1. A owes B $300, of whicli $200 is due in 3 months, and $100 in 6 months: when will the payment of $300 equitably discharge the debt ? Process. s^ jg entitled to the use of $200 for 3 $200 X 3 = $600 months, which equals the use of $600 for 1 $100 X 6 = $600 month, and to the use of $100 for 6 months, $300 ) $1200 which equals the use of $600 for 1 month ; 4 and, hence, he is entitled to the use of $300 Ans. 4 mos. ^^"^il it equals the use of $600 + $600, or $1200, for 1 month. It will take $300 as many months to equal the use of $1200 for 1 month, as $300 is contained times in $1200, which is 4. Hence, the payment of $300 in 4 months will equitably discharge the debt. 212 COMPLETE ARITHMETIC. Proof. — In paying the $200 in 4 months, A gains the use of $200 for 1 month, and in paying the $100 in 4 months, he loses the use of $100 for 2 months, which equals the use of $200 for 1 month. Hence, his gain and loss are equal. 2. A owes a merchant $200 due in 4 months, and $600 due in 8 months: what is the equated time for the pay- ment of both debts? 3. A owes B $1200, of which $300 are due in 4 months, $400 in 6 months, and the remainder in 12 months : what is the equated time for the payment of the whole? 4. A owes B $800, of which | is due in 2 months, ^ in 3 months, and the remainder in 6 months : what is the equated time for the payment of the whole? 5. A man owes $300 due in 4 months, $600 due in 5 months, and $700 due in 10 months : what is the equated time for the payment of the whole? 6. Smith & Jones bought $500 worth of goods on 4 months' credit, $700 worth on 6 months' credit, and $1000 worth on 5 months' credit: what is the equated time for the payment of the whole ? 7. A bought $2000 worth of goods, ^ of which was to be paid down, i in 3 months, ^ in 4 months, and the re- mainder in 8 months : what is the equated time for the payment of the whole? 8. AVhat is the equated time for the payment of $220, due in 30 days; $300, due in 40 days; $250, due in 60 days; and $100, due in 90 days? 9. What is the equated time for the payment of $300, due in 30 days; $250, due in 45 days; and $350, due in 60 days ? Process by Interest. Int. of $300 for 30 days, at 6% = $1.50 Int. of $250 for 45 " " 6% = 1.875 Int. of $350 for 60 " " 6%=r= 3.50 $900 $6,875 $9.00 = Int. of $900 for 60 days. .15 = " " $900 for 1 day! $6,875 H- $.15 = 45.9. Ans,, 46 days. EQUATION OF PAYMENTS. 213 The debtor is entitled to the use (1) of $300 for 30 days, which, at 6%, equals $1.50 interest; (2) of $250 for 45 days, which equals $1,875 interest; (3) of $350 for 60 days, which equals $3.50 interest. Hence, he is entitled to the use of $900, the sum of the debts, until the interest thereon, at 6%, equals the sum of $1.50 + $1,875 + $3.50, which is $6875. The interest of $900 for 1 day is $.15; and since $6,875 ^$.15 = 45.9, it will take 45.9 days for $900 to yield $6,875 interest. The equated time for payment is 46 days. Note. — When the fraction of a day in the equated time is more than i, it is counted as a day ; when it is less than ^, it is disre- garded. 10. What is the equated time for the payment of $520, due in 45 days ; $340, due in 60 days ; and $640, due in 90 days ? 11. What is the equated time for the payment of $375, due now ; $425, due in 30 days ; $500, due in 60 days ; and $600, due in 75 days ? 12. What is the equated time for the payment of $340, due May 10, 1870; $450, due June 10; $560, due July 15; and $650, due Aug. 10? Note. — Begin with the first date (May 10), and find the exact number of days between it and each succeeding date. The equated time is counted forward from the first date. 13. What is the equated time for the payment of $1000, due June 1, 1870; $850, due July 1; $750, due Sept. 1; and $900, due Oct. 1 ? 14. What is the equated time for the payment of $75, due May 6, 1870; $115, due May 26; $220, due June 25; $315, due July 16; and $350, due July 30? PRINCIPLES AND RULES. 342. The time between the contraction of a debt and its payment is called the Term of Credit, or Time of Credit. 343. Principles. — 1. The payment of a sum of money BEFORE it is due is offset by keeping an equal sum of money an equal time after it is due. 2. The use of any sum of money is measured by its interest for the tims. 214 COMPLETE ARITHMETIC. 344. Rules. — To equate the time of several debts or payments, 1. Multiply each debt or payment by its time of credit, and divide the sum of Hie products by the sum of the debts or pay- ments. Or, 2. Compute the interest of each debt or payment for its time of credit, and divide the sum of the interests by the interest of the sum of the debts or payments for one month or one day. Notes. — 1. As the result will be the same at any rate, the interest may be computed at that rate which is most convenient. 2. The correctness of each of the above methods has been called in question by a number of authors, who commend the following as " the only accurate rule " : *^Find the present worth of each of the given amounts due ; then fnd in what time the sum of these present ivorths will amount to the sum of all the payments. The inaccuracy of this so-called "accurate rule" is easily shown. The raetliods given above are both strictly accurate, and they are in general use. (See appendix.) 345. When partial payments are made on a debt before it is due, the time for the payment of the balance of the debt is proportionately extended. 15. A owes a merchant $200, due in 12 months, without interest; in 4 months he pays $50 on the debt, and in 8 months, $50 : when in equity should he pay the balance ? Process. In paying $50 in 4 months, A loses $50X8 = $400 its use for 8 months, and in paying $50 X 4 ^ $200 $50 in 8 months, he loses its use for 4 $200 — $100 = $100 ) $600 months, and hence he loses the use of 6 $400 H- $200, or $600, for 1 month. To offset this loss, he is entitled to keep the balance ($100) 6 months after its maturity. 16. A owes B $300, due in 8 months : if he pay $200 in 5 months, when should he pay the balance? 17. A man bought a horse, agreeing to pay $150 in 6 EQUATION OF ACCOUNTS. 215 months, without interest : if he pay $50 down, when should he pay the balance? 18. A owes B $600, payable in 6 months, but, at the close of 3 months, he proposes to make a payment sufficiently large to extend the time for the payment of the balance 6 months. How large a payment must he make ? 19. A owed B $1500, due in 12 months, but in 4 months paid him $400, and in 6 months $500 : when in equity ought the balance to be paid? 20. Clark and Brown bought March 10, 1870, a bill of goods amounting to $2500, on 4 months' credit; but they paid $650 Apr. 7; $500 Apr. 30; and $350 May 20. When ought they to pay the balance? 346. Rule. — Multiply each payment by the time it ivas paid before it ivas due, and divide the sum of the products by the bal- ance unpaid. EQUATION OF ACCOUNTS. 347. JEquation of Accounts is the process of find- ing the equated time for the payment of the balance of an account*; or the time when the balance was due. Case I. 7?Lccoian.ts ooiataining only X)eTDit Items. PROBIiEMS. 1. A bookseller bought of Wilson, Hinkle & Co. the following bills of goods, on 4 months' credit : Feb. 3, 1870, a bill of $450. 24, '' " 500. Mch. 25, *' " 750. Apr. 20, '^ " -600. What is the equated time of maturity ? 216 COMPLETE ARITHMETIC. Process. . Due June 3, 1870, $450 X 00 =^ " 24, " 500X21 = 10500 " July 25, " 750 X 52 --= 39000 " Aug. 20, " 600 X 78 = 46800 $2300 ) $96300 ( 41.8 days. The equated date of maturity of the above bills is 42 days from June 3, 1870, which is July 15, 1870. Notes. — 1. The date of maturity of each bill is found by counting 'forward 4 months from the date of purchase. The same result would be obtained by finding the average or equated date of purchase, and counting forward 4 months. 2. The equated time of maturity may also be found by beginning at the last date, and taking the exact number of days between each preceding date and the last date for a multiplier. The equated date is then found by counting back from the last date. 2. Murray & Co. bought of Smith & Moore goods as follows : Apr. 15, 1869, a bill of $400, on 3 mo. credit. May 20, '' '' 245, on 4 '' June 25, '' *' 375, on 4 *' Sept. 15, *' '' 625, on 3 " What is the equated time of maturity ? 3. A merchant has the following charges against a cus- tomer : May 9, 1870, $340, on 4 mo. credit. June 6, '' 530, on 4 '' July 8, '' 213, on 3 ** Aug. 30, " 150, on 4 " What is the equated time of maturity? 4. J. O. Bates & Co. bought of Smith & Brown several bills of goods, as follows : March 3, 1868, a bill of $250, on 3 mo. credit. April 15, '* " 180, on 4 *' June 20, '' ** 325, on 3 '' Aug. 10, *' '' 80, on 3 '' Sept. 1, " " 100, on 4 *' What is the equated date of maturity? How much would pay the account Dec. 1, 1868? EQUATION OF ACCOUNTS. 217 348. Rule.— To find the equated time of maturity for the debit items of an account, First find the maturity of each item or hill, and then, counting from the first date for the time of credit, find the equated time as in the equation of payments. The date of the equated time is found by counting forivard from the first date. Notes.— 1. The equated time may be found by interest, as in tlie Equation of Payments. (Art. 344, Kule 2.) 2. The sum of the debit items draws interest from the equated date of maturity to the date of payment. Case II. A^ccounts coiataining Toothi Debits a,iad Credit.-^. 5. What is the equated date of maturity of each side of the following account? IJ7 John Smith in account with John Jones. Cr. I.s6^-. \TimeofCred. 1S68. Apr. 3, To Mdse. $220 I 3 mo. July 1, By Cash $200 June 1, (( 125 ' 4 " Oct. 3, u 150 July 15, u 200 i 4 " Dec. 20, « 300 Aug. 24, li 140 i 6 " i Oct. 1, « 190 6 " i Process. Credits. Due July 1,1868,$200X 00 = Oct. 3, " 150 X 94^=14100 Dec. 20, " 300X172 ^516 00 $650 ) $65700 101 Credits are due 101 days from July 1, which is Oct. 10. Debits are due 141 days from July 3, 1868, which is Nov. 21. Note. — Each side of the account may be equated without refer- ence to the other, as is done above, or the first or last date of the ac- count may be made a common starting-point for both sides. C.Ar.— 19. Debits. Due Julv 3, 1868, $220 X 00- = Oct. 1, " 125 X 90 = = 11250 Nov. 15, " 200 X 135 := - 27000 Feb. 24, 1869, 140 X 236 = = 33040 Apr. 1, " 190 X 272- ^51680 $875 ) $122970 141 218 COMPLETE ARITHMETIC. 6. The above account, as equated, stands thus: Dr. Due Nov. 21, 1868 . . S875 Cr. Due Oct. 10, 1868 . . $650 "When is the balance of the account due? Process. I^ebits $875 ^g^Q Credits 650 42 Balance $225 $225 ) $27300 Difierence in time, 42 days. 121 Balance is due 121 days from Nov. 21, 1868, which is March 22, 1869. Suppose the account settled Nov. 21, the later date. Since the credit side of the account has been due since Oct. 10, it has been draw- ing interest for 42 days. To increase the debit side of the account by an equal amount of interest, the balance must remain unpaid 121 days. Counting /orwar Ans. 222 COMPLETE ARITHMETIC. 50. What is the product of 9 : 10 and 24 : 33? 51. What is the product of 7 : 15, 25 : 14, and 24 : 35? 52. What is the product of 12 : 25, 15 : 24, and 16 : 21 ? DEFINITIONS, PRINCIPLES, AND RULES. 352. Matio is the relation between two numbers of tho same kind expressed by their quotient. 353. The two numbers compared are called the Terms of the ratio. The first term is the Antecedent, and the second term the Consequent. The two terms form a Couplet. 354. The value of a ratio is the quotient obtained by dividing the antecedent by the consequent. When the antecedent is greater than the consequent, the value of the ratio is greater than 1 ; when the antecedent is less than tlie con- sequent, the value is less than 1. 355. The ratio of two numbers is expressed by placing a colon (:) between them; as, 5:12. The colon is called the Si^n of Ratio. Note. — The sign of ratio is the sign of division with the hori- zontal line omitted. 356. A ratio is also expressed in the form of a fraction, the antecedent being made the numerator and the conse- quent the denominator. Thus, 5 : 12 = y2. Note. — Several American authors divide the consequent by the antecedent, thus reversing the positions of dividend and divisor, as indicated by the sign of division. The great majority of mathe- matical writers divide the antecedent by the consequent. Ratios are either Simple or Compound. 357. A Simple Hatlo is the ratio of two numbers; as 5 : 8, or f : f. Note. — A simple ratio, having one or both of its terms fractional, is called by several authors a Complex Ratio. RATIO. 223 358. A Compound JRatio is the product of two or more simple ratios ; as, (5 : 6) X (f •' 10). It may be expressed in three ways, as follows : (5 : 6) X (8 : 9) X (f : 10); or I X I X i; or 8 : 9J^ 359. An Inverse Hatio is a ratio resulting from an inversion of the terms of a given ratio. Thus, 5 : 7 is the inverse of 7 : 5. It is also called a Beciproeal Ratio. 360. Principles. — 1. The two terms of a ratio must be like numhers. 2. Tlie antecedent equals the consequent multiplied by the ratio. 3. The consequent equals the antecedent divided by the ratio. 4. If the product of the two terms of a ratio be divided by either term, the quotient will be the other term. 5. A ratio is multiplied by multiplying the antecedent or dividing tJie consequent by a number greater than 1. 6. A ratio is divided by dividing the antecedent or multiplying the consequent by a number greater than 1. 7. A ratio is not changed, by multiplying or dividing both of its terms by the same number. 8. The product of two or more ratios equals the ratio of their products. 361. Rules. — 1. To reduce a simple ratio to its lowest terms, Divide both terms by their greatest common divisor. (Pr. 7.) 2. To reduce a simple ratio with fractional terms to one witli integral terms, Midtiply both terms by the least common multiple of the denominators of the fractions. (Pr. 7.) 3. To find the product of two or more simple ratios, MuUiply the antecedents together for an antecedent and the 'con- sequents together for a consequent. (Pr. 8.) Note. — The process may be shortened by cancellation. 224 COMPLETE ARITHMETIC. PROPORTION. MENTAL EXERCISES. The ratio of 12 to 6 is equal to the ratio of 14 to 7, since the value of each ratio is 2. 1. What two numbers have a ratio to each other equal to the ratio of 15 to 5 ? 24 to 12? 2. What two numbers have a ratio to each other equal to 6: 24? 7:21? 11:44? 3. What two numbers have a ratio to each other equal to 45: 15? 12:60? 72:24? 4. To what number has 10 a ratio equal to the ratio of 30 to 15? 14 to 28? 5. To what number has 16 a ratio equal to 11 :33? 6. To what number has 12 a ratio equal to 6:30? 24:16? 20 to 15? 7. 12 is to 60 as 5 is to what number? 8. 13 is to 39 as 15 is to what number? 9. 14 is to 42 as 25 is to what number? 10. 56 is to 8 as 63 is to what number? 362. The equality of two ratios is expressed by placing a double colon (::) between them. Thus, 5 : 10 =: 7 : 14 is written 5 : 10 : : 7 : 14, and is read 5 is to 10 as 7 is to 14. 11. Read 8 : 40 : : 12 : 60, and show that the two ratios are equal. 12. Read 27 : 9 :: 63 : 21, and show that the two ratios are equal. 13. Read 5 : 2J : : 25 : 12^, and show that the two ratios are equal. DEFINITIONS AND PRINCIPLES. 363. A I*roj)Ortion is an equality of ratios. 364. The first ratio of a proportion is called the First Couplet, and the second ratio the Second Couplet. SIMPLE PROPORTION. 225 365. The first and third terms of a proportion are the Antecedents, and the second and fourth terms, the Conse- quents. Note, — The antecedents of a proportion are the antecedents of its ratios, and the consequents are the consequents of its ratios. 366. The first and fourth terms of a proportion are the Extremes, and the second and third terms, the Means. The four terms of a proportion are called Proportionals, and the last is the fourth proportional to the other three in their order. 367. Three numbers are in proportion when the ratio of the first to the second equals the ratio of the second to the third ; as, 8 : 12 : : 12 : 18. The second number is called a mean proportional. 368. Proportions are either Simple or Compound. A Shnj^le Proj^ortion is an equality of two simple ratios. A Compound Proportion is an equality of two ratios, one or both of which are compound. SIMPLE PROPORTION. Cai^e I. A-Tiy Term fo"and, -wlien. tlie otliei' Tliree Tenns are given. 369. The proportion 4 : 8 : : 6 : 12 may be written 4 : 8 = 6 : 12, or f = 3^ (Art. 356) ; and multiplying the two equal fractions by 12 and 8, their denominators, we have 4 X 12 = 6 X 8. Hence, the following Principles. — 1. The product of the extremes of a propor- tion equals the product of the means. Hence, 2. If the product of the extremes of a proportion he divided by either mean, the quotient tvill be the other mean. 3. If the product of the two means of a proportion be divided by either extreme, the quotient will be the other extreme. 226 COMPLETE ARITHMETIC. -WRITTEN PROBLEMS. Find the missing term in the following proportions 14. 21 : 7 : : 36 : -- 22; 1 : 1 : : 1 : — 15. 15: 40 : : 18 : — 23. *:|::-:| 16. — : 24 : : 8 : 32 24. -:2i::i:f 17. — : 9 : : 60 : 18 25. i:-::i:i 18. 45: 30 : : - : 24 26. $5: $45:: 61b. : — 19. 2.5 : 62.5::—: 3.25 27. $.75: $3 :: — : 56 oz. 20. 7.2 : — :: 4.7 : 9.4 28. 16 men : 96 men :: 15 days 21. .25 : - :: 2.5 : 7.5 29. 8 horses : 14 horses :: ^ : - 370. Rules. — 1. To find either extreme of a simple pro- portion, Divide the product of the two means by the other extreme. 2. To find either mean of a simple proportion, Divide the product of the two extremes by the otJier mean. Case 11. The Solntion of T>ro"bleixis by Sinaple I'roportion. 371. The solution of a problem by proportion consists of two parts, viz. : 1. The arranging of the three given terms, called the Statement. 2. The finding of the fourth term by Case I. 372. If the required answer be made the fourth term of a proportion, the given number of the problem, which is of the same kind as the answer, will be the third term, since the two terms of a ratio must be like numbers. (Art. 360.) 373. Of the two remaining numbers given in the problem, the greater will be the second term when the answer is to be greater than the third term, and the less will be the second term when the answer is to be less than the third term, otherwise the two ratios can not be equal. SIMPLE PROPORTION. 227 'WRITTEN PROBLEMS. 30. If 15 yards of cloth cost $24, what will 40 yards cost? STATEMENT. Siiice the cost of 40 yards is to be ,^ , .„ 1 ff,.-,, A the answer, make $24, the cost of 15 lo vd. : 40 yd. :: ^24 : Ans. ', • , . c 40 yards, the third term of a proportion ; 15) $960 ^"d since 40 yards will cost more than $64 Arts. 1'^ yards, the fourth term is to be greater than the third, and hence the second term must be greater than the first. Make 40 yards tlie second term and 15 yards the first, giving the proportion 15 yd. : 40 yd. :: $24 : cost of 40 yard.s, which, by Case I, is found to be $64. 31. If 45 sheep cost $565, what will 140 sheep cost? 32. If 13 tons of hay cost $97.50, what will 1\ tons cost ? 33. If 70 acres of land cost $1875, what will 320 acres cost? 34. If 120 acres of land cost $3000, how many acres can be bought for $4500 ? 35. If 4 lb. 6 oz. of butter cost $1.75, what will 17 J pounds cost? 36. If a man's pulse beat 75 times in a minute, how many times will it beat in 8 hours ? 37. If a clock ticks 120 times in a minute, how many times does it tick in 9J hours? 38. If a comet move 4° 20' in 15 hours, how far will it move in 5 days? 39. If a garrison of 160 men consume 24 barrels of flour in 6 weeks, how many barrels will supply it one year? 40. If 24 barrels of flour will supply 160 men 6 weeks, how many barrels will supply 360 men the same time? 41. If a vertical staff 3 feet high casts a shadow 5 feet long, how long a shadow will a pole 120 feet high cast at the same time? 42. If a pole 20 feet high casts a shadow 12 feet long, how high is the tree whose shadow, at the same time, is 90 feet long? 228 COMPLETE ARITHMETIC. 43. If I of a farm is worth $4500, what is J of it worth? 