Van Antwerp, 0ragg SlCo. Cincinnati S. NewYork. •-^-» THE UNIVERSITY OF ILLINOIS LIBRARY . The Frank Hall collection of arithmetics! presented by Professor Ht L. Rietz of the University of Iowa* 513 Ms miw&ms:. usrar * ; / I WHITE'S GRADED-SCHOOL SERIES A COMPLETE ARITHMETIC UNITING MENTAL AND WRITTEN EXERCISES IN A NATURAL SYSTEM OE INSTRUCTION By eCwHITE, M.A., LL.D. YAN ANTWERP, BRAGG & CO. CINCINNATI NEW YORK McGuffey’s Revised Readers and Speller. Me Guffey’s Revised Charts. White’s Arithmetics. Harvey's Language Course. Eclectic Geographies. Eclectic Penmanship. Eclectic History of the United States. Thalheimer’s Historical Series. 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. ELECTROTYPKt) AT THE FRANKLIN TYPE FOUNDRY, CINCINNATI. BLRCTIC PRESS! VAN ANTWERP, BRAGG & CO,, CINCINNATI. &3Jc2.\ IAV. s$fc 6>P'X wnmncs uatuttj PREFACE. This work is called a Complete Arithmetic, because it em¬ braces all the subjects which properly belong to a school arith metic. It is designed to be a complete text-book for pupils who have a knowledge of the fundamental operations with in¬ tegral numbers, including denominate numbers. The work is characterized by ttye same features as the lower books of the series, viz.: 1. It combines Mental 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 faithfully embodies the inductive method of teaching. The w T ritten 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 generalizations. All definitions which are deducible from the processes, and, with few exceptions, all principles and rules, are placed after the problems—a feature peculiar to this Series. 3. It is specially adapted , both in matter and method, to the grade of pupils for which 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- (iii) IV PREFACE. cially meets the wants of Graded Schools. Not more than twenty pages are in any sense a repetition. The repeated matter con¬ sists 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, sim¬ ple and compound, more than sixteen discuss new topics. A sim¬ ilar difference is observable in the treatment of Common Fractions, Decimal Fractions, United States Money, etc. Among the added articles worthy of special mention are those on Denominate Frac¬ tions, the Metric System, Longitude and Time, and Foreign Ex¬ change. In the number of problems, tbe author lias aimed to bit the golden mean between a paucity and an excess, and the greatest pains has been taken to make them sufficiently 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 given to Principles. A clear comprehension of the principles PREFACE. v 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 treatment of Per¬ centage. Over eighty pages are devoted to this subject 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, will 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. 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, Ohto, July, 1870. N. B.—The steadily increasing use of this Three-book Series of Arithmetics has, from time to time, demanded not only the correction of all discovered errors, but such other slight revisions as have been made necessary by changes in business usages, laws, and values. This edition is believed to be fully up with the present condition of business. - ■ Cincinnati, Ohio, Jan. 10, 1884. SUGGESTIONS TO TEACHERS. / . 1. The Mental Problems should be made a thorough drill in analy* sis; 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 complete elucidation. Many of the written problems may also be solved mentally, thus increasing the drills in analysis. 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 ascertain the accuracy and neatness of each pupil’s work. The explanations of the written pro¬ cesses, given by the pupil, should be both analytic and inductive. A part of the mental problems should also be solved as written prob¬ lems, thus making the induction of the written process easy. 3. The Definitions should be deduced and stated by the pupils under the guidance of the teacher, and this can usually be done in connec¬ tion with the solution of the problems. See Int. Arith., p. 5, Sug. 3. When the definitions are placed before the problems, as in the appli¬ cations 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 possi¬ ble, and each should be proved 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 con¬ stant 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 understand 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. N. B.— See the author’s “Manual of Arithmetic” for other sugges¬ tions, methods of teaching, models of analysis, illustrative solutions, etc. CONTENTS SECTION I-VI.— The Fundamental Rules. PAGE Notation and Numeration . 10 Addition.13 The Addition of Two Columns . 15 Subtraction . . . .17 Multiplication . Abbreviated Processes Division Abbreviated Processes SECTION VII.— Properties op Numbers. Divisors and Factors . . 32 Greatest Common Divisor . Cancellation . . . .35 Least Common Multiple SECTION VIII.— Fractions. Notation and Numeration Reduction of Fractions Addition of Fractions Subtraction of Fractions . Multiplication of Fractions Division of Fractions . Complex Fractions Numbers Parts of Other Num bers .... Review of Fractions . 43 . 46 . 53 . 55 . 57 SECTION IX.— Decimal Fractions. Numeration and Notation Reduction of Decimals Addition of Decimals 73 79 82 Subtraction of Decimals . Multiplication of Decimals Division of Decimals SECTION X. —United States Money. Notation and Reduction . Addition and Subtraction . Multiplication and Division 90 91 92 Abbreviated Methods Aliquot Parts Bills . SECTION XI. —Mensuration. Surfaces.100 | Solids . SECTION XII. —Denominate Numbers. Reduction.107 Denominate Integers and Mixed Numbers .... 107 Denominate Fractions The Metric System . Metric Tables (vii) PAGE 20 22 25 28 37 40 61 66 66 68 83 84 85 93 95 97 104 110 119 120 CONTENTS, • • • Vlll SECTION XIII.— Compound Numbers. PAGE PAGE Addition and Subtraction . 125 Longitude and Time • • 130 Multiplication and Division 128 SECTION XIV. —Percentage. The Four Cases of Percentage . 137 Six Per Cent Method • 175 Review of the Four Cases . 144 Method by Days • 179 Applications of Percentage . 146 Partial Payments . • 181 Profit and Loss...» 146 The Problems in Interest 185 Commission and Brokerage . 149 Review of Problems a 190 Capital and Stock . 154 Present Worth and Discount 192 Insurance. 159 Bank Discount • 194 Life Insurance.... 163 Notes, Drafts, and Bonds • 198 Taxes. 164 Exchange • 201 Customs or Duties . 168 Annual Interest . • 205 Bankruptcy . 170 Compound Interest • 208 Interest . 171 Equation of Payments, • 211 General Method 172 Equation of Accounts . • 215 SECTION XV. —Ratio and Proportion. Patio....... 220 Partnership • • 234 Proportion. 224 Simple Partnership » • 235 Simple Proportion 225 Compound Partnership • • 237 Compound Proportion. 230 Problems for Analysis . • • 239 SECTION XVI.- Involution and Evolution. Involution. 246 Geometrical Explanation • • 255 Another Method of Involution . 248 Cube Root • • 257 Evolution. 249 Geometrical Explanation • • 262 Square Root .... 251 Mensuration, involving Inv. & Ev. 264 SECTION XVII. General Review. Test Problems .... 268 | Test Questions • • 281 APPENDIX. Notation. 287 Geometrical Progression . 300 Proofs by Excess of 9’s . 287 Alligation .... . 303 Circulating Decimals . 289 Duodecimals .... . 306 Tables of Denominate Numbers 291 Permutations .... . 308 Legal Rates of Interest. 295 Annuities .... . 309 Life Insurance . . . . 295 Rules of Mensuration . . 309 Equation of Payments. 296 Foreign Exchange . 312 Arithmetical Progression . 297 Answers. . 317 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 principles 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 Number is a unit or a collection of units. 4. An Integer is a number composed of whole or inte¬ gral units; as, 5, 12, 20. It is also called a Whole Number. 5. Numbers are either Concrete or Abstract. A Concrete Number is applied to a particular thing or quantity; as, 4 stars, 6 hours. "When a concrete number expresses the denominate units of cur¬ rency, weight, or measure, it is called a Denominate Number. (Art. 174.) An Abstract Number 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. 6. A Problem is a question proposed for solution. 7. An Example is a problem used to illustrate a process or a principle. 8. A Pule is a general description of a process. 9. An Arithmetical Sign is a character denoting an operation to be performed with 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 not written. 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 following 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 fourteen million forty-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 PRINCIPLES. 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. Numeration 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 of reading numbers. In Arithmetic, the term notation is used to denote the 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. A bar over a letter, as D, M, multiplies its value by one thousand. 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 Naughty 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 in unit’s place. 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 which 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 thou¬ sands, ten-thousands, hundred-thousands; the orders of millions’ period, 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 multiplied by 10 by each removal of it one order to the left, and is divided by 10 by each removal 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 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 a washstand, $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 -f 45608 + 304867 + 87609 = what? 15. Add four thousand and fifty-six; sixty-three thousand seven hundred; 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, 33809; 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: Territory ceded by England, 1783, Louisiana, as acquired from France, 1803, Florida, as acquired from Spain, 1821, Texas, as admitted to the Union, 1845, Oregon, as settled by treaty, 1846, California, etc., as conquered from Mexico, 1847, Arizona, as acquired from Mexico by treaty, 1854, Alaska, as acquired from Kussia by purchase, 1867, Square miles. 815615 . 930928 59268 . 237504 280425 . 649762 27500 . 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 oFmore col¬ umns in this manner. 21. Add 67, 58, 43, 36, and 54. Process. 67 Add thus : 54 -f o CO 84, + 6 = 90; 90 -f 40 - 130, + 58 3 == 133; 133 + 50 = 183, + 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. Sufficient 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 66. 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, -{-. It is called 'plus. 25. The Sign of Equality is two short horizontal parallel lines, =. It is read equals or is equal to. Thus, 7 -f- 8 = 15 is read 7 plus 8 equals 15. 26. Lilze 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. The sum is of the same kind 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 the 9’s.” SUBTRACTION. 17 SECTION IY. SUBTRACTION. MENTAL PROBLEMS. 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 Euroj>e 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 65 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 45003 plus 13478, take their difference. 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. When the subtrahend is a part of the minuend the difference is called the Remainder. (Art. 41.) The 31inuend is the greater number. The Subtrahend is the number subtracted. 29. The Sign of Subtraction is a short horizontal line, —. It is called minus or less . Thus, 12 — 5 is read 12 minus 5 or 12 less 5. 30. Principles. —1. The minuend, subtrahend, 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 difference 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 ARITHMETIC. 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 $48, 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 WHITTEN 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. Multiplication is a process of taking one num¬ ber as many times as there are units in another. The 3£ultiplicand is the number taken or multiplied. The Multiplier is the number denoting how many times the multiplicand is taken. The Product 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 Sign of Multiplication is an oblique cross, X- It is read multiplied by. WLp n p] p p-pH W.wppn Imx-mimbers, 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 multiplier is usually written after the sign; when it is written before it, the sign is read times. ,tt _ Vi {OS'*} y j 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 method of finding the sum of several equal numbers. 34. Principles. —1. The Multiplicand may be either con¬ crete or abstract. 2. The multiplier 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. Thus, 4 X 3 = 3 X 4. 5. The multiplicand equals the product divided by the multi- • 6. The multiplier equals the product divided by the multipli¬ 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 Multiplier 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? WRITTEN PROBLEMS. 6. Multiply 648 by 100. Process : 648 X 100 = 64800. The an nexing of a cipher to a rmm - ber removes the significant figures one place to the left, and hence increases their value 10 times; the annexing of two ciphers removes yrtA . 1 ^ . /] A 1 U &fcs. r ^ $ < *'V'' f A / •Wi c/\ i 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. f 35. Principle.— The removal of a figure one order to the left multiplies its value by 10 (Art. 21). 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 II. The Multiplier 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 3^ hours? In 12^- hours? 13. If a workman earn $40 a month, how much will he earn in 2^ months? In 12J months? 14. At 36 cents a yard, what will 25 yards of cloth cost ? 33^ yards? 15. At 24 cents a dozen, what will 12£ dozens of eggs cost? 16| dozens? WRITTEN PROBLEMS. 16. Multiply 459 by 33 J. Process. Since 33£ is I of 100, 33^ times 459 = ^ of 100 3 ) 45900 times 459 = 1 of 45900. Or, multiply the multi- 15300 Prod. plicand by 100, and divide the product by 3. 17. Multiply 486 by 3-J; by 33^-. 18. Multiply 1688 by 12 J; by 25; by 50. 24 COMPLETE ARITHMETIC. 19. Multiply 40648 byl6|; by 33*; by 333*. 20. Multiply 3468 by 25; by 125; by 250. 21. Multiply 4086 by 16| by 166|; by 333-J. 22. Multiply 10366 by 50; by 33*; by 66§ 37. Principle. —If the multiplier he multiplied hy a given number, and the resulting product he divided hy the same num¬ ber, the quotient will he the true product. 38. Rule.—T o multiply by a convenient part of 10, 100, 1000, etc., Multiply hy 10, 100, 1000, etc., and divide the product by the number of times the multiplier has been increased . Case III. The ^Multiplier a little less than 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 934 by ioo 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.—T o multiply by a number a little less than 10, 100, 1000, etc., Multiply hy 10, 100, 1000, etc., and subtract from the product the multiplicand multiplied hy the difference be¬ tween the multiplier and 10, 100, 1000, etc., as the case may he. Case IV. The Multiplier 14, 15, 16, etc., or 31, 51, 61, etc. 28. Multiply 7856 by 14; by 41. 1st Process. 7856 X 14 31424 2d Process. 7856 X 41 31424 109984, Product. 322096, P'oduct. 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. Rules. —1. To multiply by 13, 14, 15, etc., Multiply bv the units’ term , and add the successive products after the first , which is units , to the successive terms of the multiplicand. 2. To multiply by 31, 41, 51, etc., Multiply by the tens’ term , 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.—3. 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 $72 ? 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 ? 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 Dividend 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; or it is one of the equal parts of the dividend. The j Remainder is the part of the dividend which is left undivided. 42. The Sign of Division is a short horizontal line between two dots, -f-. It is read divided by. Thus, 16 -i- 4 is read 16 divided by 4. Division is also expressed by writing the dividend above and the divisor below a short horizontal line. Thus, ^ i s rea( i 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 written. 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 multiplication. 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 jian 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 multiplying of both divisor and dividend by the same number does not change the quotient. 5. The dividing of both dividend and divisor by the same num¬ ber does not change the quotient. ABBREVIATED PROCESSES. Case I. The Divisor 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. Process. 4510 45, Quotient. 7. Divide 3852 by 100. Process. 38152 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. The 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 divides its value by 10 (Art. 21). 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. Tine Divisor ending with, one or more Ciphers. 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? WRITTEN PROBLEMS. 16. Divide 71400 by 3400. Process. 34|00)714|00(21 68 34 34 First divide both divisor and dividend by 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. 45|00) 588I64(13 45 138 135 3 Remainder , 364 First divide both dividend and divisor by 100, which, in the case of the dividend, leaves a remainder of 64. Next divide 588 by 45, leaving a remainder of 3, which is 3 hundreds since the dividend (588) is hundreds. The first remainder is 64 units which, annexed to the 3 hundreds, give 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 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 the dividend. 2. Divide the new dividend thus formed by the new divisor, and the result will be the quotient. 3. Annex the figures cut off from the dividend to the remainder, if there be one, and the result will express the true remainder. Case III. The Divisor a. convenient part of lO, lOO, etc. 22. At 3J cents apiece, how many lemons can be bought for 90 cents? For 240 cents? Suggestion. —Since 10 is 3 times 3£, 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 16| 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 ? WRITTEN EXERCISES. 26. Divide 4375 by 125. Process. 4375 8 351000 35, Quotient. 27. Divide 13600 by 333£. Process. 13600 3 401800 40, Quotient. 800-f-3 = 266§ Bern. 28. Divide 6250 by 33J; by 50. DIVISION. 31 29. Divide 4365 by 250; by 166§. 30. Divide 15300 by 16f; by 3331. 50. Principle. —The multiplying of both divisor and divi¬ dend by the same number does not change the quotient. 51. Rule. — To divide by a convenient part of 10, 100, 1000, etc., Multiply the divideTid by the number denoting how many times the divisor is contained in 10, or 100, or 1000, etc., and divide the product by 10, or 100, or 1000, etc. Case IV. The Divisor a Composite Number. 31. Divide 18315 by 45. Process. Illustrative Process. 45 = 5 X 9 5)18315 9)3663 407, Quotient. 5 ) 18315 - 4 - 45 = 3663 - 4 - 9 9)3663-=- 9= 407-4-1 407-4- 1= 407 Since 45= 5 X 9, 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 Rem.) =.1 4 ) 230 .... 3 (2d “ )=3X2= .... 6 57 .... 2 (3d “ ) = 2 X 8 X 2 = 32 True Remainder = 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. -f 3 qt. -f- 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 55; by 96. 40. Divide 47633 by 90; by 110. 52. Principle. — The division of both divisor and dividend by the same number does not change 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 remainder, except the first, by all the divisors preceding its own. The sum of these products and the first re¬ mainder will be the true remainder. SECTION VII. / PROPERTIES OF NUMBERS. DIVISORS 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.—S ince 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 0 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. 2)126 3)_63 3)_21 7 126 = 2 X 3 X 3 X 7. Divide 126 by 2, a prime divisor; next divide the quotient 63 by 3, a prime divisor, and then divide the quotient 21 by 3, a prime divisor. The prime factors are 2, 3, 3, and 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 j 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 Number has other divisors besides itself and one. Every composite number is the product of two or more numbers. 56. Two or more numbers are prime to each other, or rela¬ tively prime, 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 Common Factor, Thus, 5 is a common factor of 15 and 20. 59. Whole numbers are either Even or Odd. An j Even Number 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. A. If a composite number composed of two factors be divided by one factor, 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. Rules. — 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 the last quotient are the priine factors. 2. To find the common factors of two or more numbers, Resolve the 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. Dividend, 4 X t X 0 X 12 Divisor, 4 X # X 0 = 12 . Instead of forming the prod¬ ucts, indicate the multiplica¬ tion by the proper sign, and 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 bj 4 times 12. 35. Divide 6 X 8 X 20 by 4 X 20. 36. Divide 5 X 7 X 11 X 13J by 7 X 13^. 36 COMPLETE ARITHMETIC. 37. Divide 12 X 16 X 28 by 9 X 24 X 21. Process. 8 4 Since dividing the factor of a number divides the num- 1% X X 8X4 32 her, cancel 12 in the divi- 9 x H X %l ~ 9 X 3 =T 7 = 1 *y dend and divide 24 in the % 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 -f- 27 — l/ 7 . 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 16f-f-(20 X 44 X 50 X 49) = wbat? 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 16 days, working 10 hours a day, how many men can do it in 20 days, work¬ ing 8 hours a day ? DEFINITION, PRINCIPLES, AND RULE. 62. Cancellation is the omission of one or more of the equal factors of divisor and dividend. It is used 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 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. Note.—W hen 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. WRITTEN EXERCISES. 11. What is the greatest common divisor of 126 and 210? , Process by Factoring. 126 =$X$X3X# 210 = gX$X5Xtf 2X3X7 = 42, G.C.D. to 126 and 210 will be their greatest common divisor. Resolve 126 and 210 into their prime factors. Since every divisor of a num¬ ber is a prime factor, or the product of two or more prime factors, the prod¬ uct of all the prime factors common 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 ? Divide 528 by 288, and 288 by the first remainder 240, and 240 by the sec¬ ond remainder 48; and, there being no remainder, 48 is the greatest com¬ mon divisor of 288 and 528. Since 48, the greatest divisor of itself, is a divisor of 240, it is the G. C. D. of 48 and 240. Process by Dividing. 288 ) 528 (1 288 240) 288 (1 240 48 ) 240 (5 240 48= G.C.D.oi 288 and 528. Since 48 is a common divisor of 48 and 240, it is a divisor of 288, their sum; and since every common divisor of 240 and 288 is a divisor of 48, their difference, 48, the greatest divisor of itself, is the G. C. D. of 240 and 288. Since 48 is a common divisor of 240 and 288, it is a divisor of 528, their mm; and since every common divisor of 288 and 528 is a divisor of 240, their difference, 48, the gi'eatest common divisor of 240 and 288, is the G. C. D. of 288 and 528. Note. —Let the pupil show, in like manner, that the last divisor, in the solution of problems 19 and 20, is the greatest common divisor required. What is the greatest common divisor of 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? 27. $260 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? LEAST COMMON MULTIPLE. 39 DEFINITIONS, PRINCIPLES, AND RULES. 