T ' so apl¥L&.BB0fMiiis. ■^;:3r J > ^ :J|i =J^ LIBRARY OF THE University of California. Keceived (•A^cJLr ■ /^9 7 • '^ Accession No. ^ ^/J Q J"^ . Class No. Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archivaorg/details/elementaryarithmOOfrenrich FRENCH'S MATHEMATICAL SERIES. iLEMENTARY JkElSHffilSl© FOR THE SLATE; IN WHICH METHODS AND EULES ARE BASED UPON PRINCIPLES ESTABLISHED BY INDUCTION. BY JOHN H. FKEl^CH, LL.D. If Principles are' uunderstood,, I?f*2fes^«7'a. useless. HARPER & BROTHERS. 1869. FRENCH'S ARITHMETICS. This Series consists of Five Books, viz.: I.—FIUST I.JSSSOKS JJ^ KTTMBFiltS. II. — EJLEMENTjLMT jLMITMMETIC. III. — M ENTA Ij a. MITSMETIC In Press. IV. — COMMON SCHOOL AMITMM ETIC. v. — ACADEMIC AMITSMETIC In Preparation. Tlie Publlsliers present this Series of Text-Books to American Teacliers, fully believing tliat tliey contain many new and valu- able features tliat will especially commend them to the practical wants of the age. The plan for the Series, and for each book embraced in it, was fully matured before any one of the Series was completed ; and as it is based upon true philosophical principles, there is a har- mony, a fitness, and a real progressiveness in the books that are not found in any other Series of Arithmetics published. Entered, according to Act of Congress, in the year 1867, by HAEPER & BEOTHERS, In the Clerk's Office of tlie District Court of the United States for the Southern District of New York. PREFiiCB, THE object of this book, designed especially for beginners in Written Arithmetic, is twofold, viz. : 1st. To give to young learners a good foundation for the study of the Science of Numbers, by basing all Methods of Operation upon Principles ; and, 2d. To f^ive them as much knowledge as possible of the business affairs of ife, by the introduction of business transactions stated in Correct business language. The plan of the work differs, in most of its essential points, from that of other works of a like grade. To these points of difference — and it is confidently believed of superiority — the attention of parents and teachers is particularly invited. General Divisions. — Chapters. — The work is divided into six chapters, the first one of which is devoted to Integers ; the second, to Decimals ; the third, to Compound Numbers ; the fourth, to Frac- tions; the fifth, to Percentage, and the sixth, to Miscellaneous Review Problems. Integers and Decimals are but parts of the same class of numbers, the latter being an extension of the decimal scale to the right of the decimal point, or below ones. They are both subject to the same laws, and all operations upon them are based upon the same princi- ples. Therefore, in the natural order of arrangement of subjects, the proper place for Decimals is immediately after Integers. Compound Numbers differ from Integers and Decimals only in the scales of increase and decrease, which, in the latter, are uniform and decimal, while in the former, they are irregular and varying. The new facts to be learned in Compound Numbers are, the scales or tables, and their application to the processes of Addition, Subtraction, Multiplication, and Division. Only two of the denominations given in this book— 5i or 5.5 yards are 1 rod, and 30^ or 30.25 square yards are 1 square rod— are Mixed Numbers, and these are as well expressed decimally as fractionally. There is, therefore, no good reason why Fractions should precede Compound Numbers, no knowledge of the former being necessary in studying the latter, while the advantages of the reverse order of arrangement are obvious. A knowledge of the preceding chapters prepares the pupil, on reaching Fractions, to comprehend the new facts to be learned, viz. : the Notation of Fractions, the General Principles of this class of numbers, and the application of these principles to the operations upon Fractional Numbers. In the general arrangement of the work, and also in its details, the fact has never been lost sight of, that only a small portion of all the children who commence the study of Arithmetic go through their text-book ; and that a child should be taught first that which it is most desirable and important for him to know ; so that, whenever he leaves school, the knowledge he has acquired will be of practical value to him in after life. IV PREFACE. Subdivisions.— Sections and Cases. — Each chapter except the last is divided into sectious, and wherever necessary, the sections are subdivided into cases. The subjects of corresponding sections in the first four chapters are similar. For example, Section I. in each chap- ter is Notation and Numeration ; Section II. is Addition, etc. The cases in the several Sections correspond to each other, wherever the nature .of the subjects will admit. Especial attention is invited to the following points in the several chapters and sections : Chapter I. — The first Method of Addition will familiarize pupils with the reason for the " carrying process," and also accustom them to add the reserved tens of the sum of any column to the first figure of the next column, instead of the last. All the cases in Multiplication and Division are based upon a few general principles, readily understood, and hence easily remembered. Long Division precedes Short Division, because it is simpler. In the former, all the partial results — quotient figure, partial dividend, partial product, and partial remainder— are written down ; while in the latter, the quotient figure only is written, the memory being taxed to form all the combinations, and retain all the other partial results in the process. Long Division is a general process, while Short Division is a contraction, limited in its application. The partial results written in the former are really Visible Objects, while in the latter they become Abstract Numbers. The natural order of mental development, Perception before Memory, has therefore been observed, in placing Long Division before Short Division. The divisor is written at the right of the dividend. This arrange- ment is as convenient for Short Division as that of placing the divisor at the left of the dividend ; while in Long Division, the quotient is written under the divisor, and the factors of the partial dividends are thus brought nearer together, and therefore in a more convenient position for multiplication. A section embracing the simplest cases in Measurement is intro- duced into this chapter, because, 1st. The subject is interesting to children, and is readily understood by them as soon as they have passed over the fundamental rules; and, 2d. The cases here given are the basis of the objective method used in illustrating some of the principles of Decimals. Chapter II. — The Diagram of Decimal Notation, the Table of Values of Decimal Numbers, and the Decimal Notation and Nume- ration Table, when thoroughly understood, give to pupils a clearer comprehension of Decimals than they can obtain without these aids. The reason for placing the decimal point in the product in Multi- plication is derived from the principles of Measurement ; and that for placing the decimal point in Division, from a general principle of Division. These reasons are strictly philosophical, and easily under- stood, and are entirely independent of Fractions. The divisions of the dollar being decimal, and all computations in U. S. money being based upon the same principles as f)ecimals, the subject of U. S. or Federal Money is embraced in this chapter, and the necessity for separate principles and rules is obviated. PREFACE. V Chapter III.