44. If I of a yard of silk cost $2.10, what will 16i yards cost? 45. If 6^ tons of hay cost $58.75, how many tons can be bought for $173.90? 46. At the rate of 5 jDcaches for 8 apples, how many apples can be bought for 5 dozen peaches? 47. If 12 men can mow 20 acres of grass in a day, how many acres can 25 men mow ? 48. If 9 men can build a wall in 15 days, how long will it take 5 men to build it? STATEMENT. The 15 days is the third term, _ f. , - ■, 1 since the answer is to be in days. 5 men : 9 men :: lo days : Ans. ^^ . , ^ ^ _ , , . . , 9 If it take 9 men lo days to build 5 ) ]^35 a wall, it will take 5 men more 27 days Ans. than 15 days, and hence the an- swer, or fourth term, is greater than the third term, and consequently the second term must be greater than the first term. The proportion is 5 men : 9 men : : 15 days : 27 days. Note. — The principle involyed in this class of problems may thus be stated : The greater the cause, the less the time required to produce a given effect; and, conversely, the greater the time, the less the cattse required. 49. If a quantity of provisions will supply a garrison of 90 men 125 days, how long will the same provisions supply 150 men ? 50. If 15 men can harvest a field of wheat in 12 days, how many men can harvest it in 5 days? 51. Divide 90 into two parts whose ratio is equal to the ratio of 4 and 5. r (4 + 5) : 90 : : 4 : Smaller part. Proportions, i , , , ^ ,,^ . ^ , ^ (4 4- 5) : 90 : : o : Greater part. Note. — These proportions are based on the principle that when four numbers are in proportion, the sum. of the first and second toyns is to the sum of the third and fourth terms as the first tei-m is to the ilnrd, or as the second term is to the fourth. 52. Divide 640 into two parts proportional to 8 and 12. To 9 and 11. SIMPLE PROPORTION. 229 53. An estate worth $9600 was divided between two heirs in proportion to their ages, which were 15 years and 17 years respectively: how much did each receive.'^ 54. Two men, 150 miles apart, are approaching each other, one traveling 2 miles to the other 3 : how far will each travel before they meet? Jg^^'For additional problems see Problems for Analysis, p. 239 PRINCIPLES AND RULE. 374. Principles. — 1. The ratio of two like causes equals the ratio of their effects. Conversely, 2. The ratio of two like, effects equals the ratio of their causes. 3. Tlie ratio of two like causes equals the inverse ratio of their times. Conversely, 4. TJie ratio of the times of two like causes equals the in- verse ratio of tlie causes. 5. The two terms of each couplet of a simple proportion must be like numbers. (Art. 360, Pr. 1.) 6. The fourth term of a proportion equals the product of the second and tliird terms divided by iJie first term. (Art. 370. ) 375. Rule. — To solve a problem by simple proportion, 1. Take for the third term the number which is of the same kind as the answer sought, and make the other two numbers the first couplet, placing the greater for the second term, when the answer is to be greater than the third term; and the LF.Sfi for the second term, when the answer is to be less than the third term. 2. Divide the product of the second and third terms by the first term, and the quotient will be the fourth term, or answer. Notes. — 1. When the terms of the first couplet are denominate numbers, they must be reduced to the same denomination. 2. The process of finding the fourth term may be shortened by can- cellation. The proportion 15 : 45 :: 27.5 : — may be completed thus : V 97 fS ^^•* ^"^^ X$ 4$ 27.5 27.5X3^=82.5 3. The process of solving problems bv simple proportion is also called " The Rule of Three.''' 230 COMPLETE ARITHMETIC. COMPOUND PROPORTION. Case I. R.ed.'uctiorx of CoinpourLcT. Ratios and lProj>oi'tions to Simp-le Ones. 1. Reduce the compound ratio 20:80 4:3 6:8 ratio in its lowest terras. Process. to a simple 20 4 6 80 3 8 20X4X6:80X3X8 480 : 1920 1 : 4, Ans. Or: 1 : 4, Am. A compound ratio is the product of two or more simple ra- tios (Art. 358), and the product of two or more simple ratios is found by multiply- ing the antecedents together for an antecedent, and the consequents for a consequent (Art. 361). Hence, the compound ratio given is equal to 20 X 4 X 6 : 80 X 3 X 8, or 480 : 1920, which, by dividing both terms by 480, is reduced to 1 : 4. The process may be shortened by canceling the factors common to the product of the antecedents and the product of the consequents. Note. — The process may be explained directly by changing each ratio to the fractional form, thus : 20 : 80 }- 1X1 = ^0 X ^ X ^ ^ X 1:4. $0 X ^ i 4 Reduce the following compound ratios to simple ratios in their lowest terms : 2. 6 10 16 8 12 15 7:20 40:21 12:16 82 : 45 6. Reduce ] q .' fj [ : : 16 : 15 to a simple proportion. Su(K*ESTiON. — Reduce the compound ratio to a simple ratio. COMPOUND PROPORTION. 231 Reduce the following compound proportions to simple proportions I 21 : 10 j • • 4^ • ^^ 48:336") 5:8 ,^ : : 12 : 56 12:5 3 9. $192 : $216 24:36 3:4 10. 6:9 12: 3 15:36 ,„{ ^5 : 27 •• M2 : 6 11. tion, What is the fourth term 5:8 ::13 : of the compound propor- 10 12 9 25 Process. 0:tM ^ 10 : 3 : Or, : 13 : — 1 3:: 13 3 ^10 $4 03 13 39, Am. 3X13=:= 39, Ans. An inspection of the second process shows that the four numbers on the right of the vertical line are the factors of the product of the means, and that the three numbers on the left are the factors of the first extreme. By canceling the factors common to dividend and divisor, the fourth term is found directly. Find the fourth term of these compound proportions; 20 : 48 ) (5:9 12. ^ 36 : 15 .^ : : 25 : — 14. ^ 2.5 : 7.5 ^ • : 6 : — 10:4 4:10 13. f 16:35 J 21 :8 19:6 I 12 : 45 : : 16 : — 15. 21:7 25 : 10 4 : 6i r 15:12j 1.. ::35 376. Rules. — 1. To reduce a compound ratio to a simple ratio, Multiply the antecedents together for an antecedent, and the consequents for a consequent. 2. To reduce a compound proportion to a simple propor- tion, Reduce the compound ratio, or each compound ratio, if there are two, to a simple ratio. 232 COMPLETE ARITHMETIC. Case II. Xh.e Solutioii of IProbleiTis by Com.pou.nd. IProportion. 16. If 2 men can mow 16 acres of grass in- 10 days, working 8 hours a day, how many men can mow 27 acres in 9 days, working 10 hours a day? Since the answer required Statement. is to be a number of men. 16 acres 9 days 10 hours 27 acres 10 days 8 hours ;} 2 men : Ans. 2 : 3 : : 2 men : 3 men, Ans. Or. ^10 10 10 $ make 2 men the third term. If the mowing of 16 acres requires 2 men, the mowing of 27 acres will require more than 2 men, and hence the first ratio is 16 acres : 27 acres, the greater number be- ing the second term. Tf 10 days require 2 men, 9 days will require more than 2 men, and hence the second ratio is 9 days : 10 days, the 3 greater number being the sec- ond term. If working 8 hours a day requires 2 men, working 10 hours a day will require less than 2 men, and hence the third ratio is 10 hours : 8 hours, the less number being the second term. This statement gives 3 men for the fourth terra. Note. — In determining which number of each ratio of the com- pound ratio is to be the second term, reason from the number in the CONDITION. 17. If 12 men can build 50 rods of wall in 15 days, how many men can build 80 rods in 16 days ? 18. If it cost $30 to make a walk 10 feet wide and 90 feet long, how much will it cost to make a walk 8 feet wide and 225 feet long? 19. If 6 men can excavate 576 cubic feet of earth in 8 days of 9 hours each, how much can 8 men excavate in 9 days of 10 hours each ? COMPOUND PROPORTION. 233 20. If 7 horses eat 35 bushels of oats in 25 days, how- many bushels will 15 horses eat in 21 days? 21. If a man walk 120 miles in 6 days of 10 hours each, how many miles will he walk in 16 days of 8 hours each ? 22. If 1500 bricks, each 8 in. long and 4 in. wide, will make a walk, how many slabs of stone, each 2 ft. long and 1 ft. 4 in. wide, will be required for the same purpose? 23. If the interest of $250 for 9 months is $11.25, what is the interest of $650 for 7 months? 24. If it cost $84 to carpet a room 36 ft. long and 21 ft. wide, what will it cost to carpet a room 33 ft. long and 27 ft. wide? 25. If it cost $120 to build a wall 40 ft. long, 14 ft. high, and 1 ft. 6 in. thick, what will it cost to build a wall 180 ft. long, 21 ft. high, and 1 ft. 3 in. thick? 26. If 4 men can dig a ditch 72 rd. long, 5 ft. wide, and 2 ft. deep in 12 days, how many men can dig a ditch 120 rd. long, 6 ft. wide, and 1 ft. 6 in. deep in 9 days ? 27. If 16 men can excavate a cellar 50 ft. long, 36 ft. wide, and 8 ft. deep in 10 days of 8 hours each, in how- many days of 10 hours each can 6 men excavate a cellar 45 ft. long, 25 ft. wide, and 6 ft. deep ? 28. If 30 men can dig a ditch 40 rd. long, 6 ft. wide, and 3 ft. deep in 9 days, working 8 hours a day, how many men can dig a ditch 15 rd. long, 4 J ft. wide, and 2 ft. deep in 12 days, working 6 hours a day? Note. — For additional problems, see Problems for Analysis. PRINCIPLES AND RULE. 377. Principles. — 1. A compound proportion, used in the solution of a problem, has only one compound ratio. 2. The order of the terms of each ratio composing the com- pound ratio, is determined as in simple proportion. 3. The fourth term of a compound proportion is equal to the product of all the factors of the second and third terms, divided by the product of tJie factors of the first term. r.Ar.— CO 234 COMPLETE ARITHMETIC. 378. Rule. — 1. Take for the third term the number which is of the same kind as the answer sought, and arrange the first and second terms of each ratio composing the compound ratio as in simple proportion. 2. Reduce the compound ratio to a simple ratio, and divide the product of the second and third terms of the residting pro- portion by the first term. The quotient will be the fouHh term, or answer sought. Or, Divide the product of all tJie factors of the second and third terms of the compound proportion by the product of the fa/itors of the first term, shortening the process by cancellation. . Notes. — 1. The terras of each ratio composing the compound ratio are arranged precisely as they would be if the answer depended wholly on them and the third term. 2. The process of solving problems bv compound proportion is also called ''The Double Rule of Three.'' PARTNERSHIP. 379. A Partnership is an association of two or more persons for the transaction of business. A partnership is organized and regulated by a contract, called arti- cles of agreement. (Art. 240.) 380. A partnership association is called a Company, Firm, or House, and the persons associated together are called Partners. 381. The money or property invested in the business by the partners is called Capital, Joint-stock, or Stock in Trade. When a partner furnishes capital but does not assist in conducting the business, he is called a Silent Partner. 382. Partnership is either Simple or Compound. In Shnple Partnership the capital of the several partners is invested an equal time. In Compound JPartnership the capital of the sev- eral partner^ is invested an unequal time. PARTNERSHIP. 235 SIMPLE PARTNERSHIP. PROBLEMS. 1. A, B, and C entered into partnership in business for 2 years ; A put in $3600, B $2400, and C $2000, and their net profits were $3000. What was each partner's share ? I. Process by Proportion. $3600, A's capH. $8000 2400, B's capH. $8000 2000 , C's cap'l. $8000 $8000, Entire capital. $3600 : : $3000 : $1350, A's share of profits. $2400 :: $3000 : $900, B'.s " $2000 : : $3000 : $750, C's " $3000, Entire profits. Since the capital of the several partners was employed an equal time, their shares of the profits are proportional to their capitals. Hence, the entire capital is to each partner's capital as the entire profits are to his share of the profits. II. Process by Percentage. $3000 -^ $8000 = .37 h $3600 X -37* ^ $1350, A's share. $2400 X .37* = $900, B's " $2000 X .37J = $750, C's " Or: Since the profits were equal to .372, or 37^^ % of the entire capital, each partner's share of the profits was equal to 37J % of his capital. $3600-^ $8000 -r.45. AS per cent. of the cap $2400 -f- $8000 = .30, B's i( (< ( $2000 -4- $8000 = .25, C's n a I $3000 X -45 = $1350, A's share of the profits. $3000 X .30 := $900, B's a <( a $3000 X .25 = $750, Cs- « " li III. Process by Fractional Parts. $3000 -^ $8000 = t§^^ = |. f of $3690 = $1350, A's share. f of $2400 = $900, B's " I of $2000 = $760, O's " Since the profits were equal to f of the ^entire capital, each part- ner's share of the profits was equal to I of his capital. 236 COMPLETE ARITHMETIC. Or: $3600 ^ $8000 = /^, A's part of the capital. $2400 -^ $8000 = T^V, -^'s " $2000 -^ $8000 =- i, C's " " " • /^ of $3000 = $1350, A's share of the profits. T% of $3000 =$900, B's *' " I of $3000 = $750, C's " " Note. — Let the following problems be solved by proportion and by either of the other methods, which the teacher or pupil may prefer. 2. A and B were partners in business ; A put in $5000 and B $4000, and their profits in three years were $4500 : what was each partner's share of the profits? 3. A, B, and C formed a partnership in business ; A put in $8000, B $4500, and C $3500, and their loss the first year was $3200: what was each partner's share? 4. A, B, and C are partners, and B has invested | as much capital as A, and C | as much as B : if their profits amount to $6300, what will be each partner's share? 5. The capital of two partners is proportional to 4 and 3; their profits are $10000 and their expenses $2300: what is each partner's share of the net profits? 6. A, B, and C form a partnership, A's capital being $4000, B's $6400, and C's $5600; they make a net gain of $3200, and then sell out for $2000: what is each partner's share of the gain ? Of the proceeds of the sale ? PRINCIPLES x\ND RULE. 383. Principles. — 1. The gain or loss of a partnership is shared by the partners in proportion to the use of the capital invested by them, which is its partnership value. 2. When the time is equal, the use of tlie capital of the several partners is in proportion to its amount. Hence, 3. In a simple partnership, the gain or loss is shared by the partners in proportion to the amounts of their capital. 384. Rule. — To divide the gain or loss of a simple part- nership. Divide the gain or loss among the several partners in proportion to the atnounts of capital invested by them. COMPOUND PARTNERSHIP. 237 Notes. — 1. The above principles and rule are applicable only when the several partners devote equal time or render equal service in car- rying on the business. The division of profits or losses is usually settled by the terms of the contract. 2. The problems in bankruptcy (Art. 272) may also be solved by the above methods. COMPOUND PARTNERSHIP. 7. A and B formed a partnership ; A put in $3000, and, at the close of the first year, added $2000; B put in $4000, and, at the close of the second year, took out $2000 ; at the close of the third year, the profits amounted to $3450. What was each partner's share? I. Process by Products. $3000X1= »00 $5000X2 = $10000 $13000, A's capital for 1 year. $4000 X 2 = $8000 $2000 X 1 = _$2000 $10000, B's capital for 1 year. $13000 + $10000 = $23000, Entire capital for 1 year. $23000 : $13000 : : $3450 : $1950, A's share of profits. $23000 : $10000 : : $3450; $1500, B's " Since A had $3000 invested for 1 year and $5000 for 2 years, the use of his capital was equivalent to the use of $13000 for 1 year. Since B had $4000 invested for 2 years and $2000 for 1 year, the use of his capital was equivalent to the use of $10000 for 1 year. Hence the profits, amounting to $3450, should be shared by them in proportion to $13000 and $10000. II. Process by Interest. Int. of $3000 for 1 yr. = $180 " " $5000 for 2 yr. = $600 $780, Int. of A's capital. Int. of $4000 for 2 yr. = $480 " '* $2000 fori yr. = $120 $600, Int. of B's capital. $780 + 600 = $1380, Int. of entire capital. $1380 : $780 : : $3450 : $1950, A\s share. $1380 : $600 : : $3450 : $1500, B'a " 238 COMPLETE ARITHMETIC. Since the use of capital is represented by its interest for the time, the use of A's capital is represented by $780, and the use of B's by $600. Hence, the profits ($3450) should be shared by them in pro- portion to $780 and $600. Note. — The ratio of the interests will be the same whatever be the rate per cent. ; and hence the interest may be computed at any rate. 8. A, B, and C enter into a partnership for 4 years, A putting in $6000 and B $8000. At the close of the second year, A took out $2000 and B put in $2000 ; and, at the close of the fourth year, they divided $8890 as net profits. What was the share of each? 9. A and B entered into a partnership in business for 3 years, A's invested capital being $3500 and B's $4500. At the end of the first year they each took out $1000, and B was received as a partner with a capital of $2500. At the end of the third year they dissolved partnership, dividing $5000 as net profits. What was each partner's share? 10. A, B, and C entered into business as partners, each putting in $5000 as capital. At the end of 2 years A took out $1000, B $2000, and C $3000, and, at the end of the fourth year, they closed the business with a loss of $3600. What was the loss of each? PRINCIPLE AND RULES. 386. Principle. — The value of capital in compound partner- ship depends jointly on its amount and the time of its investment. 386. Rules. — To divide the gain or loss of a compound partnership, 1. Multiply the amount of capital invested by each partner by the time of its investment, and taking the product as the partnership value of his capital, proceed as in simple partner- ship. Or, 2. Find the interest of each partner's capital for the time of its investment, at any rate per cent. ; and taking the interest thus found as the partnership value of his capital, proceed as in simple partnership. PROBLEMS FOR ANALYSIS. 239 PROBLEMS FOR ANALYSIS. Note. — These problems are here given to afford an additional drill in analysis and, if needed, in proportion. For the latter pur- pose, the teacher can select as many problems as may be necessary. MENTAL PROBLEMS. 1. If 7 pounds of sugar cost 91 cents, what will 20 pounds cost? 2. If 12 yards of muslin cost $1.02, what will 20 yards cost ? 3. If -| of a yard of silk cost $fi, what will f of a yard cost? 4. If "I of a barrel of flour cost $d^, what will -f of a barrel cost? 5. If f of a pound of coffee cost 15 cent:*, what will 3^ pounds cost? 6. A man sold a watch for $120, which was f of what it cost him : how much did it cost ? 7. If 40 yards of carpeting, j of a yard wide, will cover a floor, how many yards of matting, IJ yards wide, will cover a floor of equal size ? 8. Two men, traveling in the same direction, are 60 miles apart; the one in advance travels 5 miles an hour, and the other 7 miles an hour : in how many hours will the latter overtake the former? 9. If a vertical staff 3 feet long casts a shadow 2 feet in length, how long a shadow will a tree 90 feet high cast at the same time of day? 10. If a steeple 200 feet high casts a shadow 150 feet long, what is the height of a pole which, at the same time of day, casts a shadow 80 feet long? 11. If 5 men can do a piece of work in 12 days, how long will it take 6 men to do it? 12. If 8 men can do a piece of work in 15 days, how many men can do the same work in 10 days? 240 COMPLETE ARITHMETIC. 13. If 9 men can do a piece of work in 4f days, how long will it take 7 men to do it? 14. If 3 pipes will empty a cistern in 30 minutes, how many pipes will empty it in 10 minutes ? 15. If a quantity of provisions will supply 15 men 20 days, how long will it supply 50 men ? 16. If it require 12 days of 10 hours each to do a piece of work, how many days of 8 hours each will be required to do the same work? 17. If 5 men can do f of a piece of work in a day, how long will it take one man to do the entire work? 18. If 8 men can do | of a piece of work in 3 days, how long will it take 4 men to do the entire work? 19. If 20 men earn $120 in 4 days, how much will 5 men earn in 8 days? 20. If 6 men can mow 30 acres of grass in 3 days, how many acres will 9 men mow in 5 days? 21. If 5 horses eat 40 bushels of oats in 3 weeks, how many bushels will supply 12 horses 10 weeks ? 