65. A Divisor of a number is a number that will ex¬ actly divide it. A Common Divisor of two or more numbers is a number that will exactly divide each of them. The Greatest Common 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 prime factors of a number, is a divisor of that 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 the 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 either of two numbers and their difference is a common divisor of the two numbers. 67. Rules. —1. To find the greatest common divisor of two or more numbers by factoring, Resolve the given numbers into their 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 multiple 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. WRITTEN EXERCISES. 11. What is the least common multiple of 12, 18, and 30? Process by Factoring. 12 = $ X $ X 3 18 = 2 X $ X $ 80 = 2X 3 X $ 2X2X3X3X5 = 180, L. C. M. Resolve the numbers into their prime factors, and select all the different factors, re¬ peating each as many times as it is found in any number. The factor 2 is found twice in 12; the factor 3, twice in 18; and the factor 5, once in 30. The product of 2 X 2 X 3 X 3 X 5 is the least common multiple required, since it is the least number which contains all the prime factors of 12, 18, and 30. 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 2X3X5X7X2 = 420, L. C. M. 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 7 X 2 = 420) is the least number that contains each of them, it is the least common multiple of 12, 15, 42, and 70. Process by Division. !) 12 15 42 70 3)6 15 21 35 5)2 5 7 35 7)2 1 7 7 2 1 1 1 What is the least common multiple of 21. 12, 18, 30? 26. 30, 45, 48, 80, 120? 22. 8, 28, 70 ? 23. 9, 20, 15, 36? 24. 15, 24, 25, 30? 25. 18, 21, 27, 36? 27. 16, 30, 40, 50, 75 ? 28. 15, 27, 35, 42, 70? 29. 8, 28, 20, 24, 32, 48? 30. 2, 3, 4, 5, 6, 7, 8, 9? DEFINITIONS, PRINCIPLES, AND RULES. 68 . A Multiple of a number is any number which it will exactly divide. Note. —Every number is an exact divisor of its multiple. C. Ar.—1. 42 COMPLETE ARITHMETIC. A CoiriWlOTl Multiple of two or more numbers is any number which each of them will exactly divide. The Least (Jotyityiou Multiple of two or more numbers is the least number which each of them will exactly divide. Note. —The following definitions may be preferred: A Multiple of a number is the product arising from taking it two or more times. Or, A Multiple of a number is any number of which it is a factor. A Common Multiple of two or more numbers is a multiple of each of them. The Least Common Multiple of two or more numbers is the least 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, talcing each the greatest number of times it is found in any number. The product of the different factors, thus selected, wiU be 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, writing 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 the last result¬ ing numbers will be the least common multiple required. Note. —If no two of the given numbers have a common divisor, their product will be the least common multiple. FRACTIONS. 43 SECTION VIII, FRACTIONS. HALVES SIXTHS NUMERATION AND NOTATION. 1. If an apple be divided into two equal pieces, what part of the whole will one piece be? 2. If an apple 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 (-g-), 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 | express? What does the figure 7, below the line, denote? The figure 5, above the line? Read the following fractions, and tell, in each case, what each figure denotes: 7 & 1 * 5 10. § 13. A Ifi - 9 - 2 0 & 5 O. -g- 11. f 14. A 17. 1? 9. $ 12. i 15. A 18. ff 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 | 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. f 30. U 25. | 27. A 29. if 31. « 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 variety of common fractions. (Art. 112.) $4 FRACTIONS. cn n. 45 r r - ‘ . 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 when its numerator is less than its denominator; equal to 1 when its numerator equals its denominator; and more than 1 when its numerator is greater than its denominator. 75. Common Fractions are Proper or Improper. A Proper Fraction is one whose numerator is less than its denominator; as, f, f. An Improper Fraction is one whose numerator is equal to or greater than its denominator. The value 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 > f § 5 i A Complex Fraction is one having a fraction in 2 5 5 51 one or both of its terms; as, g> ~ 4 5 -g- H A Mixed Number is an integer and a fraction united; as, 5£, 16£. 77. The fraction J- may be considered as expressing 3 fifths of 1 unit, or 1 fifth of 3 units; and hence the numer- 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 fraction may be considered an expressed division, the numerator being the dividend, the denominator the divisor, and the fraction itself the quotient. REDUCTION OF FRACTIONS. Case I. “Whole or Mixed. TsTumlDers reduced to Improper ^Fractions. 1 . How many thirds in an apple? In 4 apples? 7 ap- nles? 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 3it? 4-f? 12-|? 16§? 6 . How many tenths in 5 t V? 8 t 3 7 ? 12 t \? 15^%? WRITTEN PROBLEMS. 7. Reduce 225 to sevenths. Process. 225 _7 1575 , Ans. 7 2251 to sevenths. Process. 225 ^ _7 1580 , Ans. 7 8 . Reduce 324 to ninths. 324J to ninths. 9. Reduce 4&J-J- to 15ths. 65|| to 15ths. 10. Reduce 54^ to 20ths. 135|^ to 30ths. 11. Reduce 63 to an improper fraction. 12. Reduce 74 J A to an improper fraction. 13. Reduce 2063 ^ 1° an improper fraction. REDUCTION OF FRACTIONS. 47 14. Reduce 145* to an improper fraction. Reduce to an improper fraction, 15. 137* 17. 600£f 19. 208* 16. 408* 18. 365* 20. 607* 78. Rules. — 1. To reduce an integer to a fraction, Mul¬ tiply the integer by the given denominator, and write the denomi¬ nator under the product 2. To reduce a mixed number to a fraction, Multiply the integer by the denominator of the fraction, to the product add the numerator, and write the denominator under the remit. Case II. Improper Fractions reduced to Whole or Mixed Numbers. 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 t many units in 36 ninths? 63 ninths? 75 ninths? 25. How many units in - 2 ^? *-? - 6 ^? *-? 26. How many units in f-J ? fj? - 1 * ? - 2 * 3 - ? WRITTEN PROBLEMS. 27. Reduce -**• to a whole number. Process : *«= 256 -4- 16 = 16, Ans . 28. Reduce - 2 * 3 - to a mixed number. Reduce to a whole or a mixed number, 29. 3 24 12 ' 5 6 7 1 2 _ 1 _ 2 80. 81 . 82. 3 4 ’ 1 8 ’ 1 Tf’ o_ 8 ’ 5. 6 > 3 4> 7 1 6 7 9 i 83. 84. 85. 3 S’ 4 5’ 2 3’ 5 6 ’ 9_ 1 0 ’ 3 T> J_3 2 0 ’ 5 6 ’ 1 1 TU 1 1 ¥TT 4 2 1 86 . Reduce f, T 7 ^-, and f J to equivalent fractions hav¬ ing the least common denominator. Process. 5 7 1 1 2 1 J TZ 2? 3Z 60 42 44 63 Z6 ¥6 ZZ 96 The least common multiple of 8, 16, 24, and 32 is 96, and hence 96 is the least common de¬ nominator. Change the fractions to 96ths. 5 6 0. 7_ J Z6 > T6 4 2 . 96 > 1 1 Z¥ 44 . 21 - 6 3 Z 6 > 3 2 THT * Reduce to the least common denominator, 00 • 4 9’ 11 1 2’ 1 7 3S’ If 90. 7 1 2’ 8 2 1’ 1 1 28’ 1 7 4 2 Oo oo h 3 S’ 5 S’ i 91. 2 5’ 1 1 12’ 1 1 10’ 29 3¥» 3 1 6~0 89. 10’ 8 1 5’ 1 1 2 0’ 1 3 3¥ 92. 5 9’ 1 4 3S’ 2 2 45’ 1 2 6 3’ 1 0 1 TIT DEFINITIONS, 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. —The multiplication of both terms of a frac¬ tion by the same number does not change its value. REDUCTION OF FRACTIONS. 51 86, Rules. —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 multiple of the denominators by the de¬ nominator of each f raction, and multiply both terms by the quotient. Case V. Compound Fractions reduced to Simple 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 ^ ^ of 4 ? ^ of £ ? 96. What is \ of |-? \ of f ? 97. What is -J- of ^ ? 4 of f ? § of f ? 98. What is y of i? 7 of |? f of | ? 99. What is | of f? § of -§-? •§ of -J? 100. What is | of -f? f of f ? f of t 7 j? 101. What is i of 12? i of 12£? } of 13* ? Solution .—4 of of 12 , which is 4, -f } of or §, which is $ or 4 -|- \ — 4|. Hence, | of 13| is 4 102. What is \ of 17-i? i of 21|? \ of 33^? 103. What is | of 12? | of 12£? f of I 64 ? 104. What is f of 22^ ? f of 25J ? i of 37£? 105. What is | of 33£? £ of 42£? ^ of 62£? WRITTEN PROBLEMS. 106. Reduce § of of 7-J- to a simple fraction. Process. 5 0 f _3_ of 74 = ^ — 2^. — 5, Ans. 9 10 2 9 X 15 X 2 270 6 Or: 4 of T s j of 7| = £X_J_X_£$_5 ' 9X15X 2 6 3 52 COMPLETE ARITHMETIC. Reduce to a simple fraction, 107. | of J of f 108. f of } of 2J 109. f of ji of ll- HO. f of f of 2| 111 . 112 . 113. 114. of f of f of 3i j of of j- of 4f 3 5 9 7 ,J 3 _4 1 of f Of 2f Of 2| f of y 9 u of yy of 3|- 87. Rules. —To reduce a compound fraction to a simple fraction, 1. Multiply the numerators together for a numerator , cmd the denominators together for a denominator. Or, 2. Indicate the continued midtiplication 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 -=-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 , and $ 7 T ° T °- each to whole or mixed numbers. 120. Reduce §, T 7 y, and || each to 30ths. 121. Reduce 12|, 18J, and 33J each to 12ths. 122. Reduce T 2 - of of 2^3 of 24 to a simple fraction. 123. Reduce |, and ^ to a common denominator; to the least common denominator. 124. Reduce f, 5|-, and f of f to their least common denominator. Suggestion. —First reduce 5| and of f to simple fractions. 125. Reduce f, f of f, and -f of 6 | to their least common denominator. 126. Reduce | of 2 -|, of 3, and ^ of -J of 6 y to their least common denominator. 127. Reduce -J of |, § of 2-J, and -J of 13| to their least common denominator. 128. Reduce -f- of 5|, 2 J of 34, and 2 T 3 y 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 J, •§, and ^? 3. A man traveled ^ of his journey the first day, and \ of it the second day: what part of the journey did he travel in the two days? 4. How many twelfths in J and ^? \ and ^-? 5. A owns f of a vessel, and B f of it: what part of the vessel do both own ? 6. What is the sum of J- and |? f and f? 7. § and ^? f and y^? -§ and -J? and J? 8. -f and ^ and J ? j- and T 9 7 ? f and | ? 9. ^ and 5J? 2J and 6J? 5^- and 6J? 8J and 9f ? 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 yf> and ? Process: -1 4- 1A 4-1® — 9+16 + 10 — 35 — 112 ^ ns . 23 1 23 1 23 23 - 23 23 12 . What is the sum of -ff, |-J, and T ^? 13. What is the sum of f§, J-^, and f J? 14. What is the sum of T y^, and ? 15. What is the sum of f, T 7 2 , and -1-J-? Process. f + T2 + TS — H + tt + *i = H= Hi Am. Since unlike fractional units can not be added, reduce the fraetions-f* -fa, and H 1° a common denominator, and then add the resulting frac¬ tions. 54 COMPLETE ARITHMETIC. 16. 17. 18. 19. 20 . Add f and and yq. ib and and 3. 8 ’ 1 3 1 8 • 1 3 1 5* 31 4 2' _o 6 ’ 3 5 ’ 1 0 ’ A* ib and 26. Add J, f of f, and f of f of 2J. 21 . 22 . 23. 24. 25. 9 1 0’ 9 2 3 2ir> j o > 4 1 Q~U- o 6 ’ A 9’ and and tJ* 1 3 ±5 J2.0 o-rifl 2 IF’ 2 7’ 8 f> , uu 3* 1 _7_ _7 anf l 8 ’ 12 ’ 18 ’ ^4 * and i 12’ _ 1 _ 1 5’ 1 28"’ 1_ ■JO' Process. 2 n f 3 — 2 ? OI x — x of of } = Since § of f = f, and f of | of 2£- = f, the sum of f -f- § of 31215 - 30116125 - 131 1 + 5 + J — loTJo Tfo — -*-40 f + y of of 2}=} + } + }. 27. Add | of |, | of fg- of 2J, and J. 28. Add J of 2^-, f of J, and f of J of 6. 29. Add t 3 4 , y of 5, and -f- of f of -§. 30. Add A- of 2, § of f, 4 of 4 of T 5 o, and 3k 31. Add 33}, 37}, 55J, and 66f. Process. 33| x \ 37 \ xx i=;r: 3 9 oo ¥ x5 - °°3 TS 193 x , Ans. The sum of 33^, 37|, 55|, and 66§, equals the sum of i + y +f + f added to the sum of 33 -f 37-(- 55 -f- 66. 4 + x + f + f = 2 x 3 j or 2|. Write the \ under the frac¬ tions and add the 2 with the integers. The sum is 193i. 32. Add 39}, 56J, 88}, and 104ft. 33. Add 45, 87f, 66§, and 75}. 34. Add 121 i6f, 18J, 30}, 33J, and 62}. 35. Add -§, f of f, 16f, and 48J. 36. Add $5.12}, $3.18}, $8.25, and $3.81}. 37. Add }, ft, } of 5}, and 65ft. 38. Add }, }, J-, ft, and ft 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. 55 89. Rules. — 1 . To add fractions, Reduce t he fractions to a co mmon denominator , add the numerators of the new fractions, and under the sum write the common dejiominator. 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 inn proper fractions and then added. SUBTRACTION OF FRACTIONS. 1. A boy spent f- of his money for a slate: what part of his money has he left? 2. How much is f less § ? f less f ? f less f ? 3. How much is less less T 5 2 ? less T 9 2 ? 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.—C hange f and ^ to twelfths. 5. How much is J less | ? f less § ? f less \ ? 6. f less f ? f less -§ ? ^ less J ? f less | ? 7. ^ less f ? § less f f less f ? § less y ? 8. | less \ ? J less f ? ^ less f ? j less ^ ? 9. ^ less } ? yj less J ? f less ? f less f ? 10. 5£ less 3| ? 6| less 4^? 9f less 7£? 12^ less 6^? 11. Why can not f be subtracted from f without first re¬ ducing the fractions to a common denominator ? WRITTEN PROBLEMS. 12 . Subtract 1 9 fpnm 2 7 TS Irom To- Process : 2 7 35 li 35 27 — 19 35 = A Jns - 13. Subtract from |«; from Jf; || from 14. Subtract yC- from -JJ. Process : = f § — f § = , Ans. 56 COMPLETE ARITHMETIC. How much is 15. 14 _ - v 18. 33 _ _ 11 ? 21. 10 _ ¥7 - 9 ? 47 - 16. 1 3 T7 -A? 19. 19 _ ¥7 -11? T ¥ • 22. 23 _ ¥(J _ 11 ? 54 ’ 17. -A? 20. -i — 12 4 ? “ T7 • 23. 29 _ 70 1 3 ? 45 ’ 24. From J of f take \ of J of f. Process : | of f — f f of f of f = | f — i = Ans. 25. From f of f take § of -§ of -§. 26. From f of 7 take % of f of 7. 27. From § of £ of f take f of J of f. 28. From -f of f of 2^ take 29. From 340J take 247|. Process. 340f * 247| M 92^, Ans . First subtract the fractions and then the integers. Since is greater than -fa, add §§ to making §§, and then subtract from §§, writing the difference, If, under the fractions, and adding 1 (|f) to the 7 units before subtracting the integers. 30. 93J — 46J = ? 33. 241 J — 153yV = ? 31. 56| — 37|- = ? 34. $2.33J—$1.62i = ? 32. 108f—90f = ? 35. $3.12| —$2.48§=? 36. What fraction added to f will make -j-^? 37. What number added to 6f will make 16|? 38. From the sum of § and | take their difference. 39. From f + -f take J- — -J. ~~ ' 40. From f ~h | + t 7 o ta ^ e f °f 1£* 41. From J -J- f take ^ — § of §. 42. From a cask containing 45^ gallons of sirup, a grocer sold one customer 16f gallons and another 21f gallons: how many gallons remained unsold ? 43. A man bequeathed of his property to his wife, T 5 ^ 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 •§ of a factory, sold § of his share: what part of the factory did he still own? MULTIPLICATION OF FRACTIONS. 57 45. Two ninths of a pole is in the mud, f- of it in the water, and the rest of it in the air: what part of the pole is in the air ? 46. The part of a pole broken off by the wind was f of the whole pole, and f of the part still standing was above the ground: what part of the pole was in the ground? PRINCIPLES AND RULES. 90. Principles. —1. The minuend and subtrahend must de¬ note like fractional units. Hence, 2. Fractions must have a common denominator before their difference can be found. u ■ — 91. Rules.— 1. To subtract fractions, Reduce the fractions to a common denominator , subtract the numerator of the subtra¬ hend from the numerator of the minuend , and under the differ¬ ence write the common denominator. 2. To subtract mixed numbers, Subtract first the fractions , and then the integers , and unite the results. Notes.—1. Compound fractions must be reduced to simple fractions before they can be subtracted. 2. When mixed numbers are small they may be reduced to im¬ proper fractions, and then subtracted. MULTIPLICATION OF FRACTIONS. Case I. Fractions multiplied, "by Integers. 1. How much is twice 2 ninths of an inch? 4 times 2 ninths of an inch? 2. If a basket hold f of a bushel, how many bushels will 8 baskets hold? 10 baskets? 3. How much is 8 times f ? 10 times f ? 20 times f ? 4. 6 times f? 8 times J? 9 times ^? 12 times -J? 5. 7 times {-£? 9 times T 4 -? 8 times ^< 5 ? U times §? 6 . 6 times 5^? 9 times 6 J? 7 times 12-J-? 10 times 7-f? 7. 8 times 12^-? 6 times 16J? 5 times 33^? 7 times 30i ? 58 COMPLETE ARITHMETIC. 8. Why does multiplying the numerator of T \ by 3 multiply the fraction by 3 ? 9. Why does dividing the denominator of ^ by 3 mul¬ tiply the fraction by 3 ? 10. In how many ways may a fraction be multiplied by an integer? WRITTEN PROBLEMS. Multiply 11. by 9. 12. if by 10. 13. by 24. 14. H by 45. 15. -fa by 12. 16. by 16. 17. by 60. 18. Hi by 25. 19. 62J by 36. 20. 45| by 80. 21. $5.18f by 32. 22. $66f by 52. PRINCIPLE AND RULES. 92. Principle. —A fraction is multiplied by multiplying its numerator or dividing its denominator. 93. Rules. —1. To multiply a fraction by an integer, Multiply the numerator or divide the denominator. 2. To multiply a mixed number by an integer, Multiply the fraction and the integer separately, and add the products. Case II. Integers multiplied by- Fractions. 23. If a ton of hay cost $16, what will f of a ton cost? f of a ton ? 24. If an acre of land is worth $50, what is i an acre worth ? | of an acre ? 25. What is i of 42? f- of 42? £ of 42? 26. What is f of 56 ? f of 56 ? J of 56 ? 27. -f of 63? * of 84? H of 99? $ of 56? Solution.— ^ of 56 = 6§, and | of 56 = 7 times 6§ = 43§. 28. | of 66? i of 66? H of 66? f of 74? MULTIPLICATION OF FRACTIONS. 59 29. What is 16 xf? 50 Xf? 42 Xf? Solution. —Since f = § of 1, 16 X f = f of 16 X 1 — I of 16 = 12. 30. 57 Xf? 75X|? 87 X *? 95 X*? 76 X|? 31. 47 X|? 68 Xi? 75 X f? 83 X*? 100 X A? 32. Show that the product of an integer by a fraction equals the fraction of the integer. WRITTEN PROBLEMS. 33. Multiply 654 by . Process. 12 ) 654 654 Since T r 2 = 7 times or of 7, 54j Q r . 7 . the product of 654 Xfi=7 times 7 12 ) 4578 T J 2 0 f 654^ or ^ 0 f 7 times 654. 3811 , Ans. 381 1 2 34. 66 by 1 9 • 37. 784 by 13 40* 40. 757 by 1 T* 35. 58 by 1 3 TT- 38. 648 by 2 5 2 6* 41. 908 by } of 21 36. 92 by 7 22* 39. 564 by 3 3 40* 42. 588 by T5 °f 43. Multiply 256 by 27£. 406 by 33f. Suggestion. —Since 256 X 27$ 256 x 27 + 256 X f , first mul tiply by the integer and then by the fraction, and add the products. 44. 66 by 8}. 47. 645 by 12J. 50. 745 by 60|. 45. 72 by 9f. 48. 465 by 18f. 51. 385 by 45 x 4 46. 96 by8 T %. 49. 406 by 33J. 52. 708 by 60f. PRINCIPLE AND RULES. 94. Principle. —The product of an integer by a fraction equals the fraction of the integer. Thus, 5 X| = t of 5. 95. Rules. —1. To multiply an integer by a fraction, (1) Divide the integer by the denominator, and multiply the quotient by the numerator. Or, (2) Multiply the integer by the numerator, and divide the product by the denominator. 2. To multiply an integer by a mixed number, Multiply by the integer and the fraction separately, and add the products. 60 COMPLETE ARITHMETIC. Case III. Fractions multiplied ‘by UTractions. 53. What is } of 1? } of J? -| of f ? 54. | of |? | off? f of }? f of f? 55. What is f X f ? f X f ? |Xf? Suggestion.— f X f = f of f; | X y = I of f, etc. 56. fX*t fX}}? }X*? 57. What is } of 12}? } of 13}? } of 13}. Solution.— | of 13^ = j of 12 + 1 °f 1| = 4 + A — 4jz > add I of 131 — ^ times 4 r \ = 8f. 58. f of 16f ? f of 22}? f of 42}? ^ of 62}? 59. } of 37}? f of 42}? } of 65}? -fy of 100}? 60. Show that f X f- = f of f. WRITTEN PROBLEMS. 61. Multiply -}§ by f. IIX Process. 3 — T — 13 X 3_I ., 15X4 Ans. 62. 13 Ev 11 nr D y T3* 66 . 63. 10 Ev 2 2 TT °y 2 T- 67. 64. 19 Ev 15 F5 °y 3¥' 68 . 65. 10 Ev 3 9 if °y nr- 69. Since f = } of 3, the product of rt X f = 1 of 3 times }f = } of 13 X 3 _ 13 X 3 15 15 X 4 1 3 SO* f by i of £• £ °f A £by| by A- by £• 2 i by A of 70. 2} by 3}. 71. 4} by 5}. 72. 6} by 3f. 73. 10} by 2-J. 74. What will f of a yard of cloth cost at $f a yard? At a yard? 75. What will 5} pounds of flour cost at 4} cents a pound ? At 61 cents a pound ? 76. What will 2} pounds of tea cost at $lf a pound? At $1} a pound ? 77. What is the cost of 35 barrels of flour at $6} a barrel? At $7} a barrel? DIVISION OF FRACTIONS. 61 78. A man owned ^ of a ship which was sold for $13250: what was his share of the money? 79. What is the product of § of 2 \, f of -fa of and 2-g- ? 80. What will 12-J- pounds of butter cost at 18f cents a pound ? At 22^ cents a pound ? PRINCIPLE AND RULES. 96. Principle. — The product of a fraction by a fraction equals the fraction of the fraction. Thus, f X f = f of f. 97. Rules. —1. To multiply a fraction by a fraction, Multiply the numerators together, and also the denominators. 2. To multiply a mixed number by a mixed number, Reduce the mixed numbers to improper fractions, and proceed as above. Notes.—1. Mixed numbers may be multiplied together by first mul¬ tiplying the integers ; next multiplying each integer by the fraction united with the other integer; next multiplying the two fractions ; and then add¬ ing the four products. Thus, 18§ X 12^ = 18 X 12 + 18 X \ + 12 X f + | X hut in most cases it is shorter to reduce the mixed numbers to improper fractions. 2 . Cases I and II may be included in Case III, by changing the integer to the form of a fraction. Thus, f X 5 = f X f> an d 8 X f = 8 v 3 T A z- 3. The process of multiplying fractions may be shortened by can¬ cellation. Compound fractions need not be reduced to simple frac¬ tions, since f X I of If = f X 1 X rr • DIVISION OF FRACTIONS. Case I. Fractions divided Toy Integers. 1 . If a man can do of a piece of work in 3 days, how much can he do in 1 dav? %/ 2. A man divided f of a farm equally between 4 sons: what part of the farm did each receive? 3. If 5 yards of muslin cost £ of a dollar, what will 1 yard cost? 62 COMPLETE ARITHMETIC. 4. If 10 oranges cost -§ of a dollar, what will 1 orange cost? 5 . If 8 bushels of oats cost $2§, what will 1 bushel cost ? 6 . If ff of a melon be divided into 5 equal parts, what will each part be? 7. Why does dividing the numerator of f by 4 divide the fraction by 4? 8 . Why does multiplying the denominator of f by 4 divide the fraction by 4? 9. In how many ways may a fraction be divided by an integer ? WRITTEN PROBLEMS. 10. Divide by 6. 1 2 2l> • ° Process. 12-f-6 25 = &, Ans - Or: if-+-6 12 25 X 6 2 2 V Giyipp 12 * “I ■ 12 12 oince ■ -L — ss, 1 2 2S 12 + 6 25 ’ or 12 25X6 + 6 = f of Or, since to divide a number by 6 is to find f of of it, if 6 — e 12 iof if 12 -f- 6 25 -, or 25X6 Divide 11 . ff by 8. 12. -if by 7. 13. ff by 11. 14. if by 15. 15. if by 20. 16. iff by 25. 17. 2f by 8. 18. 5-J by 12. 19. 6J- by 10. PRINCIPLE AND RULES. 98. Principle. —A fraction is divided by dividing its nu¬ merator or multiplying its denominator. 99. Rules.— 1. To divide a fraction by an integer, Di¬ vide the numerator or multiply the denominator. 2. To divide a mixed number by an integer, (1) Reduce the mixed number to an improper fraction and divide as above; or, (2) Divide the integral part and then the fraction , and unite the quotients. DIVISION OF FRACTIONS. 63 Case II. Integers divided by Fractions. 20. How many times is -J 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. --. - 2x. 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 J contained in 8? J in 12? § in 9? | in 15? | in 9? £ in 12? 25. How many times is | contained in 12? -J in 15? 26. Show that 8 ~~ f = . o WRITTEN PROBLEMS. 27. What is the quotient of 25 -s- £ ? Process : 25 -5- £ = — ^ = 28f, Ans . Note. —It will be noticed that the integer is multiplied by the denominator of the fraction and the product divided by its nu¬ merator. What is the quotient of 28. 21-j-*? 31. 100-f-|^? 34. 75-r-6£? 29. 42-s-tt? 32. 96--££? 35 * 120 -*-3£? 30. 72 -8-If? 33 - 125-*-ft? 36. 225-s-5}? 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 divided, "by- Fractions. 37. How many times is -f of an inch contained in | of an inch ? -J of an inch in -f of an inch ? 38. How many times f in f ? -f in -| ? J- in ^°- ? 39. How many times -f in -J? | in | in 4^? tj- in ¥? A m*? *inH» 40. How many times is | contained in | ? | in -§? Suggestion. —Change the fractions to twelfths. 41. How many times -§ in -f? f in f- ? -J in -J? | in T \? 49 in 4 9 3. in _7_ ? __3_ i n 14? 