— The Tables of Compound Numbers are arranged in the order in which they will be the most readily comprehended by young pupils ; and only those denominations in actual use are given. A few tables, such as Troy Weight, Surveyors' Measure, etc., being of limited use, are omitted. The Tables of the Metric System are given, because they are legal- ized by act of Congress ; but they are not made prominent, because they are not yet in use. Chapter IV. — The term Similar Fractions is used in place of Fractions having a cornmon denominator. The simplicity and com- prehensiveness of the term should secure its general adoption. The only important application of the subject, Common Multiple, is in the reduction of dissimilar to similar fractions. It is therefore pre- sented in this chapter. The subjects of Least Common Multiple, and Common Divisors, not being essential to an elementary work, have been omitted. The cases in Multiplication and Division are different from those in any similar work, and the methods are also new and superior. The method of Division is based upon the same general principle as is Division of Decimals. The applications of Cancellation to Multiplication and Division are made the subject of a separate section. Chapter V. — One general case is given, embracing all the general principles of Percentage; and to this case all the methods for com- putations in Insurance, Commission, Profit and Loss, Stocks, Banking, and Interest, are referred. The method for Interest is new, and its simplicity, absolute accu- racy, and general application, make it superior to any heretofore presented. Chapter VI.— The problems in this chapter embrace applications of all the principles and methods of computation contained in the previous chapters of the book. Inductions.— Each chapter* as well as many of the sections and cases, commences with Illustrations which form Visible Objects. Then follow, in the natural order. Concrete and Abstract Numbers. Illustrations.— The cuts, maps, and diagrams, all of which are new, are intended not only to aid the pupil in acquiring a clear under- standing of the subjects they illustrate, but also to educate his eye, cultivate his taste, and teach him some useful fact or principle. • Examples and Problems. — Care has been taken to use these term^ — so often used indiscriminately — in accordance with their sig- nification. In the induction to a Case or Method, one or more examples are solved, and the solution is inductively explained. These examples, except in Currency, Compound Numbers, and Percentage, contain only abstract numbers, because a general principle should not be deduced from a special or limited application. VI PREFACE. The problems are derived from actual business tranactions, and are all new. The facts stated in them have been obtained from reliable authorities, and the business transactions are in accordance with business customs. Each chapter closes with a section of Review Problems, designed to test the pupil's knowledge of all the previous chapters. Definitions. — The definitions, being intended for young minds, are stated in the inductive form. They are brief, accurate, and com- prehensive. Oral Exercises.— It is a conceded fact that children leam methods and processes of computation more readily than they learn combinations. Many persons go on through life performing all their numerical computations by counting. They never learn to step more than a one at once. The Oral Exercises, if practiced according to the directions given, will break up the counting plan — pupils will learu to step from given parts to required results without hesitation, and will soon become rapid and accurate computers. Tables of Combinations. — The Addition, Subtraction, Multi- plication, and Division tables are presented in a new, and, it is believed, a more attractive form than the solid pages of figures that have greeted the eyes of children from time immemorial. Principles and Rules. — Principles are deduced immediately from the inductive examples, and are followed by problems which require the pupil to apply the principles. He is thus made familiar with reasons for the processes, before the rules are given ; and whenever he applies a rule in solving a problem, the why is as familiar to him as the how. Rules without principles are soon forgotten ; while if principles are understood, rules are useless. Methods of Proof. — Self-reliance is one of the most important things that can be taught to children. To do their work correctly, and to feel sure that they are correct, they must be drilled in combi- nations until they add, subtract, multiply, and divide without making any mistakes. As methods of proof generally retard children in reaching this most desirable degree of accuracy, they have been omitted from this work. Teachers' Manual. — The last twelve pages of the book are devoted to notes, remarks, suggestions, and hints to teachers ; and to this Manual frequent references are made in the body of the work. Teachers should not fail to consult the Manual, whenever reference is made to it. The many new and valuable features of the book, its superior typography and beautiful illustrations, its great number of practical problems drawn directly from business life, and its adaptation to the wants of graded schools, and to the capacity of beginners in schools of any grade, will, it is hoped, secure for it the attention and careful examination of intelligent, progressive teachers. iVBE CONTENTS. CHAPTER I.— INTEGERS. PAGE Section I.— Notation and Numeration 9 Section II. — ^Addition 17 Section III. — Subtraction 31 Section IV. — Multiplication 45 Section V. — Division 64 Section VI. — Measurements 83 Section VII. — Problems in Integers 89 CHAPTER II. -DECIMALS. Section I. — ^Notation and Numeration 94 Section II. — Addition 101 Section III. — Subtraction 105 Section IV, — Multiplication 108 Section V. — Division 113 Section VI.— United States Money 121 Section VII. — Problems in Decimals 125 CHAPTER III. -COMPOUND NUMBERS. Section I. — Notation and Reduction 129 Section II. — Addition 149 Section III. — Subtraction 151 Section IV. — Multiplication 155 Section V. — Division 157 Section VI. — Problems in Compound Numbers 159 YIU CONTENTS CHAPTER IV.-FRACTIONS. PAGB Section I. — Induction and Notation 163 Section II.