22. If 8 men can dig a ditch 40 rods long in 6 days, how long will it take 12 men to dig a ditch 60 rods long? 23. If the interest of $50 for 9 months is $6, what would be the interest of $150 for 1 yr. 6 mo.? 24. A school enrolls 180 pupils, and the number of boys is f of the number of girls : how many pupils of each sex are enrolled in the school ? 25. A lady paid $130 for a watch and chain, and the cost of the watch was f more than the cost of the chain : what was the cost of each ? 26. A tree 120 feet in height was broken into two parts by falling, and | of the shorter part equaled } of the longer : what was the length of each part ? 27. A person giving the time of day, said that | of the time past noon equaled the time to midnight : what was the hour of day ? 28. A person being asked the time of day, said that ^ of the time past midnight equaled the time to noon : what was the hour of day ? PROBLEMS FOR ANALYSIS. 241 29. What is the hour of day when f of the time pa?t noon equals | of the time to midnight ? 30. What is the time of day when | of the time past noon, multiplied by 4^, is equal to the time to midnight ? 31. What is the time of day when f of the time to noon is equal to f of the time past midnight? 32. A man being asked his age said, 10 years ago my age was J of my present age : what was his age ? 33. A son's age is f of the age of his father, and the sum of their ages is 80 years : what is the age of each ? 34. Ten years ago the age of A was f of the age of B, and 10 years hence the age of A will be f of the age of B : what is the age of each ? 35. At the time of marriage a wife's age was f of the age of her husband, and 10 years after marriage her age was Y^^ of the age of her husband : how old was each at the time of marriage ? 36. f of A's age equals f of B's, and the difference be- tween their ages is 10 years : how old is each ? 37. Twice the age of A is 20 years more than the age of B, and 10 years more than the age of C, and the sum of their ages is 120 years : what is the age of each? 38. A man bought a horse and carriage for $275, and ^ of the cost of the carriage -|- $100 was equal to ^ of the cost of the horse : what Avas the cost of each ? 39. A man bought a horse, saddle, and bridle for $150 ; the cost of the saddle was ^ of the cost of the horse, and the cost of the bridle was ^ of the cost of the saddle : what was the cost of each ? 40. A man and his two sons earned $140 in a month ; the man earned twice as much as the elder son, and the elder son earned twice as much as the younger : how much did each earn ? 41. Two men bought a barrel of sirup, one paying $20 and the other $30: what part should each have? 42. Two men hired a pasture for $40, and one put in 3 cows and the other 5 cows : how much ought each to pay ? C.Ar.— 21. 242 COMPLETE ARITHMETIC. 43. A and B rented a pasture for $72; A puts in 40 sheep and B 8 cows : if 4 sheep eat as much as one cow, how much ought each to pay? 44. Two men, A and B, agreed to build a wall for $300; A sent 5 men for 4 days, and B 5 men for 6 days : how much ought each to receive? 45. A and B engage to plow a field for $81 ; A furnished 3 teams for 5 days, and B furnished 4 teams for 3 days : how much should each receive ? 46. A man can do i of a piece of work in a day, and a boy can do -J- of it in a day: in how many days can both of them, working together, do it ? 47. A and B together can build a wall in 8 days, and A can build it alone in 12 days : how long will it take B to build it? 48. A can do a piece of work in 6 days, and B in 8 days : if they both work together 3 days, how long will it take B alone to complete the work ? 49. John can saw a pile of wood in 6 days, and, with the assistance of Charles, he can saw it in 4 days : how long will it take Charles to saw it alone ? 50. A man can do a piece of work in 5 days and a boy in 8 days ; the man w^orks 2 days alone and is then assisted by the boy: how long will it take both to complete the work ? 51. A and B can do a piece of w^ork in 10 days, and A, B, and C in 8 days : how long will it take C alone to do the work? 52. A and B can do |^ of a piece of work in a day, and A can do twice as much in a day as B : how long will it take B alone to do it? 53. A can make a fence in ^ of a month, B in I of a month, and C in -J- of a month : in what time can all three together build it? 54. A can do a piece of work in 4 days, B in 5 days, and C in 6 days : in what time can they together do it? 55. A, B, and C can do a piece of work in 4 days, A PROBLEMS FOR ANALYSIS. 243 and C in 8 days, and B and C in 12 days : how long will it take A and B together to do it? dQ. a and B did a piece of work, and f of what A did equaled | of what B did : if B received $18, how much did A receive? 57. A man spent f of his money and then earned ^ as much as he had spent, and then had $21 less than he had at first: how much money did he have at first? 58. At what time between one and two o'clock will the hour and minute hands of a watch be together? 59. At w^hat time between two and three o'clock are the hour and minute hands of a watch together ? At what time between four and five o'clock? 60. A man has 2 watches, and a chain worth $20 ; if he put the chain on the first watch it will be worth f as much as the second watch, but if he put the chain on the second watch it will be worth 2J times the first watch : what is the value of each watch? WRITTEN PROBLEMS. 61. A father bequeathed $14535 to two sons, giving the younger ^^ as much as the elder : what was the share of each ? 62. An estate was so divided between two heirs that | of the share of the elder was equal to f of the share of the younger, and the diflTerence between their shares w^as $362 : what was the share of each ? 63. An estate worth $27520 was divided between two daughters in proportion to their ages, which were 14 and 18 years respectively : how much did each receive ? 64. A man paid $8100 for 2 farms, and | of the cost of the larger farm was equal to y% of the cost of the smaller : what was the cost of each ? 65. A earns $15.50 as often as B earns $12.40, and in a certain time they together earn $697.50: how much did each earn? 66. The fore wheels of a carrrage are each 9^ feet in cir- 244 COMPLETE ARITHMETIC. cumference, and the hind wheels are each 12^ feet in cir- cumference : if each fore wheel revolve 9500 times in going a certain distance, how many times will each hind wheel revolve ? 67. If it take 13200 steps of 2 ft. 9 in. each to walk a certain distance, how many steps of 1 ft. 10 in. each will it take to walk the same distance ? 68. If $75 yield $10.80 interest, what principal will yield $89.28 interest in the same time? 69. If the interest of $475 is $118.75, what would be the interest of $850 for the same time and at the same rate? 70. If the interest of a certain principal for a certain time at 5 per cent, is $120.50, wiiat would be the interest of the same principal for the same time at 12 per cent. ? 71. A broker sold 90 shares of railroad stock and gained $315: how much would he have gained if he had sold 245 shares ? 72. If a gain of 15 per cent, on a certain investment yields $2347.50, what would a gain of 24 per cent, on the same investment yield ? 73. If the commission for selling 3050 pounds of butter at 30 cents a pound is $45.75, what would be the com- mission for selling 7500 pounds at 35 cents a pound? 74. If the annual dividend on $40325 worth of mining stock is $3226, what is the dividend on $70680 of the same stock? 75. If 6 ranks of wood, each 60 ft. long and 6 ft. high, are worth $337.50, w^hat is the value of 15 ranks of wood, each 45 ft. long and 9 ft. high? 76. If it cost $110 to dig a cellar 40 ft. long, 27 ft. wide, and 4 ft. deep, how much will it cost to dig a cellar 36 ft. long, 30 ft. wide, and 5 ft. deep? 77. If 45 men can do a piece of work in 15 days, by working 8 hours a day, in how many days can 30 men, working 9 hours a day, do the same work? 78. If 5 men can cut 45 cords of wood in 6 days, how mauv cords can 8 men cut' in 15 davs? PROBLEMS FOR ANALYSIS. 245 79. If 4 men dig a trench in 15 days of 10 hours each, in how many days of 8 hours each can 5 men perform the same work? 80. A and B are partners in business ; A's capital is equal to f of B's, and their profits are $3250 : what is the share of each? 81. A and B are partners ; f of A's capital is equal to f of B's, and their loss in business is $2150: what is the share of each? 82. A, B, and C are partners in business ; A's capital is twice B's and three times C's, and their profits in business are $4675 : what is the share of each ? 83. A and B, trading in partnership 2 years, make a profit of $5460 ; during the first year A owned | of the stock, and during the second year B owned J of it : what is each partner's share of the profits? 84. A and B, trading in partnership 2 years, make a profit of $2400 ; A's capital the first year was 2|- times B's, and the second year it was 1|- times B's : what is each part- ner's share of the profits ? S5. A and B traded in partnership 3 years; A's stock the first year was $5000, the second year $6000, and the third year $7000 ; B's stock the first year was $7000, and the last two years $5000; their loss was $1750. What was the loss of each ? 86. A mechanic agreed to work 80 days on the condition that he should receive $1.75 and board for every day that he worked, and that he should pay 75 cents a day for board when he was idle ; his net earnings for the time were $80 : how many days did he work? 87. A piece of carpeting containing 135 yards was cut into 3 carpets, and -^ of the number of yards in the first carpet was equal to ^ of the number of yards in the second carpet, and to f of the number of yards in the third carpet : what was the number of yards in each carpet? 246 COMl'LETE ARITHMETIC. SECTION XVI. INYOLUTIO]^ AJ^D EYOLUTIO]:^ I. INVOLUTION. 387. The first power of 4 is 4 ; the second power of 4 is 4x4, which is 16 ; the third power is 4 X 4 X 4, which is 64 ; the fourth power is 4 X 4 X 4 X 4, which is 256 ; etc. 1. What is the second power of 5? Of 6? 8? 10? 2. What is the third power of 3 ? Of 4 ? 5 ? 6 ? 10 ? 3. What is the fourth power of 2 ? Of 3 ? 4? 10 ? 4. What is the second power of 1 ? 2? 3? 4? 5? 6? 7? 8? 9? 5. What is the third power of 1 ? 2 ? 3 ? 4 ? 5 ? 6 ? 7? 8? 9? 6. What is the second power of ^ Of f ? i? 1 ? G • WRITTEN PROBLEMS. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. What What What What What What What What What What s the s the s the s the s the s the s the s the s the s the second power of 406? third power of 42? fourth power of 24? fifth power of 16? second power of 6.5? third power of . 42 ? fifth power of .6? third power of Jf . second power of || ? fourth power of j| ? Remark. — The power to which a number is to be raised may be denoted by a little figure, called an exponent, placed at the right of the upper part of the figures expressing the number. Thus, 24^ denotes the second power of 24; 16^ denotes the third power of 16, etc. INVOLUTION. 247 Raise the following numbers to the powers indicated by e exponents : 17. 6232 22. .0452 27. mr 18. 105^ 23. (if)3 28. mr 19. 34.62 24. ar 29. .005'' 20. .016^ 25. (161)2 30. 2.04» 21. 1.4^ 26. (3i)4 31. (I)* DEFINITIONS AND RULE. 388. The Power of a number is the product obtained by taking the number one or more times as a factor. 389. The First Power of a number is the number itself. 390. The Second Poiver of a number is the product obtained by taking the number twice as a factor. It is also called the Square of the number, since the area of a geometrical square is rep- resented by the product obtained by taking the number of linear units in one of its sides twice as a factor. 3X3^ 391. The Third Poiver of a number is the product obtained by taking the number three times as a factor. It is also called the Cube of the number, since the capacity of a geometrical cube is represented by the product obtained by tak- ing the number of linear units in one of its edges three times as a factor. 3X 3 ^ 27. 392. The Exponent of a power is a small figure i:)laced at the right of the number, to show how many times it is to be taken as a factor. It denotes the degree of the power. 248 COMPLETE ARITHMETIC. The first power contains the number once as a factor, and the expo- nent is ^ ; the second power, or square, contains the number twice as a factor, and the exponent is 2 ; tlie tliird power, or cube, contains the number three times as a factor, and the exponent is ^ ; etc. 393. Involution is the process of finding the powers of numbers. 394. Rule. — To raise a number to a given power, Mul- tiply the number by itself as many times less one as there are units in the exponent of the given power. The last product will be the required power. 395. ATs^OTHER METHOD OF INVOLUTION. 29. What is the square of 53 ? Process. 53 = 50 + 3, and 53^ = (50 + 3)2 50 + 3 Parts added. 50 + 3 502 ^ 2500 50X3 + 32 ==(50 + 3)X3 2(50X3)-- 300 502 + 50 X 3 ^ (50 ^ 3) X 50 32 = 9 502 + 2 (50 X 3} + 32 = (50 + 3)2 = 532 = 2809 An inspection of the above process will show that the square of 53 is equal to the square of the 5 tens, plus twice the product of the 5 tens by the 3 units, plus the square of the units. In like manner, it may be shown that the square of any number, composed of tens and units, is equal to The square of the tens, plus twice the product of the tens by the units, plus the square of the units. 30. What is the square of 45? (402 =1600 Process: 452= ^i^^X^^Z^^? ( 2025, Ans. 31. What is the square of 67? Of 75? 32. What is the square of 82 ? Of 38 ? 33. What is the square of 93? Of 125? Suggestion.— 125 = 120 + 5. 34. What is the square of 115? Of 124? EVOLUTION. 249 35. What is the cube of 53 ? The cube of 53 = (50 + 3)^ ^50^ + 3 (502X 3) + 3 (50 X 3^) -f 3^ as may be shown by multiplying 50^ + 2 (50 X 3) + 3^ by 50 + 3. In like manner, it may be shown that the cube of any number, composed of tens and units, is equal to The cube of the tens, plus three times the product of the square of the fens by the units, plus three times the product of the tens by the square of the units, plus the cube of the units. 36. What is the cube of 45 ? r 403 =. 64000 3 (402 X 5) = 24000 Process: 45^ ^ < 3 (40 X 5^) = 3000 53 = 125 *- 91125, Am. 37. What is the cube of 23? Of 32? 38. What is the cube of 24? Of 43? 39. What is the cube of 33? Of 54? 40. What is the cube of 51 ? Of 35 ? 41. What is the cube of 45? Of 52? 42. What is the cube of 41 ? 52 ? 43. What is the cube of 75? 80? II. EVOLUTION. MENTAL PROBLEMS. 1. What are the two equal factors of 16? Of 25? 49? 2. Of what number is 81 the second power or square? 3. What are the three equal factors of 8 ? Of 27 ? 125 ? 4. Of what number is 125 the third power or cube ? One of the two equal factors of a number is called its second or square root; one of its three equal factors, its third or cvhe root; one of its four equal factors, its fourth root, etc. 5. What is the square root of 25 ? 6. What is the cube root of 8 ? 7. What is the cube root of 216? 8. What is the fourth root of 16? 9. What is the square root of 1 ? 36? 49? 64? 81? Of 49? 64? 81? Of 27 ? 64? 125? 512? 1000? Of 81? 256? 625? 4? 9? 16? 25? 250 COMPLETE ARITHMETIC. 10. What is the cube root of 1 ? 8 ? 27 ? 64 ? 125 ? 216? 343? 512? 729? 11. What integers between 1 and 100 are perfect squares? 12. What integers between 1 and 1000 are perfect cubes ? 13. Show that the square root of a perfect square ex- pressed by two figures, can not exceed 9. 14. Show that the cube root of a perfect cube expressed by three figures, can not exceed 9. DEFINITIONS. 396. The Root of a number is one of the equal factors which will produce it. 397. The First Hoot is the number itself. 398. The Second Moot is one of the two equal factors ,of the number. It is also called the Square Root. 399. The Third Moot is one of the three equal factors of the number. It is also called the Cube Boot. A number is the second power of its square root; the third power of its cube root ; the fourth power of its fourth root ; etc. 400. A JPerfect Power is the product of equal factors. It has an exact root. 401. An Imperfect Power is a number which is not the product of equal factors. Its root is called a Surd. 402. The Madical Sign is a character, ] , placed before a number to show that its root is to be taken. 403. A small figure placed above the radical sign is called the Index of the root. Thus, 1^^ denotes the first root of 25 ; 1^25 denotes the second root of 25 ; f^25, the third root of 25, etc. SQUARE ROOT. 251 "When the square root is indicated, the index is usually omitted. 1^16 and Vl6 alike denote the square root of 16. Note. — The root of a number may also be indicated by a frac- tional exponent. Thus, 16- denotes the square root of 16 ; 16*, the cube root of 16, etc. ; 16^ denotes the cube root of the square of 16. 404. EvoltttiOfl is the process of finding the roots of numbers. Note. — Evolution is the reverse of involution. SQUARE ROOT. Tlie Division of th.e ]>f\xm'ber' into [Periods. 405. The smallest integer composed of one order of fig- ures is 1, and the greatest is 9 ; the smallest integer com- posed of two orders is 10, and the greatest is 99, and so on. The squares of the smallest and greatest integers com- posed of one, two, three, and four orders, are as follows : 12= 1 92=. 81 102= 100 992= 9801 1002 = 10000 9992 =: 997001 10002 = 1000000 99992 ^ 99880001 A comparison of the above numbers with their squares shows that the square of a number contains twice as many orders as the number, or twice as many orders less one. 406. Hence, if a number be separated into periods of two orders each, beginning at the right, there will he a§ mamj orders in its square root as there are periods in the numher. 1. How many orders in the square root of 2809? Suggestion.— First divide the number into periods of two orders each, thus: 2809. 2. How many orders in the square root of 36864? 3. How many orders in the square root of 345744? 252 COMPLETE ARITHMETIC. 4. How many orders in the square root of 87616? 5. How many orders in the square root of 5308416 ? 6. How many orders in the square root of 5475600? 7. How many orders in the square root of 14440000? 407. The squares of the smallest and greatest number of units, tens, hundredths, and thousandths, are as follows : V= 1 92= 81 102^ 100 902= 8100 1002 ^ 10000 9002 ^ 810000 10002 = 1000000 90002 := 81000000 A comparison of the above numbers with their squares shows that the square of units gives no order higher than tens; that the square of tens gives no order lower than hundreds, nor higher than thousands ; that the square of hundreds gives no order lower than ten-thousands, nor higher than hundred-thousrinds, etc. 408. Hence, if a number be separated into periods of two orders each, the left-hand period will contain the square of the left-hand or first term of the square root ; the first two left-lmnd periods will contain the square of the first two terms of the square root, etc. 8. What is the tens' term of the square root of 2025 ? Ans. — The left-hand period of 2025 is 20 ; the greatest square in 20 is 16, and the square root of 16 is 4. Hence, the tens' figure of the square root of 2025 is 4. 9. What is the hundreds' term of the square root of 87616? Of 345741? 10. What is the left-hand term of the square root of 16129? Of 336400? 11. What is the left-hand term of the square root of 87616? 12. What are the first two terms of the square root of 16129? SQUARE ROOT. 253 "WRITTEN PKOBLEMS. 13. What is the square root of 3364? Process. Since 3364 is composed of two periods, SSfi4 I 58 ^'^ square root will be composed of two 52 -— 25 orders. (Art. 406.) 