2 i n 3 ? 2. i n A? “iZ. i 0 m "o • 8 1 2 • 1 0 111 T 5 • 3 111 7 * 5 111 8 * 43. Show that the quotient of two fractions having a com¬ mon denominator, equals the quotient of their numerators. WRITTEN PROBLEMS. 44. Divide J Process : ,1- Since 7X5 » and | 3X8 \ 7 > 8 3 7 7X5 . 3 X 8 _ 7 X 5 40 a 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, arid the product of the denominators for the denominator, is the saihe as reducing the frac¬ tions to a common denominator, and dividing the numerator of the dividend by the numerator of the divisor. 7X5 Note. —That | -4- f = n may also be thus explained: f times |, and since f-4-| = 8X3 7X5 8 > 8 = 4 of What is the quotient of 45. 46. if 47. 9 TV 48. H--?-A? I 4 9 T~5 • . ? II • If 49. 50. 51. 52. 3* 54 16| -5- 24? 3|? 6|-v-12|? 3|? 7 X 5_7 X 5 8 8X3* = 3 53. 54. |of^-v-|of 4? 55. |L.of3|-^|of2J? 56. ||-i 0 ffof3|? DIVISION OF FRACTIONS. 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 $f, how many bushels can be bought for S6-J-? For $9f ? 59. If 13 yards of silk cost S17-J-, how many yards can be bought for $48f ? For $62^-? 60. If a man walk 3f% miles an hour, in how many hours will he walk 20^ miles? 61. At $33^ an acre, how many acres of land can be bought for $841f ? 62. By what must f be multiplied that the product may be 26J ? 63. Divide the product of 6^- multiplied by 3^ by the quotient of 4^ -r- 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. Be- d uce the fractions to a comm on denominator, and divide the numerator of the dividend by the numerator of the divisor. Or, 2. Invert the terms of the divisor, and then multiply the numerators together and also the denominators. Or, 3. Multiply both dividend and divisor by the 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, the l. c. m., t iV— 15 -s- 14 = ; multiplying by 6, the l. c. m., 6f-r-5| = 40 -4- 33 = 1 gtj-. Compound fractions should first be reduced to simple fractions. 2. It is not 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.—6. 66 COMPLETE ARITHMETIC. COMPLEX FRACTIONS. 4 64. Reduce the complex fraction to its simplest form. T Process : | = f -s- f = = xf> ^ ws - 7 0 X O Reduce to the simplest form 65. 3 8 69. iet -4 CO • 5 of 6 6 U1 7 77. A 4- 8 3 T¥ 9 T¥ 25 f oi 't 3 _ 8 3 T¥ 66. f 70. 25 74. *of2* 78. 5 T ¥ 24 let 3 _L ¥ 67. 15 5 71. 12* 5 75. 3 of A 8 UA 9 5 79. 7 fx ¥ 1 9 ¥ 1 2 4 6 68. 72. 5 ¥ 76. 6, _ 1 1 80. 2 9. 3 ¥ H 12-j 4 n ** 1 4 T 01 ¥¥ 3 * ¥ 5 9* 103. A complex fraction is simply an expressed division, the numerator being the dividend and the denominator 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 PROBLEMS. 1 . If ^ of a barrel of flour cost $3, what will a barrel cost ? 2. If -J- of a ream of note paper cost 75 cents, what will a ream cost ? 3. Charles gave Henry 7 marbles, which were -J- of all he had: how many marbles had Charles ? FRACTIONS. 67 4. 15 is } of what number? 5 . 16 is ^ of what number? 6 . 12^ is J of what number? 7 . 16J is y 1 ^ of what number? 8 . 22f is -J- of what number? 9. 24 is f of what number? Solution.— If 24 is f of a number, ^ is | of 24, which is 12. If 12 is ^ of a number, f is 5 times 12, or 60. Hence, 24 is f of 60. 10. 27 is -J of what number? 11. 45 is of what number? 12. 64 is of what number? 13. 27^- is -J of what number? 14. 46f is of what number? 15. 37J- is -§ of what number? 16. 87-J- is ^ °f what number? 17. 45 is y of how many times 9? 18. 63 is } of how many times 12? 19. 80 is of how many times 20? 20. 108 is }} 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. \ is what part of f? J of J? } of } ? -J of -J? 97 1 1 ? 2 3 ? 3 5 ? 4 n f 9 ? 5 Af 11? L • • Uf 01 ¥ * ¥ 01 ¥ • ¥ 01 ¥ • T 01 T¥ • "6 01 T2 * 28. | of 11? f of 4? f- of 10? f of 8? j- of 10? 29. 5i of 16|? 6| of 331? 121 of 37|? 33i of 16|? 30. 3| of 6|? 5i of 2J? 21 of 3i? 6J 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 . 68 COMPLETE ARITHMETIC. REVIEW OF FRACTIONS. MENTAL PROBLEMS. 1. A boy having gave for a knife: how much money had he left? 2. If ^ be added to a certain fraction, the sum will be : what is the fraction ? 3. A laborer spends -§ of his wages for board and -J 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 J- 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 -f- of his age equals the age of his son: how old is the son ? 7. A man bought a cow for $33J 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 16f years, and Mary’s age is |- of Jane’s: how old is Mary? 10. A man owning f of a mill sells J- 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 J- of what remained: what part of a pound had he left? 12. A wife is 35 years of age, and her age is f of the age of her husband: how old is her husband? 13. The difference 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 f 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 yy of what number? 21. A man is 45 years old, and f- of his age is -f- of the age of his wife: how old is his wife? 22. Samuel is § as old as Charles, and Harry, 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 62 years: what is A’s age ? 25. A farmer’s sheep are in 4 fields; the first contains -J of all, the second |, the third and the fourth 52 sheep: how many sheep in the 4 fields? 26. A saddle cost $35, and f of the cost of the saddle was J- of the cost of a bridle: wdiat 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, ff of which is f 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 -| of the value of a house equal 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 equal f of B’s, and both together have $340, how much has each? 70 COMPLETE ARITHMETIC. 32. If | of A’s age is f of B’s, and -J of B’s is 20 years, what is the age of each ? 33. If -f- of a yard of velvet cost $>2f, what will of a yard cost? 34. How many pounds of honey, at $f a pound, can be bought for $3? 35. How many bushels of apples, at $f a bushel, can be bought for $16§ ? 36. If a barrel hold 2f bushels, how many barrels will be required to pack 55 bushels of apples? 37. If lb. of sugar cost $1, how much will 49 \ lb. cost ? 38. If f of a yard of silk cost $1^, how many yards can be bought for $10J? 39. If 3f yards of cloth cost $5^, what will 6J 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-J- 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 lOf 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 w^ork 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 •J as much as B: what part of the field can each mow in 1 day? How long will it take each to mow the field? 49. IIow is the value of a proper fraction affected by REVIEW PROBLEMS. 71 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, affected by adding the same number to both of its terms? By subtracting the same number? (Illustrate.) WRITTEN PROBLEMS. 51. Add |, |, of and 3f 52. From -§ of If take f of ff. 53. From the sum of 27f and 20f take their difference. 54. Multiply by 35; 35 by -ff; by ff; 3f by 2f. 55. Divide ff- by 32; 32 by ff; ff by f; 4f by 3f. 56. # + * = what? xi — t 7^? fix*? 57. Multiply 2045f by 35; 806 by 84|; 30f by 16J-. 58. Divide 347f by 15; 692 by 21f; 19f by 16f. 59. A farm is divided into five fields, containing respect¬ ively 21f A., 34-J A., 45 J A., 56f A., and 29f A.: how many acres in the farm? 60. There are 30f sq. yd. in a square rod: how many square rods in 786f sq. yd.? 61. A man travels 5f miles an hour: how long will it take him to make a journey of 75| miles? 62. At $8f a ton, how many tons of hay can be bought for $108f? 63. If ^ of an acre of land cost $68, what will 12f acres cost ? 64. If f of a yard of velvet cost $8f, how many yards can be bought for $196f? 65. If a number be diminished by f of f-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 ff of his income: what is his income ? 68. A man bequeathed to his wife $4860, which was f§ of his estate: what was the value of the estate ? 72 COMPLETE ARITHMETIC. 69. A graded school enrolls 208 boys, and of the pupils are girls: how many pupils are enrolled in the school ? 70. A man owning 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 -§ 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 ^ of a section of land; B, ^ of a section; and C, T % as much as both A and B: what part of a sec¬ tion does C own? 73. A bought | of a factory for $21840, and sold f of his share to B, and J- 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 f of B’s: how many acres does each own ? 75. A stock of goods is owned by three partners, A own¬ ing f, B 3 ^, and C the remainder; the goods were sold at a profit of $6160: what was each partner’s share? 76. -§ 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, 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 which he wishes to cut an equal number of coats, pants, and vests: how many of each can he cut if they contain 3f, 2J, and 14 yards respectively? 79. An estate was divided between two brothers and a sister; the elder brother received •§ of the estate, the younger j^-, and the sister the remainder, which was $450 less than the elder brother received: what part of the estate did the sister receive? What was the value of the estate? What was each brother’s share? DECIMALS. 73 SECTION IS. 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 y^? 3. If a hundredth of a unit be divided into ten equal parts, what is one part? What is y 1 ^ of -yj-g-? 4. What part of a tenth is a hundredth ? What part of a hundredth is a thousandth? 5. How do the fractions y|y, and yg 3 0 -y compare with each other in value? -j^, yjy, and y&W? 106 . Since the fractional units, tenths, hundredths, thou¬ sandths, etc., decrease in value like the orders of integers, they can be expressed 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., and placing a period at the left of the order of tenths. Thus, -fa is written .5; y§y is written .05; yy 5 0 y is written .005, etc. Copy and read (6) ( 7 ) (8) ( 9 ) (10) (11) .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? .63? .78? .84? .69? .39? C.Ar—7. In .45? 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 -j 1 ^? In yfy? .34? .42? 16. How many thousandths in y§-§ 7 ? In y^ftnr? .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 be expressed decimally. Express decimally (22) (23) (24) (25) (26) 3 4 4 5 7 5 1 8 TIT 1 OUT 100T ITT TTTT A 6 TOOT 6 3 1 0 0 O' TTTT 208 1 COT . 6 ttt 1 4 iooo 2 15 lTTT T?TT 355 lTTT 8 TTT 56 TTT 40 7 lTOT TFOT 4 3 TTTT 1 2 40 500 1 0 6 5 TOT TOT lTTT TTTT lTTT 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) (30) (31) (32) .246 .0635 .00647 .0307 .0246 .00635 .000647 .03007 .708 .3464 .04056 .030007 .0708 .03464 .004056 .034005 .3425 .32875 .32453 .450605 109 . When a decimal fraction is expressed decimally, the right-hand figure is written name of the decimal. Thus, Express decimally (33) (34) Tihr 6 1OO00 -iwu 3 3 i oxnnr 8 40 5 unnnr i Tnnr 3042 1 000 0 3 a 6 TT700 50 0 7 1 00 01 Express decimally (37) 7 tenths; 24 hundredths; 29 thousandths; 405 thousandths; 65 millionths; 5064 millionths; 40056 millionths. in the order indicated by the 3 2 5 100000 lto written .00325. (35) (36) 7 L 29 100000 10 0 OTTUTF 3 7 6 0 9 1 0 OWO 1 ao 0 ootf 208 4 04 5 1 0 0 0 OlT TUoooinr 3056 lFOOOTF 3 3 0 3 3 TU 0 0 070 38045 100000 2 0 4 0 5 6 ltfOOOFCT (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. 76 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 63 % or 6.5; 253 - 5 - 3 - 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. Read 45.6; 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 56 T fo 5 604 T f -fa ; 400^°^. 56. Write decimally 207^^; 2560 T o%°o 5 oW 57. Write 300 units and 348 millionths. DEFINITIONS, PRINCIPLES, AND RULES. 111 . A Decimal Fraction is a fraction whose de¬ nominator is ten or a product of tens. The word decimal is derived from decern, a Latin word meaning ten. It is applied to this class of fractions because they arise from the division of a unit into tenths, as tenths, hundredths, thou¬ sandths, etc. Such a division of a unit is a decimal division, and the resulting parts of the unit are decimal parts. Note. —The decimal denominators are 10, 100, 1000, etc. They are powers of ten. (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, T 3 y, y 1 ^. DECIMAL FRACTIONS. 77 3. By omitting the denominator and writing the fraction in the decimal form, or decimally; as, .3, .012. The de¬ nominator is understood. Note. —Three tenths, r 3 o, and .3, each express the same decimal fraction, which is the thing expressed, and not its expression. A decimal fraction can be expressed orally , and is so expressed when read or dictated. When expressed in words, written or oral, the decimal form is not used. It is an error to teach that a decimal fraction depends on the manner of its expression. 113. The Decimal Point is a period placed at the left of the order of tenths, to designate the decimal orders. 114. A Complex Decimal is a decimal ending at the right with a common fraction; as, . 6 |-, .033J. 115. A Mixed Decimal Number is an integer and a decimal written together as one number. It is called more simply a Mixed Decimal. 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 following table gives the names of a few in¬ tegral and decimal orders, and shows the relation between them: m 000000000 . 00000000 Integral Orders. Decimal Orders. 117. Principles.— 1 . The denominator of a decimal frac¬ tion is 1 with as many ciphers annexed as there are decimal places in the fraction. 78 COMPLETE ARITHMETIC. 2. Ten units of any decimal order equal one unit of the next order at the left. Hence, 3. The removal of a decimal figure one place to the right divides its value by 10, and its removal one place to the left multiplies its value by 10. 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 would be 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 plac,e 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 five ten-thousandths. 59. Forty thousand thirty-four millionths. 60. Two thousand four hundred-thousandths. 61. Six hundred fifteen ten-millionths. 62. Six hundred units and fifteen ten-thousandths. 63. Fifteen units and fifteen thousandths. 64. Three hundred thousand three hundred thirteen hun¬ dred-millionths. 65. Five million eighty-five ten-millionths. 66 . Twelve hundred-thousandths. 67. Four hundred units and four hundred and sixty-five millionths. /- 68 . Twenty-five units and twenty-five thousandths. 69. Five thousand units and five thousandths. 70. Three hundred and seventy-five units and three hun¬ dred and seventy-five billionths. REDUCTION OF DECIMALS. 79 71. Thirty thousand forty-six hundred-thousandths. 72. One million forty-five billionths. 73. Eighty thousand and forty units and three hundred and six ten-thousandths. 74. Fifteen thousand units and fifteen ten-thousandths. 75. Seventy-five units and five thousand and forty-three millionths. 76. One million units and one millionth. REDUCTION OF DECIMALS. Case I. Decimals reduced to Dower or Higher Orders. 1. How many tenths in 6 units? In 15 units? In 24 units ? 2. How many hundredths in 5 tenths ? In .6? .8? .7? 3. How many thousandths in .06? In .24? .47? .55? 4. How many tenths in .60? In .70? .90? .600? .700? .800? .5000? 1.50? 5. How many hundredths in .240? In .420? .560? .4500? .8500? .35000? .0700? WRITTEN PROBLEMS. 6 . Reduce .875 to millionths. Process : .875 = .875000 7. Reduce .0674 to ten-millionths. 8 . Reduce .075 to hundred-thousandths. 9. Reduce 62.7 to thousandths. 10. Reduce 5.33 to ten-thousandths. 11. Reduce 3. to hundredths. 12 . Reduce 45. to ten-thousandths. 13. Reduce .04500 to thousandths. Process : .04500 = .045 14. Reduce 5.24000 to hundredths. 80 COMPLETE ARITHMETIC. 119. Principles. —1. Annexing ciphers to a decimal frac¬ tion multiplies both of its terms by the same number, and hence does not change its value. (Art. 85.) 2. Cutting off ciphers from the right of a deci mal fr actipxk. divides both of its terms by the same number, and hence does not change its value. (Art. 81.) Note. —The annexing of decimal ciphers to an integer does not change its value. Thus, 12. = 12.0, or 12.00; that is, 12 units = 120 tenths = 1200 hundredths, etc. Case II. Decimals reduced to Common Fractions. 15. How many fifths in T \? -j%? .2? .8? 16. How many fourths in y 2 ^-? -j^nr • *25? .50? .75? 17. How many twentieths in -jVf? iVf? *20? -25? .55? .75? .95? WRITTEN PROBLEMS. 18. Reduce .625 to a common fraction in its lowest terms. Process: .625 = x % 2 ^ = = f, Ans. Reduce to common fractions in lowest terms 19. .125 20. .75 21. .075 22. .0625 23. .1625 24. .2250 25. .004 26. .5625 27. .0125 28. .3525 29. 3.525 30. 37.75 31. 62.025 32. 37.625 33. 56.371 34. 247.331 35. 16.66f 36. 214.00^ 120. Rule. —To reduce a decimal to a common fraction, Omit the decimal point and supply the denominator, and then reduce the fraction to its lowest terms. Note. —When the denominator is written the fraction is both deci¬ mal and common. REDUCTION OF DECIMALS. 81 Case III. Common. Fractions reduced to Decimals. 37. How many tenths in -J? In f? f? 38. How many hundredths in f? f? f? 39. How many hundredths in -^g-? A? A? 40. How many hundredths in A? A? A? A? WRITTEN PROBLEMS. 41. Change yfy to a decimal. Process. Since — tI? °f 3, and since 3 = 3.000 125 ) 3.00 (.024, Ans. (Art. 119, Note), T \ z of 3 = As of 3.000 = 2 50 .024. Or, ylx —tts of 3 units, and 3 units 500 = 3000 thousandths, and of 3000 tliou- 500 sandths = 24 thousandths = .024. Reduce to decimal fractions 42. 5 "S' 4^ 00 • 32 T5 54. 1 3 TO 60. i^A 43. 9 T6 - 49. 8 7 ft 55. 7 ToTT 61. 25yf 44. 3 To 50. 1 2 1 2T 56. 2 3 2lT(7 62. Q71 3 ^'t<7 45. 2 5 3 2 51. A 57. 4 T2T0 63. 5 irtr o 4G. 64 T2 5" 52. A 58. 1 Tiro 64. 23 3 017 47. 80 TT5 53. 1 9 M 59. 2 1 4817 65. 1 4 11 1 121. Rule.—T o reduce a common fraction to a decimal, Annex decimal ciphers to the numerator and divide by the denominator, and point off as many decimal places in the quo¬ tient as there are annexed ciphers. Notes. —1. When a sufficient number of decimal places is obtained, the remainder may be discarded, or the quotient may be expressed as a mixed decimal. 2. When the denominator of a common fraction in its lowest terms contains other prime factors than 2 and 5, the process will not terminate. 3. When the quotient repeats the same figure, or the same set of figures, as in problems 63, 64, and 65, it is called a Repeating Decimal, or a Circulating Decimal, and the figure or figures repeated are called a Rcpetend. (Art. 431.) 82 COMPLETE ARITHMETIC. ADDITION OF DECIMALS. 1 . Add 16.25, 48.037, 90.0033, and .864. Since only like orders can be added (Art. 27), write the figures of the same order in the same column. Since ten units of any order equal one unit of the next higher order, begin at the right and add as in simple numbers. Place the decimal point at the left of the 1 tenth. 2. Add .375, 80.06, 45.0084, .00755, and 84.635. 3. Add 84.08, 16.075, 2.9, 1.96, 1.003, and 5.0008. 4. Add $15.34, $65,048, $9,083, $12., $16.66}, $18.06, $95.37|, and $35.75. 5. Add 26.371, 19.081, 23.042}, 38.5, 6.00}, and 7^. 6 . Add 256 thousandths, 3005 millionths, 207 ten-thou¬ sandths, 34 ten-millionths, and 94 hundred-thousandths. 7. Add fifteen thousandths, eighty-one ten-thousandths, fifty-six millionths, seventeen ten-millionths, and two hun¬ dred and five hundred-thousandths. 8 . How many rods of fence will inclose a field, the four sides of which are respectively 46.6 rd., 50.65 rd., 24.33} rd., and 27 rd. ? 9. Five bars of silver weigh respectively .75 lb., 1.15 lb., .86} lb., 1.34 lb., and .9 lb.: what is their total weight? 10. The average amount of rain in San Francisco in the winter months is 11.25 inches; in the spring, 8.81 inches; in the summer, .03 inches; and in the autumn, 2.75 inches: what is the amount for the year? 122. Rule. —To add decimals, 1. Write the numbers so that figures of the same order shall stand in the same column. 2. Add as in the addition of integers, and place the decimal point at the left of the tenths’ order in the amount. Note.— If a mixed decimal does not contain as many decimal places as either of the other numbers, change the terminal common fraction to a decimal, and continue the division until the requisite number of decimal places is secured. Process. 16.25 48.037 90.0033 •864 155.1543, Ans. SUBTRACTION OF DECIMALS. 83 SUBTRACTION OF DECIMALS. 1. From 47.625 take 28.7. 1st Process. 47.625 28.700 18.925 2d Process. 47.625 28.7 18.925 Reduce the decimals to a like order (Art. 119), and since units can only be taken from like units, write the numbers so that figures of the same order shall stancTTn 2. From 46.7 take 29.825. the same column; and since ten units of any decimal order equal one unit of the next higher order, subtract as in simple numbers. Place the decimal point at the left of the tenths’ order. 2d Process. 46.7 29.825 16.875 1st Process. 46.700 29.825 16.875 Note.— A comparison of the two processes shows that it is unnec¬ essary to fill the vacant orders with ciphers. 3. From 4.05 take 2.0075. 4. From .6J take .0087^-. 5. From 12. take .0005. 6 . From six tenths take six thousandths. 7. From forty-four thousandths take forty-four millionths. 8 . From 301 ten-thousandths take 4005 millionths. 9. From 50065 ten-millionths take 1307 billionths. 10. A man walked 33.7 miles the first day and 28.75 miles the second: how much farther did he walk the first day than the second ? 11 . The average amount of rain at Cincinnati in the summer months is 13.7 inches, and in the winter months it is 11.15 inches: what is the difference? 12. The mean height of the barometer at Boston is 29.934 inches, and at Pekin it is 30.154 inches: what is the difference? 123. Rule.— To subtract decimals, 1. Write the numbers so that figures of the same order shall stand in the same column. 2. Subtract as in the subtraction of integers , and place the decimal point at the left of the tenths' order in the remainder. 84 COMPLETE ARITHMETIC. MULTIPLICATION OF DECIMALS. 1. How much is 7 times T *g? 7 times T 4 g? 8 times 2. How much is 8 times yj-g ? 8 times y^-g ? 6 times yf-g ? 3. What is the product of ^ X T V txt X A? t 8 tt X T V 4. What is the product of y 1 ^ X -^g-g? t 4 o X rgir? 5. What is the product Of -y^-g by yyg? yjg by yf-g? 6 . What is the denominator of the product when tenths are multiplied by units? Tenths by tenths? Tenths by hundredths ? Hundredths by hundredths ? Hundredths by thousandths ? 7. What is the denominator of the product of any two fractions whose denominators are powers of 10? WRITTEN PROBLEMS. 8 . Multiply .625 by .23. Process. .625 .23 1875 1250 .14375 Since .625 = Amy, and .23 = T 2 g 3 g, .625 X -23 = T 6 ^ 2 o 5 g X T 2 o 3 g = tWoVV = .14375. Hence, .625 X .23 = .14375. Since thousandths multiplied by hundredths produce hun¬ dred-thousandths, the product contains Jive decimal places, or as many as both of the factors. Multiply 9. 6.5 by .75 14. 4.36 by .27 19. .085 by • O CO - 10. .043 by 6.5 15. 64. by .032 20 . 2.56 by 250. 11 . .0432 by 5.4 16. 30.3 by .018 21 . 3.24 by .33^ 12 . .048 by 24. 17. .056 by 24. 22 . 5.75 by 8 3 13. 5.6 by .056 18. 50. by .08 23. 16J by .045 24. Multiply sixteen thousand by sixteen thousandths. 25. Multiply 205 millionths by 46 thousandths. 26. Multiply 6.25 by 10. By 100. Since the removal of a decimal figure one place to the left multiplies its value by 10 (Art. 117, Pr. 3), the removal of the decimal point one place to the right multiplies 6.25 by 10, and the removal of the point two places to the right multiplies 6.25 bv 100. 0 Process. 6.25X 10 =62.5 6.25 X 100 = 625. DIVISION OF DECIMALS. 85 27. Multiply 3.406 by 100. By 1000. 28. Multiply .00048 by 1000. By 100000. 29. Multiply .0000256 by 10000. By 1000000. PRINCIPLES AND RULES. 124. Principles. —1. The number of decimal places in the product equals the number of decimal places in both factors. 2. Each removal of the decimal point one place to the right, multiplies the decimal by 10. 