— Reduction 166 Section III. — Addition 171 Section IV.— Subtraction 174 Section V. — Multiplication 177 Section VI. — Division 181 Section VII. — Cancellation 184 Section VIII. — Problems in Fractions 186 CHAPTER V. -PERCENTAGE. Section I. — Notation and Numeration 189 Section II. — General Applications 190 Section III. — Commission 191 Section IV. — ^Insurance 192 Section V.— Profit and Loss 193 Section VI. — Interest 195 Section VII.— Problems in Percentage 199 CHAPTER VI. Miscellaneous Problems 201 Manual for Teachers 209 top. ElEMENfm 'ARITHMETIC. CHAPTER I. INTEGERS. SECTION I. * 1. In writing numbers, ten characters, called figures, are used. The first figure, 0, is called a cipher or rmught^ and denotes nothing or the absence of number. The other nine figures are used to represent the first nine numbers. 7. Seven. 9. Nine. 10 INTEGERS. To express numbers greater than nine, two or more of these ten figures must be combined. 2. In writing numbers, every ten ones taken together are called a ten. Ten is written 10 Two tens are called twenty, written, 20 Three tens a ' u thirty, SO Four tens a u forty, Jfi Five tens u ii fifty. 50 Six tens u ' u sixty, 60 Seven tens u ii seventy. 70 Eight tens u ii eighty, 80 Nine tens u u ninety, 90 When two figures are written together to express a number, the left-hand figure expresses tens, and the right-hand figure ones. Sixteen consists of 1 ten and 6 ones, written 16 Twenty-four u " 3 tens u 4 " a n Thirty-two a " 3 " u 2 " a 32 Forty-nine u a ^ u a 9 " a JiS . Fifty-five (( a 5 a u 5 " (( 55 Sixty-seven u a (5 u a 7 " 11 67 Seventy-three a u ly a u 3 " a 73 Eighty a u 3 u u " a 80 Ninety-one u u 9 u ii 1 one, (See Manual, 91 page 214.) JEXEMCISES. 1. Write in words the following numbers : 14, 25, 37, 42, 56, 69, 71, 88, 93. 2. Express by figures the following numbers: twelve, twenty-eight, thirty-five, forty-one, fifty-nine, sixty-three, seventy-six. 3. Write in words, 17, 29, 30, 48, 52, 65, 70, 81, 99. 4. Express by figures, forty-three, sixty-six, ninety-five, fifteen, eighty-six, thirty-eight, fifty-seven, sixty-one. 5. Write in figures, twenty, forty, eleven, thirty-six, ninety- four, eighty-nine, forty-six, seventy-five. NOTATION AND NUMERATION. 11 6. Write in words, 13, 45, 51, 78, 83, 97. 7. Write in words, 60, 91, 79, 84, 27, 83. 8. Express by figures, twenty-two, thirty-nine, fifty-four, ninety-six, twenty-seven, sixty-two, fifty-three, seventy-four. 9. Write in figures, eighty-seven, ninety-two, thirty-four, twenty-six, seventy-two, sixty-eight, forty-four, ninety-eight. 10. Write in words, 19, 7)^171, 64, 23, 82, 58. 3* In writing numbers, every ten tens taken together are called a hundred. One hundred is written 100 Twenty tens are two hundred, written 200 Thirty tens " three hundred, ' ' SOO Forty tens " four hundred, ' ' 400 Fifty tens " five hundred, ' ' 500 Sixty tens " six hundred, ' ' 600 Seventy tens " seven hundred, ' ' 700 Eighty tens " eight hundred, ' ' 800 Ninety tens " nine hundred, ' ' 900 When three figures are written together to express a number, the left-hand figure expresses hundreds, the second or middle figure, tens, and the right-hand figure, ones. Thus, two hundred forty-three consists of 2 hun- dreds, 4 tens, and 3 ones, and is written 243. The numbers in the first column below consist of hundreds, tens, and ones, as shown in the second col- umn, and are written as in the third column. One hundred forty-nine, 1 hundred, 4 tens, and 9 ones, I49 Four hundred sixty-two. 4 hundreds , 6 " "2 " 462 Five hundred twenty. 5 2 " "0 " 520 Six hundred seventy. 6 7 " "0 " 670 Seven hundred five, 7 " "5 " 705 Eight himdred four. 8 " "4 " 8O4 Nine hundred twelve. 9 Iten, " 2 (See Manual, " 912 page 214.) 12 INTEGERS. exehcis ES. 11. Write in words the numbers 247, 356, 528, 646, 935. 12. Express by figures, one hundred seventy-three, four hundred ninety-one, seven hundred sixty-four, and nine hun- dred eighty-two. 13. Write in words, 617, 121, 745, 514, 311. 14. Express by figures, four hundred nineteen, nine hundred thirty-nine, three hundred thirty-three, eight hundred eleven. 15. Write in words, 560, 310, 290, 420, 750. 16. Write in figures, one hundred thirty, six hundred forty, eight hundred eighty, four hundred ten. 17. Write in words, 208, 906, 301, 606, 807. 18. Express by figures, eight hundred two, one hundred nine, four hundred three, seven hundred five. 19. Write in words, 293, 780, 519, 103, 612, 999. 20. Express by figures the numbers which consist of 8 hun- dreds, 2 tens, and 4 ones ; 2 hundreds, 1 ten, and 8 ones ; 5 hundreds, 7 tens, and ones ; 9 hundreds, 5 tens, and 4 ones ; 3 hundreds, tens, and 7 ones. 4* In writing numbers, every ten hundreds taken to- gether are called a thousand. Thousands are written thus : One thousand, 1000 Two thousand, 2000 Three thousand, 3000 Four thousand, Ji,000 Five thousand, 5000 Six thousand, 6000 Seven thousand, 7000 Eight thousand, 8000 Nine thousand, 9000 In any number written with more than three figures, the figure at the left of hundreds expresses thousands. Thus, 3579 consists of 3 thousands, 5 hundreds, 7 tens, and 9 ones ; and expresses the number three thousand five hundred seventy-nine. ' 5* Every ten thousands taken together are called a ten-thousand. 6* Every ten ten-thousands taken together are called a hundred-thousand. NOTATION AND NUMERATION. 13 When a figure stands at the left of thousands, it ex- presses ten-thousands ; and when a figure stands at the left of ten-thousands, it expresses hundred-thousands. Ten thousand is written 10000 Two ten-thousands are twenty thousand, written 20000 Three ten-thousands are thirty thousand, " 30000 Eight ten-thousands are eighty thousand, " 80000 One hundred-thousand is written 100000 Two hundred thousands are written 200000 Five hundred thousands " " 500000 7. Every three figures in any number counting from the right are called a Period. Periods of figures are separated from each I | other by commas. ^^La^^^ The first or right-hand period con- 8 7 4,235 sists of ones, tens, and hundreds ; and the second period of ones, tens, and hundreds of thousands. Thus, Eighteen thousand five hundred thirty-six is written Thirty-two thousand eight " Forty-seven thousand two hundred " Sixty thousand four hundred twenty " Two hundred forty thousand " Four hundred eight thousand five hundred " Five hundred thousand three hundred sev- enty-five " Six hundred fifty-two thousand ten " Seven hundred forty-four thousand six " Eight hundred fifty-three thousand five hundred seventy-six " " 853^576 (See Manual, page 214.) EXEMCISJES. 21. Write in words, 8,000; 5,400; 2,560; 1,644; 3,739. 22. Write in words, 6,944 ; 3,405 ; 4,094 ; 7,010 ; 6,009. 23. Express by figures, five thousand three hundred fifty- six, seven thousand two hundred forty, one thousand nine hundred three, four thousand fifty-six. t : 1 s : £ : : i 1 3 1 J ^ § 5 5 § ivritt( 3n 18,536 u 32,008 u Jft,200 u 60,^20 u 21^0,000 u 1^08,500 u 500,375 u 652,010 u 7U,006 14 INTEGERS. 