5 \/ 2 = 10 )~864 The left-hand period 33 contains the 108 X 8 := 864 square of the tens' term of the root. (Art. ■ 408.) The greatest square in 33 is 25, and the square of 25 is 5. Hence, 5 is the tens' term of the root. The square of a number composed of tens and units is equal to the square of the tens plus twice the product of the tens by the units, plus the square of the units. (Art, 395.) Hence, the difference be- tween 3364 and the square of the 5 tens of its root, is composed of tivice the product of the tens of the root by the units, phis the square of the units. But the product of tens by units contain no order lower than tens, and hence the 86 tens in the 864, the difierence, contains ticice the product of the tens by the units. Hence, if the 86 tens be divided by twice the 5 tens of the root, the quotient, which is 8, will be the units' term of the root. If the 8 units be annexed to the 10 tens, used as a trial divisor, and the result, 108, be multiplied by 8, the product will be twice the prod- uct of the tens of the root by the units, plus the square of the units. 108X8 = 2 (5X8) + 82. Proof.— 58 X 58 = 3364. 14. What is the square root of 625 ? Of 4225 ? 15. What is the square root of 576? Of 7744? 16. What is the square root of 1444 ? Of 6241 ? 17. What is the square root of 3025? Of 7569 ? 18. What is the square root of 133225 ? Process. 19. What is the square root of 133225 j 365 210681? -9_ — 20. What is the square root of ^ ^.^ ^^ ^ of 419904? 66X6 = 396_ ^^^^^''- . 36X2 = 72) 3625 21. What is the square root of 725X5— 3^25 94249? Of 492804? : 254 COMPLETE ARITHMETIC. 22. What is the square root of 57600? Of 40960000? 23. What is tlie square root of Process. 10.4976? 10.4{)7t) I 3.24 24. What is the square root of 1— 176.89? ^^olTo-^'tA 25. What is the square root of. 6.2X2— _1.24_ 062'!^ 3.2X2 = 6.4) .257 6 .oozo . 6.44 X -04 = .2576 2^- What is the square root of ^~^ .451584? Of .008836? 27. What is the square root of 586.7? Suggestion. — Point thus 5^6.70, and carry the root to three deci- mal places by annexing periods of decimal ciphers. 28. What is the square root of 75.364? Of 5.493? 29. What is the square root of 263.85? Of 13467? 30. What is the square root of |ff ? Of -^^-fy? 31. What is the square root of 272^ ? Of 1040^^ 32. What is the square root of 2 ? Of 3 ? Of 5 1 _9 9 PRINXIFLES AND RULE. 409. Principles. — 1. The square root of a number contains as many orders as there are periods of two orders each in the number. 2. The left-hand period of a number contains the square of the first term of its square root. 3. The square of a number, composed of tens and units, is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units. 410. Rule. — To extract the square root of a number, 1. Begin at the units^ order and separate the number into periods of two orders each. 2. Find the greatest perfect square in the left-hand period, and place its square root at the right for the first or highest term of tJw root. 3. SubtmLct the square of the term of the root found from the SQUARE ROOT. 255 left-hand 'period, and to the difference annex the second period for a dividend. 4. Take twice the term of the root found for a trial divisor, and the dividend, exclusive of its right-hand figure, for a trial diviclend. The quotient {or the quotient reduced) will he the next term of the root. 5. Annex the second term of the root to the trial divisor, and multiply the residt by the second term, and subtract the product from the dividend. 6. Annex the third period to tJie remainder for the next divi- dend, and divide the same, exclusive of tlie right-hand figure, by twice the terms of the root found; and continue in like manner until all the periods are used. Notes. — 1. The left-hand period may contain but one order. 2. Twice the term or terms of the root, as the case may be, is called a trial divisor, since the next term of the root is obtained from the quotient. The term of the root sought is sometimes less than the quotient, since the dividend may contain a part of the square of the next term of the root. The true divisor is the trial divisor with the next term of the root annexed. 3. If the number is not a perfect square, the exact root can not be found. The exact root may be approximated by annexing periods of decimal ciphers. Since the square of no one of the nine digits ends with a cipher, the operation may be continued indefinitely. 4. In pointing off a decimal, or a mixed decimal number, begin with the order of units. If there be an odd number of decimal places, annex a decimal cipher. 5. When both terms of a common fraction are not perfect squares, the exact square root can not be found. An approximate root may be obtained by multiplying both terms of the fraction by tlie denom- inator, and extracting the root of the resulting fraction. Thus, A'l?. = V-7?^ = ]7,» "early. 40 ^ 402 40' 6. The square root of a perfect square may be found by resolving it into its prime factors, and taking the product of one of every two of those that are equal. GJ-eonietrical Explanation. 411. The area of a square surface is found by squaring the length of one side ; and, conversely, the length of the side is found by extracting the square root of the number (k noting the area. «* 256 COMPLETE ARITHMETIC. Let the annexed diagram represent a square surface whose area is 625. Required the length of one side. Since the number denoting the area contains two periods, there are two terms in the square root; and since the greatest square in the left-hand period is 4, the tens' term of the root is 2. (Art. 409.) Hence the length of the side of the square is 20 plus the units' term of the root. What is the units' term? ^ Taking from the given surface a square whose ¥iHe is 20 and whose area is 400, there remains a surface whose area is 625 — 400, or 225. This surface consists of two equal rect- angles, each 20 in length, and a small square, the length of whose side equals the width of each rectangle. What is the width of each rectangle ? Since the two rectangles contain most of the surface whose area is 225, their width may be found by dividing 225 by their joint length, which is twice 20, or 40. The quotient is 5, and hence the width of each rect- angle is 5, and their joint area is 40 X 5, or 200. Removing the two rectangles, there remains the small square, whose side is 5 and whose area I ^m is 25, the diiference between 225 and 200. Hence, 5 is the units' term of the root, and the length of the side of the square is 20 -f 5, or 25. Adding the area of the several parts, we have 202 _|_ 20 x 5 X 2 + 5^ = 400 ^—~ + 200 4- 25 =r 625. It is seen that the square whose side is 20, represents the square of the tens of the root; the two rectangles, twice the product of the tens by the units ; and the smaller square, tlie square of the units. sax 5 ji ^m ^m s^^ i^ 1 CUBE ROOT. 257 Note.— The entire length of the surface whose area is 225, is twice the side of the square removed, plus the side of the smaller square (20 X 2 + 5 = 45j, and this multiplied by 5 gives an area of 225. cu:be root. Th-e Division, of the ISTviiTiber into I*ex*iods. 412. The cubes of the smallest, greatest, and an interme- diate number, composed of one, two, and three orders, are as follows : V = 1 93 r= 729 43 = 64 103 := 1000 993 = 970299 44^ = 85184 1003 == 1000000 9993 = 997002999 444^ = 87528384 A comparison of the above numbers with their cubes shows that the cube of ar number contains three times as many orders as the number, or three times as many orders less two or less one. 413. Hence, if a number be separated into periods of three orders each, there will be as many orders in its cube root as there are periods in the number. 1. How many orders in the cube root of 91125? Suggestion. — First point off the number into periods of three orders each; thus, 91125. 2. How many orders in the cube root of 84604519? 3. How many orders in the cube root of 912673? 4. How many orders in the cube root of 48228544? 5. How many orders in the cube root of 2357947691 ? 414. The cubes of the smallest and greatest number of units, tens, and hundreds are as follows : V = 1 93 = 729 103 == 1000 903 ^ 729000 1003 ^ 1000000 9003 = 729000000 A comparison of the above numbers with their cubes shows that the cube of units gives no order higher than c.Ar.— 22 258 COMPLETE ARITHMETIC. hundreds ; that the cuhe of tens gives no order lower than thousands nor higher than hundred-thousands; and that the cube of hundreds gives no order lower than millions nor higher than hundred-millions. Hence, if a number be separated into periods of three orders each, tlw left-haiid period will contain the cube of the first term of the cube root; the first two left-hand periods will contain the cube of the first two terms of the cube root, etc. 6. What is the tens' terra of the cube root of 91125? 7. What is the tens' term of the cube root of 912673? 8. What is the hundreds' term of the cube root of 48228544? 9. AVhat is the first term of the cube root of 529475129? 10. What is the first term of the cube root of 257259456? WRITTEN- PROBLEMS. 11. What is the cube root of 262144? Process. ^ gj^^^ 262144 is composed of two pe- , 262144 [64 riods, its cube root will be composed of 6' =216 4 two orders (Art. 413). The left-hand 62 X 3 = 108 ) 461 44 period, 262, contains the cube of the tens' 64 '^ = 262144 term of the root (Art. 414). The greatest cube in 262 is 216, the cube root of which is 6 ; hence, 6 is the tens' term of the root. How is the units' term to be found? The cube of a number, composed of tens and units, is equal to the cube of the tens, plus three times the product of the square of the tens by the units, plus three times the product of the tens by the square of the units, plus the cube of the units (Art. 395). Hence, the difference between 262144 and the cube of the 6 tens of its cube root, is composed of three times the product of the square of the tens of its root by the units, plus three times the product of the tens by the square of the units, plus the cube of the units. But since the square of tens gives no order lower than hundreds (Art. 407), the 461 hundreds of the difference (46144) contains three times the product of the square of the tens by the units. Hence, if the 461 hundreds (rejecting the two right-hand figures) be divided by three CUBE ROOT. 259 times the square of the 6 tens of the root, the quotient, which is 4, will be the units' term of the root. Cube 64, and subtract the result from 262144. There is no remainder, and hence 64 is the cube root sought. Note. — Instead of cubing 64, the parts which compose the differ- ence, 46144, may be formed and added, thus: ^ 602X4X3 = 43200 60 X 42 X 3 = 2880 43= 64 46144 11. What is the cube root of 42875? Of 91125? 12. AVhat is the cube root of 117649? Of 185193? 13. What Is the cube root of 274625? Of 405224? 14. What is the cube root of 704969? Of 912673? 15. What is the cube root of 48228544? Process. 48228544 j 364, Cube root. '6^ =27 32 X 3 -- 27 ) 212 363 ^ 46656 74, Trial quotients. 362X3 = 3888) 15725 _ 364 '^ = 48228544 Since the two right-hand figures of each dividend are rejected, only the first figure of each period need be brought down and an- nexed to the difference. The quotient obtained by dividing 212 by 27 is 7, which is too large for the second term of the root, since the cube of 37 is more than 48228, the first two periods. The second difference is found by subtracting the cube of 36, the first two terms of the root, from 48228, the first two periods of the number. 16. What Is the cube root of 3048625 ? Of 34328125 ? 17. What is the cube root of 41063625 ? Of 43614208 ? 18. What is the cube root of 27270901 ? Of 515849608 ? 19. What is the cube root of 185193? 128024064? 20. What is the cube root of 103823? Of 27054036008? 21. What is the cube root of 15.625 ? Of .074256? 22. What is the cube root of 97.336 ? Of .015625 ? 260 COMPLETE ARITHMETIC. 23. What is the cube root of 56.47 ? Of 12.3456? Suggestion. — Point from units' order, and fill decimal periods, thus: 0(3.470, and 12.345G06. 24. What is the cube root of .000042875 ? Of 67.917312 ? 25. What is the cube root of 9 ? Of 31 ? Of 50 ? Suggestion. — Annex periods of decimal ciphers and carry the root to three decimal places. 26. What is the cube root of 2 ? Of 20 ? Of 200 ? 27. What is the cube root of yVA" '^ ^^ Hrh ? 28. What is the cube root of llff ? Of 37^^? 29. A cubical box contains 19683 cubic inches : what is the length of its edge ? 30. A block of granite in the form of a cube, contains 41063.625 cubic inches : what is the length of its edge? 31. A cubical bin holds 100 bushels : what is the length of its edge ? 32. If 6 ranks of wood, each 128 ft. long, 3 ft. wide, and 6 ft. high, were piled together in the form of a cube, what would be the height of the pile ? PRINCIPLES AND RULE. 415. Principles. — 1. The cube root of a number contains as many orders as there are periods of three figures each in the number. 2. The left-hand period of a number contains the cube of the first term of its cube root; the tivo left-hand periods contain the cube of the first two terms of the cube root, etc. 3. The cube of a number, composed of tens and units, is equal to the cube of the tens, plus three times the product of the square of the tens by the units, plus three times the "product of the tens by the square of the units, plus the cube of the units. 416. Rule. — To extract the cube root of a number, 1. Begin at the units' order and separate tJie number into periods of three orders each. 2. Find the greatest cube in the left-hand period, and place its cube root at tlie right for Uie first term of tlie root. CUBE ROOT. 261 3. Subtract the cube of the first term of the root from the left- hand period, and to the difference annex tJie first figure of the next period for a dividend. 4. Take three times the square of the first term of the root for a trial divisor, and the quotient for the second term of the root. Cube the root now found, and, if the residt is not greater than the two left-hand periods, subtract, and to the difference annex the first figure of the next period for a second dividend. If the cube of the root found is greater than the two left-hand periods, diminish the second term of the root. 5. Take three times the square of the two terms of the root found for a second trial divisor, and the quotient for the third term of the root. Cube the three terms of the root found, and subtract the result from the three left-hand periods, and continue the operation in like manner until all the terms of the root are found. Notes. — 1. Tlie quotient obtained by dividing the dividend by the trial divisor may be too large, since three times the square of the next figure of the root may be a part of the dividend. Usually the term of the root sought is the quotient, or one less than the quotient. 2. When a dividend does not contain the trial divisor, write a cipher for the next term of the root. Take three times the square of the root thus formed for a trial divisor, and to the dividend annex tlie two remaining figures of the period, and the first figure of the next period for a new dividend. 3. If the number is not a perfect cube, the root may be approx- imated by annexing periods of decimal ciphers, thus adding decimal terms to the root. Sufficient accuracy is usually secured by continuing the root to two or three decimal platfes. 4. When both terms of a common fraction are not perfect cubes, the cube root may be found approxiiilately by multiplying both terms of the fraction by the square of the denominator, and extracting the root of the resulting fraction. The error Avill be less than one di- vided by the denominator of the root. 5. The above methods of extracting the square or cube root of numbers, is a general method by Avhich any root may be extracted. The fourth root, for example, is found by dividing the number into periods of four figures each, then taking the fourth root of the left- hand period for the first term of the root, four times the cube of this first term for a trial divisor, and the remainder with the first term of the next period annexed, for a dividend, etc. 6. The cube root of a perfect cube may be found by resolving it into its prime factors and taking the product of one of every three of those that are equal. 262 COMPLETE ARITHMETIC. Greoinetrical Explanation of th.e Process of Ex- tracting th.e Cu-be Root. 417. The solid contents of a cube are found by cubing the length of its edge, and, con- versely, the length of the edge is found by extracting the cube root of the number denoting the solid contents. Let the annexed cut represent a cube whose solid contents are 15625. Required the length of the edge. Since the number denoting the solid contents contains two periods, there will be two terms in the cube root, and since the greatest cube in the left-hand period is 8, the tens' term of the root i.s 2 (Art. 414). Hence, the length of the edge of the cube is 20 plus the units' term of the root. What is the units' term ? Process. 15625^25 2^=_8_ 6 20^X3 = 1200 ) 7625 20^X5X3= 6000 20 X 52 X 3 = 1500 53 = 125 " 7625 Taking from the given cube a cube whose edge is 20 and whose capacity is 8000, there remains a solid whose capacity is 15625 — 8000, which is 7625. An inspection of the an- nexed cut shows that this solid contains Hiree equal rectan- CUBE ROOT. 263 gular solids, whose inner face (20 2) is equal to the face of the removed cube and whose thickness equals the units' term of the root. What is the thickness of each of these rectangular solids? Since they compose only a part of the solid whose solid contents are 7625, their thickness can not be greater than the quotient obtained by dividing 7625 by the area of their joint inner faces, which is 20 ^ X 3, or 1200. The quotient is 6, which is at least one greater than the thickness of each of the three rectangular solids, since 26^ is greater than 15625, the solid contents of the given cube. Try 5 for the thickness. 25^ = 15625, and hence 5 is the required thick- ness, and the length of the edge of the given cube is 20 -j- 5, or 25. The correctness of this result may also be shown by find- ing the solid contents of the several parts of the given cube. The solidity of the cube removed is, as shown above, 20^ =: 8000. The joint solidity of the three adjacent rectangular solids is 202 X 5 X 3 = 6000. Removing these three rectan- gular solids, there remain three other rectangular solids, whose solidity is 20 X 5^, or 500 each, and whose combined solidity is 500 X 3, or 1500. Removing these three rectangular solids, there remains the small cube, whose solidity is 53 = 125. fYpI Adding the solidity of the sev- eral parts, we have 8000 + 6000 -f 1500 + 125 = 15625, which is the solidity of the given cube. ,, :::::::". It is seen that the cube whose edge is 20, represents the cube of the tens of the root; the three ad- 'V-' 264^ COxMPLETE ARITHMETIC. jacent rectangular solids represent three times the product of (lie square of tens by the units; the three smaller rectangular solids, three times the product of the tens by the square of the units; and the smaller cube, tJie cube of the units. APPLICATIONS OF INVOLUTION AND EVOLU- TION. I. TO THE KIGHT-ANGLED TRIANGLE. 418. The HypOtemise of a right-angled triangle is the side opposite the right angle. The other two sides are called the Base and the Perpendicidar. (Art. 155.) 419. Principles. — 1. The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the otJwr two sides. This principle, which may be proven by geometry, is illustrated by the annexed diagram. 2. The square of the base or the per- pendicular of a right-angled triangle is equal to the square of the hypotenuse less the square of the other side. (J PROBLEMS. 1. The base of a right-angled triangle is 8, and the per- pendicular 6 : what is the length of the hypotenuse ? Solution. — Since the square of the hypotenuse equals the square of the base plus the square of the perpendicular, the hypotenuse equals VS^ -]- 6^ =■- Vl^ = 10. 2. The hypotenuse of a right-angled triangle is 20 inches and the base is 16 inches: what is the perpendicular? 3. The hypotenuse of a right-angled triangle is 45 feet and the perpendicular is 27 feet : what is the base ? APPLICATIONS OF INVOLUTION AND EVOLUTION. 265 4. A rectangular field is 192 yards long and 144 yards wide : what is the lengtli of the diagonal ? 5. The foot of a ladder is 18 feet from the base of a building, and the top reaches a window 24 feet from the base : what is the length of the ladder? 