125. Rules. —1. To multiply one decimal by another, Multiply as in the multiplication of integers, and point off as many decimal places in the product as there are decimal places in both multiplicand and multiplier. Note. —If there be not enough decimal figures in the product, supply the deficiency by prefixing decimal ciphers. 2. To multiply a decimal by 10, 100, 1000, etc., j Remove the decimal point as many places to the right as there are ciphers in the multiplier. Note. —If there be not enough decimal places in the product, supply the deficiency by annexing ciphers. DIVISION OF DECIMALS. 1. How many times are 5 tenths contained in 10 tenths? 7 tenths in 35 tenths? 2. How many times are 7 hundredths contained in 21 hundreths? 7 hundredths in 35 hundredths? Q Whnt i~ 9.39 27. 99 7 o. u nai is Tir — TTr r too - ~ uni • To' 4. What is . 8 - 1 -.4? .21--.07? .084--.012? 5. What is -- yfy ? toW ? Suggestion. —Reduce the fractions to a common denominator. . 25 9 • 1 0 0 0 • 15 ? nnnr* 6 . What is .3-4-.15? .25--.125? . 12 --. 012 ? 7. Of what order is the quotient when tenths are divided by tenths? Hundredths by hundredths? Thousandths by thousandths ? 86 COMPLETE ARITHMETIC. 8 . Of what order is the quotient when any order is divided by a like order ? When any number is divided by a like number? WRITTEN PROBLEMS. 9. Divide 8.05 by .35 Process. .35 ) 8.05 ( 23., Ans. 35 hundredths are contained in 805 hun- 7 0 dredths, a like number, 23 times, and hence 1 05 8.05 —;— .35 = 23. The quotient is units. 105 10. Divide 80.5 by .35 Process. By annexing a decimal cipher to 80.5, .35 ) 30.50 ( 230., Ans. w } 1 j c } 1 d oes no t change its value (Art. 119), the dividend and divisor are made like num- 10 5 10 5 0 bers, and hence their quotient is units. 80.50 .35 = 230. 11. Divide .805 by .35 Process. .35 ) .805 ( 2.3, Ans. 70_ 105 105 Since .35 and .80, the first partial dividend, are like numbers, the first quotient figure (2) denotes units; and if the first figure denotes units, the second must denote tenths. Hence. .805 -4- .35 = 2.3 The pointing in all the cases in the division of decimals, may also be explained on the principle, that the dividend is the product of the divisor and quotient, and hence it must contain as many decimal places as both divisor and quotient. In the 9th example, the divisor and dividend contain an equal number of decimal places, and hence there are no decimal places in the quotient. In the 10th example, the divisor contains one more decimal place than the dividend, and hence a decimal place must be added to the dividend before the division is possible. In the 11th example, the divisor contains two decimal places and the dividend three , and hence the quotient contains one decimal place. DIVISION OF DECIMALS. 8T Divide 12 . 32.4 by 1.8 25. 6.241 by .0079 13. 2.56 by .64 26. 67.5 by .075 14. .288 by .036 27. .675 by 75. 15. 82.5 by 2.75 28. 6.75 by 750. 16. 62.5 by .025 29. 256. by .075 17. 9. by .45 30. .256 by 250. 18. 4.53 by .0302 31. .0025 by 50. 19. .3 by .0125 32. 25. by .00125 20 . .625 by 12.5 33. .001 by 100. 21 . .0256 by .32 34. 100 . by .001 22 . 17.595 by 8.5 35. .045 by 900. 23. 3.3615 by 12.45 36. $13.50 by $.37|- 24. .031812 by 4.82 37. $12. by $.06J 38. Divide twenty-four thousandths by sixteen millionths. 39. Divide seventy-eight by thirty-four thousandths. 40. Divide fifteen millionths by six hundredths. 41. Divide 45.7 by 10. By 100. Process. Since the removal of a decimal figure one ^ y ^ gy place to the right divides its value by 10 45 7 ioo_ 457 (Art. 117, Pr. 3), the removal of the decimal point one place to the left divides a decimal by 10 , and the removal of the point two places to the left divides it by 100 . 42. Divide 483.75 by 100. By 1000. 43. Divide 54.50 by 100. By 10000. 44. Divide .005 by 1000. By 100. PRINCIPLES AND RULES. 126. Principles. — 1 . Since the dividend is the product of the divisor and quotient , it contains as many decimal places as both divisor and quotient. Hence, 2 . The quotient must contain as many decimal places as the number of decimal places in the dividend exceeds the number of decimal places in the divisor. Hence, 88 COMPLETE ARITHMETIC. 3. When the divisor and dividend contain the same number of decimal places, the quotient is units. 4. The dividend must contain as many decimal places as the divisor before division is possible. 5. Each removal of the decimal point one place to the left divides a decimal by 10. 127. Rules. —1. To divide one decimal by another, Divide as in the division of integers, and point off as many decimal places in the quotient as the number of decimal places in the dividend exceeds the number in the divisor. Rotes.—1. When the divisor contains more decimal places than the dividend, supply the deficiency in the dividend by annexing deci¬ mal ciphers. 2. When the quotient has not enough decimal figures, supply the deficiency by 'prefixing decimal ciphers. 3. When there is a remainder, the division may be continued by annexing ciphers, each cipher thus annexed adding one decimal place to the dividend. Sufficient accuracy is usually secured by carrying the division to four or five decimal places. 2. To divide a decimal by 10, 100, 1000, etc., Remove the decimal point as many places to the left as there are ciphers in the divisor. REVIEW PROBLEMS. 1. Reduce yf-g- to a decimal. 2. Reduce -25W to a decimal. 3. Change .325 to a common fraction. 4. Change .0045 to a common fraction. 5. From the sum of 67.5 and .54 take their difference. 6. From the sum of 64.5 and .015 take their product. 7. Multiply 6.25 + .075 by 6.25 —.075. 8. Divide .0512 by .032 X .005. 9. From 25.6 .064 take 32.4 X .015. 10. What is the value of $5.33 X 2.5 -j-.075? *11. What is .08J x 1.2i-r-.0061 X .016? 12. Multiply 15 millionths by 7 million. 13. Divide 16 ten-millionths by 25 thousandths. 14. Divide 205 millions by 41 ten-thousandths. s;: See “Note” on page 386 UNITED STATES MONEY. 89 SECTION X. UNITED STATES MONET. PRELIMINARY DEFINITIONS. 128. United States 31oney is the legal cur¬ rency of the United States. It is also called Federal 129. The denominations used in business and ac¬ counts, are dollars, cents, and mills. A dollar equals 100 cents, and a cent equals 10 mills. The figures denoting dol¬ lars are separated from those denoting cents by a decimal point, called a Separatrix, and they are preceded by the character, 8, called the Dollar Sign. 130. The first two figures at the right of dollars denote cents, and the third figure denotes mills. The two figures denoting cents express hundredths of a dollar, and the figure denoting mills expresses tenths of a cent, or thousandths of a dollar. The three figures denoting cents and mills may be read together as so many thousandths of a dollar. Notes. —1. United States Money consists of Coin and Paper Money. Coin is called Specie Currency or Specie, and paper money is called Paper Currency. 2 . The principal gold coins are the double eagle ($ 20 ), eagle ($10), half-eagle, quarter-eagle, three-dollar piece, and dollar. The silver coins are the dollar, half-dollar, quarter-dollar, and dime. The smaller coins are the five-cent piece, three-cent piece, two-cent piece, and cent, the first two being made of copper and nickel, and the last two of bronze, an alloy of copper, tin, and zinc. C. Ar.—8. 90 COMPLETE ARITHMETIC. 3. Gold and silver coins are alloyed, to make them harder and more durable. The gold coins contain 9 parts of gold and 1 part of copper; and the silver coins contain 9 parts of silver and 1 part of copper. Nickel and copper coins are made in the proportion of 1 part nickel to 3 parts copper. 4. Paper money consists of notes issued by the United States, called Treasury Notes, and bank notes issued by banks. 131. NOTATION AND REDUCTION. 1. Express in words, $75.50; $105.08; $1000.45; $15080.; $.87; $.375; $5. 2. Express in words, $37,507; $250,075; $80,005; $.075; $2080.375; $100,058; $.065. 3. Read decimally, $70.25; $140.05; $387.60; $560.09; $84.37; $.08. 4. Read decimally, $.255; $16,455; $300,056; $475,005; $1005.375; $240,061; $.005. WRITTEN PROBLEMS. 5. Write, in figures, ten dollars fifty cents. 6 . Write forty dollars sixty cents five mills. 7. Write 100 dollars 37 cents 4 mills. 8 . Write 430 dollars 5 cents; 25 dollars 5 mills. 9. Write 75 cents 6 mills; 6 cents 5 mills. 10. Write 10 mills; 10 cents 4 mills. 11. How many cents in $25? $100? $350? 12. How many mills in $47 ? $150 ? $165 ? 13. How many mills in $.75? $.625? $.017? 14. How many cents in $5.37? $16.85? $40.08? 15. How many mills in $.37^? $4.62J? $10? 16. Reduce 1500 cents to dollars. 17. Reduce 15000 mills to dollars. 18. Reduce 450 mills to cents. 19. Reduce $25.08 to mills. 20 . Reduce $100.01 to cents; to mills. UNITED STATES MONEY. 91 ADDITION AND SUBTRACTION. 1 . A man paid $7.50 for a pair of boots, and $5.50 for a hat: how much did he pay for both? 2. A lady paid $15 for a shawl, $5.75 for a hat, $2.25 for a pair of gloves, and $4 for a pair of gaiters: what was the amount of her purchases? 3. A drover bought cows at $36.50 a head, and sold them at $40 a head: how much did he gain ? 4. A man bought a coat for $24.25, and a vest for $4.50, and handed the merchant three $10 bills: how much money did he receive back? 5. A mechanic earns $20 a week, and his family expenses amount to $16.75 a week: how much has he left? 6 . A bookseller bought a set of maps for $17, and a set of charts for $6.50, and sold both sets for $28.50: how much did he gain ? WRITTEN PROBLEMS. 7. What is the sum of $.65, $15.44, $60.62J, $100, $94.05, and $.87J? 8 . From $100.15 take $62,371 9. To the sum of $308.60 and $190,125 add their dif¬ ference. 10. From the sum of $2750. and $1680.62^- take their difference. 11. A merchant’s sales for a week were as follows: Monday, $125.60; Tuesday, $98.50; Wednesday, $190.30; Thursday, $215.; Friday, $175.80; Saturday, $247.90: what was the amount of his sales for the week? 12. A man exchanged three city lots, valued respectively at $900, $1200, and $750, for a farm valued at $3075, pay¬ ing the difference in money: how much money did he pay ? 13. A man receiving a salary of $1600 a year, pays $325 for house rent, $450.80 for provisions, $200.60 for clothing, and $245 for all other expenses: how much has he left? 92 COMPLETE ARITHMETIC. 14. A man deposits in a bank, at different times, $75, $230.80, $180.40, and $95, and he draws out $40, $87.50, $331.45, $20.15, and $18.60: what is his hank balance? 132. Rule. —To add or subtract sums of money, Write units of the same denomination, in the same column , add or sub¬ tract as in simple numbers , and separate dollars and cents by a decimal point, and prefix the dollar sign. MULTIPLICATION AND DIVISION. 1. A mechanic earns $2.50 a day: how much will he earn in 6 days? 10 days? 2. What will 8 barrels of flour cost, at $7.25 a barrel? At $6.50 a barrel? 3. What will 20 yards of carpeting cost, at $1.75 a yard? At $2.25 a yard? 4. A drover paid $38.70 for 9 sheep: what did they cost apiece ? 5. A man paid $42 for 8 tons of coal: what did it cost per ton? 6 . If a man earn $39 in 6 days: how much will he earn in 10 days? In 20 days? 7. At 25 cents a dozen, how many dozens of eggs can be bought for $4.50? WRITTEN PROBLEMS. 8 . A farmer sold 45 hogs at $22.45 apiece: how much did he receive for them? 9. A miller sold 237 pounds of flour, at $7.62-|- a barrel: what amount did he receive? 10. A man sold a farm of 260 acres, at $33^- per acre: what was the amount received? 11. A farm containing 125 acres was sold for $5093.75: what was the price per acre? 12. How many carriages, at $125 apiece, can be bought for $8000 ? For $7500 ? LEDGER COLUMNS. 93 13. At $12.37^ a ton, how many tons of hay can be bought for $4653? For $1163.25? 14. A farmer sold 3 hogs, weighing respectively 278, 309, and 327 pounds, at $.07^- a pound: how much did he receive ? 15. A farmer sold in one year 536 pounds of butter, at 30 cts. a pound; 1200 pounds of cheese, at 16-J cts.; and 19 tons of hay, at $8.75 a ton: how much did he receive? 16. A grocer bought 540 pounds of coffee for $81, and 420 pounds of tea for $525; he sold the coffee at 18 cts. a pound, and the tea at $1.60 a pound: how much did he gain ? 133. Rules. — 1. To multiply or divide sums of money by an abstract number, Multiply or divide as in simple numbers, separate dollars and cents in the result by a decimal 'point, and prefix the dollar sign. 2. To divide one sum of money by another, Reduce both numbers to the same denomination , and divide as in simple numbers. ABBREVIATED METHODS. LEDGER COLUMNS. 134. A Ledger is a book in which business men keep a summary of accounts. The items on a ledger page often make long columns o figures, which are foo ted with absolute accu¬ racy. 135. Let the pupil foot the following ledger columns by adding two or more columns at once, being as careful to obtain accurate results as he would be in actual business. (See Art. 22.) COMPLETE ARITHMETIC. 94 ( 1 ) ( 2 ) ( 3 ) ( 4 ) $ 1.25 $ 19.50 $ 75.50 $ 1912.88 8.14 20.00 184.30 806.40 2.75 12.45 111.10 1000.00 .65 14.52 43.95 1250.86 .75 25.48 263.55 943.82 8.37 40.50 100.00 607.55 12.50 8.60 90.00 400.33 4.65 9.35 7.15 148.67 .83 .65 13.48 249.50 7.16 .73 2.75 2040.00 10.28 .84 52.30 4508.70 1.20 12.10 900.25 3406.30 .95 .86 625.80 1280.75 .48 .93 314.87 1300.00 13.47 2.95 64.50 877.77 23.00 14.63 49.87 620.14 3.08 9.82 302.58 8.60 6.15 12.60 10.10 7.45 24.92 19.30 100.98 13.33 .83 22.33 78.60 286.45 .92 9.81 44.50 1300.80 .45 8.76 77.88 1440.00 14.86 12.57 320.65 986.70 5.80 18.19 19.10 87.80 7.26 7.63 8.50 137.40 12.00 14.60 436.75 1500.00 5.00 4.85 135.20 885.73 4.37 9.63 44.88 236.40 6.45 12.83 65.90 13483.86 17.83 18.10 6.01 11456.20 2.65 7.63 7.83 88.00 1.50 2.20 4.22 24.30 .85 .35 3.25 16.50 12.20 .75 .85 9.85 4.65 8.50 .62 100.00 8.15 4.65 1.25 40.60 Suggestion. —The partial footings obtained by each summary, should be written upon a separate piece of paper. This will permit the re-adding of any column or set of columns, as the case may be, without the trouble of re-adding the preceding columns, and it will also avoid the defacing of the page by erasures and corrections. ALIQUOTS. 95 ALIQUOT PARTS. 136 . When the price of an article is an aliquot part of a dollar, the cost of any number of such articles may be found more readily than by multiplying. 137 . The aliquot parts of a dollar commonly used in busi¬ ness, are: 50 cts. = \ of $1.00 12| cts. — i — j of $1.00 25 « — i — 4 of 1.00 6\ “ _ i — TS of 1.00 20 U - 1 — 3 of 1.00 331 « — 1 — ? of 1.00 10 u — l — TIT of 1.00 16f “ — 1 — -S' of 1.00 The following aliquot parts of aliquot parts of a dollar are frequently used: 25 cts. = | of 50 cts. 16f cts. = \ of 33^ cts. 12\ “ = \ of 50 “ 12| “ = \ of 25 “ 6£ “ = \ of 50 “ 6\ “ = l of 25 “ MENTAL PROBLEMS. 1. What will 56 pounds of grapes cost, at 12^- cts. a pound ? Solution. —At $1 a pound, 56 pounds will cost $56, and at 12| cts., which is | of $1, 56 pounds will cost £ of $56, which is $7. 2. What will 120 spellers cost, at 25 cts. apiece? At 33^ cts. ? 3. What is the cost of 96 dozens of eggs, at 16f cts. a dozen ? At 20 cts. ? At 25 cts. ? 4. What will 240 pounds of sugar cost, at 12^ cts. a pound? At 16f cts.? At 20 cts.? 5. At 16| cents a dozen, how many dozens of eggs can be bought for $15? Solution. —At 16f cents a dozen, $1 will buy 6 dozens of eggs, and $15 will buy 15 times 6 dozens, or 90 dozens. 6. At 12^- cts. a pound, how many pounds of lard can be bought for $12? For $25? 7. How many pounds of butter, at 33^ cts. a pound, can be bought for $15? For $33? 96 COMPLETE ARITHMETIC. 8. At 6J cts. a quart, how many quarts of currants can be bought with 30 quarts of cherries, at 10 cts. a quart? WRITTEN PROBLEMS. 9. What will 348 yards of carpeting cost, at $1.62|- cts. a yard? Process. $1.62| = $1 + 50 cts. + 12£ cts. $348 = cost at $1 a yard. | 174 = “ “ 50 cts. a yard. 4 43.50 = “ “ 12J“ $565.50 = “ “ $1.62| “ 10. What will 1600 bushels of oats cost, at 37^- cts. a bushel? At 45 cts. a bushel? At 62^ cts.? 11. What will 2464 bushels of wheat cost, at $1.25 a bushel? At $1.37i? At $1.62£? 12. What will 1250 yards of carpeting cost, at $1.37^ a yard? At $1.50? At $1.87-4? 13. What will 640 bottles of ink cost, at 87^- cents a bottle? At 62J cts.? At 75 cts.? 14. At 25 cts. a dozen, how many dozens of eggs can be bought for $42 ? For $105? For $60.50? 15. At 33J cts. a yard, how many yards of cloth can be bought for $750? For $120? 16. What will 5 lb. 10 oz. of butter cost, at 35 cts. a pound ? Process. $ .35 = cost of 1 lb. $L75 == “ “ 5 “ .175 = “ “ 8 oz. lb.) .044 = “ “ 2 “ (4 lb.) $1,969 = “ “ 5 lb. 10 oz. 17. What will 9 lb. 13 oz. of cheese cost, at 15 cts. a pound ? At 18 cts. ? At 20 cts. ? 18. What will 16 gal. 3 qt. of sirup cost, at $1.75 a gallon? At $1,621? At $1.90? BILLS. 97 19. What will 7 bu. 3 pk. 4 qt. of cherries cost, at $4.25 a bushel ? At $3.50 ? At $4.50 ? 20. What will 2 pk. 7 qt. of chestnuts cost, at $3.50 a bushel? At $2.75? At $2.62^? At $3.12J? DEFINITION AND RULES. 138 . An Aliquot Part of a number is any integer or mixed number which will exactly divide it. 139 . Rules. —1. To find the cost of a number of articles when the price is an aliquot part of a dollar, Find the cost at $1, and take such part of the result as the price is of $1. 2. To find the number of articles which can be purchased for a given sum of money when the price is an aliquot part of a dollar, Find the number of articles that can be purchased for $1, and multiply the result by the given sum of money. BILLS. 140 . Each of the following bills should be neatly made out on paper, in proper form, and receipted. Thomas Knight, 1869 (1) Cincinnati, O., Jan. 1, 1870. Bought of Baker, Smith & Co. a u u u 18, 48 lb. Castile Soap, @ 16|c. . $8.00 “ 25 “ Starch, @> 61 1.56 30, 65 “ Sugar, @i 15 . . . 9.75 “ 33 gal. Vinegar, @r 20 . . 6.60 12, 16 lb. Rio Coffee, @ 23 3.68 5 “ Star Candles, @ 20 1.00 “ 56 “ Butter, @ 33j . 18.67 15, 10 “ Cheese, @ 15 1.50 $50.76 Received Payment, Baker, Smith & Co. per Coons. C.Ar.—9. 98 COMPLETE ARITHMETIC. James Cooper & Bro., ( 2 ) St. Louis, March 3, 1870. To Charles Camp & Co., Dr. To 37 bis. Flour, Ex., @ $4.50 . . . . $ " 23 “ “ Fy., @ 5.25 “ 25 “ Green Apples, 2.12| .... “ 14 bxs. Lemons, @ 7.50 “ 5 “ Raisins, @ 4.75 .... $ Received Payment , What is the amount due ? (3) Cleveland, O., Nov. 24, 1869. Dr. William Jones, To Charles C. Wilhelm, Dr. To 24 Days’ Work, @ $2.75 . . . . $ “ 21 lb. Nails, @ 6| “ 540 ft. Pine Lumber, @ 4.50 per 100 . “ 4 M. Shingles, @ 8.33^- . . . _ By Cash, Oct. 16, . . . . . . $25 <' “ “ 23,.44 il “ 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 @ 83.30; Apr. 12, 7 pair ladies’ slippers @ 81.33|; 3 pair calf boots @ 85.62^. 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^; 9 boxes steel pens @ 8.87-J-; 8 reams note paper @ 83.50; 5 dozen spellers @ 82.33-J-; and 2 dozen copy books @ 81.80. They sold Dudley & Bro. 3 sets outline maps @ 88.25, and paid them 815 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 @ 81.62-J-; 8 handkerchiefs @ 45c.; 2 pair kid gloves @ 81.87J. 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 Bill 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. When 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 affixing 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 COMPLETE ARITHMETIC. promissory note or due-bill, the fact may be added to the words “Received Payment” as in Bill 3. 144. A Hill 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 XI. MENSURATION I. SURFACES.— Definitions. 145. A Line is length. _ _ 146. A Straight Line is a line having the same direction throughout its whole exent. _ Note.—T he word line is commonly used to denote a straight line. 147. An Angle is the divergence of two lines which have a common point. The common point is called the vertex. Thus the divergence of the lines B A and B 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 Bight 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 BD is an obtuse angle, and the angle ABC is an acute angle. The line A B is an oblique line. 150 . A Surface is that which has length and width, but not depth or thickness. 151 . A j 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 Rectangle is a plane ^ - ^ figure bounded by four straight lines - BH IBS and having four right angles. ■ H 153 . A Square is a rectangle with its four sides equal. A Square Inch is a square i inch, each side of which is an inch in length. The figure represents a square inch of c 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. linch. 154 . A Triangle is a plane fig¬ ure bounded by three straight lines and having three angles. 102 COMPLETE ARITHMETIC. Sh' o5 'a •3 p Cj u QJ Pi .Base. 155. A Right-angled Tri¬ angle 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 Arc. 157. The Diameter 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 3}) times the diameter. 158. The Area 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- 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 PROBLEMS. 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 1 foot wide ? 3 feet wide ? 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 the floor of a room 20 by 15 ft. ? 30 by 24 ft. ? 50 by 30 ft. ? Note.—T he dimensions 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 ? WRITTEN PROBLEMS. 11. How many square feet in a floor 37J by 23 ft.? Process: 37| sq. ft. X 23 = 862| sq. 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 18J ft. and 101 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. by 208 ft. ? 16. A garden containing 3267 square yards is 49J yards wide: how 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^ 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 20 ft. in diam¬ eter? 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 the other side. 3. To find the area of a triangle, Multiply the base by one half the altitude. 4. To find the circumference of a circle, Multiply the diam¬ eter by 3.1416. 5. To find the area of a circle, Multiply the circumference by one fourth of the diameter. 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. 163. A Rectangular Solid is a body bounded by six rectangular surfaces. 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, and whose lateral surface is uniformly curved. 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. WRITTEN PROBLEMS. 8. How many cubic feet in a block of granite 16 ft. long, 8 ft. wide, and 5 ft. thick? Process. 16 cu. ft. _8 128 cu. ft. _5 640 cu. ft. A block 16 ft. long, 1 ft. thick, and 1 ft. wide, con¬ tains 16 cu. ft.; and a block 16 ft. long, 1 ft. thick, and 8 feet wide, contains 8 times 16 cu. ft., or 128 cu. ft.; and a block 16 ft. long, 8 feet wide, and 5 ft. thick, contains 5 times 128 cu. ft. = 640 cu. ft. Hence, 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 12 \ 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 foundation wall? 13. A pile of wood, containing 840 cu. ft., is 30 ft. long and 3J 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. 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 it? 167 . Rules. — 1 . To find the solid contents of a rectan¬ gular solid, Multiply the length , width , and thickness together. 2. To find the length, width, or thickness of a rectangular solid, Divide the solid contents by the product of the other two dimensions. 3. To find the entire surface of a cylinder, Multiply the circumference of the base by the altitude , and to the product add the areas of the two bases. 4. To find the solid contents of a cylinder, Multiply the area of the base by the altitude. REDUCTION. 107 SECTION XII. DENOMINATE NUMBERS. REDUCTION. Case I. Reduction, of* Denominate Integers and IVIixed Numbers. 1. How many mills in 9 cents? In 12-J- 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 8J shillings? 10^ s. ? 33^ 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 72 d. ? 144 d. ? 25.2 d. ? 34.8 d. ? 52.92 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. ? 