24. Express by figures, nine thousand eight hundred, two thousand six, eight thousand fifty, five thousand, six thousand eight hundred nineteen, one thousand one. 25. Write in words, 15,380 ; 26,506 ; 37,081 ; 40,269 ; 98,274. 26. Write in words, 6,793; 72,400; 80,560 ; 63,004; 50,041. 27. Write-in words, 33,000; 40,900; 90,209; 19,040; 20,007. 28. Express by figures, sixty thousand, fifty thousand twenty, nineteen thousand three hundred one, fifty-six thou- sand eleven. 29. Write in words, 132,041 ; 270,405 ; 320,500 ; 400,385. 30. Write in words, 574,000 ; 629,005 ; 700,044 ; 803,000 ; 957,503 ; 793,461 ; 809,051 ; 907,200. 31. Write in words, 503,020 ; 482,070 ; 600,002 ; 855,480. 32. Write in words, 100,905 ; 350,240 ; 904,306 ; 100,040. 33. Express by figures, one hundred five thousand four hun- dred seventy, two hundred thousand, five hundred forty thou- sand seventy-two, seven hundred forty-seven thousand two hundred. 34. Express by figures, two hundred fifty thousand three hundred sixty-three, four hundred sixty thousand twenty, seven hundred ten thousand, eight hundred one thousand four. 35. Express by figures, two hundred thousand six hundred forty, three hundred five thousand two hundred ninety-four, six hundred eighty thousand five, nine hundred thousand six hundred. a § 8. The third period of figures consists of ones, tens, and hundreds of millions. Thus, ^7^ ^ 5~0~8 ,294 In any full period the right- hand figure is ones, the middle figure is tens, and the left-hand figure is hundreds. Thus, in any number consisting of three full periods, there are ones, ones of thousands, and ones of millions ; tens, tens of thousands, and tens of millions ; and hundreds, hundreds of thousands, and hundreds of millions. NOTATION AND NUMERATION, Two million five hundred thousand eighty is written 2, 500, 080 Thirty-four million three hundred twenty- four thousand five hundred eighty-six " " 34,324,586 Forty million forty-four thousand twelve " " 40,044^012 One hundred twenty-nine million three hundred seventeen thousand five hun- dred " Six hundred fifty million two hundred thousand seventy " Nine hundred three million fifty thousand five hundred ninety-four, " Three hundred million three thousand thirty " 129,317,500 650,200,070 903,050,594 300,003,030 9. The writing of mimbers in figures is Notation. 10. The reading of numbers which are expressed by figures is Numeration. 11, The place which any figure occupies in a number determines the value expressed by it in that number. The values of the different places in their order are shown by the following RATION TABLE. ten, hundred, thousand, ten-thousand, hundred-thousand, million, ten-million, hundred-million. ones, tens, hundreds, thousands, ten-thousands, hundred-thousands, millions, ten-millions. (See Manual, page 214.) NOTATION AND N 10 ones 10 tens 10 hundreds 10 thousands 10 ten-thousands 10 hundred-thousands 10 millions 10 ten-millions 1 ten 1 hundred 1 thousand 1 ten-thousand 1 hundred-thousand 1 million 1 ten-million 1 hundred-million UME are 1 " 1 " 1 " 1 " 1 " 1 " 1 " 1 is 10 " 10 " 10 " 10 " 10 " 10 " 10 " 10 16 INTEGEES. MEVIEW EXERCISES. 36. Write in words, 4,650 ; 738 ; 450,840 ; 93,066 ; 8,050,800; 1,005; 4,000,800. 37. Write in words, 37,098,430 ; 502,000 ; 730,900. 38. Write in words, 85,700,035 ; 6,000,030 ; 13,006,400 ; 45,000,000; 6,413; 2,578,034. 39. Write in words, 73,059,209 ; 10,765,291 ; 8,010; 70,045; 8,050,000. 40. Express by figures, seven hundred six thousand two hundred ninety-one, seventy-four thousand seven hundred four, nine thousand ninety, eighty thousand twelve. 41. Express by figures, eight million five thousand three hundred ninety-four, six million eight, seven million, three hundred thousand seven hundred twenty. 43. Express by figures, nine million two hundred seventy, two hundred three thousand four hundred five, twenty thou- sand six hundred seven, eighteen million nineteen. 43. Write in words, 4,080,306; 332,107,003; 14,200; 500,007; 60,572; 536,000. 44. Write in words, 34,000,709; 1,702,050; 605,400,300; 9,600,309. 45. Write in words, 12,065,587 ; 40,080,276 ; 7,200,000 ; 15,009,820; 3,031,504. 46. Express by figures, four million two hundred fifty-nine, seven million two hundred two thousand five hundred, eighty thousand, four hundred thousand two hundred fifty-six. 47. Express by figures, eight hundred million seven hun- dred seven thousand five hundred six, sixteen thousand six- teen, seven million five thousand forty-four, twenty-nine million forty-one thousand. 48. Express by figures, two hundred four thousand two hundred seventy, fifty thousand thirty-three, one million three hundred six, seven hundred fifty thousand nine. 49. Write in figures, ten million twenty-five thousand four hundred, six thousand one, three million two thousand, six- teen million ninety thousand five. 50. Write in words, 7,019,003 ; 506,427, 711 ; 243,424 ; 736,378; 9,999; 300,003,303. ADDITION. 17 SECTION II. i:isri>TJCTio:Nr. (See Manual, page 215.) 12. Heee is a picture of some boys and girls in an orcliard gathering apples. John has 1 apple in each hand ; Harry has 2 apples in one hand and 1 in the other ; Mary has 3 apples in one hand and 1 in the other; and Fanny has 4 apples in her lap and 1 in her hand. 1. How many apples has John ? How many has Harry ? How many has Mary ? How many has Fanny ? 2. How many apples have John and Harry together ? 3. How many apples have Mary and Fanny ? 4. If John and Harry give their apples to Mary, how many apples will she have ? 5. If Harry and Mary give their apples to Fanny, how many will she have ? 18 INTEGERS. 6. If all the children put their apples into Mary's basket, how many apples will be in the basket ? 13. When two or more numbers are united to form one number, the process is Addition. * 14. The result thus formed is the Amount or Sum, and the numbers to be united are the Parts. The amount or sum must contain as many ones as all the parts taken together. 7. What is the amount of 11 cents, 5 cents, and 8 cents ? 8. What is the amount of 8 pencils, 6 pencils, and 10 pencils ? 9. What is the sum of 4 walnuts, 1 walnut, 2 walnuts, and 9 walnuts ? 10 What is the sum of 2 days, 4 days, 3 days, and 6 days ? 11. Add 8 peaches, 4 peaches, 3 peaches, and 7 peaches. 12. Add 5 roses, 12 roses, 7 roses, and 6 roses. 13. Martha has a 5-cent piece, a 3-cent piece, a 2-cent piece, and a 10-cent piece. What sum of money has she ? 14. A laborer worked four weeks, earning 8 dollars the first week, 6 dollars the second, 4 dollars the third, and 7 dollars the fourth. What amount of money did he earn 15. Add 3 and 7 and 5 and 8 and 2. 16. Add 9 and 4 and 6 and 1 and 5. 15. This sign +, written between numbers, signifies that they are to be added. It is called Ptus, or the Sign of Addition. . 