6. Two boys start from the same point, and one walks 96 rods due north, and the other 72 rods due east : how far are they apart? 7. A flag pole 180 feet high casts a shadow 135 feet in length : what is the distance from the top of the pole to the end of the shadow? 8. A boy in flying his kite let out 240 feet of string, and the distance from where he stood to a point directly under the kite was 208 feet : how high was the kite, supposing the string to be straight? 9. A rectangular field is 84 rods long and 63 rods wide : what is the side of a square field of the same area? 10. A farm is 125 rods square, and a rectangular farm, containing the same number of acres, is 100 feet in length : what is its width ? 420. Rules. — 1. To find the hypotenuse of a right-angled triangle. Extract the square root of the sum of the squares of the other two sides. 2. To find the base or the perpendicular of a right-angled triangle. Extract the square root of the difference between the squares of the hypotenuse and the other side. II. TO THE CIRCLE. • 421. Principles. — 1. The area of a circle is equal to the square of its diameter midtiplied by .7854. 2. The areas of two circles are to each other as the squares of their diameters. Note. — The above propositions can be proven -by geometry. The area of a circle also equals its circumference multiplied by one fourth of its diameter. (Art. 161.) C.Ar.— 23. 266 COMPLETE ARITHMETIC. PROBLEMS. 11. The diameter of a circle is 15 inches : what is its area? 12. A circular pond is 100 feet in diameter : how many- square yards does it contain ? 13. A circular room has an area of 78.54 square yards: what is its diameter? 14. How many circles, each 3 inches in diameter, will equal in area a circle w^hose diameter is 2 feet? 15. How many circles, each 15 inches in diameter, will equal in area a circle whose diameter is 5 feet? 16. A horse, tied to a stake by a rope, can graze to the distance of 40 feet from the stake: on how much surface can it graze ? 17. A horse, tied to a stake, can graze on 218^ square yards of surface : to what distance from the stake can it graze ? 18. How many circles, each 3 inches in diameter, contain the same area as a surface 2.5 feet square? III. TO THE SPHERE. 422, Principles. — 1. The surface of a sphere is equal to the square of the diameter multiplied by 3.1416. 2. The solidity of a sphere is equal to the cube of the diameter multiplied by .5236. 3. Two spheres are to each other as the cubes of their diame- ters. Note. — The surface of a sphere may also he found bij multiplying the circumference by the diameter; and the solidity by multiplying the surface by one sixth of the diameter. (Arts. 474, 475.) PROBLEMS. 19. What is the surface of a sphere whose diameter is 1 inches ? 20. How many square miles on the surface of the earth, its mean diameter being 7912 miles? APPLICATIONS OF INVOLUTION AND EVOLUTION. 267 21. How many cubic miles in the solidity of the earth ? 22. How many cubic inches in a cannon ball whose diameter is 7 inches? 23. How many balls 2 inches in diameter, equal in solid- ity a ball whose diameter is 8 inches? 24. The diameter of the earth is about 4 times the diam- eter of the moon : how many times larger than the moon is the earth? 25. The diameter of Jupiter, the largest planet, is about 85000 miles, and the diameter of the sun is about 850000 miles : how many times larger than Jupiter is the sun ? 26. The surface of the planet Mercury contains about 28274400 square miles : what is its diameter ? 27. The planet Uranus contains about 18816613200000 cubic miles : what is its diameter ? Suggestion.— Divide the solidity by .5236, and extract the cube root of the quotient. 28. A brass ball contains 904.7808 cubic inches: what is its diameter? 29. A square and a triangle contain an equivalent area, and the base of the triangle is 36.1 inches, and its altitude is 5 inches : what is the side of the square ? 30. One of the mammoth pines of California is 110 feet in circumference : what is its diameter ? 31. How many cubic feet in a portion of the above tree 100 feet in length, supposing its mean circumference to be 94ifeet? 32. The mean distance of the earth from the sun (new value) is about 91400000 miles, and it revolves in its orbit in 365^ days : what is its mean hourly motion ? 33. The mean distance of Mercury from the sun (new value) is about 35400000 miles, and it revolves in its orbit in 87.9 days: what is its mean hourly motion? 34. The diameter of the moon is about 2000 miles : how does the extent of the moon's surface compare with that of the earth? 268 COMPLETE ARITHMETIC. SECTION XVI. GENERAL REVIEW. Note. — The following problems are selected from several sets used in the examination of pupils for promotion to high schools, and in the examination of teachers. MENTAL PKOBLEMS. 1. If 3 apples are worth 2 oranges, how many oranges are 24 apples worth? 2. How long will it take a man to lay up $60, if he earn $15 a week and spend $9? 3. i of 74| is J of what number? 4. "I of 45 is f of how many times 10? 5. A has 20 cents ; and | of what A has is f of what B has: how many has B? 6. If -J of a yard of cloth cost 63 cents, what will f of a yard cost? 7. If 3 yards of muslin cost 13|^ cents, what will f of a yard cost? 8. The difference between f and | of a number is 10 : what is the number? 9. What fraction is as much greater than f as f is less? 10. A piece of flannel lost | of its length by shrinkage in fulling, and then measured 30 yards : what was its length before fulling? 11. A horse cost $90, and -j\ of the price of the horse equals f of 3 times the cost of the saddle : what did the saddle cost? 12. If to my age you add its half, its third, and 28 years, the sum will be three times ray age : what is my age ? 13. A boy being asked his age, said that f of 80 was f of 10 times his age: what was his age? GENERAL REVIEW. 269 14. A boy gave f of his money for a sled, ^ of it for a hat, and then had 7 cents left: how many cents had he at first? 15. I of my money is in my purse, | in my hand, and the remainder, which is 25 cents, is in my pocket: how much money have I ? 16. A boy having f of a dollar, gave |- of his money to John and ^ of the remainder to James : what part of a dollar did James receive? 17. A farmer sold f of his sheep and then bought f as many as he had left, when he had 40 sheep : hoAV many had he at first? 18. John lost f of his money and spent ^ of the re- mainder, and then had only 10 cents: how much money had he at first? 19. A man sold a horse for $60, which was |^ of f of its cost: how much was lost by the bargain? 20. A man sold a horse for $130, which was f more than it cost him : what was the cost of the horse ? 21. A sold B a horse for -J- more than its cost, and B sold it for $80, losing |- of its cost : how much did A pay for the horse ? 22. At $f a bushel, how many bushels of corn may be bought for $8? 23. If J of a bushel of wheat cost $|, what part of a bushel can be bought for $|? 24. If $18f will purchase f of a load of corn, what part of it will $16| purchase? 25. If 2-}y pounds of cheese cost 3| dimes, what part of a pound can be bought for 1 dime ? 26. How many bushels of coal at 12^^ cents a bushel can be bought for $15 ? 27. What part of 7 bushels is | of a peck ? 28. What part of a pound of gold is .25 of an ounce? 29. What part of | of a gallon is | of a pint ? 30. From f of a day take -J of an hour. 31. If a staff 5 feet long cast a shadow 2 feet long at 12 270 COMPLETE ARITHMETIC. o'clock, what is the height of a steeple whose shadow, at the same hour, is 80 feet? 32. If a five cent loaf weigh 10 ounces when flour is $4 a barrel, what ought it to weigh when flour is $5 a barrel? 33. If 20 bushels of oats will feed 40 horses 80 days, how long will 180 bushels feed them ? 34. If a horse eat 2 bushels of oats in 6 days, in how many days will 2 horses eat 18 bushels ? 35. If 3 men can mow 18 acres of grass in 4 days, how many men can mow 9 acres in 3 days? 36. A garrison of 20 men is supplied with provisions for 12 days : if 12 men leave, how long will the provisions serve the remainder? 37. A man bought a watch and chain for $80, and the chain cost ^ as much as the watch: how much did each cost? 38. A has 1^ times as many cents as B, and they together have 40 cents : how many has each ? 39. A pole 120 feet high fell and broke into two parts, and f of the longer part was equal to the shorter : how long was each part? 40. A and B together own 824 sheep, and A has If times as many as B : how many has each ? 41. A, B, and C rent a pasture for $42 ; B pays half as much as A, and C half as much as B : what does each pay? 42. A and B own a farm ; A owns f as much as B, and B owns 40 acres more than A: how many acres does each own? 43. f of A's money is | of B's, and f of B's is | of C's, which is $81 : how much have A and B each ? 44. If a man can reap f of an acre of wheat in a day, how much can 6 men reap in 10 days? 45. A makes a shoe in | of a day ; B makes one in f of a day: how many shoes can both make in a day? 46. A can mow an acre of grass in f of a day, and B in •| of a day : how long will it take both together to mow an acre ? GENERAL REVIEW. 271 47. A can mow a field of grass in 5 days, and B in 4 days: how long will it take both, working together, to mow it? 48. A can build a house in 20 days, but, with the assist- ance of C, he can do it in 12 days : in what time can C do it alone ? 49. A alone can build a certain wall in 6 days, B alone in 10 days, and C alone in 15 days : in how many days can they all together build it? 50. A, B, and C can do a job in 20 days ; A and B can do it in 40 days ; and A and C in 30' days : in how many days can each do it alone ? 51. A broker bought rail-road stock at 80 and sold it at 70 : what per cent, did he lose ? 52. A broker bought stock at 70 and sold it at 90 : what per cent, did he gain ? 53. A merchant bought 40 yards of cloth for $90 : at how much a yard must he sell it to gain 33^ per cent. ? 54. For how much must tea costing 90 cents, be sold to gain 12^ per cent. ? 55. A man bought a hat for $5 and sold it for $6 : what per cent, did he gain ? 56. I sell cloth at $2.50 a yard and gain 25 per cent.; for how much must I sell it to lose 20 per cent. ? 57. A man earned a certain sum of money, and, after adding to it $12.50, found that what he then had was 133J per cent, of what he earned : how much did he earn ? 58. A man sold a watch for $90, and gained 50 per cent. : what per cent, would he have gained if he had sold it for $75? 59. f of the price received for an article is equal to f of its cost : what is the gain per cent. ? 60. Two men, A and B, engaged in trade with different capitals ; A lost 33^ per cent, of his capital, and B gained 50 per cent, on his, when each had $600: with what capital did each begin trade? 61. How much grain must I take to mill to bring away 2 bushels after the miller has taken 10 per cent, for toll ? J 272 COMPLETE ARITHMETIC. 62. At what rate per cent. , simple interest, will $1 double itself in 8 years? 63. The interest on a certain sum for 4 years was ^ the sum : what was the rate per cent. ? 64. Two men start from two places 495 miles apart, and travel toward each other ; one travels 20 miles- a day, and the other 25 miles a day : in how many days will they meet? 65. A owes f of B's income, but, by saving -^ of B's income annually, he can pay his debt in 5 years, and have $50 left: what is B's income? 66. C and D are traveling in the same direction ; C is 18 miles ahead of D, but D travels 7 miles while C travels 4 : how many miles from the place of starting will D have traveled when he overtakes C? 67. If a man traveling 14 hours a day, perform half a journey in 5 days, how long will it take to perform the other half, if he travel 10 hours a day ? 68. If a man can do a piece of w^ork in 9f days, working 8 hours a day, how long will it take, if he work 6 hours a day? 69. A is 20 years old ; the sum of the ages of B and C equals 4 times A's age ; C's age is ^ of A's and B's to- gether : what is the age of each ? 70. A hare is 30 rods before a hound, but the hound runs 7 rods while the hare runs 5 : how far must the hound run to catch the hare? 71. A hare starts 50 leaps before a hound, and leaps 4 times while the hound leaps 3 times ; but 2 of the hound's leaps equal 4 of the hare's : how many leaps must the hound take to catch the hare? 72. If a steamer sails 9 miles an hour down stream, and 5 miles an hour up stream, how far can it go down stream and back again in 14 hours? 73. A steamer sails a mile down stream in 5 minutes, and a mile up stream in 7 minutes : how far down stream can it go and return again in one hour? GENERAL REVIEW. 273 74. A pipe will fill a cistern in 4 hours, and another will empty it in 6 hours : how long will it take to fill it when both pipes run ? 75. At what time between six and seven o'clock are the hour and minute hands of a clock together ? ^ WRITTEN PROBLEMS. 76. The minuend is 1250, and the remainder 592 : what is the subtrahend ? 77. The quotient is 71, the divisor 42, and the remainder 15 : what is the dividend ? 78. If a certain number be multiplied by 22, and 64 be subtracted from the product, and the remainder be divided by 4, the quotient will be 50: what is the number? 79. What will be the cost of 3760 lbs. of hay, at $8.50 a ton? 80. At $24.50 per acre, how many acres of land can be bought for $3560.75? 81. Add I, I, 1 of I, and | of 2i 82. From 17^ take f of 6 J, and multiply the remainder by |. 83. Multiply f of f by |^ of f, and divide the product by A. 84. Divide f of 6-1 by | of 7^. 85. What number multiplied by 28f will produce 145? 86. From the sum of 215| and 125 J take their differ- ence. 87. Multiply f -f I of -I by I - 1 of f . 88. Divide 2^ by 3^, and multiply the quotient by 3J. 89. What must 8|f be multiplied by that the product may be 3? 90. A man bought ^^ of a section of land for $2880, and sold f of it at $10 an acre, and the rest at $12 an acre: how much did he gain? 91. A merchant owning |- of a ship sells -f of his share for $16800 : what is the value of the whole ship, at this rate, and what part of the ship has he left? 274 COMPLETE ARITHMETIC. 92. Add 9 thousandths, 3 hundredths, and 7 units. 93. From 15 ten-thousandths take 27 milliouths, and multiply the difference by 20.5. 94. Multiply 160 by .016, and divide the product by .0025. - 95. Multiply 15 thousandths by 15 hundredths, and from the product take 15 millionths. 96. Divide 256 thousandths by 16 millionths. 97. Multiply 625 by .003, and divide the result by 25. 98. Change ^|^ to a decimal, and divide the result by 2^. 99. Change yf^- to a decimal, and divide the result by 5000. 100. Reduce .625 of a pound Troy to lower integers. 101. What decimal of a rod is .165 of a foot? 102. What will 63 thousandths of a cord of wood cost, at S2.25 per cord? 103. How many minutes will there be in the month of February, 1880? 104. How many seconds are there in the three summer months ? 105. How many steps, 2 ft. 4 in. each, will a person take in walking 2\ miles? 106. How many times will a wheel, 12 ft. 6 in. in cir- cumference, turn round in rolling one mile? 107. How many acres in a street 4 rods wide and 2J miles long? 108. How many yards of carpeting, f of a yard wide, will it take to cover a parlor, 18^ feet long and 15 feet wide ? 109. How many grains in 14 ingots of silver, each weigh- ing 27 oz. 10 pwt.? 110. How many square feet of lumber in 40 boards, each 12 feet long and 7^ inches wide? 111. AVhat will a board 20 feet long and 9 inches wide cost, at $30 a thousand? 112. What will it cost to lay a pavement 36 feet long and 9 feet 6 inches wide, at 40 cents a square yard? GENERAL REVIEW. 275 113. A pile of wood, containing 10 cords, is 20 feet long and 8 feet wide : how high is it ? 114. What is the value of a pile of wood 40 feet long, 8 feet wide, and 5^ feet high, at $5.30 a cord? 115. How many sacks, holding 2 bu. 3 pk. 2 qt. each, can be filled from a bin containing 366 bu. 3 pk. 4 qt. of wheat ? 116. A lady bought 6 silver spoons, each weighing 3 oz. 3 pwt. 8 gr., at $2.25 an ounce, and a gold chain, weighing 14 pwt., at $1.25 a pwt. : what was the cost of both spoons and chain? 117. How many bricks will it require to build a wall 2 rods long, 6 feet high, and 18 inches thick, each brick being 8 inches long, 4 inches Avide, and 2^ inches thick? 118. Cincinnati is 7° 49' west of Baltimore: when it is noon at Baltimore, what is the hour at Cincinnati? 119. New York is 75 degrees of longitude west of Lon- don : when it is noon at New York what is the hour at London ? 120. Boston is 71° 4' 9" W. longitude, and Cleveland is 81° 47' W. : when it is 4 P. M. at Cleveland, what is the hour at Boston ? 121. What part of a rod is 2 ft. 9 in.? 122. Reduce 5 fur. 8 rd. to the decimal of a mile. 123. Reduce f of a square yard to the fraction of an acre. 124. From | of a pound Troy take f of an ounce. 125. Reduce f of a quart to the fraction of a bushel. 126. A regiment lost 8 % of its men in a battle, and 25% of those that remained died from sickness, and it then mustered 621 men: how many men were in the regiment at first? 127. A quantity of sugar was bought for $150, and sold for $167.50: what was the gain per cent.? 128. A merchant bought 500 yards of cloth for $1800 : for how much a yard must he sell it to gain 25 % ? 129. A man sold a piece of cloth for $24, and thereby 276 COMPLETE ARITHMETIC. lost 25 % ; if he had sold it for $34, wouhl he have gained or lost, and what per cent. ? 130. I sold goods at 20 ^ gain, and, investing the pro- ceeds, sold at 20 % loss : did I gain or lose by the opera- tion, and what per cent. ? 131. Sold 2 carriages, at $240 apiece, and gained 20 % on one and lost 20 % on the other : how much did I gain or lose in the transaction? 132. A man bought a horse for $72, and sold it for 25 % more than cost, and 10 % less than he asked for it : what did he ask for it? 133. A merchant marked a lot of goods, costing $5800, at 30 % above cost, but sold them at 10 % less than the marked price : how much and what per cent, did he gain ? 134. What must I ask for cloth, costing $4 a yard, that I may deduct 20 % from my asking price and still make 20%? 135. A man bought stock at 25 % below par and sold it at 20 % above par : what per cent, did he make ? 136. A fruit dealer lost 33J per cent, of a lot of apples, and sold the remainder at a gain of 50 ]3er cent. : required the per cent, of gain or loss. 137. I bought 63 kegs of nails, each keg containing 100 lbs., at 4|- cents a pound, and sold \ of them for what \ of them cost: what per cent, did I lose on the part sold? 138. I bought $128.25 worth of goods; kept them on hand 6 months when money was worth 8 % interest, and then sold them at a net gain of 6 % : for how much were they sold? 139. When money was worth 9% interest, I bought $800 worth of goods, kept them 4 months and then sold them for $959. 10 : what per cent, on the cost did I gain ? 140. A house valued at $3240 is insured for | of its value, at f % : what is the premium ? 141. I pay $19.20 premium for insuring my house for \ of its value, at 1^ % '- what is the value of my house? GENERAL REVIEW. 277 142. A capitalist sent a broker $25000 to invest in cotton, after deducting his commission of 2^%: how much cotton, at 5 cents a pound, did the broker purchase ? 143. x\n agent received $502.50 to purchase cloth, after deducting | % commission : how many yards did he buy at $1.25 a yard? 144. What is the interest of $125.50 for 7 months and 10 days, at 7 % ? 145. What is the interest of $50000 for one day, at 8%? 146. What is the interest of $75.50 from June 12, 1869, to Aug. 6, 1870, at 71 % ? 147. A man loaned $800 for 2 years and 6 months, and received $90 interest : what was the rate per cent. ? 148. At what rate per cent, will $311.50 amount to $337.40 in 1 yr. 4 mo. ? 149. What sum of money will yield as much interest in 3 years, at 4^ per cent., as $540 yields in 1 yr. 8 mo., at 7%? 150. The amount of a certain principal- for 3 years, at a certain rate per cent., is $750, and for 5 years $1250: what is the principal, and what is the rate per cent. ? 