12J- oz.? 16. How many ounces of gold in 7 lb. ? 12J 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 3? 8-J 3 ? 14.5 3? 19. How many drams in 15 3 ? 12f3? 11.5 3? 20. How many ounces in 9 lb ? 5.5 lb? 10.5 lb? 21. How many inches in 8J ft.? 15|- ft.? 33L 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. ? 16f 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. ? 12| pk. ? 30^ 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.? WRITTEN PROBLEMS. 31. Reduce 5£ 6 s. 3d. to pence. 1275 d. to pounds. 12 )1275 Process : 5 £ 6 s. 3d. 20 106 s. 12 Process : 20 ) 106 3 d. 5 £ 6 s. 1275 d., Ans. 5£ 6s. 3d., Ans. REDUCTION. 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 degrees. 40. Reduce 30684 sec. to higher denominations. 41. How many pence in £45? In £237§? 42. How many perches in 95 A. ? 320J 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 365J 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 nu mber of the ^highest denomination by the number of iniits of the next lower which equals a unit of the higher, and to the product add the number of the lower denomi¬ nation, if any. 2. Proceed in like manner with this and each successive 110 COMPLETE ARITHMETIC. result thus obtained, until the number is reduced to the 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 own 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 II. Reduction of Denominate Fractions. 1. What part of a peck is T *g- of a bushel? bu. ? Solution. — T h bu. = T a g of 4 pk. = x 4 g pk. or \ pk., and bu. = 3 times \ pk. = f pk. Hence, T 3 g bu. = § pk. 2. What part of a quart is of a peck ? ^ pk. ? 3. What part of a day is T 2 - of a week ? T 8 7 w. ? 4. What part of an hour is ^ of a day? A d* ? 5. What part of an inch is of a foot ? ^ 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 x 3 0 of a gallon? .06 gal. ? REDUCTION. Ill 12. What part of a foot is § of an inch? Solution. — | in. = f of T J Z ft. = ^ ft. 13. What part of a week is of a day? ^ d.? 14. What part of an hour is -fa of a minute? ^ T °- min. ? 15. What part of a gallon is f of a pint? f 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 ft. = fa 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 f 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.? WRITTEN PROBLEMS. 25. Reduce r s of a day to the fraction of a minute. Process : 7 d = 7_X 24 h = 7 X 24 X 60 18000 ' 18000 * 18000 min. 14 . — min. 25 26. Reduce - 5 -^ 75 - of a pound avoirdupois to the fraction of a dram. 27. Reduce \\ of a yard to inches. 28. Reduce 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 of a rod to the fraction of a league. 112 COMPLETE ARITHMETIC. 36. Reduce |-f of a degree to the fraction of a circum¬ ference. 37. Reduce -J-J- of a day to minutes. 38. Reduce of a week to hours. 39. Reduce f 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 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 ^ of an hour to the fraction of a day. 49. Reduce .012 of a mile to yards. 50. Reduce f 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 the reduction of denominate 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 arc no units of a lower order to add, and in reduction ascending there are no remainders. Case III. Reduction of Denominate Fractions to Lower Integers. 1. How many months in | of a year? J of a year? | of a year? 2. How many hours in f of a day? J of a day? i-J of a day? 3. How many minutes in IT of an hour? of an hour? yu of an hour? 4. How many yards in of a rod? -§ of a rod? 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? .33^ 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 week? 12. How many pecks and quarts in .55 of a bushel? WHITTEN PROBLEMS. 13. Reduce ^ of a day and .415 of an hour each to integers of lower denominations. Process. Process. 7 V 24 A15 h. & da. = T V of 24 h. = h. = 10| h. _60 16 24.900 min. \ h. = i of 60 min. = — min. = 30 min. _§9 2 54.000 sec. ■fo da. = 10 h. 30 min. .415 h. = 24 min. 54 sec. Reduce to integers of lower denominations 14. | of a mile. 15. of a week. 16. of a lb. Troy. 17. of a rod. 18. of an acre. 19. { of a cord. 20. .85 of a lb. avoir. 21. .325 of a ton. 22. .08^ of a yard. 23. .9375 of a gallon. 24. .5625 of a cwt. 25. .0135 of a cord. 170. Rule.—T o 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 reached. 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. He cl notion of Integers of Lower Denominations to Fractions of Higher Denominations. 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. = of a yd., and 30 in. = §£ yd. = f yd. Hence, 2 ft. 6 in. = •§ 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. WHITTEN PROBLEMS. 11. Reduce 15 w. 5 da. to the fraction of a common year. Process. 15 w. 5 da. = 110 da. y r - = ft y r -> ^s. 12. Reduce 1 yd. 2 ft. 6 in. to the fraction of a rod. 13. Reduce 1 pk. 2 qt. 1-J- 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 answer 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 is reached. Thus, in the 16th problem above, the 1.2 pt. = .6 qt.; and 3.6 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 Compound Denominate Number is com¬ posed of units of several denominations; as, 5 bu. 3 pk. 7 qt. It is also called a Compound Number. Note.—E very 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. JKeasuve is the representation of extent, capacity, or amount. TVeight 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: I. Currency, { 1 . 2 . Coin, | f 1. Paper Money. / 1 2. United States Money, English Money. 1. Lines | L Lon S Measure, and arcs, 1 % Circular Measure II. Measure, 1. Of extension, 2. Surfaces: Square Measure. _ 3. Capacity, 1. Cubic Measure, 2. Wood Measure, 3. Dry Measure, 4. Liquid Measure. 2. Of duration: Time Measure. { 1. Avoirdupois Weight, 2. Troy Weight, 3. Apothecaries Weight. Note. —For tables see appendix. 176. The Reduction of a denominate number is the process of changing it from one denomination to another without altering its value. 177. Reduction is of two kinds, Reduction Descending and Reduction Ascending. Reduction Descending is the process of changing a denominate number from a higher to a lower denomi¬ nation. Reduction 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 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 16f cents a pound? 5. If a man spend ^ 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 4j 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. wido? 118 COMPLETE ARITHMETIC. 16. How many bricks, 4 by 8 in., will it take to pave a walk 16 feet wide and 6^- 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, 8 feet high, and 2J 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 masonry 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 round well 3J ft. in diameter and 20 ft. deep? 25. A round 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 does it contain? 28. A speculator bought 3^ sections of land at $4.50 an acre, and sold them at $6.25 an acre: how much did he gain? ' n 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 System is a system of weights and measures expressed 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 em¬ ployed, 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 in the arts. 179 . The Meter is the primary unit of the system. It is the length of a bar of metal kept at Paris as a standard. The meter was intended to be the ten-millionth part of the distance from the equator to the north pole, but subsequent measurements of this quadrant show that its length is a little more than ten million meters. A standard meter, copied from the one at Paris, is kept by each nation that has adopted the metric system. The standard meter of the United States is kept at Washington. Its length is 39.37 inches. The Titer ( le'-ter ) is the unit of the measures of ca¬ pacity. 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 greatest density. 180 . The meter, liter, and gram are each multiplied by 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, Veka (10), Hekto (100), Kilo (1000), and Myria (10000); and the subdivisions are named by prefixing the Latin words, Deci (10th), Centi (100th), and Midi (1000th). 120 COMPLETE ARITHMETIC. METRIC TABLES. 182 . —I. Measures of Length. The Unit is a Meter = 39.37 inches. Denominations Abbreviations. Values. Myriameter . . . . Mm. = 10000 meters. Kilometer . . . . Km. = 1000 “ Hektometer. . . . Hm. = 100 “ Dekameter . . . . Dm. = 10 “ Meter .... = 1 meter. Decimeter . . . . dm. .1 “ Centimeter . . . . cm. = .01 “ Millimeter . . . . mm. zzz .001 “ Decimal Scale. 00000.000 Ten units of anv 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 dekameter, tens, etc. The correspondence between the metric denominations and those of United States Money is also noticeable. The millimeter cor¬ responds to mills; the centimeter to cents; the decimeter to dimes ; the meter 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. —The spelling, pronunciation, and abbreviations employed are those recommended by the Metric Bureau, Boston, and the American Metrological Society. The equivalents of the metric units are those legalized by Congress. THE METRIC SYSTEM. 121 183.—II. Measures of Surface. The Unit is an Ar, a Square Dekameter = 3.95 sq. rd. Denominations. Abbreviations. Values. Hektar.Ha. = 10.000 sq. m. At (are) .a. = 100 sq. m. Centar.ca. = 1 sq. m. Decimal Scale. a Since 100 units of each denomination in the above table equal one of the next higher, each occupies two orders of figures. The centars correspond, in this respect, to cents, which occupy two places. The above table is used in measuring land. The primary unit for the measuring of small surfaces is a square meter. S-i a 24 d principal. 3 d year’s interest. Amount due at the end of the third year. 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 the given principal for one interval of time; then, taking this amount as a new principal, find the amount for the second interval, and so continue for the entire time. The dif¬ ference between the last amount and the principal is the com¬ pound interest for the time. C.Ar.—18. 210 COMPLETE 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 Showing 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. YUS. 3 PER CENT. 4 PER CENT. 5 PER CENT. 6 PER CENT. 7 PER CENT. 8 PER CENT. I 1.03 1.04 1.05 1.06 1.07 1.08 o 1.0009 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 1.713824 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 1.468534 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 If) 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 EQUATION OF PAYMENTS. 211 340. The amount of $1 for the given time and rate, multiplied by the given principal, 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 Payments 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. PROBLEMS. 1. A owes B $300, of which $200 is due in 3 months, and $100 in 6 months: when will the payment of $300 equitably discharge the debt ? Process. $200X3= $600 $100X6= $600 $300 ) $1 200 4 Ans., 4 mos. A is entitled to the use of $200 for 3 months, which equals the use of $600 for 1 month, and to the use of $100 for 6 months, which equals the use of $600 for 1 month ; and, hence, he is entitled to the use of $300 until 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, -J- 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. What 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% = 3.50 $900 $6,875 $9.00 = Int. of $900 for 60 days. .15 = “ “ $900 for 1 day. $6,875 -r- $.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 $6 875. The interest of $900 for 1 day is $.15; and since $6,875 -j- $.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 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 time . 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 the 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 find in what time the sum of these present worths will amount to the sum of all the payments. The inaccuracy of this so-called “accurate rule” is easily shown. The methods 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. $50 X 8 = $400 $50 X 4 = $200 $200 — $100 = $100 ) $600 6 In paying $50 in 4 months, A loses its use for 8 months, and in paying $50 in 8 months, he loses its use for 4 months, and hence he loses the use of $400 -(- $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 was paid before it was due , and divide the sum of the products by the bal¬ ance unpaid. EQUATION OF ACCOUNTS. Note. —Equation of Accounts, especially Case II, needs to be studied only by advanced classes. It is seldom used in ordinary business. 347. Equation of Accounts is the process of finding the equated time for the payment of the balance of an account, or the time when the balance was due. Case I. Accounts containing only Debit Items. 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. Dae June 3, 1870, $450 X 00 = “ “ 24, “ 500 X 21 = 10500 “ July 25, “ 750 X52 = 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, << u 180, on 4 a a June 20, (< it 325, on 3 it a Aug. 10, <( a 80, on 3 (( u Sept. 1, << a 100, on 4 a a 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 maturity for the debit items of an account, First find th / maturity of each item or bill, 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 forward from the first date. Notes. —1. The equated time may be found by interest, as in the Equation of Payments. (Art. 344, Rule 2.) 2. The sum of the debit items draws interest from the equated date of maturity to the date of payment. Case II. Accounts containing "both Debits and. Credits. 5. What is the equated date of maturity of each side of the foliowing account? Dr. John Smith in account with John Jones. Or. 1868 . Time of Cred . 1868 . Apr. 3, To Mdse. $220 3 mo. July 1, By Cash $200 June 1, U 125 4 “ Oct. 3, « 150 July 15, a 200 4 “ Dec. 20, u 300 Aug. 24, u 140 6 “ Oct. 1, u 190 6 “ Process. Debits. Due July 3,1868, $220 X 00 = Oct. 1, “ 125 X 90 = 11250 Nov. 15, “ 200 X 135 = 27000 Feb. 24,1869, 140X236 = 33040 Apr. 1, “ 190 X 272 = 51680 $875 ) $122970 141 Debits are due 141 days from July 3, 1868, which is Nov. 21. Credits. Due July 1,1868, $200 X 00 = Oct. 3, “ 150 X 94 = 14100 Dec. 20, “ 300 X 172 = 51600 $650 ) $65700 101 Credits are due 101 days from July 1, which is Oct. 10. Note.—E ach 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. 218 COMPLETE ARITHMETIC. 6. The above account, as equated, stands thus: Dr. Cr. Due Nov. 21, 1868 . . $875 Due Oct. 10, 1868 . . $650 When is the balance of the account due? Process. Debits.$875 Credits.650 Balance.$225 Difference in time, 42 days. 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 forward 121 days from Nov. 21, the balance is found to be due March 22, 1869. $650 _42 $225 ) $27300 121 7. Suppose that the debit and credit sides of an account when equated stand as follows: Dr. Or. Due Nov. 21, 1868 . . $650 Due Oct. 10, 1868 . . $875 What would be the equated time of payment for the bal¬ ance ? Process. Credits.$875 Debits. 650 Balance.$225 Difference in time, 42 days. $875 _42 $225) $36750 163 Balance is due 163 days prior to Nov. 21, 1868, which is June 11, 1868. Suppose the account settled Nov. 21, as before. The credit side, having been due since Oct. 10, has been drawing interest for 42 days. That the debit side of the account may be increased by an equal amount of interest, the balance must be regarded as due 163 days prior to Nov. 21. EQUATION OF ACCOUNTS. 219 8. The debit and credit sides of an account when equated stand as follows: Dr. Cr. Due June 5, 1870 . . $1285 Due July 1, 1870 . . $1000 What is the equated time of payment for the balance ? 9. At what time did the balance of the following equated account begin to draw interest: Dr. Cr. Due July 12, 1870 . . $450 Due Sept. 1, 1870 . . $800 10. When will the balance of the following account begin to draw interest, the debit items having a credit of 3 months ? Dr. It. Hill & Co., in account with O. Cooke. Cr. 1870. 1870. July 10 To Mdse. $120 Nov. 20 By Cash $350 “ 30 a 450 Dec. 25 “ Mdse. 250 Aug. 30 (( 380 1871. Sept. 9 (( 560 “ 30 u 400 Jan. 1 “ Cash 750 349 . Rule. —To find the equated time for the payment of the balance of an account, 1. Find the equated time for each side of the account. 2. Multiply the side of the account which falls due first by the number of days between the dates of the equated time of the two sides, and divide the product by the balance of the account. 3. The quotient will be the number of days to the maturity of the balance, to be counted forward from the later equated date when the smaller side of the account falls due first, and back¬ ward when the larger side falls due first. Notes. —1. When an account is settled by cash, each side of the account is increased by its interest from maturity to the date of set¬ tlement, and the difference between the two sides thus increased by interest, is called the Cash Balance. Instead of adding the accrued interest to each side, the balance of interest may be found and added 220 COMPLETE ARITHMETIC. to or subtracted from the balance of items, according as the two bal¬ ances fall upon the same or upon opposite sides of the account. Thus, in problem 6 above, the balance of interest, which is the interest of $650 for 42 days, falls on the credit side, and the balance of items on the debit side. The cash balance is $225 — $3.90, which is $221.10. 2. The cash balance may be found directly, without equating the account, by finding the interest of each item from its maturity to the date of settlement, and taking the difference between the sums of the debit interests and credit interests for the balance of interest. When the balance of interest and the balance of items fall on the same side, the cash balance is their sum; when they fall on opposite sides, the cash balance is their difference. SECTION XV. RATIO AND PROPORTION. RATIO. 350 . The relation between two numbers expressed by their quotient, is called their Ratio. The ratio of 6 to 2 is 6 a- 2, or 3; and the ratio of 2 to 6 is 2 — 6, or MENTAL EXERCISES. 1. What is the ratio of 8 to 4? 24 to 8? 45 to 15? 2. What is the ratio of 6 to 12 ? 12 to 36? 16 to 64? 3. What is the ratio of 42 to 14? 14 to 42? 12 to 30? 4. What is the ratio of 50 to 15? 15 to 50? 80 to 25? 5. What is the ratio of 36 to 16 ? 60 to 25 ? 70 to 40 ? 6. What is the ratio of 45 to 60 ? 18 to 45 ? 75 to 45 ? 7. What is the ratio of $33 to $11? $20 to $50? $45 to $36? $50 to $150? 8. What.is the ratio of 16 lb. to 40 lb.? 28 lb. to 13 lb.? 9. What is the ratio of T 9 ^ to T s 7 ? T 3 y to -j^-? yf to -^-? 10. What is the ratio of \ to J? y to J? £ to ^? 11. What is the ratio of J to |? §• to §? f to f ? 12. What is the ratio of 5 to J-? \ to 4? } to 2^-? RATIO. 221 WHITTEN EXERCISES. 351 . The ratio of two numbers is expressed by placing a colon between them. The ratio of 4 to 10 is denoted by 4 : 10, and the ratio of | to | by | The expression 4 : 10 is read the ratio of 4 to 10. 13. Express the ratio of 7 to 15. 12 to 35. 35 to 17. 14. Express the ratio of 2.5 to 7.5. 3.4 to .62. 15. Express the ratio of f to -§. f to 5^. 2^ to f. 16. What is the value of the ratio of 112 to 35? Process : 112 : 35 = 112 -s- 35 = 3b Ans. What is the value of 17.216:81? 21. -A-': |? 25. 6 qt.: 3 pk. ? 18. 129 : 215 ? 22. 150:16}? 26. 5 lb. 12 oz.: 17 lb. 4 oz.? 19. 14.3:6.5? 23. 12^:30^? 27. 2 ft. 6 in.: 12 ft. 6 in. ? 20. 1.44:3.2 ? 24. 34^: 5J ? 28. 15 pk.: 12 bu. 2 pk. ? 29. Reduce 24 : 60 to its lowest terms. Process : 24 : 60 = ■§$ = $ = 2 : 5, Ans. Reduce the following ratios to their lowest terms: 30. 35 : 84. 31. 63 : 108. 32. 121 : 220. 33. 105 : 140. 34. 81 : 189. 35. 105 : 195. 36. 169 : 65. 37. 256 : 112. 38. 225 : 120. 39. Reduce £ : f- to an equal ratio with integral terms. Process : f : £ = : t 9 j = 10 : 9. Reduce the following ratios to equal ratios with integral terms : 40. 2 J . 3 • ¥* 43. 7 • 5 ¥ * ¥• 46. i-.io. 41. 5 T . 1 1 . x?. 44. J1 • 7 TW * TJ' 47. Ol . 5 ■“3 * 6* 42. TT . 5 • 8* 45. 13 . 17 1 5 • Tor* 48. 14 : 5}. 49. Multiply 10 : 21 by 14 : 15. Process- I 10:21 — $}. 14:15 — ff. * l Hence, (10 : 21) X (14 : 15) = X = f, Ans. 222 COMPLETE ARITHMETIC. 50. What is the product of 9 : 10 and 24:33? 51. What is the product of 7 : 15, 25 : 14, and 24 : 35? 52. What is the product of 12 : 25, 15 : 24, and 16 : 21 ? DEFINITIONS, PRINCIPLES, AND RULES. 352. j Ratio is the relation between two numbers of the same kind expressed by their quotient. 353 . The two numbers compared are called the Terms of the ratio. The first term is the Antecedent, and the second term the Consequent. The two terms form a Couplet. 354 . The value of a ratio is the quotient obtained by dividing the antecedent by the consequent. When the antecedent is greater than the consequent, the value of the ratio is greater than 1; when the antecedent is less than the con¬ sequent, the value is less than 1. 355 . The ratio of two numbers is expressed by placing a colon (:) between them; as, 5:12. The colon is called the Sign of Ratio. Note. —The sign of ratio is the sign of division with the hori¬ zontal line omitted. 356. A ratio is also expressed in the form of a fraction, the antecedent being made the numerator and the conse¬ quent the denominator. Thus, 5 : 12 = T \. Note. —Several American authors divide the consequent by the antecedent, thus reversing the positions of dividend and divisor, as indicated by the sign of division. The great majority of mathe¬ matical writers divide the antecedent by the consequent. Ratios are either Simple or Compound. 357 . A Simple Hatio is the ratio of two numbers; as 5 : 8, or f : f. Note. —A simple ratio, having one or both of its terms fractional, is called by several authors a Complex Ratio. RATIO. 223 358 . A Compound Ratio is the product of two or more simple ratios; as, (5 : 6) X ($ : 10). It may be expressed in three ways, as follows: (5 : 6) X (8 : 9) X (f : 10); or^X^Xfj or 8 : 9. 6 9 10 | . 10 . 359 . An Invevse Ratio 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 Reciprocal Ratio. 360 . Principles. —1. The two terms of a ratio must be like numbers. 2. The 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 the 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 with integral terms, Multiply both terms by the least common multiple of the denominators of the fractions. (Pr. 7.) o. To find the product of two or more simple ratios, Midtiply the antecedents together for an antecedent and the con¬ sequents together for a consequent. (Pr. 8.) Note.—T he 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 : 12J, and show that the two ratios are equal. DEFINITIONS AND PRINCIPLES. 363. A _P ropovtion 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 Simple Proportion 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. Case I. Any Term found, when the other Three Terms are given. 369. The proportion 4 : 8 :: 6 : 12 may be written 4 : 8 = 6 : 12, or f = 1 ^- (Art. 356) ; and multiplying the two equal fractions by 12 and 8, their denominators, we have 4X12 = 6x8. 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 be divided by either mean, the quotient will 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 :: 15. 15 : 40 16. — : 24 17. — : 9 : 18. 45 : 30 19. 2.5:62.5 20. 7.2 : — : 21. .25 : — : 36: — : 18 : — : 8 : 32 60: 18 : — : 24 : —: 3.25 4.7 : 9.4 2.5 : 7.5 99 _2. . 3 » . 5 . • Q • A • • ft • 3 OQ 3.2.. _ . Z 40. j . -g- . . . f 24. — : 24 ¥ 2 1 ¥ 3 T 95 i. . .. x . l • 3 * # * 5 # 26. $5 : $45 :: 6 lb. 27. $.75 : $3 :: — : 56 oz. 28. 16 men : 96 men :: 15 days : — 29. 8 horses : 14 horses :: f : — 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 other mean. Case II. The Solution of Problems by Simple Proportion. 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 WHITTEN PROBLEMS. 