16. This sign = written between numbers or sets of numbers, signifies that they are equal to each other. It is called the Sign of Equality. Thus, 6 + 7 + 12 = 25 is read, 6 plus 7 plus 12 equal 25. 17. Bead 7 + 3 + 5 = 15. 18. Bead 29 = 10 + 9 + 6 + 4. 19. Read 12 + 5 + 4 = 13 + 8. 20. 16 eggs + 5 eggs + 11 eggs + 9 eggs = how many eggs ? 21. 8 hats + 21 hats + 6 hats + 10 hats = how many hats? ADDITION. 19 22. What is the sum of 7 chairs + 12 chairs + 6 chairs + 10 cliairs + 1 chair ? 23. What is the amount of 19 dollars + 10 dollars + 7 dol- lars + 6 dollars + 5 dollars ? 24. 18 + 7 + 6 + 9 = how many ? (See Manual, page 215.) 17» ADDITION TABLE. rtJO 123456789 W|o 000000000 KJ0123456789 0|5 555555555 0123456789 5 6 7 8 9 10 11 12 13 14 -(0 123456789 1)1 111111111 123456789 10 cJO 123456789 0]6 666666666 6 7 8 9 10 11 12 13 14 15 oJO 123456789 *]2 222222222 ^012345678 9 7|7 777777777 23456789 10 11 7 8 9 10 11 12 13 14 15 16 oJO 123456789 33333333333 q(0 123456789 o]8 888888888 3 4 5 6 7 8 9 10 11 12 8 9 10 11 12 13 14 15 16 17 -(0123456789 4|4 444444444 q(0 123456789 »]9 999999999 4 5 6 7 8 9 10 11 12 13 9 10 11 12 13 14 15 16 17 18 OMAJL EXEMCISES, 1. — 1. Count to 100 in this manner ; and 1 are 1, 1 and 1 are 2, 2 and 1 are 3, and so on. 2. Count to 100, thus ; 0, 1, 2, 3, 4, 5, and so on. 2. — ^1. Count by 2's to 100, in this manner, and 2 are 2, 2 and 2 are 4, 4 and 2 are 6, and so on. 2. Count by 2's to 100, thus ; 0, 2, 4, 6, 8, and so on. 3. Commence with 1, and count by 2's to 101, thus ; 1 and 2 are 3, 3 and 2 are 5, 5 and 2 are 7, and so on. 4. Count by 2's from 1 to 101, thus ; 1, 3, 5, 7, 9, and so on. 3. — 1. Count by 3's from to 102, thus ; and 3 are 3, 3 and 3 are 6, 6 and 3 are 9, and so on. 2. Count by 3's from to 102, thus ; 0, 3, 6, 9, 12, and so on. 3. Commence with 1 and count by 3's to lOO, thus ; 1 and 3 are 4, 4 and 3 are 7, 7 and 3 are 10, and so on. 4. Count by 3's from 1 to 100, thus ; 1, 4, 7, 10, i3, and so on. 5. Commence with 2 and count by 3's to 101, thus ; 2 and 3 are 5, 5 and 3 are 8, 8 and 3 are 11, and so on. 6. Count by 3's from 2 to 101, thus ; 2, 5, 8, 11, 14, and so on. 20 INTEGERS. 4. — 1. Commence with and count by 4's to 100, thus ; and 4 are 4, 4 and 4 are 8, 8 and 4 are 12, and so on. 2. Count by 4's from to 100, thus ; 0, 4, 8, 12, 16, and so on. 3. Commence with 1 and count by 4's to 101, thus ; 1 and 4 are 5, 5 and 4 are 9, 9 and 4 are 13, and so on. 4. Count by 4's from 1 to 101, thus ; 1, 5, 9, 13, 17, and so on. 5. Commence with 2 and count by 4's to 102, thus ; 2 and 4 are G, 6 and 4 are 10, 10 and 4 are 14, and so on. 6. Count by 4's from 2 to 102, thus ; 2, 6, 10, 14, 18, and so on. 7. Commence with 3 and count by 4's to 103, thus ; 3 and 4 are 7, 7 and 4 are 11, 11 and 4 are 15, and so on. 8. Count by 4's from 3 to 103, thus ; 3, 7, 11, 15, 19, and so on. 5. — 1. Commence with and count by 5's to 100, thus ; and 5 are 5, 5 and 5 are 10, 10 and 5 are 15, and so on. 2. Count by 5's from to 100, thus ; 0, 5, 10, 15, and so on. 3. Commence with 1 and count by 5's to 101. (See Manual, page 215.) 4. Commence with 2 and count by 5's to 102. 5. Count by 5's from 3 to 103. 6. Count by 5's from 4 to 104. 6. — 1. Count by 6's from to 102, thus ; and 6 are 6, 6 and 6 are 12, 12 and 6 are 18, and so on. 2. Count by 6's from to 102, thus ; 0, 6, 12, 18, 24, and so on. 3. Commence with 1 and count by 6's to 103. 4. Count by 6's from 2 to 104. 5. Count by 6's from 3 to 105. 6. Count by 6's from 4 to 100. 7. Count by 6's from 5 to 101. , 7, — 1. Commence with and count by 7'8 to 105, thus ; and 7 are 7, 7 and 7 are 14, 14 and 7 are 21, and so on. 2. Count by 7's from to 105, thus ; 0, 7, 14, 21, 28, and so on. 3. Commence with 1 and count by 7's to 106. 4. Commence with 2 and count by 7's to 100. 5. Commence with 3 and count by 7*8 to 101. 6. Count by 7's from 4 to 102. 7. Count by 7's from 5 to 103. 8. Count by 7's from 6 to 104. 8. — 1. Commencing with 0, count by 8's to 104, thus ; and 8 are 8, 8 and 8 are 16, 16 and 8 are 24, and so on. 2. Count by 8's from to 104, thus ; 0, 8, 16, 24, 32, and so on. 3. Commencing with 1, count by 8's to 105. 4. Commencing with 2, count by 8's to 106. 5. Commencing with 3, count by 8's to 107. 6. Count by 8's from 4 to 100. 7. Count by 8'8 from 5 to 101. 8. Count by 8'8 from 6 to 102. 9. Count by S's from 7 to 103. ADDITION. 21 9. — 1. Commencing with 0, count by 9's to 108, thus ; and 9 are 9, 9 and 9 are 18, 18 and 9 are 27, and so on. 2. Count by 9's from to 108, thus ; 0, 9, 18, 27, 36, and so on. 3. Commencing with 1, count by 9's to 100. 4. Commencing with 2, count by 9's to 101. 5. Commencing with 3, count by 9's to 102. 6. Count by 9's from 4 to 103. 7. Count by 9's from 5 to 104. 8. Count by 9's from 6 to 105. 9. Count by 9's from 7 to 106. 10. Count by 9's from 8 to 107. C^SE I. The aum of all the figures of any place not more than 9. 18. ExiiMPLE. What is the sum of 2,344 and 3,152 ? Explanation. — Since these parts are too solution. large to be added mentally, we write them 2,344 1 ^^^^^ one under the other, writing the ones of ^>1^^ ; one part under the ones of the other, the 5,496 Sum. fens under tens, the hundreds under hun- dreds, and the thousands under thousands. The sum of 2 ones »nd 4 ones is 6 ones, which we write under the ones; the sum of 5 tens and 4 tens is 9 /ens,. which we write under the tens ; the sum of 1 hundred and 3 hun- dreds is 4 hundreds, which we write under the hundreds; and the sum of 3 thousands and 2 thousands is 5 thou- sands, which we write under the thousands. The result, 5,496, is the sum required. 19. We can add apples to apples, dollars to dollars, pens to pens, or hours to hours ; but we can not add apples to dollars, nor pens to hours. For 4 apples + 9 dollars — neither 13 apples nor 13 dollars. Again, we can add ones to ones, tens to tens, or hun- dreds to hundreds ; but we can not add ones to hun- dreds, nor tens to thousands. For 4 tens + 9 thou- sands = neither 13 tens nor 13 thousands. Hence, 22 INTEGEES 20» General Principles o/ Addition, I. Only numbers expressing the same hind of things can he added. II. Only figures occupying the same place in different numbers can be added ; that is, ones must be added to ones, tens to tens, hundreds to hundreds, thousands to thousands, and so on. (See Manual, page 215.) PROBIjEMS. Find the sum of the numbers in each of the first ten problems. (1) (2) (3) (4) (5) (6) 62 26 34 452 281 504 24 72 145 37 612 283 C^) (8) (9) (10) 235 men 612 men 141 men 2,413 books 146 books 30 books 5,241 miles 306 miles 2,432 miles 31,410 dollars 1,245 dollars 26,332 dollars 11. James paid 12 cents for a slate, and 15 cents for a writ- ing-book. How many cents did he pay for both ? 12. Myron found 25 plums under one tree in the garden, and 13 ptums under another. How many plums did he find under both trees ? 13. In a village school are 56 boys and 43 girls. How many pupils in the school ? 14. One day a lady traveled 42 miles by railroad and 16 miles by stage. How many miles did she travel ? 