151. A note for $500, dated Oct. 8, 1864, and bearing interest at 9 %, is indorsed as follows: Nov. 4, 1865, $30; Jan. 30, 1866, $250. What was due July 1, 1866? 152. What is the present worth of a note of $1320, due in 3 years and 4 months, without interest, money being worth 6 % ? What is the discount ? 153. What is the true discount of $236, due in 3 years, at 6 % ? 154. What is the bank discount on $125, j)ayable in 90 days, at 8 % ? 155. AVhat is the difference between the true discount and the bank discount of $359.50, for 90 days, without grace, at 12%? 156. For what sum must I give my note at a bank, pay- able in 4 months, at 10^, to get $300? 157. I borrow of A $150 for 6 months, and afterward I 278 COMPLETE ARITHMETIC. lend him $100: how long may he keep it to balance the use of the sum he lent me? 158. A owes B $800, of which $50 is due in 2 months, $100 in 5 months, and the remainder in 8 months : what is the equated time for the whole sum? 159. A man owes $300 due in 5 months, and $700 due in 3 months, and $200 due in 8 months : if he pays ^ of the whole in 2 months, when ought the other half to be paid? 160. I have sold 50 bushels of wheat for A, and 60 bushels for B, receiving $150 for both lots: if A's wheat is worth 20 % more than B's, how much ought I to pay each ? 161. Two men divided a lot of wood, which they pur- chased together for $27; one took 5^ cords, the other 8: what ought each to pay? 162. If 8 men cut 84 cords of wood in 12 days, working 7 hours a day, how many men will cut 150 cords in 10 days, working 5 hours a day? 163. If 16 horses consume 84 bushels of grain in 24 days, how many bushels of grain will supply 36 horses 16 days? 164. If the wages of 24 men for 4 days are $192, what will be the wages of 36 men for 3 days? 165. If 4 men in 7f days earn $53f , how much will 7 men earn in :^ of a day ? 166. A and B traded in company and gained $750, of which B's share was $600; A's stock was $1200: what was B's stock? 167. A and B formed a partnership for 1 year, and A put in $2000 and B $800 : how much more must B put in at the close of 6 months to receive one-half of the profits? 168. A and B engage in trade; A puts in $200 for 5 months, B $300 for 2 months ; they draw out capital and profits to the amount of $1389: what was each man's share? 169. What is the square root of 41616? Of 420.25? 170. What is the cube root of 46656? Of 42.875? 171. A certain window is 30 feet from the ground: how far from the base of the building must the foot of a ladder 50 feet long be placed to reach the window? GENERAL REVIEW. 279 172. Two men start from the same point; one travels 52 miles north and the other 39 miles west: how far are they apart? 173. A house is 40 feet high from the ground to the eaves, and it is required to find the length of a ladder which will reach the eaves, supposing the foot of the ladder can not be placed nearer to the house than 30 feet. 174. How many rods of fence will inclose 10 acres in the form of a square ? 175. A floor is 24 feet long and 15 feet wide: what is the distance between tw^o opposite corners? 176. A room is 20 feet long, 16 feet wide, and 12 feet high : what is the distance from one of the lower corners to the upper opposite corner? 177. How many cubic feet in a stone 8 feet long, 5^ feet wide, and 3^ feet thick? 178. How many square feet on the surface of a stone 6 feet long, 4 feet wide, and H feet thick? 179. There is a circular field 40 rods in diameter : w^hat is its circumference? How many acres does it contain? 180. The area of a circle is 470.8f square inches : what is the length of its diameter? 181. How many iron balls 2 inches in diameter, will weigh as much as an iron ball 8 inches in diameter? 182. How many cubical blocks, each edge of which is -J- of a foot, are equivalent to a block of wood 8 feet long, 4 feet wide, and 2 feet thick? 183. How many bushels of wheat will fill a bin 8 feet long, 5 feet wide, and 4 feet deep? 184. How many gallons of w^ater will a cistern contain which is 7 feet long, 6 feet wide, and 11 feet deep? 185. How many gallons of water will fill a circular cis- tern 6 feet deep and 4 feet in diameter? 186. Divide $1000 among A, B, and C, and give A $120 more than C, and C $95 more than B. 187. A can mow 2 acres in 3 days, and B 5 acres in 6 days : in how many days can they together mow 9 acres ? 280 COMPLETE ARITHMETIC. 188. A sold cloth to B and gained 10 per cent. ; B sold it to C and gained 10 per cent. ; C sold it to D for $726 and gained 1 per cent. : how much did it cost A ? 189. A man steps 2 feet 8 inches, and a boy 1 foot 10 inches; but the boy takes 8 steps while the man takes 5: how far will the boy walk while the man walks 3 J miles ? 190. A father bequeathed j\ of his estate to his eldest son, f of the remainder to his second son, and the rest to his youngest son ; by this arrangement the eldest's share was $1300 more than the youngest's: what was the share of each son ? 191. If 7 bushels of wheat are worth 10 bushels of rye, and 5 bushels of rye are worth 14 bushels of oats, and 6 bushels of oats are Avorth $3.13, how many bushels of wheat will $50 buy? 192. In a company of 90 persons there are 4 more men than women and 10 more children than men and women together : how many of each in the company ? 193. Divide $630 among 3 persons so that the second shall have f as much as the first, and the third -^ as much as the other two together. 194. A and B can do a piece of work in 12 days, B and C in 9 days, and A and C in 6 days : how long will it take each alone to do it? 195. A and B perform together -^j^ of a piece of work in 2 days, when, B leaving, A completes it in ^ a day: in what time can each do it alone? 196. C and D engage in trade with different sums of money ; C loses 40 per cent, of his capital, and D gains 57 per cent, on his, when their capitals are equal : how much greater was C's capital than D's when they began business ? 197. A man walks 100 miles in 2 days, and ^ of the distance walked the first day, added to ^ the distance walked the second day, equals half the distance walked the first day : how far did he walk each day ? 198. How far from the end of a stick of timber 30 feet long, of equal size from end to end, must a handspike be TEST QUESTIONS. 281 placed so that 3 men, 2 at the handspike and 1 at the end of the stick, may each carry ^ of its weight ? 199. Two trees stand on opposite sides of a stream 40 feet wide ; the height of one tree is to the width of the stream as 8 is to 4, and the width of the stream is to the height of the other as 4 is to 5: what is the distance between their tops? 200. A cistern is filled by two pipes, one of which will fill it in two hours, and the other in 3 hours ; it is emptied by three pipes, the first of which will empty it in 5 hours, the second in 6 hours, and the third in 7^ hours : if all the pipes be left open, in what time will it be filled? TEST QUESTIONS. 1. What is a number? In how many ways may numbers be repre- sented? Name them. 2. What is the difference between numeration and notation? Be- tween the Arabic notation and the Roman? Between the English numeration and the French ? 3. What is the simple value of the figure 5 in 452? What is its local value? How is the local value of a figure affected by its re- moval one order to the left? One order to the right? How is the value of a number affected by annexing a cipher? Why? 4. How many units are there in the sum of two or more integers? Why in addition are like orders of figures written in the same column ? Why in adding numbers do we begin at the right hand? 5. Why are the minuend, subtrahend, and difference like numbers? Show that the adding of 10 to a term of the minuend, and 1 to the next higher term of the subtrahend, increases the minuend and sub- trahend equally. 6. Why must the multiplier be an abstract number? When the multiplicand is concrete, what is true of the product ? Why ? How is the product affected by dividing either the multiplicand or the multiplier? Show that annexing two ciphers to an integer, and dividing the result by 4, gives the product of the number multiplied by 25. 7. What kind of number is the quotient when both divisor and dividend are like numbers? When the divisor is abstract and the dividend concrete? What is the difference between short division and long division? Of what order is the first left-hand figure of the quotient? C.Ar.-24. 282 COMPLETE ARITHMETIC. 8. How is the quotient affected by multiplying or dividing both dividend and divisor by the same number? By multiplying the divi- dend by any number greater than 1 ? On what principle may the four fundamental rules be reduced to two? 9. Name all the prime numbers from 1 to 20 inclusive. Show that two composite numbers may be prime with respect to each other. What is meant by the factors of a number? The prime factors? Show that the common factor of two or more numbers is a factor of their sum. 10. Why is the factor of a number its divisor? How is a number affected by the canceling of a factor? On what principle may the common factors of a dividend and a divisor be canceled ? 11. When is a divisor a common divisor? What is the greatest common divisor of two or more numbers? Show that the common divisor of two numbers is a divisor of their sum and diff'erence. In how many ways may the greatest common divisor of two or more numbers be found? 12. How many multiples has every number? What is a common multiple? What is the least common multiple of two or more numbers? In how many ways may the least common multiple of two or more numbers be found ? 13. What is the difference between a divisor and a multiple of a number ? Between the terms factor, divisor, and measure ? Is 2i a divisor of 5? Is 12J a multiple of 5? Is 12} a multiple of 6]-? 14. What is a fraction? In what two ways may a fi'action be ex- pressed? When a fraction is expressed by words, which word or words denote the denominator? 15. What is the difference between the unit of a fraction and a frac- tional unit? Which term of a fraction denotes the size of the fractional unit? When is the value of a fraction equal to 1? Greater than 1? Less than 1 ? 16. Show that the division or multiplication of both terms of a fraction by the same number, does not change its value. How is the value of a proper fraction affected by adding the same number to both of its terms? By subtracting the same number from both of its terms? 17. On what principle is a fraction reduced to lower terms? To higher terms? On what principle are two or more fractions reduced to a common denominator? 18. In what two ways may a fraction be multiplied by an integer? Why? In what two ways may a fraction be divided by an integer? In what three ways may a fraction be divided by a fraction? Wiiy must fractions have a common denominator before they can be added or subtracted ? TEST QUESTIONS. 283 19. What is a decimal fraction? Is the fraction fifteen-hundredtks a decimal fraction? In Avhat two Avays may it be expressed by figures? Which is called the decimal form? What is the denominator of a decimal fraction? 20. What is meant by decimal places? What is the name of the third decimal order from units? The sixth? The ninth? How is a decimal read ? 21. How is the local value of a decimal figure affected by its re- moval one order to the right ? One order to the left ? How is the value of a decimal affected by annexing decimal ciphers? Why? By prefixing decimal ciphers? Why? 22. Plow is a decimal reduced to a common fraction? A common fraction to a decimal? Wliy can decimals be added and subtracted like integers? 23. Why does the product contain as many decimal places as both multiplicand and multiplier? Why does the dividend contain as many decimal places as both divisor and quotient? 24. How is a decimal divided by 10, 100, etc.? How is a decimal multiplied by 10, 100, etc.? Why are numbers denoting sums of money added and subtracted like decimals? 25. What is a rectangle? How is its area found ? What is a circle? How is its area found? 26. What is a right-angled triangle ? How is its area found? 27. What is a rectangular solid ? What is the difference between an edge and a face of a solid? 28. Show that the product of the three dimensions of a rectangular solid represents its volume or solid contents. How are the contents of a cylinder found? 29. Is every concrete number denominate? Give examples. What is the difference between a simple denominate number and a compound number? Give examples. 30. What do denominate numbers express ? What is the difference between reduction descending and reduction ascending? 31. How are denominate fractions reduced from a higher to a lower denomination? From a lower to a higher? How is a denominate number reduced to the fraction of a higher denomination ? Give an example. 32. What is the Metric System ? What is the primary unit of the system? What is its length in inches? What is a liter? What is a gram. 33. How are the multiples of the meter, liter, and gram named ? How are the subdivisions named ? 34. What is the difference between simple addition and compound addition? In what respect are the processes alike? 284 COMPLETE ARITHMETIC. 35. When are compound numbers of the same kind ? Give exam- ples. How is a compound number divided by another of the same kind? 36. What part of tlie equator passes beneath the vertical rays of the Bun every hour? What part of the tropic of Cancer? What part of any parallel situated between the polar circles? 37. Why is the time of day earlier at New York than at St. Louis ? When the difference in longitude between two places is given, how is the difference in time found? 38. What is meant by 5 per cent, of a number? What is the dif- ference between the terms rate per cent, and rate? Give examples. 39. What four numbers are considered in percentage? Define each. Give the four cases of percentage and the formula for each. 40. What is the difference between the cost and the selling price of an article? Give the four formulas in profit and loss. 41. What is meant by commission? What is the difference between a factor and a broker ? Give the four formulas in commission and brokerage. 42. What is the difference between the market value and the par value of capital? When is capital at a premium? When is it at a discount? 43. What is the difference between a dividend and an assessment? How is the rate of dividend found ? 44. What is insurance ? What is fire insurance ? What is the premium? Give the formulas covering the four cases in insurance. 45. What is life insurance ? How is the premium computed ? What is a mutual insurance company ? 46. What is the difference between a poll tax and a property tax? How is a property tax assessed ? How is the rate of tax determined ? 47. What is an income tax? An excise tax? From what kind of taxes is the internal revenue of the United States derived ? 48. What are customs or duties? What is the difference between specific, duties and ad valorem duties ? What is a tariff? 49. What is interest? What is the rate of interest? 50. How is the interest of any principal for one year, at any rate per cent., found ? Give the formula for the general method of com- puting interest. Give the formula for the six per cent, method. 51. How many methods are there of finding the time between two dates? Which is called the method by days? 52. On what principle is the United States Rule for partial pay- ments based? What rule is used when a note runs less than a year? 53. What quantities are considered in interest? State the five problems in interest, and give the formula for each. TEST QUESTIONS. 285 54. What is discount? What is the difference between true dis- count and interest? Between true discount and bank discount? Be- tween bank discount and interest? 55. What is meant by days of grace? When does a note with grace become due? How is a note not drawing interest discounted by a bank ? How is a note drawing interest discounted ? 56. What is a promissory note ? What is its face ? Who is an in- dorser? When is a note negotiable? When is a note not negotiable? 57. What is a draft ? What are the names of the three parties named in a draft ? What is meant by the acceptance of a draft ? By its protest? 58. What is a bond ? What is a coupon ? When bonds are quoted at 108, what are they worth? Name the three principal classes of United States Bonds. 59. What is annual interest ? When annual interest is not paid when due, what kind of interest does it draw until paid? 60. What is compound interest ? In what respect does compound interest differ from annual interest ? 61. On what principle is the common method of finding the equated time of several debts or payments based ? What is meant by the equa- tion of accounts ? 62. Define ratio. In how many and what ways may the ratio of two numbers be expressed ? What are the two terms of a ratio called ? Which is the dividend ? When is the value of a ratio less than one? When is it greater than one? G3. Why must the two terms of a ratio be like numbers? Why is the value of a ratio not changed by multiplying or dividing both of its terms by the same numbers? 64. What is a compound ratio? How is a compound ratio reduced to a simple ratio? 65. What is a proportion ? How many ratios in a simple propor- tion ? When is a proportion called simple? When is it comjDound? How many terras in a simple proportion ? 66. Which terms are called the extremes, and which the means? To what is the product of the extremes equal? 67. How can a missing mean be found ? Why ? A missing ex- treme? Why? If the second term of a proportion is greater than the first term, how will the fourth term compare with the third ? 68. In stating a problem in proportion, which number is made the third term ? Why ? What is the relation between the ratio of like causes and the ratio of their effects ? 69. How may a compound proportion be reduced to a simple propor- tion? How may the fourth term of a compound proportion be found? 286 COMPLETE ARITHMETIC. 70. What is the difference between a simple partnership and a com- pound partnership ? On what does the partnership value of capital depend ? 71. What is the difference between the power of a number and its root? Give examples. What is the difference between involution and evolution? 72. What is the difference between a perfect power and an imper- fect power? Give examples. When is a root called a sm^d f 73. To what is the square of a number composed of tens and units equal ? To what is the cube of a number composed of tens and units equal ? 74. How many orders in the square of any number? How many orders in the square root of any number? How many orders in the cube of any number? How many orders in the cube root of any number ? 75. How is the first term of the square root of any number found ? The second term? How is the first term of the cube root of any number found? The second term ? 76. To what is the square of the hypotenuse of a right-angled tri- angle equal? The square of the base or perpendicular? 77. How may the area of a circle be found ? When the area is given, how may the diameter be found? What is the relation be- tween the areas of two circles? 78. How is the surface of a sphere found? Its solidity? What is the relation between the solid contents of two spheres? APPENDIX. NOTATION. 423. In the decimal system of notation, with ten for its base, ten figures are used ; in a system with twenty for its base, twenty figures would be needed ; in a system with five for its base, only five figures (1, 2, S,% 0) would be needed; and, generally, a system of notation requires as many different figures as there are units in its base. 424. In a system with five for its base, 24 would express fourteen; 124 would express thirty-nine; 1120 would express one hundred and sixty. EXERCISES. 