30. If 15 yards of cloth cost $24, what will 40 yards cost? Since the cost of 40 yards is to be the answer, make $24, the cost of 15 yards, the third term of a proportion; and since 40 yards will cost more than 15 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 the second term and 15 yards the first, giving the proportion 15 yd. : 40 yd.:: $24 : cost of 40 yards, which, by Case I, is found to be $64. STATEMENT. 15 yd. : 40 yd. :: $24 : 40 15) $960 $64, Arts. Ans. 31. If 45 sheep cost $565, what will 140 sheep cost? 32. If 13 tons of hay cost $97.50, what will 7J 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 17J 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 f of a farm is worth $4500, what is f of it worth? 44. If of a yard of silk cost $2.10, what will 16| yards cost? 45. If 6J tons of hay cost $58,75, how many tons can be bought for $173.90? 46. At the rate of 5 peaches 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? The 15 days is the third term, since the answer is to be in days. If it take 9 men 15 days to build a wall, it will take 5 men more 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. STATEMENT. 5 men : 9 men :: 15 days : Ans. _9 5)135 27 days, Ans. Note. —The principle involved 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 cause 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. Proportions, { (4 -{- 5) : 90 :: 4 : Smaller part. (4 + 5) ; 90 :: 5 : Greater part. Note. —These proportions are based on the principle that when four numbers are in proportion, the sum of the first and second terms is to the sum of the third and fourth terms as the first term is to the third, 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 $9600 was divided between two heirs in proportion to their ages, which were 15 years and 17 years respectively: how much did each receive? 54. Two men, 150 miles apart, are approaching each other, one traveling 2 miles to the other 3: how far will each travel before they meet? 6^“For additional problems see Problems for Analysis p. 239. PRINCIPLES AND RULE. 374 . Principles. — 1 . The ratio of two like causes equals the ratio of their effects. Conversely, 2. The ratio of two like effects equals the ratio of their causes. 3. The ratio of two like causes equals the inverse ratio of their tunes. Conversely, 4. The ratio of the times of two like causes equals the in¬ verse ratio of the causes. 5. The two terms of each couplet of a simple proportion must he like numbers. (Art. 360, Pr. 1.) 6. The fourth term of a 'proportion equals the product of the second and third terms divided by the first term. (Art. 370.) 375 . Rule. —To solve a problem by simple proportion, 1. Take for the third term the number which is of the same kind as the answer sought, and make the other two numbers the first couplet, placing the greater for the second term, when the answer is to be greater than the third term; and the less for the second term, when the answer is to be less than the third term. 2. Divide the product of the second and third terms by the first term, and the quotient will be the fourth term, or answer. Notes.—1. When the terms of the first couplet are denominate numbers, they must be reduced to the same denomination. 2. The process of finding the fourth term may be shortened by can¬ cellation. The proportion 15 : 45 : : 27.5 : — may be completed thus: 27.5 U = 82.5. 4$ 3 27.5 27.5X3 = 82.5 3. The process of solving problems by simple proportion is also called “ The Ride of Three.” 230 COMPLETE ARITHMETIC. COMPOUND PROPORTION. • Case I. RecLiaction of Compound Ratios and Proportions to Simple Ones. 1. Reduce the compound ratio ratio in its lowest terms. Process. to a simple Or: 20 4 6 80 3 8 20X4X6:80X3X8 480 : 1920 1 : 4, Ans. 20 : 4 : $ . % 0 : $ 4 4 1 : 4, Ans. 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 1 4:3 \ 6:8 J —. 20 V 4 V 6 — is A j A j ?0Xi■ :: 13 : — ? ( 12 : 25J Process. 0 : M : 0 3 :: 13 : — %!%:%%% _ 1 : 3:: 13 : — _3_ 39, Ans. Or: $ M tn 03 tn 13 3 X 13 = 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: /* 13. { 20 : 48' 36 : 15 10 : 4 - :: 25 : — i4 -l '5:9 2.5 : 7.5 4 : 10 16 : 353 21 : 8 9 : 6 12 : 45 y :: 16 : — 15.- f 2 i:7 3 25 : 10 4:6i (15 : 12 J 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. The Solution, of Problems by Compound Proportion.. 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 ? Statement. 16 acres 9 days 10 hours 27 acres 10 days 8 hours ‘ U s i 2 men : Ans. 2 : 3 :: 2 men : 3 men, Am. Or: Since the answer required is to be a number of men, 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. If 10 days require 2 men, 9 days will require more than 2 men, and hence the second ratio is 9 days : 10 days, the 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 term. 0 10 U $ % 3 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, bow 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 32 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^ 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 the factors of the first term. C.Ar.—20. 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 resulting pro¬ portion by the first term. The quotient will be the fourth term, or answer sought. Or, Divide the product of all the factors of the second and third terms of the compound proportion by the product of the factors of the first term, shortening the process by cancellation. Notes. — 1. The terms 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 by 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. 0 382. Partnership is either Simple or Compound. In Simple Partnership* the capital of the several partners is invested an equal time. In Compound Partnership the capital of the sev¬ eral partners 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 cap’l. $8000 : $3600 :: $3000 : $1350, A’s share of profits. 2400, B’s cap’l. $8000 : $2400 :: $3000 : $900, B’s “ “ 2000 , C’s cap’l. $8000 : $2000 :: $3000 : $750, C’s “ “ $8000, Entire capital. $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 $3000 -r- $8000 = .37£ $3600 X -37£ = $1350, A’s share. $2400 X -37£ = $900, B’s “ $2000 X .37£ = $750, C’s “ Or: $3600 -f- $8000 = .45, A’i $2400 -f- $8000 = .30, B’i $2000 -r- $8000 = .25, C’i $3000 X -45 = $1350, A’i $3000 X -30 = $900, B’. $3000 X -25 = $750, C’. Percentage. Since the profits were equal to -37£, or 37 \ % of the entire capital, each partner’s share of the profits was equal to 37£ % of his capital. per cent, of the capital, u u « u u u share of the profits, u a u u << « III. Process by Fractional Parts. $3000 -f- $8000 = = f. f of $3600 = $1350, A’s share. | of $2400 = $900, B’s “ | of $2000 = $750, C’s “ Since the profits were equal to f of the entire capital, each part¬ ner’s share of the profits was equal to f of his capital. 236 COMPLETE ARITHMETIC. Or: $3600 -f- $8000 = 2 9 o> A’s part of the capital. $2400 -T- $8000 = T 3 ^, B’s “ “ “ $2000 = $8000 = C’s “ “ “ -f-Q of $3000 = $1350, A’s share of the profits. x % of $3000 =$900, B’s “ “ “ 4 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 $20000: what is each partner’s share of the gain ? Of the proceeds of the sale ? PRINCIPLES AND 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 the 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. Bule. —To divide the gain or loss of a simple part¬ nership, Divide the gain or loss among the several partners in proportion to the amounts of capital invested by them. * Revised. 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. $3000 X 1= $3000 $5000 X 2 = $10000 $13000, A’s capital for 1 year. $4000X 2= $8000 $2000X 1= $2000 $10000, B’s capital for 1 year. $13000 -f- $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 for 1 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’s “ 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 and B entered 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 C 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? 9 PRINCIPLE AND RULES. 385. 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. Problems marked * are simplified in this edition. 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 J of a yard of silk cost what will £ of a yard cost? 4. If J of a barrel of flour cost $5£, what will £ of a barrel cost? 5. If £ of a pound of coffee cost 15 cents, 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, 1£ 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 4| 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 j 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 f- 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? [Sug.: § t. past n. -f- J t. past n. = 12 h.] 28. A person being asked the time of day, said that J of the time past midnight equaled the time to noon: what was the hour of day ? PEOBLEMS FOE ANALYSIS. 241 29. What is the hour of day when f of the time past noon equals f of the time to midnight? Suggestion: If f t. past n. = § t. to m., t. past n. = 4 t. to m. Hence, t. to m.-f-i t. to m. = 12 h. *30. What is the time of day when § of the time past noon is equal to twice the time to midnight? 31. What is the time of day when § of the time to noon is equal to J- 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 A’s age was f of B’s age, and the sum of their ages was 70 years: what is the present age of each? *35. At the time of marriage a wife’s age was f of the age of her husband, and the sum of their ages was 48 years: how old w r as each 20 years after marriage? 36. f of A’s age equals 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 J of the cost of the horse equals J of the cost of the car¬ riage : what was 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 a month; the man earned twice as much as the elder son, and the elder son twice as much as the younger: how much did each earn? 41. Two men bought a barrel of syrup, 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 \ of a piece of work in a day, and a boy can do 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 4 days and a boy in 8 days ; the man works 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 work 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 -J- 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 ^ of a month, and C in jr 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 6 days: how long will it take each, working alone, to do it? 56. A and B did a piece of work, and f of what A did equaled f 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? Note. —For solution, see page 286. 59. At what 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 bought two watches and a chain for $140; and the first watch cost twice as much as the chain, and the second watch four times as much as the chain: what was the cost of each? 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 difference between their shares was $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 f of the cost of the larger farm was equal to ^ 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 carriage arc 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, what would he 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, what 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 many cords can 8 men cut in 15 days? 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 f of it: what is each partner’s share of the profits? 84. A and B, trading in partnership 2 years, make each year a profit of $1200; A’s capital the first year was 2^ times B’s, and the second year it was 1J times B’s: what is each partner’s share of the profits ? 85. 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? Sug.—A mount of loss-v-$2.50 (loss each idle day) = no. idle days. 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 § of the number of yards in the third carpet: what was the number of yards in each carpet? 246 COMPLETE ARITHMETIC. SECTION XVI. INVOLUTION AND EVOLUTION. 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 ofl? 2? 3? 4? 5? 6? 7? 8? 9? 5. What is the third power ofl? 2? 3? 4? 5? 6? 7? 8? 9? 6. What is the second power of f? Off? £? f? i ? A ? 6 * 6 ' WRITTEN PROBLEMS. 7. What is the second power of 406? 8. What is the third power of 42 ? 9. What is the fourth power of 24? 10. What is the fifth power of 16? 11. What is the second power of 6.5? 12. What is the third power of .42? 13. What is the fifth power of .6? 14. What is the third power of ^-f. 15. What is the second power of ? 16. What is the fourth power of f-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 2 denotes the second power of 24; 16 3 denotes the third power of 16, etc. INVOLUTION. 247 Raise the following numbers to the powers indicated by the exponents: 17. 623 2 18. 105 3 19. 34.6 2 20. .016 3 21. 1.4 4 22. .045 2 23. (If) 3 24. (J) 4 25. (16f) 2 26. (3i) 4 27. (6J) 8 28. (if) 2 29. .005 5 30. 2.04 3 31. (|) 4 DEFINITIONS AND RULE. 388 . The Fower of a number is the product obtained by taking the number one or more times as a factor. 389 . The First Fower of a number is the number itself. 390 . The Second Fower 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. 391 . The Third Fower 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. 3 x 3 x 3 = 27. 392. The Exponent of a power is a small figure placed 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 * 1 ; the second power, or square, contains the number twice as a factor, and the exponent is 2 ; the third power, or cube, contains the number three times as a factor, and the exponent is 3 ; etc. 393. Involution is the process of finding the powers of numbers. 394. Rule.—T o 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. ANOTHER METHOD OF INVOLUTION. 32. What is the square of 53 ? Process. 53 = 50 + 3, and 53 2 = (50 + 3) 2 50 + 3 50 + 3 50X3 + 3 2 = (50 + 3) X 3 50 2 + 50 X 3 = (50 + 3) X 50 Parts added. 50 2 = 2500 2(50X3)= 300 3 2 =_9 = 2809 50 2 + 2 (50 X 3) + 3 2 = (50 + 3) 2 = 53 2 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. 33. What is the square of 45? ( 40 2 =1600 Process: 45 2 = j ^(40X5)= 400 I 2025, Ans. 34. What is the square of 67? Of 75? 35. What is the square of 82 ? Of 38 ? 36. What is the square of 93 ? Of 125 ? Suggestion.—125 = 120 + 5. 37. What is the square of 115? Of 124? EVOLUTION. 249 38. What is the cube of 53? The cube of 53 = (50 + 3) 8 = 50 3 + 3 (50 2 X 3) + 3 (50 X 3 2 ) -f 3 3 , as may be shown by multiplying 50 2 + 2 (50 X 3) + 3 2 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 tens by the units, plus three times the product of the tens by the square of the units, plus the cube of the units. 39. What is the cube of 45 ? Process: 45 3 = ( 40 3 = 64000 3 (40 2 X 5) = 24000 3 (40 X 5 2 ) = 3000 5 3 = 12 5 ^ 91125, Ans. 40. What is the cube of 23? Of 32? 41. What is the cube of 24? Of 43? 42. What is the cube of 33 ? Of 54 ? 43. What is the cube of 51 ? Of 35 ? 44. What is the cube of 45? Of 52? 45. What is the cube of 41 ? 55? 46. 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 cube root; one of its four equal factors, its fourth root , etc. 5 What is the square root of 25 ? Of 49 ? 64? 81 ? 6. What is the cube root of 8 ? Of 27 ? 64 ? 125 ? 7. What is the cube root of 216? 512? 1000? 8. What is the fourth root of 16? Of 81? 256? 625? 9. What is the square root of 1 ? 4 ? 9 ? 16 ? 25 ? 36? 49? 64? 81? 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 Hoot of a number is one of the equal factors which will produce it. 397 . The First Hoot is the number itself. 398 . The Second Hoot is one of the two equal factors of the number. It is also called the Square Root. 399 . The Third Hoot is one of the three equal factors of the number. It is also called the Cube Root 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 Ferfect Fower is the product of equal factors. It has an exact root. 401 . An Imperfect Fower is a number which is not the product of equal factors. Its root is called a Surd. 402 . The Hcidical Sign is a character, y" , 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, 1^25 denotes the first root of 25 ; ^25 denotes the second root of 25 ; 1^25, the third root of 25, etc. SQUARE ROOT. 251 When the square root is indicated, the index is usually omitted. V 16 and V 16 alike denote the square root of 16. Note. —The root of a number may also be indicated by a frac¬ tional exponent. Thus, 16 2 denotes the square root of 16; 16^, the cube root of 16, etc.; 16^ denotes the cube root of the square of 16. 404 . J Evolution is the process of finding the roots of numbers. Note. —Evolution is the reverse of involution. SQUARE ROOT. The Division of the IN - umber 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 the greatest integers com¬ posed of one, two, three, and four orders, are as follows: 1 2 = 1 10 2 = 100 100 2 = 10000 1000 2 =1000000 9 2 = 81 99 2 = 9801 999 2 = 998001 9999 2 =99980001 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 be as many orders in its square root as there are periods in the number . 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 the greatest number of units, tens, hundreds, and thousands, are as follows: l 2 = 1 10 2 = 100 100 2 = 10000 1000 2 =1000000 9 2 = 81 90 2 = 8100 900 2 = 810000 9000 2 =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-thousands, 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-hand 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 PROBLEMS. 13. What is the square root of 3364? Process. 3364 I 58 5 2 =_25 5X2 = 10) 864 108X8= 864 Since 3364 is composed of two periods, its square root will be composed of two orders. (Art. 406.) The left-hand period 33 contains the 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 twice the product of the tens of the root by the units , plus 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 difference, contains twice 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. 108X 8=2 (5X 8) +8 2 . Proof. — 58 X 58 = 3364. 14. What 15. What 16. What 17. What 18. What 133225? 19. What 210681? 20. What 419904 ? 21. What 94249 ? Of is the square root of 625? Of 4225? is the square root of 576? Of 7744? is the square root of 1444? Of 6241? is the square root of 3025? Of 7569? is the square root of is the square root of is the square root of Process. 133225 I 365 9 3 X 2 = 6 ) 43 2 66 X 6 = 396 36X 2 = 72) 362 5 is the square root of 725 X 5 = 3625 492804 ? 254 COMPLETE ARITHMETIC. 22. What is the square root of 57600? Of 40960000? 23. What is the square root of 10.4976? 24. What is the square root of 176.89? 25. What is the square root of .0625? 26. What is the square root of .451584? Of .008836? 27. What is the square root of 586.7? Suggestion.— Point thus 586.70, 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 -§-§-J? Of 31. What is the square root of 272^? Of 1040^? 32. What is the square root of 2 ? Of 3 ? Of 5 ? PRINCIPLES 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 the root. 3. Subtract the square of the term of the root found from the Process. 10.4976 | 3.24 9 3X2 = 6)1.49 6.2 X 2 = 1.24 3.2X2= 6.4) .257 6 6.44 X .04 = .2576 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 dividend. The quotient (or the quotient reduced ) will be the next term of the root. 5. Annex the second term of the root to the trial divisor, and multiply the result by the second term, and subtract the product from the dividend. 6. Annex the third period to the remainder for the next divi¬ dend, and divide the same, exclusive of the 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 the denom¬ inator, and extracting the root of the resulting fraction. Thus, 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. G-eometrical 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 denoting 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 side 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 is 25, the difference between 225 and 200. Hence, 5 is the units’ term of the root, and the length of the side of the square is 20 -j- 5, or 25. Adding the area of the several parts, we have 20 2 -J- 20 X 5 X 2 -f- 5 2 = 400 L — J -f 200 -f 25 = 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, the square of the unitSo 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 -f- 5 = 45), and this multiplied by 5 gives an area of 225. CUBE ROOT. The Division of the Number into Periods. 412 . The cubes of the smallest, the greatest, and an inter¬ mediate number, composed of one, two, and three orders, are as follows : l 3 = 1 9 3 = 729 4 3 = 64 10 3 = 1000 99 3 = 970299 44 3 = 85184 100 3 = 1000000 999 s = 997002999 444 3 = 87528384 A comparison of the above numbers with their cubes shows that the cube of a 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 he 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, 9il25. 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 the greatest number of units, tens, and hundreds are as follows: 1» = 1 9 3 =7 729 10 3 = 1000 90 3 = 729000 100 3 = 1000000 900 3 == 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 cube 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, the left-hand 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’ term 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. What 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? Since 262144 is composed of two pe¬ riods, its cube root will be composed of two orders (Art. 413). The left-hand period, 262, contains the cube of the tens’ 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 Process. 262144[64 6 3 = 216 4 6 2 X 3 = 108 ) 46144 64 3 = 262144 CUBE ROOT. 259 times the square of the 6 tens of the root, the quotient, which is 4, will be the units’ teinn 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: 60 2 X 4X3 = 43200 60X4* X 3= 2880 4 3 —_64 46144 11 . What is the cube root of 42875? Of 91125? 12. What 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 [ 364 , Cube root. 3 3 — 27 74, Trial quotients. 3 2 X 3 — 27 ) 212 36 3 = 46656 36 2 X 3 = 3888 ) 15725 364 s = 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 17. What is the cube root of 18. What is the cube root of 19. What is the cube root of 20. What is the cube root of 21. What is the cube root of 22. What is the cube root of 3048625 ? Of 34328125 ? 41063625? Of 43614208? 27270901 ? Of 515849608 ? 185193? 128024064? 103823? Of 27054036008? 15.625? Of .074256? 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: 56.470, and 12.345600. 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"? Of ? 28. What is the cube root of 11-g-J? Of 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 two 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 the number into periods of three orders each. 2 . Find the greatest cube in the left-hand period, and place its cube root at the right for the first term of the 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 the first figure of the next period for a dividend. 4. Take three times the square of the first term oj the root for a trial divisor, and the quotient for the second term of the root. Cube the root now found, and, if the result 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. The 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 the 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 places. 4. When both terms of a common fraction are not perfect cubes, the cube root may be found approximately by multiplying both terms of the fraction by the square of the denominator, and extracting the root of the resulting fraction. The error will 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 which 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. Oeometrical Explanation, of 1 fhe Process of Ex tracting the Cube Root. 417 . Tlie 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. Process. I5625|25 2 s = 8 20 2 X 3 = 1200 ) 7625 20 2 X 5 X 3 = 6000 20 X 5 2 X 3 = 1500 5 3 = 125 7625 6 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 is 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 ? 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 three 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 2 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 3 is greater than 15625, the solid contents of the given cube. Try 5 for the thickness. 