15. An orchard consists of 53 winter apple-trees and 14 fall apple-trees. How many trees are in the orchard ? 16. A builder paid 610 dollars for a city lot, and built upon it a house which cost him 2,085 dollars. How much did the house and lot cost ? 17. What is the sum of 542 + 36 ? 18. 21 -f- 45 -f 32 = how many? 19. 6,132 + 31 + 36 = how many? 6,199. ADDITION. 23 20. What is the sum of 123 + 231 + 312 + 123 + 201 ? 989. 21. What is the sum of four hundred one thousand nine hundred fifty, twenty-four thousand twenty-four, and two thousand and four ? 4^7, 978. 22. A farmer harvests from five fields of wheat, 151 bushels, 204 bushels, 120 bushels, 312 bushels, and 211 bushels. How^ many bushels of wheat did he harvest ? 998. 23. In January a laborer deposited in the savings bank 12 dollars, in February 30 dollars, in March 13 dollars, in April 11 dollars, in May 21 dollars, and in June 12 dollars. How many dollars did he deposit in the six months ? 99 dollwrs. 24. One week in May one dairyman furnished to a cheese factory 2,432 pouncte of milk, another dairyman 4,145 pounds, and another 3,221 pounds. How many pounds were furnished by the three dairymen ? 9, 798. 25. The amount of cheese manufactured at the same factory in June was 12,147 pounds, in July 13,410 pounds, in August 22,221 pounds, and in September 11,211 pounds. How many pounds were manufactured in the four months ? 58,989. 26. At a cotton factory 1,465,207 yards of cloth were made in 1864, and 1,532,492 yards in 1865. How many yards were made in the two years ? 2,997,699. 27. A grocer bought four hogsheads of sugar, weighing 1,154 pounds, 1,213 pounds, 1,301 pounds, and 1,231 pounds. How many pounds did they all weigh ? 4,899. 28. Long Island consists of three counties. Kings County contains 72 square miles. Queens County 410 square miles, and Suffolk County 1,200 square miles. How many square miles in the island ? i^ 682. 29. A fruit-grower sold 123 barrels of apples to one man, 31 barrels to another, 103 to a third, 30 barrels to a fourth, and 112 barrels to a fifth. How many barrels of apples did he sell ? S99^ 30. The mason work of a new school-house cost 1,220 dol- lars, the carpenter work 1018 dollars, and the painting and glazing 430 dollars. How many dollars did the school-house cost? ^2,668. 24 INTEGERS. CA.SE3 II. The sum of all the figures of any place more than 9. FIRST METHOD. 21. Ex. What is the sum of 28, 76 and 39 .^ Explanation. — IsL Writing the numbers. — first step. AVe write ones under ones, and tens under ^^ tens, and below the last number we draw two ' ^ parallel horizontal lines, far enough apart to — allow us to write figures between them. — 2d. Adding the Numbers. — Adding i;he ones, we find the sum to be 23, or 3 ones and 2 tens. We second step. write the 3 ones below the lower line as the 28 ones of the required sum, and, since we must 76 add the 2 tens to the tens of the given num- ?5 bers, we write them in tens' place, between the ?_ two hues. Adding the tens, we find the sum ^ to be 14, or 4 tens and 1 hundred. As there are no hundreds in the given numbers to which solution. to add this 1 hundred, we write the 4 28) tens and the 1 hundred below the lower 76 [- Parts. line, as tens and hundreds of the required _39 ) sum. The result, 143, is the sum re- 2 quired. (See Manual, page 215.) 143 FMOBJOJEMS. 31. A farmer has 46 sheep in one flock and 38 in another. How many sheep has he ? 84. 32. A merchant sold 13 yards of calico to one lady, 14 yards to another, and 16 yards to another. How many yards did he sell to the three ladies ? 4^. 33. Two wood-choppers worked together through the win- ter. One of them chopped 174 cords of wood and the other 167 cords. How many cords did both of them chop ? Sj^I, ADDITION. 25 <&&,)£:ji —MEADOW^ -..:ri-.-^...../i; :i: :|o;|, ; 3d!--c^ r^etcfl r,§(S <3» d* * as*"'"" rrS pRCHARO.o*'"" b^ 34. On this map of a farm, how many acres of wood- land on both sides of Wil- low Pond ? (See Manual p. 215.) 35. How many acres of pasture on both sides of Stony Brook ? 36. How many acres are in the two meadows ? 37. How many acres of tilled land does the farm contain ? 71. How many acres are on the east side of Willow Pond and Stony Brook ? 39. How many acres on the west side ? 40. How many acres in the farm in- cluding Willow Pond ? 256. 41. The orchard is 47 rods long and 34 rods wide. How many rods long is the stone fence around it ? 162. 42. The meadow north of the orchard is 66 rods long and 58 rods wide. How many rods of fence around it ? 43. The whole pasture is 110 rods long and 64 rods wide. How many rods of rail fence on the three sides, as shown on the map ? 28^. 44. The yard and garden are 34 rods long and 19 rods wide. In front is a picket fence, and on the other three sides is stone fence. How many rods long is the stone fence ? 45. How many rods in the fences which inclose the yard and garden ? 46. The lengths of the different fences inclosing the farm are shown on the map. How many rods of these are stone fence ? 237. 47. How many rods are rail fence ? 48. How many rods of fence of all kinds around the farm ? 49. How many rods of road in front of this farm ? C Principal Distances ] Stations. in miles. Boston, . . . Worcester, . . 44 Springfield, . 54 Pittsfield, . . 53 Albany, . . . 49 Albany, . . . Schenectady, . 17 Utica, . . . 78 Syracuse, . . 52 Rochester, . . 82 Batavia, . . . 32 Buffalo, . . . 36 INTEGERS Distances hetween Boston and St. Louis. Principal Distances Stations. in miles. Buffalo, . . . Principal Distances Stations. in miles. Chicago, . . . Dunkirk, . . 40 Joliet, ... 36 Erie, .... 48 Bloomington, . 88 Cleveland, . . Sandusky City, Toledo, . . . 95 61 51 Springfield,. . 60 Alton, ... 72 St. Louis, . . 25 Toledo, . . . Adrian, . . . 32 (See Manual, page 215.) Coldwater, . . 56 South Bend, . 69 La Porte, . . 27 Chicago, . . . 59 50. How many miles from Boston to Albany ? SOO. 51. How many miles from Albany to Buffalo ? 297. 52. What is the distance from Buffalo to Toledo ? 53. What is the distance frem Toledo to Chicago ? 54. How far is it from Chicago to St. Louis ? 281 miles. 55. How far is it from Albany to Chicago ? 835 miles. 56. What is the distance from Boston to St. Louis ? 57. One day a miller bought 1,284 bushels of wheat, and the next day 859 bushels. How many bushels did he buy in the two days? 2,143. 58. A butcher killed an ox, the quarters of which weighed respectively 136 pounds, 143 pounds, 178 pounds, and 187 pounds. What was the weight of the four quarters ? 644- 59. A grocer bought five jars of butter, containing respec- tively 33 pounds, 47 pounds, 32 pounds, 54 pounds, and 45 pounds. How many pounds of butter did the five jars contain ? 60. One month a woolen manufacturer paid out 31,587 dol- lars for stock, and 23,476 dollars for labor. How many dol- lars did he pay out during the month ? 55,063. 61. 