1. What number is expressed by 200 on a scale of five? 2. What number is expressed by 1240 on a scale of five? 3. Express forty on a scale of five. 4. Express one hundred on a scale of five. 5. Express two hundred on a scale of five. PKOOF OF THE SIMPLE RULES BY - CASTING OUT THE 9's." 425. The method of proving the elementary operations of arithmetic by "casting out the 9's" is based on the principle, that the excess of 9's m any number is equal to the excess of 9's in the sum of its digits. Take, for example, 2345. Dividing it by 9, we have the remainder 5, for the excess of 9's ; and adding the digits (2-{-S-\-4-{-5 = 14), and dividing the sum by 9, we have the same remainder. (287) 288 COMPLETE ARITHMETIC. 426. This principle may be thus explained : 2000 =: 222 X 9 + 2 300 = 33 X 9 -h 3 40 = 4x9 + 4 5= 5 2345 It is seen that 2000 is 222 times 9, with a remainder 2 ; 300 is 33 times 9, with a remainder 3 ; 40 is 4 times 9, with a remainder 4. Hence, the remainders obtained by dividing the several parts of a number, denoted by the local value of its digits, by 9, are respectively the digits of the number; and the remainder obtained by dividing the number itself by 9, equals the remainder obtained by dividing the sum of its digits by 9. Hence, The excess of 9's in any number is found by adding its digits and finding the excess of 9's in their sum. 427. Proof of Addition. Process. The excess of 9's in the first number, found by one rr 1 adding its digits, is 1 ; in the second number, 4 ; in 25(5 <' 4 the third, 7. The excess of 9's in the sum of these 358 " 7 excesses is 3, which equals the excess of 9's in 939, 939 " 3 the amount. Hence, The excess of 9^s in the sum of several numbers is equal to the excess of 9's in the sum of their excesses. 1. Add and prove 2346, 5084, 6784, 8653, and 9045. 2. Add and prove 30483, 50678, 346864, and 706037. 3. Add and prove 530902, 672084, 567084, and 1345602. Process. 3676 Excess 4 1508 " 5 2168 " 8 " 4 428. Proof of Subtraction. Since the minuend is equal to the sum of the sub- trahend and remainder, the excess of 9's in the minuend equals the excess of 9'.s in the SUM of the excesses in the subtrahend and remainder. 1. From 40603 take 27475, and prove the result. 2. From 607853 take 492097, and prove the result. CIRCULATING DECIMALS. 289 429. Proof of Multiplication. Process. 347 Excess 5 58 " _8 1041 40 1735^ 18391 Excess 4 Since 347 contains a certain number of 9's with an excess of 5, and 53 contains a certain number of 9's with an excess of 8, the product of 347 and 53 consists of the product of the number of 9's in them, plus the product of 5 and 8, the excesses of 9's. Hence, IVie excess of 9's in the. product of two numbers is equal to the excess of 9's in the product of the excesses in these numbers. 1. Multiply 45603 by 708, and prove the result. 2. Multiply 60875 by 690, and prove the result. 430. Proof of Division. Process. 347 ) 18496 ( 53 1735^ 1146 1041 105 18496 Excess 1 347 " 5 53 " 8 105 " 6 5 X 8 -f 6 ==: 46 " 1 Since the dividend equals the product of divisor and quotient, plus the re- mainder, the excess of 9's in the divi- dend is equal to the excess of 9's in the product of divisor and quotient, plus the excess in the remainder. Hence, The excess of 9'.s' in the dividend is equcd to the excess of 9's in the product of the excesses in divisor and quotient, plus the excess hi the remainder. 1. Divide 6480 by 47, and prove the result. 2. Divide 15685 by 625, and prove the result. CIRCULATING DECIMALS. 431. A Circulating Decimal is an interminate decimal, containing the same figure or set of figures, repeated in the same order indefinitely. (Art. 12L) 432. The figure or set of figures repeated is called a Bepetend. A repetend is denoted by a dot placed over the first and last of its figures; as, .5 .16 .325. f.Ar.-i'). 290 COMPLETE ARITHMETIC. 433. When a circulating decimal has no figure but the repetend, it is called a Pure Circulate; as, .325. When a circulating decimal has one or more figures before the repetend, it is called a Mixed Circulate; as, .4526. 434. A pure circulate is reduced to a common fraction by taking the repetend for the numerator, and as many 9's for the denominator as there are figures in the repetend. Proof. Let .63 be a pure circulate. Then, 63.63 =^100 times the pure circulate. ■63 = 1 time " " " Subtracting, 63. ^ 99 times " " " Hence, — — the value of '^ " 99 435. A mixed circulate is reduced to a common fraction by subtracting the terms which precede the repetend from the ivhole repetend, and taking the difference for the numerator; and, for the denominator, taking as many 9's as there are figures in the repetend, with as many ciphers annexed as there are decimal figures before the repetend. Proof. Let .45124 be a mixed circulate. Then, 45124.124 = 100000 times the mixed circulate. And, 45.124 = 100 " " " " Subtracting, 45079 = 99900 " " " Hence, fffj^ = the value of " " " 436. Pure or mixed circulates may be added, subtracted, multiplied, or divided by first reducing them to common fractions. Note. — Circulates may be added, subtracted, multiplied, or divided without tirst reducing them to common fractions; but the processes are not of sufficient practical importance to justify their explanation in a school arithmetic. In all computations, circulates are carried to enough places to avoid any appreciable error in the result, and then are treated as other decimals. TABLES. 291 437. TABLES OF DENOMINATE NUMBERS. I. CURRENCIES. 1. United States Money. The denominations are mills, cents, dimes, dollars, and ec Table. 10 rn. = 1 ct. 10 ct. = I d. 10 d. = $1 $10 = \ E. 2. English Money. The denominations are/ar- tkings (q.), j)ence (d.), shil- lings (s.), and pounds (£). Table. 4q.=--ld. Ud. = l s. 20 s. = 1 £. 1 £ = $4.84. II. MEASURES OF EXTENSION AND TIME. 1. MEASUKES OF LINES AND AKCS. Long Measure. The denominations are inches, feet, yards, rods, fur- longs, and miles. Table. 12 in. ^\ft. 3 A = lyd. 5| yd. = 1 rd. 40 rd. = 1 /wr. 8 fur. = 1 m. Also: 3 barleycorn. ,• = 1 incA. 4 mcAes = 1 hand. 3/ee< ■= 1 pace. 6 feet = 1 fathom. 3 wuVes ^rr 1 league. 60 geographic miles -y les {nearly) f 69 1 statute nu Circular Measure. The denominations are sec- onds, minutes, degrees, signs, and circumferences. Table. 60^^ = y 60' = r 30° -= 1 s. 12 s. 360° } = IC. Cloth Measure. (Little used.) 2\in.^= 1 nxiil. 4 w. =1 quarter. A. qr. = 1 yard. 5 gr. = 1 Ell Eng. 1 degree at the equator. 292 COMPLETE ARITHMETIC. 2. MEASURES OF SURFACES OR AREAS. Square Measure. The denominations are square inches, square feet, square yards, square rods (or perches), roods, acres, and square miles. Table. 144 sq. in. = 1 sq. ft. 9 sq. ft' ^Isq. yd. SO\sq .yd. = 1 P. 40 p. = li2. 4i2. = 1 A. 640 A. =: 1 sq. mi. Surveyor's Measure. Table. 7.92 in. = 1 link (/.). 2b I. = 1 rod. 4 rd. = 1 chain (ch.) 80 ch. = 1 mile. Also: 625 sq. I . -=1 P. 16 P. = 1 so. ch. 10 sq. ch. =1 A. 640 A. ■= 1 sq. mi. 1 sq. mi. =^ 1 section. 36 sect. = 1 township 3. MEASURES OF SOLID CONTENTS OR CAPACITY. Cubic Measure. The denominations are cubic inches, cubic feet, and cubic yards. Table. 1728 CM. in. = 1 cw. ft. 27 cu. ft. = 1 cu. yd. Wood Measure. Table. 16 cu. ft. = 1 cord ft. ^'^'f^-^n = lcord. 128 cu. ft. / 24| cu.ft. = 1 perch of stone. 40 cu. ft. round timber == 1 ton. 50 cu. ft. hewn timber = 1 ton. Note. — The standard bushel gallon, 231 cu. in. ; and tlie beer Dry Measure. The denominations are pints, quarts, pecks, and busJiels. Table. 2 pt. =1 qt. 8 qt. 4pk. Ipk. 1 bu. Liquid Measure. Table. 4 gills = 1 pt. 2pt. =1 qt. 4 qt. =1 gal. 3U gal. =-- 1 66;. 63 gal. = 1 hhd. 42 gal. = 1 tierce. contains 2150| cu. in. ; the liquid gallon (little used), 282 cu. in. TABLES. 293 4. measures of duration or time. Time Measure. The denominations are seconds, minutes, hours, days, years, and centuries. Table. 60 sec. -— 1 min. 60 min. -- 1 A. 24 A. =1 da. 365 da. := 1 common yr. 366 da. ^^ 1 leap yr. 365^ da. = 1 sola7' yr. 100 s. ?/7'. =; 1 century. Also : 7 da. ^-^ 1 week. 4 w. :^ I lunar mo. Calendar Months. January, 1st mo. 31 days. February, 2d u 28 or 29 March, 3d u 81 days. April, 4th " 30 " May, 5th il 31 " June, 6th ii 80 '' July, 7th il 31 " August, 8th il 31 " September, 9th ii 80 " October, 10th ,1 81 '' November, 11th ii 30 " December, 12th 11 31 " Also : A Julian year contains 13 lunar mo. 1 da. 6 h. A civil year contains 12 calendar months. A solar year contains 365 da. 5 h. 48 min. 48 sec. III. WEIGHTS. Avoirdupois Weight. The denominations are drams, ounces, pounds, hundred- weights, and tons. Table. 16 dr. r=loz. 16 oz. = 1/6. 100 lb. =^ 1 cwt. 20 cwt . - 1 T. Ako : 196 lb. flour = 1 200 lb. beef or pork = 1 barrel. 100 lb. fish 56 lb. corn or rye ^ 60 lb. tvheat > 32 lb. outs J 14 lb. iron or lead 2H stones 8 pigs = 1 quintal. =^ 1 bushel. ■= 1 stone. = 1 pig. = 1 /other. 294 COMPLETE ARITHMETIC. Troy Weight. The denominations are grains, 'pennyweights, ounces, and pounds. Table. 24 gr. = 1 pivt. 20 pwt. = 1 oz. 12 oz. =1 lb. 4 ^T. = 1 carat. Apothecaries Weight. The denominations are grains, scruples, drams, ounces, and pounds. Table. 20 gr. = 16 36=13 83=13 12 3 = 1 tb COMPARISON OF WEIGHTS. 1 ^6. Avoir. = IxVi lb. Troy = 1 yVf ^ ^poth. 1 oz. " = n% oz. " = lii 3 IV. MISCELLANEOUS TABLE. 12 thirigs are 1 dozen. 12 (^02671 (( 1 gi'oss. 12 gross « 1 great gross. 20 fAiw^rs u 1 score. 18 iTicAes a 1 cubit. 22 inches (nearly) " 1 sacred cubit. Paper . 24 sheets are 1 quire. 20 ^mVes a 1 ream. 2 reams a 1 bundle. 5 bundles n 1 bale. Books . sheet folded in 2 leaves - 's called a folio. << 4 a qwirto, or 4/0. « 8 an octa.vo, or 8ro. « 12 a duodecimo, or 12mo. « 16 a 16wio. « 24 a 24wo. Note. — In estimating the size of the leaves, as above, the double medium sheet (23 by 26 inches) is taken as a standard. LIFE INSURANCE. 295 LEGAL RATES OF INTEREST IN THE SEVERAL STATES. 438. When no rate is mentioned, the legal rate in Louisiana, except on bank interest (6%), is 5 % New York, New Jersey, Michigan, Wisconsin, Minnesota, South Carolina, and Georgia 7 % Alabama and Texas 8 % California, Oregon, Kansas, Nebraska, Nevada, and Colorado . 10 fo All the other States and District of Columbia 6 % When stipulated in the contract, the legal rate in Ohio, Florida, and Louisiana is as high as 8 % Illinois, Iowa, Michigan, Arkansas, Mississippi, Missouri, and Tennessee 10 % Minnesota, Texas, Wisconsin 12 % Nebraska 15 % Kansas 20 % Massachusetts, Rhode Island, California, Nevada, and Colorado, any per cent, agreed upon. The legal rate in England and France is 5 % Canada, Nova Scotia, and Ireland % Note. — Since the rate of interest is often changed by legislation, the above rates may not in all cases be strictly accurate. LIFE INSURANCE. 439. The rate of premium in life insurance is based on the applicant's expectation of life, as shown by life statistics or bills of mortality. The annual premium must be such a sum as, when put at interest, will amount to the sum insured at the close of average extension of life beyond the applicant's age. 440. There are two tables showing the Expectation of Life, called the Carlisle Table and the Wigglesworth Table. The former is based on bills of mortality prepared in Eng- land, and the latter is based on the mortality in the United States. Both tables are in use in this country. 296 COMPLETE AUITIIMETIC. 441. The Expectation of Life, as shown by the two tables, is as follows : n > a > n >■ n H a 1 >< w o < Is X -< 5w o -< 38.72 28.15 24 38.59 32.70 48 22.80 22.27 72 8.16 1 44.68 36.78 25 37.86 32.33 49 21.81 21.72 73 7.72 2 47.55 38.74 26 37.14 31.93 50 21.11 21.17 74 7.33 3 49.82 40.01 27 36.41 31.50 51 20.39 20.61 75 7.61 4 50.76 40.73 28 35.69 31.08 52 19.68 20.05 76 6.49 5 51.25 40.88 29 35.00 30.66 53 18.97 19.49 77 6.10 6 51.17 40.69 30 34.34 30.25 54 18.28 18.92 78 6.02! 7 50.80 40.47 31 33.68 29.83 55 17.58 18.35 79 5.80 8 50.24 40.14 32 33.03 29.43 56 16.89 17.78 80 5.51 9 49.57 39.72 33 32.36 29.02 57 16.21 17.20 81 5.21 10 48.82 39.23 34 31.68 28.62 58 15.55 16.63 82 4.93 11 48.04 38.64 35 31.00 28.22 59 14.92 19.04 83 4.65 12 47.27 38.02 36 30.32 27.78 60 14.34 15.45 84 4.39 13 46.51 37.41 37 29.64 27.34 61 13.82 14.86 85 4.12 14 45.75 36.79 38 28.96 26.91 62 13.31 14.26 86 3.90 15 45.00 36.17 39 28.28 26.47 63 12.81 13.66 87 3.71 16 44.27 35.76 40 27.61 26.04 64 12.30 13.05 88 3.59 17 43.57 35.37 41 26.97 25.61 65 11.79 12.43 89 3.47 18 42.87 34.98 42 26.34 25.19 66 11.27 11.96 90 3.28 19 42.17 34 59 43 25.71 24.77 67 10.75 11.48 91 3 26 20 41.46 34.22 44 25.09 24.35 68 10.23 11.01 92 3.37 21 40.75 33.84 45 24.46 23.92 69 9.70 10 50 93 3.48 22 40.04 33.46 46 23.82 23.37 70 9.18 10.06 94 3.53 23 39.31 33.08 47 2317 22.83 71 8.65 9.60 95 3.53 X 9.14 8.69 8.25 7.83 7.40 6.99 6.59 6.21 5.85 5.50 5.16 4.87 4.66 4.57 4.21 3.90 3.67 3.56 3.73 3.32 3.12 2.40 1.98 1.62 Note. — A comparison shows that the Wij?glesworth table has a less expectation of life than tlie Carlisle table for all ages below 50 years ; and that the latter table has a less expectation than the former for all ages from 50 to 90 years inchisive. EQUATION OF PAYMENTS. 442. In 1860, the author published a demonstration of the correctness of the common INIercantile Kule for finding the equated ti-me for the payment of several debts, due at dif- ferent times without interest. The inaccuracy of the rule hy present ivorths, commended by several authors as '^the only accurate ride'^yvas thus pointed out: ARITHMETICAL PROGRESSION. 297 " The equated time for the payment of $200, of which $100 is now due, and the other $100 is due in two years, as found by this rule, is 11.32 months. Now, the amount of $100 for 11.32 months, at 6 per cent., is $105.66; the present worth of the other $100, due in 12.679 months, is $94,038, and $105.66 + $94,038 := $199,698, whereas it ought to be $200. " It is also evident that the equated time, as found by this 'accurate' rule, will not be the same for all rates of interest. At 50 per cent, the equated time of the above example is 8 months, and the error, by the above test, $8.33^; at 100 per cent, it is 6 months, with an error of $10. "This supposed accurate rule is based upon the principle that the amount to be paid on a debt due at a future date, witiiout interest, at any time previous to this date, is the present worth of the debt at any prior date, plus the interest of the present worth up to date of payment. The incorrectness of this principle is easily shown. Suppose I owe a man $100, due in two years, without interest; how much ought I to pay in one year? "The present worth of $100, due in tAvo years (at 6 per cent.), is $89.2857, and the interest on this sum for one year is $5.3571 ; hence, the sum to be paid is $89.2857 + $5.3571 = $94.6428. The true amount to be paid, however, is the present worth of $100, due in one year, which is $94,339." Note. — The accuracy of the Mercantile Rule and the inaccuracy of the rule by Present Worths were rigidly demonstrated by Prof. A. Schuyler, in an article published in the Ohio Educational JHonthly, for 1862, p. 116. ARITHMETICAL PROGRESSIOK 443. An A ritlimetical JProgression is a series of numbers which so increases or decreases that the diiference between the consecutive numbers is constant. 444. The numbers which form the series are called Terms, the first and last terms being the ExtremeSy and the intervening terms the Means. The difference between the consecutive terms is called the Common Difference. 445. An Ascending Series is one in which the terms in- crease; as, 2, 5, 8, 11, 14, etc. A Descending Seines is on€ in which the terms decrease ; as 20, 17, 14, 11, 8, etc. 446. In an arithmetical progression five quantities are 298 COMPLETE ARITHMETIC. considered ; and such is the relation between them, that, if any three are given, the other two may be found. These quantities are: 1. The first term. 2. The last term. 3. The common difference. 4. TJie number of terms. 5. The sum of all the terms. 447. The ascending series, 2, 5, 8, 11, 14, having 5 terms, may be expressed in three forms, as follows : (1) 2 5 8 11 14 (2) 2 2+3 2+(3+3) 2+(3+3+3) 2+(3+3+3+3) (3) 2 2+3 2+3X2 2+3X3 2+3X4 A comparison of these three forms of the same series shows, that each term is composed of two parts, viz.: (1) the first term ; (2) the common difference taken as many times as there are preceding terms. Hence, 1. TJie last term of an ascending series is equal to the first term, plus the common difference taken as many times as there are terms in the series less one. Conversely, 2. The first term of an ascending series is equal to the last term, minus the common difference taken as many times as there are terms in the series less one. 3. The common difference is equal to the difference between the first and last terms, divided by the jiumber of terms less one. 4. TJw number of terms less one is equal to the difference between the first and last terms, divided by the common difference. 448. Let 3 5 7 9 11 13 be an arithmetical series, and, 13 11 9 7 5 3 be the series reversed. Then, 16 + 16 + 16 + 16 + 16 + 16 = twice the sum of the terms. and 8 -f 8 + 8 + 8 + 8 + 8 = the sum of the terms. ARITHMETICAL PROGRESSION. 299 An inspection of the above shows that the sum of the first and last terms of an arithmetical series, multiplied by the number of terms, is equal to twice the sum of all the terms. Hence, The sum of all Hie terms of an arithmetical series is equal to the product of one half the sum of the first and last terms, multiplied by the number of terms. Note. — One half of the sum of the first and last terms is equal to the average of the several terms of the series. 449. From the above principles may be deduced the fol- lowing FORMULAS. 1. Last term = first term dz (com. difference X number of terms less one). 2. First term = last term =h (com. difference X number of term^ less one). 3. Common differeme = | '«** '^'■™ -fi'"^ '^'" I ^ number L first term — last term ) of terms less one. 4. Number of term^ less on^^\^\f^~fij^, ^'^X •^ ( first term — last term J -T- common dfference. 5. Sum of terms = ^ (first term -\- last term) X number of terms. Note. — The first term of an ascending series corresponds to the last term of a like descending series, and the last term of a descending series corresponds to the first term of a like ascending series. PBOBLEMS. 1. What is the tenth term of the series 5, 7, 9, 11, etc.? 2. The first term of an ascending series is 4, the common difference 3, and the number of terms 8 : what is the last term ? 3. The last term of a descending series is 1, the common difference 4, and the number of terms 12 : what is the first term? 300 COMPLETE ARITHMETIC. 4. The extremes of an arithmetical series are 47 and 3, and the number of terms 12: what is the common differ- ence ? 5. The 1st term is 7 and the 21st term 57: what is the common difference? 6. The 4th term of a series is 21 and the 9th term is 41 : what are the four mean terms? 7. The two extremes of a series are 12 and 177, and the common difference 5 : what is the number of terms? 8. The two extremes of a series are 20 and 152, and the number of terms 45 : what is the sum of all the terms ? 9. What is the sum of all the terms of the series described in the 6th problem above? In the 7th problem? 10. How many strokes does the hammer of a clock make in 24 hours? 11. A man agreed to dig a trench 50 yards long for 2 cents for the first yard, 5 cents for the second yard, 8 cents for the third, and so on, the price of each yard being 3 cents more than that of the preceding yard : what did he receive for digging the last yard? For digging the trench? GEOMETRICAL PROGRESSION. 450. A Geometrical Profp^ession is a series of numbers which so increases or decreases that the ratio be- tween the consecutive terras is constant. The first and last terms are called the Extremes, and the intervening terms are' called the Means. 451. A geometrical progression is ctscending or descending according as the series increases or decreases from left to right. 452. In a geometrical progression five quantities are con- sidered, and these (as in arithmetical progression) are so related to each other that, any three being given, the other two may be found. GEOMETRICAL PROGRESSION. 301 These five quantities are 1. The first term. 2. The last term. 3. The common ratio. 4. The number of terms. 5. The sum of all the terms. 