25 3 = 15625, and hence 5 is the required thick¬ ness, and the length of the edge of the given cube is 20 -f- 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 3 = 8000. The joint solidity of the three adjacent rectangular solids is 20 2 X 5 X 3 = 6000. Removing these three rectan¬ gular solids, there remain three other rectangular solids, whose solidity is 20 X 5 2 , 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 5 3 == 125. Adding the solidity of the sev¬ eral parts, we have 8000 -\~ 6000 + 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- 264 COMPLETE ARITHMETIC. jacent rectangular solids represent three times the product of the square of tens hy the units; the three smaller rectangular solids, three times the product of the tens by the square of the units; and the smaller cube, the cube of the units. MENSURATION, INVOLVING INVOLUTION AND EVOLUTION. I. THE RIGHT-ANGLED TRIANGLE. 418 . The Hypotenuse of a right-angled triangle is the side opposite the right angle. The other two sides are called the Base and the Perpendicular. (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 other 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. 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 V8 2 + 6 2 = V 100 = 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 length 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 150 rods 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. THE CIRCLE. 421 . Principles.— 1 . The area of a circle is equal to the square of its diameter multiplied by .7854. Hence, 2. The diameter of a circle equals the square root of the quotient of the area divided by .7854. 3. The areas of two circles are to each other as the squares of their diameters. Note.— The above propositions can be proved by geometry. 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 whose 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. 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.—T he surface of a sphere may also be found by multiplying the circumference by the diameter; and the solidity by multiplying the surface by one sixth of the diameter. PROBLEMS. 19. What is the surface of a sphere whose diameter is 10 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 94^ feet? 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, whose diameter is about 8000 miles? 268 COMPLETE ARITHMETIC. SECTION XVII. 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. Problems marked * are simplified in this edition. MENTAL PROBLEMS. 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 . -J of 74J is f of what number? 4. | of 45 is -§ of how many times 10? 5. A has 20 cents; and f 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 13J cents, what will j- of a yard cost? 8 . The difference between f and -g- of a number is 10: what is the number? 9. What fraction is as much greater than § as -J is less? 10. A piece of flannel lost J- of its length by shrinkage in fulling, and then measured 30 yards: what was its length before fulling? 11. A horse cost $90, and t 3 q- 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 my 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, -J of it for a hat, and then had 7 cents left: how many cents had he at first ? 15. | 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 -§ of his sheep and then bought f as many as he had left, when he had 40 sheep: how many had he at first? 18. John lost f of his money and spent -J- 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 ^ more than its cost, and B sold it for $80, losing ^ of its cost: how much did A pay for the horse ? 22. At $^ a bushel, how many bushels of corn may be bought for $8? 23. If J of a bushel of wheat cost $f, what part of a bushel can be bought for $f ? 24. If $18f will purchase f of a load of corn, what part of it will $16f purchase? 25. If 2-|- pounds of cheese cost 3J 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 -f of a peck ? 28. What part of a pound of gold is .25 of an ounce? 29. What part of f of a gallon is f 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 1J 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 J- 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 j 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 f 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 f 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 133^ per cent of what he earned: how much did he earn ? 58. A man sold a watch for $90, and gained 50 per cent: w T hat per cent w r ould he have gained if he had sold it for $75? 59. f of the price received for an article is equal to | 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 lias taken 10 per cent for toll? 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 work in days, working 8 hours a day, how long will it take, if he work 6 hours a day? *69. A is 20 years of age; 4 times A’s age equals the sum of B’s and C’s ages; and C’s age is f of B’s age: what are the ages of B and C? 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 1 of the hound’s leaps equal 2 of the hare’s: how many leaps must the hound take to gain one of the hare’s leaps? To catch the hare? *72. If a steamer sails 9 miles an hour down stream, and 5 miles an hour up stream, how many hours will it be in sailing 45 miles down stream and returning? 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 -J, -J, •§• of f-, and -f of 82. From 17^- take | of 6^, and multiply the remainder by -§. 83. Multiply f of -§ by -J- of f, and divide the product b y A- 84. Divide f of 6^ by -§ of 7^. 85. What number multiplied by 28f will produce 145? 86. From the sum of 215§ and 125J take their differ¬ ence. 87. Multiply f -f f of J by f — -f of f. 88. Divide 2|- by 3J, and multiply the quotient by 3^. 89. What must 8f| be multiplied by that the product may be 3 ? 90. A man bought of a section of land for $2880, and sold -J of it at $10 an acre, and the rest at $12 an acre: how much did he gain? 91. A merchant owning -J 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 millionths, 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 yfo to a decimal, and divide the result by 2J. 99. Change yfy 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 $2.25 per cord ? 103. How many minutes were there 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 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 2^ miles long? 108. How many yards of carpeting, f of a yard wide, will it take to cover a parlor, 18J 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. What 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 5J 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 wide, 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. Keduce 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 J- 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, would 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 per cent: required the per cent of gain or loss. 137. I bought 63 kegs of nails, each keg containing 100 lbs., at 4r^ 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 -J of its value, at f %: what is the premium? 141. I pay $19.20 premium for insuring my house for f 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. An 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 7 ^ % ? 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 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 the interest is J of the prin¬ cipal : what is the principal, and what is the rate percent? 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, payable in 90 days, at 8 % ? 155. What 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 $300, 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 $53J, 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 two 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 1^ feet thick? 179. There is a circular field 40 rods in diameter: what 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 ^ of a foot, are equivalent to a block of wood 8 feet long, 4 feet wfide, 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 water 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 round cistern 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 10 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 3f miles ? 190. A father bequeathed of his estate to his eldest son, § 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 worth $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? Suggestion: ^ of (90—10)= number of men and women. 193. Divide $630 among 3 persons, so that the second shall have § as much as the first, and the third ^ as much as the other two together. [Sug. : Third’s share = £ of $630.] 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 T 9 ^ of a piece of work in 2 days, when, B leaving, A completes it in J 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 50 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 walked 100 miles in 2 days, and J of the distance walked the first day equaled \ the distance walked the second 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 J of its weight ? Suggestion. —Since the handspike is to support two thirds of the weight, or twice as much as is carried by the man at the end of the stick, it must be placed half as far from the middle of the stick, which is half the distance from the end to the middle; \ of 15 ft. is 7^ ft. The weights sustained at the two points of support, are inversely as their respective distances from the middle of the stick; 2:1 : : 15 ft : Ans. 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 ? Suggestion. —Base of right-angled triangle is 40 feet, and its per¬ pendicular 30 feet. 200. A cistern is filled by two pipes, one of which will fill it in 2 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 tt 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 rep¬ resented? Name them. 2. What is the difference between numeration and notation? Between the Arabic notation and the Roman? 3. What is the value of the figure 5 in 452? How is the value of a figure affected by its removal one order to the left? One order to the right? How is the value of a number affected by an¬ nexing 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 num¬ bers? 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 subtrahend equally. 6. Why must the multiplier be an abstract number? When the multiplicand is concrete, what is true of the product? Why? 7. What kind of number is the quotient when both divisor and dividend are like numbers? What is the difference between short division and long division? 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 difference. 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 2J a divisor of 5? Is a multiple of 5? Is 12£ a multiple of 61? 14. What is a fraction? In what two ways may a fraction 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 o 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? Why 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-hundredths a decimal fraction? In what two ways 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. How is a decimal reduced to a common fraction ? A common fraction to a decimal? Why 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. Plow 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 the equator passes beneath the vertical rays of the sun 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 k 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 chiefly 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 ? IIow 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? What is exchange? The rate of exchange? 58. What is a bond? What is a coupon? When bonds are quoted at 108, what are they worth? What are United States bonds also called? 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 ? 63. 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 compound ? How many terms 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 surd ? 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? Solution of Watch Problems, page 243. The dial of a watch is divided into 12 equal spaces, and while the hour hand passes over 1 of these spaces, the minute hand passes over 12, and hence the minute hand passes over 12 spaces to gain 11 spaces on the hour hand, or of a space to gain 1 space. But it takes the minute hand 5 minutes to pass over 1 space; and to gain 1 space on the hour hand, it will take j-f- of 5 minutes, or 5 t 5 t minutes. Hence, it will take 5 T 5 T minutes for the minute hand to overtake the hour hand between 1 and 2 o’clock (to gain 1 space), twice 5 x 5 t minutes between 2 and 3 o’clock, 3 times 5 T 5 T minutes be¬ tween 3 and 4 o’clock, etc. 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, 3, 4, 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. PROOF 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 in 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 3 -f- 4 -j- 5 = 14), and dividing the sum by 9, we have the same remainder. (287) 288 COMPLETE ARITHMETIC. 426 . This principle may be thus explained: 2345 4 2000 = 222 X 9 + 2 300 — 33 X 9 + 3 40 = 4X9 + 4 5 = 5 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. The excess of 9’s in the first number, found by adding its digits, is 1; in the second number, 4; in the third, 7. The excess of 9’s in the sum of these excesses is 3, which equals the excess of 9’s in 939, 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. 325 Excess 1 256 “ 4 358 “ 7 939 “ 3 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. 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, The 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. Process. 347 Excess 5 53 “ 8 40 1041 1735 18391 Excess 4 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 J. 347 “ 5 53 “ 8 105 “ 6 5X8 + 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 equal to the excess of 9’s in the product of the excesses in divisor and quotient , plus the excess in 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. 121.) 432 . The figure or set of figures repeated is called a Repetend. A repetend is denoted by a dot placed over the first and • • • • • • last of its figures; as, .5 .16 .325. C.Ar.—25. 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 “ u u Subtracting, 63. = 99 times 11 u « Hence, 63 99 = the value of u « 435. A mixed circulate is reduced to a common fraction by subtracting the terms which precede the repetend from the whole circulate, 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.i24 = 100000 times the mixed circulate. And, 45.i24 = 100 “ “ “ “ Subtracting, 45079 = 99900 “ “ “ “ Hence, = 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 first 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. 2. English Money. The denominations are mills, cents, dimes, dollars, and eagles. The denominations are far¬ things (q.), pence (d.), shil¬ lings (s.), and pounds (£). Table. Table. 10 m. = 1 ct. 10 ct. = 1 d. 10 d . = $1 $10 = 1 E. 4 q. = 1 d. 12 d. = 1 s 20 s. = 1 £. 1 £ = $4.8665. II. MEASURES OF EXTENSION AND TIME. 1. MEASURES OF LINES AND ARCS. Long Measure. Circular Measure. The denominations are inches, feet, yards, rods, fur¬ longs, and miles. The denominations are sec¬ onds, minutes, degrees, signs, and circumferences. Table. 12 in. = 1 ft. 3 ft. =1 yd. yd. = 1 rd. 40 rd. = 1 fur. 8 fur. = 1 m. Table. 60" = V 60' = 1° 30° = 1 s. 12s - UlC. 360° / Also: Cloth Measure. 3 barleycorns = 1 inch. 4 inches = 1 hand. 3 feet = 1 pace. 6 feet — 1 fathom. 3 miles (geog.)— 1 league. (Little used.) 2\ in. — 1 nail. 4 n. =1 quarter. 4 qr. = 1 yard. 5 qr. = 1 Ell Eng. 60 geographic miles ) ~ nl . , . •i / n > = 1 degree at the equator. 69 J statute miles ( nearly ) J J 1 292 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. = 1 sq. yd. 304 sq. yd. = 1 P. 40 P. =1 B. 4 B. =11 640 A. =1 sq. mi. Surveyor’s Measure. Table. 7.92 in. = 1 link ( l.). 25 l. =1 rod. 4 rd. = 1 chain ( ch.) 80 ch. = 1 mile. Also: 625 sq. 1. =1 P. 16 P. =1 sq. 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 cu. in. = 1 cu. ft. 27 cu. ft. — 1 cu. yd. Dry Measure. The denominations are pints, quarts, pecks, and bushels. Table. 2 pt. =1 qt. 8 qt. =1 pk. 4 pk. 1 bu. Wood Measure. Table. 16 cu. ft. = 1 cord ft. 8 cd.fi., or), 128 cu. ft. i 24f cu. ft. — 1 perch of stone. Liquid Measure. Table. 4 gills = 1 pt. 2 pt. =1 qt. 4 qt. =1 gal. Notes. — 1 . The standard bushel contains 2150f cu. in.; the liquid gallon, 231 cu. in.; and the beer gallon (little used), 282 cu. in. 2. The size of casks for liquids are variable. Barrels contain from 30 to 40 gallons, or more. The capacity of vats, cisterns, etc., is usually measured in barrels of 314 gallons. 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 h. 24 h. =1 da. 365 da. = 1 common yr. 366 da. = 1 leap yr. 365^ da. = 1 solar yr. 100 s. yr. — 1 century. Also: 7 da. = 1 week. 4 w. =1 lunar mo. Calendar Months. January, 1st mo., 31 days. February, 2d <« 28 or 29. March, 3d « 31 days. April, 4th u 30 U May, 5th u 31 a June, 6th u 30 u July, 7th a 31 a August, 8th « 31 u September, 9th u 30 a October, 10th a 31 u November, 11th u 30 u December, 12th u 31 ii 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 weights, and tons. are drams, Table. ounces, pounds, hundred- 16 dr. — 1 oz. 16 oz. = 1 lb. 100 lb. = 1 cwt. 20 cwt. — IT. Also: 196 lb. flour — 1 barrel. 200 lb. beef or pork = 1 “ 100 lb. fish 56 lb. corn or rye \ 60 lb. wheat 32 lb. oats J 14 lb. iron or lead 21^ stones 8 pigs — 1 quintal. = 1 bushel. = 1 stone. = 1 pig- = 1 jother. 294 COMPLETE ARITHMETIC. Troy Weight. The denominations are grains, permy weights, ounces, and pounds. Table. 24 gr. — 1 pwt. 20 pwt. = 1 oz. 12 oz. — 1 lb. 3 J T. gr. = 1 carat. Note. —In determining the fine¬ ness of gold, it is considered as composed of 24 parts, called carats, and the number of carats specified is the number of 24 ths of pure gold which it contains. A sixteen- carat chain contains \\ of pure gold and of alloy. Apothecaries Weight. The denominations are grains, scruples, drams, ounces, and pounds. Table. 20 gr. = 1 ^ 3 9=13 3 3 = 1 § 12 % — 1 lb Apothecaries Fluid Measure. 60 minims = 1 dram, f 3. 8/3 = 1 ounce, f £. 16 / 3 = 1 pint, 0 . COMPARISON OP WEIGHTS. 1 lb. Avoir. 1 02. “ IjVi d). Troy = l T 3 / ? lb Apoth. — 175 n ~ — oz — 175 — 1^2 a IV. MISCELLANEOUS TABLES. 12 things 12 dozen 12 gross 20 things 18 inches 22 inches ( nearly) are 1 dozen. “ 1 gross. “ 1 great gross. “ 1 score. “ 1 cubit. “ 1 sacred cubit. Paper. 24 sheets 20 quires 2 reams 5 bundles are 1 quire. “ 1 ream. “ 1 bundle. “ 1 bale. Books. A sheet folded in tt u tt tt tt u tt a u tt 2 leaves is called a folio. 4 tt a quarto or 4 to. 8 u an octavo , or 8 vo. 12 u a duodecimo, or 12wio. 16 it a 16 mo. 24 u a 24 mo. Note. —In estimating the size of the leaves, as above, the doubl® medium sheet (23 by 26 inches) is taken as a standard. LIFE INSURANCE. 295 LEGAL RATES OF INTEREST. 438. The rates of interest fixed by law in the several states, are as follows: NAME OF STATE. Legal Rate of Interest. Rate al¬ lowed by Contr’ct. Alabama . 8 % 8 % Arkansas . 6% 10% California . 10 % Any. Colorado. 10 % Any. Connecticut. 6% 6% Delaware. 6% 6% Florida . GO Any. Georgia . 7% Any. Illinois . 6% 8 % Indiana . 6% 8 Jo Iowa . 6% 10% Kansas . 7% 12% Kentucky . 6% 100 Louisiana . 5 % 80 Maine . 6% Any. Maryland . 6% 6% Massachusetts . 6% 60 Michigan . 7 % 10% Minnesota . 7% 100 NAME OF STATE. Legal Rate of Interest. Rate al¬ lowed by Contr’ct. Mississippi. 60 10% Missouri . 60 10% Nebraska. 10% 12% Nevada. 10% Any. New Hampshire... 6% 6% New Jersey. 6% 6% New York. 6% 6% North Carolina. 6% 8% Ohio. 6% 8% Oregon. 10% 12% Pennsylvania. 6% 6% Rhode Island. 6% Any. South Carolina. 7% Any. Tennessee. 6% 6% Texas. 8% 12% Vermont. 6% 6% Virginia. 6% 8% West Virginia. 6% 0% Wisconsin. 7% 10% 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, put at interest, will amount to the sum insured at the close of the 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 En¬ gland, and the latter is based on the mortality in the United States. Both tables are in use in this country. 296 COMPLETE ARITHMETIC. 441. The Expectation of Life, as shown by the two tables, is as follows : AGE. EXPECTATION BY C. TABLE. EXPECTATION BY W. TABLE. AGE. EXPECTATION BY C. TABLE. EXPECTATION BY W. TABLE. W CJ ◄ EXPECTATION BY C. TABLE. EXPECTATION BY W. TABLE. W o < EXPECTATION BY C. TABLE. EXPECTATION BY W. TABLE. 0 38.72 28.15 24 38.59 32.70 48 22.80 22.27 72 8.16 9.14 1 44.68 36.78 25 37.86 32.33 49 21.81 21.72 73 7.72 8.69 2 47.55 38.74 26 37.14 31.93 50 21.11 21.17 74 7.33 8.25 3 49.82 40.01 27 36.41 31.50 51 20.39 20.61 75 7.61 7.83 4 50.76 40.73 28 35.69 31.08 52 19.68 20.05 76 6.49 7.40 5 51.25 40.88 29 35.00 30.66 53 18.97 19.49 77 6.10 6.99 6 51.17 40.69 30 34.34 30.25 54 18.28 18.92 78 6.02 6.59 7 50.80 40.47 31 33.68 29.83 55 17.58 18.35 79 5.80 6.21 8 50.24 40.14 32 33.03 29.43 56 16.89 17.78 80 5.51 5.85 9 49.57 39.72 33 32.36 29.02 57 16.21 17.20 81 5.21 5.50 10 48.82 39.23 34 31.68 28.62 58 15.55 16.63 82 4.93 5.16 11 48.04 38.64 35 31.00 28.22 59 14.92 19.04 83 4.65 4.87 12 47.27 38.02 36 30.32 27.78 60 14.34 15.45 84 4.39 4.66 13 46.51 37.41 37 29.64 27.34 61 13.82 14.86 85 4.12 4.57 14 45.75 36.79 38 28.96 26.91 62 13.31 14.26 86 3.90 4.21 15 45.00 36.17 39 28.28 26.47 63 12.81 13.66 87 3.71 3.90 16 44.27 35.76 40 27.61 26.04 64 12.30 13.05 88 3.59 3.67 17 43.57 35.37 41 26.97 25.61 65 11.79 12.43 89 3.47 3.56 18 42.87 34.98 42 26.34 25.19 66 11.27 11.96 90 3.28 3.73 19 42.17 34.59 43 25.71 24.77 67 10.75 11.48 91 326 3.32 20 41.46 34.22 44 25.09 24.35 68 10.23 11.01 92 3.37 3.12 21 40.75 33.84 45 24.46 23.92 69 9.70 10 50 93 3.48 2.40 22 40.04 33.46. 46 23.82 23.37 70 9.18 10.06 94 3.53 1.98 23 39.31 33.08 47 23.17 22.83 71 8.65 9.60 95 3.53 1.62 Note. — A comparison shows that the Wigglesworth table has a less expectation of life than the 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 inclusive. EQUATION OF PAYMENTS. 442. In 1860, the author published a demonstration of the correctness of the common Mercantile Rule for finding the equated time for the payment of several debts, due at dif¬ ferent times without interest. The inaccuracy of the rule by present worths , commended by several authors as “the only accurate rule ”, was 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, without 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 two 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 -f- $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 Monthly , for 1862, p. 116. ARITHMETICAL PROGRESSION. 443. An Arithmetical Progression is a series of numbers which so increases or decreases that the difference between the consecutive numbers is constant. 444. The numbers which form the series are called Terms , the first and last terms being the Extremes , 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 Series is one 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. 8. The common difference. 4. The 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 24-3 2-K34-3) 2+(3+3+3) 2+(3+3+3+3) (3) 2 24-3 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. The 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 number of terms less one. 4. The 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 -f- 16 + 16 + 16 — twice the sum of the terms, and 8-[“ 8 + 8 + 8 + 84“ 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 the 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 ± ( com . difference X number of terms less one). 2. First term = last term (com. difference X number of terms less one). 3. Common difference = \ Ud term ~fi rst Urm j -4- number (first term — last term j of terms less one. . 7 - . 7 f last term — first term) 4. dumber of terms less one = ■> ^ term ] —j— common difference. 5. Sum of terms = \ (first term -J- 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. PROBLEMS. 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 Progression is a series of numbers which so increases or decreases that the ratio be¬ tween the consecutive terms is constant. The first and last terms are called the Extremes , and the intervening terms are called the Means . 