53 feet + 171 feet + 23,869 feet + 24 feet + 359,487 feet = how many feet ? 383,604 feet. ADDITION 27 SOLtTTlON. SECOND METHOD. 22. Ex. Add 346, 5,279, and 8,165. Explanation. — After writing the parts, witli ones under ones, tens under tens, and so on, ^^^ we draw one horizontal line under the last qSak number. Adding the ones, we find the sum to '■ be 20, or ones and 2 tens. We write the 1^,790 ones below the line in the ones' place of the required sum ; and the 2 tens we add with the tens of the given numbers, but without first writing it in a line by itself. The sum of all the tens is 19, or 9 tens and 1 hundred. We write the 9 tens below the line as the tens of the required sum ; and the 1 hundred we add with the hundreds of the given numbers. The sum of all the hundreds is 7, which we write below the hne in hun- dreds' place. The sum of all the thousands is 13, which we write below the hne as the thousands and ten-thousands of the required sum. The result, 13,790, is the sum required. (See Manual, page 216.) :PItOBLEMS. 63. In the first passenger car of a railroad train were 49 passengers, in the second 63, in the third 54, in the fourth 63, and in the fifth 48. How many passengers were on the train ? 63. A railroad company purchased in one day 167 cords of wood at one station, 289 cords at another, 84 cords at another, and 417 cords at another. How many cords were purchased at the four stations ? 957 cords. 64. A merchant by selling a lot of damaged goods for $587, lost $94. How much did the goods cost him ? $681. A number with the sign $ before it expresses dollars. 65. Three men engaged in business together, the first fur- nishing $3,425 dollars, the second $2,163 dollars, and the third $896. What was the amount of their capital ? $6^Jf^4. 28 INTEGERS. 66. A grain-buyer in Chicago paid $7,594 for a cargo of wheat, shipped it to New York at an expense of $2,841, and sold it so as to gain $1,565. For how much did he sell it ? 67. A merchant pays for rent of store 1,275 dollars a year, for clerk-hire 3,895 dollars, for fuel 242 dollars, for gas 437 dollars, for freight and cartage on goods 936 dollars, and for other expenses 359 dollars. What is the amount of his yearly expenses ? $7, 144- 68. At a sale of government vessels, August 10, 1865, the bark Restless was sold for $12,000, the tug Larkspur for $8,100, the side- wheel steamer Alabama for $28,000, the schooner Matthew Vassar for $7,300, and the steam packet- boat Hartford for $9,100. For how much were all these ves- sels sold? $64,500. 69. A store in a brick building rents for $365 a year, the offices in the second story rent for $162, and a daguerrean room in the third story rents for $78. How much is the whole rent of the building ? $605. 70. A merchant's cash sales on Monday were $96, Tuesday $132, Wednesday $98, Thursday $72, Friday $115, and Satur- day $149. What was the amount of his cash sales for the week ? $662. 71. One season a farmer killed six hogs which weighed 427 pounds, 329 pounds, 314 pounds, 217 pounds, 208 pounds, and 1 96 pounds. How much did they all weigh ? 1,691 pounds. 72. Seven rafts of saw-logs from Alleghany River passed Pittsburg in one day. The first raft contained 276 logs, the second 359, the third 409, the fourth 293, the fifth 318, the sixth 325, and the seventh 358. How many logs in all the rafts? 2,338 has. (73) (74) (75) (76) (77) 30,076 141 28 14,193 647,129,341 5,821 30,648 52 6,009 327,293 498 8,291 164 417 284,384 167 287 386 1,306 43,100,085 22,849 165 1,227 129 2,873 3,482 24 2,873 873 541 691 2,841 642 154,685 30,698 482 596 578 7,676 28,165 1,642 417 249 48 475 56 13,509 3,871 509 1,465,127 ADDITION. 29 23* ^ule for Addition of Integers, I. Write the nuimhers to he added with ones under ones, tens under tenSy hundreds under hundreds, and so on. n. Add the column of ones, and, if the sum does not exceed 9, place it under the ones ; but if it exceeds 9, place the right-hand figure under the ones. m. Add the column of tens, and with it the left-hand figure of the sum of the ones, and if the sum does not exceed 9, place it under the tens ; hut if it exceeds 9, place the right-hand figure under the tens. rV. Proceed in the same manrwr with each column suc- cessively, and write down the whole sum of the left-hand column. PJROBLEMS. 78. A builder received $17,525 for erecting a church, $2,485 for building a dwelling ; $580 for building a bam, and $265 for repairs on a store. How much did he receive for the four jobs? $20,855. 79. England contains 57,101 square miles, Scotland 31,324 square miles, Ireland 32,512 square miles, Wales 7,219 square miles, and the smaller British islands contain 324 square miles. How many square miles in the whole of Great Britain ? 80. What is the sum of thirty-five million eight hundred seventy-six thousand one hundred twenty, three hundred ninety-six thousand four hundred ninety-one, and five hundred forty-three thousand six hundred seven ? 36,816,218. 81. One year a farmer raised 587 bushels of wheat, 1,229 bushels of oats, 643 bushels of com, 184 bushels of rye, 259 bushels of barley, and 296 bushels of buckwheat. How many bushels of grain did he raise ? 3, 198 Imshels. 83. A man paid $3,478 for a farm, $1,117 for live stock, $635 for farming implements, $423 for grain and seeds, and $189 for repairing fences and buildings. How much was his total outlay ? $5,8Jt2, 30 INTEGERS. 83. A pork-packer in Cincinnati packed 15,287 barrels of pork in December, 13,164 barrels in January, and 9,645 barrels in February. How many barrels did he pack in the three months ? S8, 096 larreU. 84. One day five fishing-smacks entered the harbor of Mar- blehead, bringing respectively 147 barrels of mackerel, 204 barrels, 89 barrels, 246 barrels, and 94 barrels. How many barrels of mackerel did all of them bring ? 780 barrels. 85. A drover paid $5,897 for 465 head of cattle, $3,486 for 284 head, $9,784 for 587 head, and $2,563 for 108 head. How many cattle did he buy, and how much did he pay' for them ? 1, 444 head of cattle ; $21, 730, 86. A merchant buys a bale of sheeting, containing 3 pieces of 38 yards each, 4 pieces of 39 yards each, 6 pieces of 42 yards each, and 5 pieces of 40 yards each. How many pieces in the bale ? How many yards ? 722 yards. 87. At the battle of Gettysburg the loss in the Union army was 2,834 men killed and 13,790 wounded, and in the Confed- erate army 4,500 killed and 26,500 wounded. What was the whole loss in each army ? U?iion, 16,624; Confederate, 31,000. 88. What was the whole number of men killed ? 7,334. 89. What was the whole number wounded ? 40,290. 90. What was the whole loss in both armies ? 4'^,^^4- 91. The number of cattle received at the New York Cattle Market in one week was 226 by the New York and Erie Rail- road, 116 by the Hudson River Railroad, 2,669 by the Harlem Railroad, 319 by the New Jersey Central Railroad, 445 by Hudson River boats, and 26 on foot. How many cattle were received that week ? 3, 801. 