453. The ascending series, 2, 6, 18, 54, 162, 486, has 6 terms, and the first term is 2, and the common ratio or multiplier is 3. This series may be expressed in three forms, as follows: (1) 2 6 18 54 162 486 (2) 2 2X3 2X3X3 2X3X3X3 2X3X3X3X3 2X3X3X3X3X3 (3) 2 2X3 2X32 2X3^ 2X3* 2X3^ A comparison of the corresponding terms of the three forms, shows that each term of the series is composed of two factors, viz. : (1) the first term, and (2) the common ratio raised to a power whose exponent or degree is equal to the number of preceding terms. Hence, 1. The last term of a geometrical series is equal to the first term, multiplied by the common ratio, raised to a power whose degree is one less than the number of terms. Conversely, 2. The first term is equal to the last term divided by the com- mon ratio, raised to a power whose degree is one less than the number of terms. 3. The common ratio is equal to the root wh-ose index is one less than the number of terms, of the quotient of the last terra divided by Hie first term. 454. By an algebraic process it may be shown that 4. The sum of a geometrical series is equal to the product of the last term and the common ratio, less the first term, divided by the common ratio less one. 455. When the number of terms in a descending geomet- rical series is infinite, the last term is 0, and the sum of the series is equal to the first term divided by the ratio less one. 302 COMPLETE ARITHMETIC. 456. From the above principles may be deduced the fol- lowing FORMULAS. 1. Last term = first term X ratio^^~^. 2. First term = last term ■— ratio "~ ^ . 3. Matio = -[/last term -v- first term. .CI /. • (^^^ ^''"'^ X ratio) — iirst term 4. bum of series = -. — ' -^ ratio — 1 5. Sum of infinite descending series = first term -~ (ratio — 1 ). Notes. — 1. By "ratio "-i," in 1st and 2d formulas, is meant the ratio raised to a power whose degree is the number of terms less 1. The index of the root, in the 3d formula {n — 1), is the number of terms less ]. 2. In an ascending series the ratio is greater than 1, and in a de- scending series the ratio is less than 1. PROBLEMS. 1. What is the 6th term of the series 5, 10, 20? etc. 2. The first term of a geometrical series is 5, the ratio is 3, and the number of terms 7 : what is the last term ? 3. The first term of a series is 1220, the ratio ^, and the number of terms 6 : what is the last term ? 4. The last term of a series is 64, the ratio 2, and the number of terms 10 : what is the first term? 5. What is the sum of the series described in the 4th problem ? In the 3d problem ? 6. The first term of a series is 5, and the sixth term is 1215: what is the ratio? 7. The first term of a series is 10, the sixth term 2430, and the ratio 3 : what is the sum of the six terms ? 8. A father gave his son 50 cents on his 12th birthday, and agreed to double the amount on each succeeding birth- day to and including the 21st : how much did the son re- ceive on his 21st birthday? How much in all? 9. A man worked 15 days on condition that he should receive 1 cent the first day, 5 cents the second day, and so ALLIGATION. 303 on, the wages of each day being 5 times the wages of the previous day : how much did he receive ? ALLIGATION. 457. Alligation is the process of finding the average value or quality of a mixture composed of articles of dif- ferent values or qualities. It is also the process of compounding several articles of different values or qualities to form a mixture of an average value or quality. The first process is called Alligation Medial^ and the sec- ond Alligation Alternate. Note. — The term Alligation is derived from the Latin alligare, to bind or link. The term is applied to this process because some of the problems may be solved by joining or linking the numbers. Case I. 458. Several ingredients of a mixture, and their respective values given, to find their average value. PROBLEMS. 1. A farmer mixed 25 bushels of oats, at 50 cents a bushel ; 15 bushels of rye, at 80 cents a bushel ; and 30 bushels of corn, at 70 cents a bushel : Avhat was the value of a bushel of the mixture ? Process. ^A* V 95 — mo ^^"^^ *^^ *°*^^ ^^^"^ ^^ *^^ '^^ bushels 80 X 15 ^ 1200 ^^ grain mixed together was 4550 cents, 70 X 30 = 2100 the value of 1 bushel was yV of 4550 cents. 70 ) 4550 which is 65 cents. 65 cts., Ans. 2. A grocer mixed 20 pounds of cofiee worth 28 cents, 30 pounds worth 35 cents, and 50 pounds worth 33 cents : what is a pound of the mixture worth ? 304 COMPLETE ARITHMETIC. Case II. 459. The values of several articles given, to find in what pro- portion they must be compounded to make a mixture of a given value. 3. A grocer has sugars worth 16, 18, and 24 cents a pound: in what proportion must they be taken to make a mixture worth 20 cents a pound ? I. Solution by Analysis. On each pound of sugar worth 16 cents taken, there is a gain of 4 cents, and on each pound at 24 cents, there is a loss of 4 cents. Hence, these two kinds of sugar may be taken in equal quantities, or 1 pound of each. On each pound worth 18 cents there is a gain of 2 cents, and hence 2 pounds of it must be taken to offset a loss of 4 cents on 1 pound at 24 cents. Hence, the simplest proportionals are 1 lb. at 16 cts., 2 lb. at 18 cts., and 2 lb. at 24 cts. II. Another Solution. 1 lb. at 16 cts. selling for 20 cts. gains 4 cts. \ /> . 1 " 18 " " 20 " " 2 cts. i ^^'''' 1 " 24 " " 20 " loses 4 cts. . . 4 cts. loss. Taking two pounds each of the first two kinds, the loss will be 12 cents, and by taking 3 pounds of the third kind, the loss will be 12 cents. Hence, the proportionals 2, 2, 3 make the gains and losses equal. III. Solution by Linking. 20 16 n . . .■ . 4 18—1 .... 4 24 1 J . 4 4-2 = 6 2 2 (3 Note. — When only two articles of different values are given, they can be compounded in but one way ; but when more than two articles are given, they may be compounded in an infinite number of tvays. They may be combined two and two in such proportions as to make, in each case, a mixture of the required value, and then these com- pounds may be united in any proportions whatever. APPENDIX. 305 4. A merchant has teas worth $1.25, $1.40, $1.60, and $1.75: how much of each kind must be taken to make a mixture worth $1.50? Case III. 460. The values of the several ingredients of a mixture, their average value, and the quantity of one or more of ike ingredients given, to find the respective quantities of the other ingredients. 5. A grocer wishes to mix 100 pounds of coffee at 25 cts, with coffees at 22, 28, and 30 cts., making a mixture worth 27 cts. : how much of each kind must he take ? Suggestion. — Find the proportionals of the ingredients by Case II, and then muhiply each proportional by the quotient of 100 lbs. di- vided by the proportional for the coffee worth 25 cts. 6. A farmer wishes to mix 60 bushels of corn at 60 cts. , with rye at 75 cts., barley at 50 cts., and oats at 40 cts., to make a mixture worth 65 cts. : how many bushels each of rye, barley, and oats must he take ? Case IV. 461. The values of the ingredients, and the quantity and value of the mixture given, to find the quantity of each ingre- dient. 7. How much gold 16 carats fine, 18 carats fine, and 22 carats fine, must be taken to make 12 rings 20 carats fine, and weighing 4^ pwt. each ? Suggestion. — Find the proportionals by Case II, and then divide the whole quantity into parts proportional to these proportionals. 8. HoAV much sugar worth 15 cts., 17 cts., and 20 cts. must be taken to make a mixture of 200 pounds, worth 18 cts.? 9. How much water must be mixed with vinegar, worth 60 cts. a gallon, to make 90 gallons, worth 50 cts. a gallon ? C.Av.— 2(^ 306 COMPLETE ARITHMETIC. DUODECIMALS. 462. A Duodechnal is a denominate number in which twelve units of any denomination make a unit of the next higher denomination. A duodecimal may be regarded as a fraction Avhose denominator is a power of 12 ; or a number whose scale is 12. The term is de- rived from the Latin duodecim, twelve. 463. Duodecimals are used by artificers in measuring surfaces and solids. The foot is divided into primes, marked ' ; the primes into seconds (") ; the seconds into thirds ('"), etc., as is shown in the following Table. 12 fourtlis {'''^) are V^^ 12 thirds " V 12 seconds " V 12 primes " 1 ft. The accents used to mark the different denominations, are called Indices. 464. The prime denotes the twelfth of a foot ; the second, ilie twelfth of the twelfth of a foot, etc. When a duodecimal denotes the area of a surface, the foot is a square foot; the prime, t]ie twelfth of a square foot; the second, the twelfth of a twelfth of a square foot, etc. When a duodecimal denotes the contents of a solid, the foot is a cubic foot ; the prime, the twelfth of a cubic foot, etc. 465. ADDITION AND SUBTRACTION. PROBLEMS. 1. Add 12 ft. 8' 11'', 16 ft. 10' 9", and 24 ft. 6". Process: J i« ^* i"' Q'^ f 12 ft. 8" ir ;: -^ 16 ft. 10' 9' [ 24 ft. 0' 6' r 53 ft. 8' 2'' Ans. DUODECIMALS. 307 2. Add 12 ft. 9' IV A"\ 23 ft. 1" 10'", and 10' 6" 9'". 3. From 21 ft. T 10" take 15 ft. 9' 4". Process- / ^1 ft. 7^10^' PROCESS. ^i5ft^ 9/ 4// 5 ft. 10^ %'' Am. 4. From the sum of 30 ft. 8" 4'" and 14 ft. T 10'", take their difference. 466. MULTIPLICATION OF DUODECIMALS. 5. Multiply 13 ft. 7' 8" long and 6 ft. 5' wide? Process. Multiply first by b' and tlien by 6 ft., and add the partial products. ^^ ^^' Tft r Since 1XtV = tV, TVXTV=Th, g-^— g/ '^77^>7 Ti:rXTV=T7W etc., feet X primes 81 ft' 1(K 0^^ (^^ twelfths) must produce primes; o« ... 1^7^ 2^/ 4/// 4 primes by primes, seconds; seconds by primes, thirds; and, generally, the denomination of the product of any two denominations is de- noted by the sum of their indices. 6. What are the superficial contents of a board 9 ft. 7' 4" long and 10' 6" wide ? 7. What are the solid contents of a block of marble 7 ft. 6' long, 2 ft. 8' wide, and 1 ft. 4' thick? Note. — The answers to the 5th and 6th problems are in square feet and duodecimal parts of a square foot, and the answer to the 7th problem is in cubic feet and duodecimal parts of a cubic foot (Art. 464). 467. DIVISION OF DUODECIMALS. 8. Divide 87 ft. 6' 2" 4'" by 13 ft. T 8". Process. The process is the reverse of Dividend. Divisor. that in multiplication. For con- 87 ft. 6^ 2^^ 4''''^ ) 13 ft. 7^ 8''^ venience in multiplying, place the 81 ft. 10^ 6 ft. 5^ Q'nt. divisor at the right of the divi- 5 ft. 8'' 2'^ 4^^"^ dend, and the terms of the quo- 5 ft. 8^ 2'' 4''' tient below those of the divisor. 308 COMPLETE ARITHMETIC. 9. Divide 62 ft. ir' W" by 8 ft. 6' %". 10. Multiply 10 ft. 5' 8" by 3 ft. 10', and divide the product by 5 ft. 2' 10". PERMUTATIONS, 468. Permutations are the changes of order, which a number of objects may undergo, and each object enter once and but once in each result. (cha 469. The diagram at the right shows f ^« 1 ^^« • the number of permutations of 1, 2, and al ^ , 3 letters. [ ah'l ach ( ahc The letter a permits no change of order. The letter h may be placed before and after the letter a, giving two (1X2) permutations of two letters — ha, ab. The letter c may be placed before, between, and after the two letters ab; and the same for b a, giving six (1X2X3) permutations of three letters. A fourth letter, as d, may evidently occupy four different positions in each of the six combinations of these letters, giving twerity-four (1x2x3X4) permutations of four letters. In like manner it may be shown that the number of 'permu- tations of any number of objects is equal to the continued product of all the integers from 1 to the given number of objects inclusive. PKOBLEMS. 1. In how many different orders may 6 boys sit on a bench ? 2. In how many different orders may all the letters in the word permutation be written ? 3. How many permutations may be made of the nine digits ? 4. How many different combinations of eight notes each may be made of the octave? RULES OF MENSURATION. 309 ANNUITIES. 470. An Annuity is a sum of money, payable annu- ally, for a given number of years, for life, or forever. The term is also applied to sums of money payable at any regular intervals of time. 471. A Certam Annuity is an annuity that is payable for a given number of years. A Contingent Annuity is an annuity payable for an uncer- tain period, as during the life of a person. A Perpetual Annuity is one that continues forever. 472. An Immediate Annuity is an annuity whose payment begins at once. A Deferred Annuity is an annuity whose payment begins at a future time. 473. The Forborne or Final Value of an annuity is the sum of the compound amounts of all its payments, from the time each is due to the end of the annuity. The Present Value of an annuity is the present worth of the forborne or final value. Note. — The principal applications of the subject of annuities are in leases, life estates, rents, dowers, life insurance, etc. ; and the prob- lems arising are readily solved by means of tables which give the present and final values of $1 at the usual rates of interest. A full discussion of the principles involved in the construction of these tables, can not well be presented in a school arithmetic. KULES OF MENSURATION. 474. Surfaces and Lines, 1. To find the area of a rectangle, Midtiply the length by the width. 2. To find either side of a rectangle, Divide the area by the oilier side. 310 COMPLETE ARITHMETIC. 3. To find the area of a triangle, Multiply the base By one half of iJie altitude. 4. To find the area of any quadrilateral having two sides parallel, Multiply one half of the sum of the two parallel sides by the perpendicular distance between them. 5. To find the circumference of a circle, 1. Multiply the diameter by 3.1416. Or, 2. Divide the area by one fourth of the diameter. 6. To find the area of a circle, 1. Multiply the square of the diameter by .7854. Or, 2. Multiply the square of the radius by 3.1416. Or, 3. Multiply the circumference by one half of the radius. 7. To find the diameter of a circle, whose area is given, Divide the area by .7854, and extract the square root of the quo- tient. 8. To find the side of the largest square that can be in- scribed in a circle, Multiply the radius by the square root of 2. 9. To find the side of the largest equilateral triangle that can be inscribed in a square, Midtiply the radius by iJie square root of 3. 10. To find the area of an ellipse, the two diameters be- ing given. Multiply the product of the two diameters by .7854. 11. To find the surface of a sphere, 1. Multiply the circumference by the diameter. Or, 2. Multiply the square of the diameter by 3.1416. 12. To find the entire surface of a right prism or right cylinder. Multiply the perimeter or circumference of the base by the height, and, to the product, add the surface of the tivo bases. 13. To find the convex surface of a pyramid or cone, Midtiply the perimeter or circumference of the base by one half the, slant height. 14. To find the hypotenuse of a right-angled triangle, HENKLES METHOD OF WRITING DECIMALS. 311 Extract the square root of the sum of the squares of the other two sides. 15. To find the base or the perpendicular of a right- angled triangle, Extract the square root of the difference be- tween the square of the hypotenuse and the square of the other 476. Contents of Solids. 1. To find the solid contents of a rectangular solid, Mul- tiply the length, width, and thickness together. 2. To find either dimension of a rectangular solid, Divide the solid contents by the product of the other two dimensions. 3. To find the solid contents of a cylinder, Multiply the area of the base by the altitude. 4. To find the solid contents of a sphere, 1. Multiply the cube of the diameter by .5236. Or, 2. Multiply the surface by one third of the radius. 5. To find the solid contents of a cone or pyramid, Multiply the area of the base by one third of the altitude. ' 6. To find the solid contents of the frustum of a cone or pyramid, To the sum of the areas of the two bases, add the square root of their product, and multiply the result by one third of the altitude. 7. To measure timber, as planks, joists, etc., by board measure, Eind the number of square feet in one surface, and multiply the residt by the thickness in inches. HENKLE'S METHOD OF WRITING DECIMALS. 476. It is seen from the decimal scale, that the teris of any number of tenths, the hundreds of any number of hun- dredths, the thousands of any number of thousandths, etc., each falls in the order of units when the decimal is expressed decimally. Thus, 42 tenths, written decimally, is 4.2, the 312 . COMPLETE ARITHMETIC. 4 (tens) falling in units' order; 1265 huridredths, written decimally, is 12.65, the 2 (hundreds) falling in units' order; and 425 thousandths, written decimally, is .425 or 0.425, the (thousands) falling in units' order. Hence, the following Rule. — To write a decimal, Begin at tJie left and write the term corresponding to the name of the decimal, in the order of units. SCHUYLER'S CONTRACTED METHOD OF MUL- TIPLYING DECIMALS. 477. Rule. — Write the midtiplier so that its units' place shall fall under that term of the multiplicand whose order is one lower than the lowest required in the product. Multiply first by the units, if any, carrying to the nearest unit from the part rejected. In 7nultiplying by a term on either side of the unit, commence with that term of the multiplicand as far on Process. ^j^^ ^^^^^ ^^^^ ^j ^j^^ ^^^^ ^j ^^^ midtipli- 12 S45 ^^^^ ^^^^ ^'^^ units^ place of the multiplier, ^^ carrying to the nearest unit. Add the 34564 partial products, reject ike sum of the last jgg column, after carrying to the nearest unit. _ 17 Point off the required number of decimal 42.669 places in the sum. Note. — Before commencing to multiply by any term of the multi- plier, it is convenient to mark that term, also the term with which we commence to multiply in the multiplicand. THE Eclectic Series of Geographies, BY A. VON STEIN WEHR. The JPrhnart/ Geofjraphy, No, 1. The plan of this book is natural, the language simple, and the definitions and descriptions exact. Illustrated, small 4to. The Tntei'tnediate Geogrcvphy, No, 2 ; for more advanced classes. Contains the leading principles of the science, so arranged as to give correct ideas to pupils without requiring the constant aid of the teacher. Full instructions in Map Drawing. Illustrated, large 4to. The School Geography , No, S ; embraces a full Mathemat- ical, Physical, and Political description of the Earth, and is intended for the highest classes in this branch of study. Illustrated, large 4to. GRADED SCHOOL ARITHMETICS: A New Series, on an Entirely JVew JPlan, BY E. E. WHITE, M. A. 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BRINTO The series cbnsistd of three b'^oks: a Primary, an Inte and a School. Geographiy. Tiiough each buok of tb' from the omers' in scapei and treatment, tKo arrant att^jpiatmaVd is the same -a all. This uniformit JpfntatesthG; progress of the pupil, as hc.is not ' : ^ew classifica.tion, or to master a new drrang^ be passes fron^ the study of one book. oif the '^ / The teach/ r is furnished with every mea:- ' is not forceffj, at each step, to classify a mass facts, or to adjd an extended commentary o The Pnll^a^^ry Geogfra].^y contains, the -ciorice, stated! in plai j, simple la:iguage. Che Injt.ern[iedi^^ oh ^eography is' fuller sufh^ientf^J^ a. r^^hm , loui-se. The map-dra ' ] after uhe "'escrii/' .. the political divisioi^ ;^ ■. prv siinple < 0,' and* will prove an inval >iA the form, physical features, and p t-n'tte ' ' ;:ool Geography is designed tcK complete the course, anoucj t(ipics art^ niort fully treate^Jah this book than in the ai ':itl\ and it contains ft complete outline of niathematicii! a ^ ography. The sectional maps ;ire .so constructed ■le for refeiorce. ' Ishers invite "".pecial attention to the Maps, which ' ■\V'^"^\>'?' titles arid artistic execiltion, are greatly superior to those ol . ■/ A ^' '-, ai/ Y\ •e,*"'^-\;t» " ^^'^ ^^ skillfully treated that the maps re,seni^>f,. '^ise, V 0*^ ^^ projection is the pclyconic — that on whi< b >"- ^-'uri..> V)aat Survey are constructed. T1j< ^vot>d-cuj /,' iuive been -engraved by the best ar " *^-vP^ns( red to render tlte series attractive, ct'o series is i.iC ^nly one publisher ' }^^icl], iho New Geoffrai^hy— the product of '-^^'ibory ;r and Humb ''v/CTiT? p WILSON, ^■^^''-