451. A geometrical progression is ascending 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 terra 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 2X3 2 2X3 3 2X3 4 2X3 5 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 senes 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 whose index is one less than the number of terms, of the quotient of the last term divided by the first term. 454. By an algebraic process it may be shown that 4. The sum of an ascending 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 one less the ratio. 302 COMPLETE ARITHMETIC. 456. From the above principles may be deduced the fol¬ lowing FORMULAS. 1 . 2 . 3. 4. 5 . Last term —first term X ratio n ~ 1 . First term = last term-—ratio n ~ l . Ratio = n ~i/last term —first term. C1 r . (last term X ratio) — first term bum of series = - -- ratio — 1 Sum of infinite descending series = first term-±-(l — ratio). Notes. —1. By “ratio w_1 ,” 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 1. 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 -J, 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? 0. 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: what was the value of a bushel of the mixture ? Process. cts. cts. 50 X 25 = 1250 80 X 15 = 1200 70 X 30 = 2100 70 ) 4550 65 cts., Ans. 2. A grocer mixed 20 pounds of coffee worth 28 cents, 30 pounds worth 35 cents, and 50 pounds worth 33 cents: what is a pound of the mixture worth ? Since the total value of the 70 bushels of grain mixed together was 4550 cents, the value of 1 bushel was of 4550 cents, which is 65 cents. 304 COMPLETE ARITHMETIC. Case II. 459. The values of several articles given, to find in what pro¬ portion they must he 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. 1 “ 1 “ at 16 cts. selling for 20 cts. gains 4 cts. 1 « , - 18 “ “ 20 “ “ 2 cts. / C ’ ga * 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 Note. —When only two articles of different values are given, they can be compounded in but one way; but wften more than two articles are given, they may be compounded in an infinite number of ways. 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 the 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 multiply 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 4J pwt. each ? Suggestion. —Find the proportionals by Case II, and then divide the whole quantity into parts proportional to these proportionals. 8. How 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.Ar.—2<5. 306 COMPLETE ARITHMETIC. DUODECIMALS. 462. A Duodecimal 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 whose 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 fourths ( //// ) are 1"' 12 thirds u 1" 12 seconds n 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, the twelfth of the twelfth of afoot, etc. When a duodecimal denotes the area of a surface, the foot is a square foot; the prime, the 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". f 12 ft. 8 / 11" Process: ^ 16 ft. 10 / 9" ( 24 ft. O' 6" 53 ft. 8 / 2" Ans. DUODECIMALS. 307 2. Add 12 ft. 9' 11" 4"', 23 ft. 7" 10"', and 10' 6" 9'". 3. From 21 ft. 7' 10" take 15 ft. 9' 4". t>^ / 21 ft. r io" Process: 1 15 ft> 9 / 4 // 5 ft. 10 / 6" Arcs. 4. From the sum of 30 ft. 8" 4'" and 14 ft. 7' 10'", take their difference. 466. MULTIPLICATION OF DUODECIMALS. 5. Multiply 13 ft. 7' 8" long and 6 ft. 5' wide. Multiply first by 5 / and then by 6 ft., and add the partial products. Since lXiV— tV» f? X tUXtV=T72T> etc., feet x primes (or twelfths) must produce ‘primes; 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). Process. 13 ft. V 8" 6 ft. 5' _ 5 ft. 8' 2" 4 /// 81 ft. W 0" 87 ft. 6' 2" 4 7// , Ans. 467. DIVISION OF DUODECIMALS. 8. Divide 87 ft. 6' 2" 4'" by 13 ft. 7' 8". Process. Dividend. Divisor. 87 ft. 6' 2" 4 /7/ ) 13 ft. 7 / 8" 81 ft. 10 / 6 ft. 5', (fnt. The process is the reverse of that in multiplication. For con¬ venience in multiplying, place the divisor at the right of the divi¬ dend, and the terms of the quo¬ tient below those of the divisor. 5 ft. 8' 2" 4 /7/ 5 ft. 8' 2" 4 /7/ 308 COMPLETE ARITHMETIC. 9. Divide 62 ft. 11" 3'" by 8 ft. 6' 9". 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. 469. The diagram at the right shows the number of permutations of 1, 2, and 3 letters. The letter a permits no change of order. The letter b may be placed before and after the letter a , giving two (1X2) permutations of two letters— ba, 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 twenty-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. PROBLEMS. 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 I 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 Certain 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. RULES OF MENSURATION. 474. Surfaces and Lines. 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 the other side. 310 COMPLETE ARITHMETIC. 3. To find the area of a triangle, Multiply the base by one half of the 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 circle, Multiply the radius by the 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 two bases. 13. To find the convex surface of a pyramid or cone, Multiply the perimeter or circumference of the base by one half the slant height. 14. To find the hypotenuse of a right-angled triangle, RULES OF MENSURATION. 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 side. 475. 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. Board Measure. 1. To measure lumber 1 inch or less in thickness, Multiply the length in feet by the width in inches, and divide the product by 12. 2. To measure lumber more than 1 inch in thickness, as planks, joists, etc., Multiply the number of square feet in one surface by the thickness in iiiches. Note. —Boards 1 inch or less in thickness are sold by the square foot, surface measure; but lumber more than 1 inch in thickness is measured by finding the number of square feet in one surface, and multiplying the result by the thickness in inches. 312 COMPLETE ARITHMETIC. FOREIGN EXCHANGE. 476. A Foreign Bill of Exchange is a draft drawn in one country and payable in another. (Art. 328). Foreign Bills are expressed in the currency of the country on which they are drawn. They are issued in sets of three, of the same tenor and date, called the First, Second, and Third of Exchange, and are sent by different mails to avoid delay in case of miscarriage. When one is paid, the others are void. 477. The Far of Exchange is the comparative value of the currencies of two countries. (Art. 329). The commercial value of foreign exchange may be higher, equal to, or lower than the par of exchange. A bill payable in sixty days costs less than a bill payable on sight, or in three days, called “short sight.” The commercial or quoted value of exchange is used in finding the cost of a foreign bill. EXCHANGE ON ENGLAND. 478. Bills between the United States and England are expressed in sterling money, and are drawn on London. They are called Sterling Bills. The legal or par value of a pound sterling is $4.8665, and this is now custom-house value. The commercial or exchange value is now quoted in dollars and cents, gold. The gold coin, whose value is £1, is called a Sovereign. PROBLEMS. 1. What will a bill on London for £448 11 s. cost in New York, when sterling exchange is quoted at 4.86§? Process. £448 11s. = £448.55. $4.86$ X 448.55 = $2182.943, cost of bill. Since £1 is worth $4.86$, £448.55 are worth $4.86$ X 448.55, which is $2182.943. FOREIGN EXCHANGE. 313 2. What will a sterling bill for £219 10s. 6d. cost in New York, when sterling exchange is quoted at 4.91J? 3. What will a bill on London for £200 12s., payable in 60 days, cost in New York, when sterling exchange is quoted at 4.85f ? Note.— In all exchange problems in this edition, the gold quo¬ tations are omitted. 4. What will a sight draft on London for £300 8s. cost in New York, when sterling exchange is quoted at 4.88? 5. What will a sight draft on London for £250 cost a merchant in Cincinnati, when sterling exchange is quoted at 4.86, and the broker’s commission is of cost of draft in New York? 6. What will be the cost of the following bill when ster¬ ling exchange is quoted at 4.85J? £1000. New York, Jan. 10, 1876. Sixty days after sight of this First of Exchange (Second and Third of same tenor and date unpaid) pay to the order of Wilson, Hinkle & Co. One Thousand Pounds, value re¬ ceived, and charge to account of August Belmont & Co. To Brown , Shipley & Co., ) London. j 7. What amount of sterling exchange can be bought for $1080.45 in gold, when sterling exchange is quoted at 4.90? Process. $1080.45 —t- $4.90 = 220.5; £220.5 = £220 10s. 8. How large a draft on London can be bought in Chicago for $2195.475, when sterling exchange is quoted at 4.86§, and the broker’s commission is \^/ 0 of cost of draft in New York? 9. How large a sight draft on London can be bought in New York for $1174.20, when sterling exchange is quoted at 4.89J? C. Ar.—27. 314 COMPLETE ARITHMETIC. 479. Rules. — 1. To find the cost of sterling exchange, Multiply the cost of £1 by the number of pounds denoting the face of the bill. Note. —When the face of the bill contains shillings and pence, reduce them to the decimal of a pound. (Art. 171.) 2. To find the amount of sterling exchange that can be bought for a given sum of United States money, Divide the given sum of money by the cost of £1 of exchange. EXCHANGE ON FRANCE. 480. The New York quotations of exchange on Paris give the number of francs and centimes which are equal in ex¬ change to $1 of United States money (gold). The centimes are usually expressed as hundredths. Quotations on Antwerp and Switzerland are also in francs. Quota¬ tions on Amsterdam are in guilders, worth about 41 cents. The value of a franc (Louis Napoleon) is $.192 nearly, $1 being equal to 5 francs and 14| centimes. The custom-house value is $.193. PROBLEMS. 1. What will be the cost of a bill on Paris for 3870 fr., when Paris exchange is quoted in New York at 5.16? Process. 3870 fr. -r- 5.16 fr. = 750; $1 X 750 = $750. Since $1 will buy 5.16 fr., it will take as many times $1 to buy 3870 fr. as 5.16 fr. is contained times in 3870 fr., which is 750. 2. What will a draft on Paris for 6475 fr. cost in New York, when Paris exchange is quoted at 5.18? 3. What will a bill on Paris for 5330 fr. cost in New York when Paris exchange is quoted at 5.12J? FOREIGN EXCHANGE. 315 4. What amount of exchange on Paris can be bought for $1500, when Paris exchange is quoted at 5.14J? Process. 5.14| fr. X 1500 = 7717.5 fr., Ans. 5. How large a draft on Paris can be bought in New York for $2432, when Paris exchange is quoted at 5.16§? 6. What will be the cost of a bill on Antwerp for 6418f fr., when exchange is quoted at 5.13J? 7. What amount of exchange on Switzerland can be bought for $650 when exchange is quoted at 5.15? 481. Rules.— 1. To find the cost of exchange on Paris, Divide the number of francs in the face of the bill by the number of francs that equal $1 of exchange . 2. To find the amount of exchange on Paris that can be bought for a given sum of money, Multiply the number of francs that equal $1 of exchange by the number denoting the given sum of money. EXCHANGE ON GERMANY. 482. Exchange on Germany is quoted at so many cents per four reichsmarks (marks). The par value of a reichsmark, the new German coin, is $.243, or about 11 francs. PROBLEMS. 1. What will be the cost of a sight bill on Hamburg for 2240 marks, when exchange is quoted in New York at .96J? Process. $.96} X 2240 -s- 4 == $539, cost. 316 COMPLETE ARITHMETIC. 2. What would be the cost of a sight bill on Berlin for 1680 marks, when exchange is quoted at .96J? 3. What would be the cost in St. Louis of a sight draft on Hamburg for 3200 marks, when exchange is quoted at .96J, and the broker’s commission is \^ 0 ? 4. What amount of exchange on Frankfort can be bought for $1752.60, when exchange is quoted at .95J? Process. $1752.60 -r- $.95J X 4 = 7360, marks. 5. What amount of exchange on Berlin can be bought for $324, when exchange is quoted at .94J? 6. What will a draft for 960 marks cost, when exchange is quoted at .95|? 7. How large a draft on Frankfort can be bought for $514.35, when exchange is quoted at .95J? 8. What amount of exchange on Hamburg can be bought for $231, when exchange is quoted at .96, and the broker’s commission is \% ? 483. Rules. —1. To find the cost of exchange on Ger- many, Multiply the cost of four marks by the number of marks in the face of the bill , and divide the result by 4. 2. To find the amount of exchange on Germany that can be bought for a given sum of money, Divide the given sum of money by the cost of four marks , and multiply the result by 4. Or, Divide the given sum of money by the cost of 1 mark. .ANSWERS TO THE WRITTEN PROBLEMS. N. B.—The last answer is given when a problem has several an¬ swers, and also when several problems are united. NOTATION. Page 11. 9. 40,605. 15. 4,014,045,000. 10. 700,007. 16. 65,000,006,050. 11. 5,005,500. 17. 850,049,000,000. 12. 60,060,060. 18. 17,070,000,700,400. 13. 700,700,700. 19. 56,000,016,000,090. 14. 560,068,000. 20. 7,000,085,000,000,204. ADDITION. Page 14. 19. 8,984,342. 26. 598. 13. 108,657. 20. 3,578,392 sq. m. 27. 732. 14. 442,555. 21. 258. 28. 803. 15. 63,077,833. Page 16. 29. 631. 16. 74,467,648. 22. 383. 30. 865. 17. 12,369 bush. 23. 512. 31. 636. Page 15. 24. 462. 32. 633. 18. 443,275 sq. m. 25. 649. SUBTRACTION. Page 18. 20. 1,116,942 sq. m. Page 19. 16. $4,075. 21. 914,054 sq. m. 26. $800. 17. 49,894,136 m. 22. 56,077,528 bush. 27. 6,890 bush. 18. 429,559. 23. $467. 28. All, 1802 A. 19. 35,965. 24. $2,330. 29. 26,956. 25. $1,032. 317 21-36 COMPLETE ARITHMETIC. Page 21. 11. 2,499,120. 12. 230,668,800. 13. 503,232,000. 14. 364,800,000,000. 15. 17,424,000 ft. 16. 1,572,480 m. 17. 25,600,000 A. 18. 28,500. 19. Gained $798. Page 23. 7. 4,560,000. Page 26. 15. 54. 16. 233, with 20 R. 17. 7, with 600 R. 18. 1, with 109,304 R. 19. 3,464. 20. 8,743. 21. 4,567. 22 . 41 cars. 23. 5m. 2,700 ft. Page 27. 24. 205 h. MULTIPLICATION. 8 . 305,000,000. 9. 347,000,000,000. 10. 88,900,000. 16. 15,300. 17. 16,200. 18. 84,400. Page 24. 19. 13,549,333£. 20. 867,000. 21. 1,362,000. 22. 691,066|. DIVISION. 25. 548, with 128 R. Page 29. 8 . 356. 9. 46, with 35 R. 10. 38, with 4,602 R. 11. 95. Page 30. 18. 9, with 200 R. 19. 24, with 800 R. 20. 9, with 10,800 R. 21. 2, with 1,600 R. 28. 125. 23. 45,766. 24. 5,666,328. 25. 40,740,411. 26. 86,772,642. 27. 61,870,306,330. 28. 322,096. Page 25. 29. 652,919. 30. 94,240. 31. 25,629,438. 32. 4,432,246. 33. 613,566. Page 31. 29. 26, with 190 R. 30. 45, with 300 R. 32. 761, with 39 R. 33. 190, with 28 R. 34. 387, with 13 R. 35. 57, with 39 R. Page 32. 36. 480 with 7 R- 37. 1,487, with 26 R. 38. 5,203 with 40 R. 39. 3,604, with 16 R. 40. 433, with 3 R. PROPERTIES OF NUMBERS. Page 33. 15. 2, 2, 2, 2, 2, 5. 16. 5, 5, 7. 17 . 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 . 18. 5, 5, 13. 19. 2, 3, 5, 11. 20. 2, 2, 3, 5, 7. 21. 2, 3, 7, 11. 22. 2, 5, 7, 7. 23. 2, 3, 3, 3, 11. 24. 2, 2, 11, 17. 25. 3, 3, 7, 11. 26. 3, 5, 7, 11. Page 34. 27. 3, 3. 28. 5,2. 29. 2, 2, 2, 2, 2. 30. 2, 5, 5. 31. 5,5. 32. 2, 2, 3, 3. Page 35. 34. 7. 35. 12, 36. 55. Page 36. 38. b 39. 8 . 40. 41. 54 cts. 42. 9 men. 318 ANSWERS. 38-53 GREATEST COMMON DIVISOR. Page 38. 20. 48. 29. 39 lbs. 12. 12. 21. 37. 30. 25. 13. 63. 22. l. 31. 120. 14. 48. 23. 252. 32. 72. 15. 28. 24. 14. 33. 7. 16. 42. 25. 192. 34. 8. 17. 128. 26. 57. 35. 5. 18. 48. 27. $52. 36. 13. 19. 4. 28. $165. 37. 1. LEAST COMMON MULTIPLE. Page 41. 18. 300. 25. 756. 12. 120. 19. $720. 26. 720. 13. 126. 20. 420. 27. 1,200. 14. 480. 21. 180. 28. 1,890. 15. 108. 22. 280. 29. 3,360. 16. 144. 23. 180. 30. 2,520. 17. 210. 24. 600. FRACTIONS. Page 46. 32. 191. 60. 5 13* 8. 2923 ~2~' 33. 9. 61. 3 2* 9. 988 'T2~* 34. I7f. 62. 5 7* 10. 4061 22 * 35. 38y. 63. 1 3 '7'* 11 . 13 3 T2~- 36. 46^. 64. 5 4* 12. 112 1 "T2 - ' 37. 109A. Page 49„ 13. 618 7 3 0 * 38. 12gf3. 65. 1 8 TT- Page 47. 39. 53jtj. Page 50. 14. 5803 4 0 * 40. 2,016f. 87. 32 66 34 69 72) 72) 72) 72* 15. 343 3 ~22~' Page 48. 88. 80 72 100 105 120) 120) 120) 120' 16. 20409 51. f 89. 42 3 2 33. 2 6 60) 60) 60) 60* 17. 2 4 6 1 3 4T * 52. If. 90. 49 32 33 34 84) 24) 24) 84- 18. HP- 53. |~f* Q1 24 55 33 58 31 v A. 62) 22) 62) 20) 60* 19. 54. 92. 4025 2898 3542 7242) 7242) 7242) 20. HP- 55. f. 1380 6363 7242) 7242* 28. 16 t V 56. fl* Page 52. 29. 27. 57. 1- 107. 2 2' 30. 17f. 58. 2§* 108. 7- 31. 39*. 59. 109. 4' 319 COMPLETE ARITHMETIC. 52-59 110. 8 Z‘ 30. 4** 42. 7f 111. 20 21- 31. 193*. 43. 1 7 6 0* 112. 4 z- 32. 288f. 44. 2 Z‘ 113. 3 2* 33. 274**. Page 57. 114. 49 ¥T)" 34. 174. 45. 8 1 63" 115. 128 8 • 35. 65if. 46. 3 20" 116. 5 3* 36. $20.37*. Page 58. 117. -tit gal. 37. 68**. 11 . 4*. 118. 12§. 38. Q 1 7 d Z0' 12 . 5** 119. $63 X V Page 55. 13. 5f. 120. 12 21 22 3 0) 3 0) 3 0 ' 13. 3 TT" 14. 7f* 121. 150 225 400 "12) 12)12" 14. 43 90" 15. 9 1 ■"1 2* 122. 3 6 65" Page 56. 16. 91 5 ■"16" 123. 15 20 14 2¥) 2T) 2T" 15. 17. 5*. 124. 5 33 2 ¥) '6 ) 6" 16. 5 3 6" 18. 4**. 125. 28 36 336 6 3) 6 3) 6 3 " 17. 1 7 ZZ’ 19. 2,250. 126. 15 30 25 12) 12) 12" 18. 11 TOO" 20. 3,648. 127. 14 48 135 24) 2?) 2?" 19. 5 ¥2) 21 . $166. 128. 10 5 450 1000 120) 120) 1 2 0 ) 20. 19 60" 22. $3,466*. 2 7 6 120" 21 . 1 3 13 2" Page 59. Page 53. 22. 3 5 1 OZ’ 34. 514. 12. 2f. 23. _7_ 3^" 35. 50*. 13. 014 ^ZZ- 24. _7_ 3 0" 36. 29*. 14. 2tVo" 25. 7 3¥" 37. 254*. 15. Iff* 26. 93 9 Z ¥0" 38. 623*. Page 54. 27. 5 Y2" 39. 465*. 16. iH- 28. 1 2 TT* 40. 432*. 17. iff- 29. 92**. 41. 1,702*. 18. Ol 7 ■“3 6* 30. 46*. 42. 522f. 19. 2*. 31. 19*. 43. 13,736*. 20. 2-b 32. 17H- 44. 572. 21 . 2f. 33. 88**. 45. 693. 22. 031 A ZZ' 34. $.70f. 46. 808. 23. O 91 Z T62" 35. $.63f. 47. 8,223f. 24. 9 5 ^36" 36. 1 3 2?" 48. 8,649. 25. *• 37. 9**. 49. 13,533*. 26. If*" 38. i*. 50. 45,196*. 27. If. 39. 119 120" 51. 17,427*. 28. 3f. 40. ItV 52. 42,745*. 29. lit* 41. HI* 320 ANSWERS. 60-78 Page 60. 34. 12. 77. 9. 62. ff. 35. 36. 78. AV 63. !?. 36. 40 79. 3§. 64- tV Page 64. 80. Iff* 65. f. 45. f. Page 71. 66. f. 46. lsV* 51. 5. 67. sV 47. f. 5 2. ts* 68. If. 48. f. 53. 411. 69. b 49. lb 54. 81. 70. 8 b 50. lyV 55. Iff. 71. 24f. 5 1. T5’ 56. Iff 72. 2211. 52. 5. 57. 504f 73. 28. 53. b 58. if 74. $||. 54. b 59. 1881 A. 7 5. 34| cts. 55. 4f. 60. 26 sq. rd. 76. $3f. 56. 2if 61. 1311 h. 77. $253§. Page 65. 62. 121T. Page 61. 5 7. 5f months. 63. $3,132 b 78. $7,729f 5 8- 15f bu. 64. 14f yds. 79. Iff* 59. 46| yds. 65. 105. 80. $2,811. 60. 6 2 3 2 h. 66. 115| m. Page 62. 61. 25f A. 67. $1,375. 11. ST* 62. 44f. 68. $14,175. 12. S 2 3* 63. 25ff. Page 72. 13. t x . 86. 251*. 87. *V 88 . 2 |. 89. X f« 90. $960. q * f Value, $32000. 1 Part left, Page 274. 92. 7.039 93. .0301965 94. 1024. 95. .002235 96. 16000. 97. .075 98. .003 99. .000008 100. 7 oz. 10 pwt. 101 . .01 C. Ar.—29. 102. $.14175. 103. 41760 min. 104. 7948800 sec. 105. 509 If steps. 106. 422§. 107. 18 A. 108. 41^ yd. 109. 184800 gr. 110. 300 sq. ft. 111. 45 cts. 112. $15.20 Page 275. 113. 8 ft. 114. $72,875 115. 130, with 1 bu. 1 pk. R. 116. $60.25 117. 6415^. 118. 28 min. 44 sec. past 11 A. M. 119. 5 P. M. 120. 42 min. 51f sec. past 4 P. M. 121 . I 122. .65 123. jtzj' 124. tVst 16. 125. 126. 900 men. 127. llf% 128. $4.50 129. Gained 337 276-280 COMPLETE ARITHMETIC. Page 276. 130. 4% loss. 131. Lost, $20. 132. $100. iq o / $986. 1 Gained, 17 %. 134. $6 per yard. 135. 60% 136. 0% 137. 25% 138. $141,382 139. 16ft% 140. $16.20 141. $2048. Page 277. 142. 487804.87 lb. 143. 400 yd. 144. $5,368 145. $11,111 146. $6,511 147. 4i% 148. 6H. 149. $466f. 150. r $600. *-8£% 151. $289,532 152. Discount, $220. 153. $36. 154. $2,583 155. $.314 156. $310.61 157. 9 months. Page 278. 158. 6 months. 159. In 6| months. 160. $75. 161. $11 and $16. 162. 24 men. 163. 126 bushels. 164. $216. 165. $1. 166. $4800. 167. $2400. 168. Profits, 169. 20.5 170. 3.5 171. 40 feet. f A, $555§. IB, $333f. Page 279. 172. 65 miles. 173. 50 feet. 174. 160 rods. 175. 23.3 —l— feet. 176. 28.28+ feet. 177. 154 cu. ft. 178. 78 sq. ft. 179. Area, 7.854 A. 180. 24.4+ in. 181. 64 balls. 182. 1728 blocks. 183. 128.57+ bu. 184. 3456 gal. 185. 564.019 gal. r A, $445. 186. \ B, $230. 1C $325. QO <1 e 6 days. Page 280. 188. $545,454 189. 4J miles. r 1st, $3250. 190. | 2d, $3900. 13d, $1950. 191. 23|xi bu. 338 ANSWERS, c W., 18 i ne / A in 5 days. 192. \ M., 22. 1 Bin 4 days. lCh.,50. 196. 2| times. r 1st, $240 i q 7 J 1st, 90 miles. 193. \ 2d, $180. 12d, 40 miles. Ud, $210. 198. 7£ feet. c A, 14§ days. 194. \ B, 72 “ Page 281. l C, lOf “ 199. 50 feet. 200. 3 hours. APPENDIX. Page 287. Page 302. 280-305 1. Fifty. 1. 160. 2. One hundred and ninety- 2. 3645. five. 3. 130. 4. 400. 5. 1300. Page 299. 1. 23. 2. 25. 3. 45. 3. 3* 4. b 5 f 4th, 127f. ' 13d, 1827Hf. 6 . 3. 7. 3640. g f $256. I $511.50 9. $76293945.31 Page 303. Page 300. 4. 4. 5. 2 b 6 . 25, 29, 33, 37. 7. 34. 8. 3870. g f 6th, 225. * \ 7th, 3213. 10. 156. 22 / Last yard, $1.49 ’ i Trench, $37.75 2. $.326 Page 305. 4 . 1 lb. of each. (1 ans.) { 300 lb. at 22 cents. 200 lb. at 28 “ 500 1b. at 30 “ c Rye, 270 bu. 6 . j Barley, 60 bu. v Oats, 60 bu. 339 305-30S COMPLETE ARITHMETIC. { 10f pwt. 16 carat. 10f “ 18 “ 321 “ 22 “ r 50 lb. at 15 cents. 8 . | 50 “ 17 “ 1100 “ 20 “ 9 . 15 gal. Page 307. 2. 36 ft. 9 / 1" 11"'. 4. 29 ft. 2 / 1" 8"'. 6. 8 sq. ft. 4' 11", 7. 26 cu. ft. 8'. Page 308. 9. 7 ft. 3'. 10. 7 ft. 8'. 1. 720. 2. 39916800. 3. 362880. 4. 40320. FOREIGN EXCHANGE. Page 312. 2. $1078.11+ Page 313. 3. $974.11+ *4. $1465.952 *5. $1216.518+ *6. $4855. 8 . £450. 9. £240. Page 314. 2. $1250. *3. $1040. Page 315. 5. 12565£ fr. 6. $1250. 7. 3347£ fr. Page 316. *2. $405.30. *3. $770.92!-. 5. 1371f marks. *6. $229.80. *7. 2160 marks. - 8. 960 marks. Revised Problems. 340 />. yy jo y 'o ft , !? b I/) cSl ^ 1 €3 Eclectic Ecluca UNIVERSITY OF ILLINOIS-URBANA 513W581C coo? A COMPLETE ARITHMETIC CINCINNATI POLITICAL EC 3 0112 017106888 OMY. ANDREWS S MANUAL OF THE CONSTITUTION. Manual of the Constitution of the T’nited States. iJe- signed for the Instruction of American Youth in the Du¬ ties, Obligations and Rights of Citizenship. Bv Israel Ward Andrews, D. D., President of Marietta College. 121110, cloth, 408 pp. While the primary object has been to provide a suitable text-book, a conviction that a knowledge of our government can not be too widely diffused, and that large numbers would welcome a good book on this subject, has led to the attempt to make this volume a manual adapted for consultation and reference, as well for citizens at large as for stu¬ dents. With this end in view the work embodies that kind uf mforma- tion on the various topics which an intelligent citizen would desire to possess. - GREGORY S POLITICAL ECONOMY. A New Political Economy. By John M. Gregory, LL.D., late President III. Industrial University. 12mo, t ^ 7 393 PP- An essentially new statement of the facts and principles of Political Economy, in the following particulars: x. The clear recognition of the three great economic facts of Wants, Work and Wealth, as the principal and constant factors of the indus¬ tries, and as constituting, therefore, the field of Ecc nomic Science. 2. The recognition of man and of the two great crystallizations of man into society and into states, as presenting three distinct fields of Economic Science, each having its own set of problems, and each its own species of quantities or factors, to be taken into account in the solution of problems. 3. A new definition and description of Value as made up of its three essential and ever-present factors forming the triangle of Value, and evidenced by the clear explanation they afford of the various fluctua¬ tions of prices. 4. The new division and- distribution of the discussion arising, out of these new fundamental facts and definitions. 5. The aid rendered to the reader and student by the diagrams and synoptical views. Van Antwerp, Bragg & Co., Publishers, CINCINNATI and NEW YORK.