92. The value of the gold and silver exported from Califor- nia in ten years commencing with 1854, was as follows : In 1854, . .$52,045,633 In 1855, . . 45,161,731 In 1856, . . 50,697,43^ In 1857, . . 48,976,697 In 1858, . . 47,548,026 In 1859, In 1860, In 1861, In 1862, In 1863, $47,640,462 42,325,916 40,676,758 42,561,761 46,071,920 What was the total value for the ten years ? $463,706,338. SECTION III. S ITS T^ ^CTIOJV, IN3DTJCTION. ^ (See Manual, page 216.) 24. 1. Of the 8 ladies in this picture, 3 are coming down the street, and the others are going up the street. How many- ladies are going up the street ? 2. Four of the ladies are walking, and the others are riding. How many are riding ? 3. In the picture are 9 horses, going up street, and the others coming down. How many horses are coming down street ? 4. All but 3 of the 9 horses are driven in teams. How many are driven in teams ? 5. There are 12 barrels in the picture, 5 of them on a cart, and the others by the store on the corner. How many are by the store ? 6. Of the 13 men shown in the picture, 6 are walking, and the others are riding. How many are riding ? 7. Of the 13 men, 10 are coming toward us, and the others are going from ii§. How many are going from us ? 32 INTEGERS. 25. When one of two numbers is taken from iha other, tlie process is Subtraction. 26. The result thus found is the Remainder^ or Dif- ference. 27. The number from which another is to be taken is the Minuend. 28. The number to be taken from another is the Subtrahend. The number of ones in the subtrahend and remain- der, taken together, must equal the number of ones in the minuend. 8. Subtract 7 books from 11 books. 9. What will be the remainder if you take 9 chairs from 16 chairs ? 10. If 6 cents be subtracted from 15 cents, what will be the remainder ? 11. A cook, having 18 eggs, used 9 on Monday, and the remainder on Tuesday. How many did she use on Tuesday ? 13. What is the difference between 17 leaves and 8 leaves ? 13. How much is the difference between 14 bushels of pota- toes and 5 bushels of potatoes ? 14. From 15 inches subtract 7 inches. 15. Subtract 9 from 19. 16. The minuend is 13, and the subtrahend 4. What is the remainder ? 29. This sign — , written between two numbers, sig- nifies that the number after it is to be subtracted from the number before it. It is called Minus, or the Sign of Subtraction. Thus, 25 — 16 — 9 is read 25 minus 16 equals 9. 17. Read 15 - 7 = 8. I 19. Read 13+ 8 = 30 - 9. 18. Read 17-6 = 11. I 20. 18 - 7 = how many ? 31. 31 brushes — 11 brushes = how many brushes ? 33. 37 words — 8 words — 5 words — 7 words = how many words ? (Sec Manual, page 216.) SUBTRACTION. 33 30i SUBTRACTION TABLE. ojS 12 3 4 5 6 7 8 9 - j 5 6 7 8 9 10 11 12 13 14 0\5 555555555 12 3 4 5 6 7 8 9 0123456789 2 3 4 5 1111 12 3 4 6 7 § 910 11111 5 6 7 8 9 ^ ( 6 7 8 9 10 11 12 13 14 15 0|6 666666666 0123456789 2{| 3 4 5 6 2 2 2 2 12 3 4 7 8 9 10 11 2 2 2 2 2 5 6 7 8 9 - j 7 8 9 10 11 12 13 14 15 16 7|7 777777777 0123456789 3J3 4 5 6 7 3 3 3 3 8 9 10 11 12 3 3 3 3 3 o j 8 9 10 11 12 13 14 15 16 17 OJ8 888888888 12 3 4 5 6 7 8 9 0123456789 4]t 5 6 7 8 4 4 4 4 9 10 11 12 13 4 4 4 4 4 ft j 9 10 11 12 13 14 15 16 17 18 9]9 999999999 12 3 4 5 6 7 8 9 0123456789 OBJLL EXJEnCISES. 1. — ^1, Subtract 1 from every number from 100 down to 1, thus ; 1 from 100 leaves 99, 1 from 99 leaves 98, 1 from 98 leaves 97, and so on. 2. Count from 100 down to 1, thus ; 100, 99, 98, 97, and so on. 2. — 1. Subtract 2 from every second number from 100 down to 0, thus ; 2 from 100 leaves 98, 2 from 98 leaves 96, 2 from 96 leaves 94, and so on. 2. Count by 2's from 100 down to 0, thus ; 100, 98, 96, 94, and so on. 3. Subtract 2 from every second number from 101 down to 1, thus ; 2 from 101 leaves 99, 2 from 99 leaves 97, 2 from 97 leaves 95, and so on. 4. Count by 2's from 101 down to 1, thus ; 101, 99, 97, 95, and so on. 3. — 1. Subtract 3 from every third number from 100 down to 1, thus ; 3 from 100 leaves 97, 3 from 97 leaves 94, and so on. 2. Count by 3's from 100 down to 1, thus ; 100, 97, 94, 91, and so on. 3. Count by 3's from 101 down to 2. (See Manual, page 216) 4. .Count by 3's from 102 down to 0. 4. — 1. Subtract 4 from every fourth number from 100 down to 0, thus ; 4 from 100 leaves 96, 4 from 96 leaves 92, 4 from 92 leaves 88, and so on. 2. Count by 4's from 100 down to 0, thus ; 100, 96, 92, 88, 84, and so on. 3. Count by 4's from 101 down to 1. 4. Count by 4's from 102 down to 2. 6, Count by 4's from 103 down to 3. 34 INTEGERS. 5. — 1. Subtract 5 from every fifth number from 100 down to 0, thus ; 5 from 100 leaves 95, 5 from 95 leaves 90, and so on. 2. Count by 5's from 100 down to 0, thus ; 100, 95, 90, 85, and so on. 3. Count by 5's from 101 down to 1. 4. Count by 5'8 from 103 down to 2. 5. Count by 5's from 103 down to 3. 6. Count by 5'8 from 104 down to 4. 6. — 1. Subtract 6 from every sixth number from 102 down to 0, thus ; 6 from 102 leaves 96, 6 from 96 leaves 90, and so on. 2. Count by 6's from 102 down to 0, thus ; 102, 96, 90, 84, and so on. 3. Count by 6's from 103 down to 1. 4. Count by 6's from 104 down to 2. 5. Count by 6's from 105 down to 3. 6. Subtract 6 from every sixth number from 100 down to 4. 7. Count by 6's from 101 down to 5. 7. — 1. Subtract 7 from every seventh number from 105 down to 0, thus ; 7 from 105 leaves 98, 7 from 98 leaves 91, and so on. 2. Count by 7's from 105 down to 0, thus ; 105, 98, 91, 84, and so on. 3. Count by 7's from 106 down to 1. 4. Count by 7's from 100 down to 2. 5. Subtract 7 from every seventh number from 101 down to 3. 6. Subtract 7 from every seventh number from 102 down to 4. 7. Count by 7'8 from 103 down to 5. 8. Count by 7's from 104 down to 6. 8. — 1. Subtract 8 from every eighth number from 104 down to 0, thus ; 8 from 104 leaves 96, 8 from 96 leaves 88, and so on. 2. Count by 8's from 104 down to 0, thus ; 104, 96, 88, 80, and so on. 3. Subtract 8 from every eighth number from 105 down to 1. 4. Count by 8's from 106 down to 2. 5. Count by 8's from 107 down to 3. 6. Count by 8's from 100 down to 4. 7. Subtract 8 from every eighth number from 101 down to 5. 8. Count by 8's from 102 down to 6. 9. Count by 8's from 103 down to 7. 9. — ^1. Subtract 9 from every ninth number from 108 down to 0, thus ; 9 from 108 leaves 99, 9 from 99 leaves 90, and so on. 2. Count by 9's from 108 down to 0, thus ; 108, 99, 90, 81, and so on. 3. Subtract 9 from every ninth number from 100 down to 1. 4. Count by 9's from 101 down to 2. 5. Count by 9's from 102 down to 3. 6. Subtract 9 from every ninth number from 103 down to 4. 7. Subtract 9 from every ninth number from 104 down to 5. 8. Count by 9's from 105 down to 6. 9. Count by 9'8 from 106 down to 7. 10. Count by 9'8 from 107 down to 8. SUBTRACTION. 35 C^SE I. No figure of the subtrahend greater than the corres- ponding figure of the minuend. 31. Ex. What is the difference between 8,397 and 3,265 ? Explanation. — Since these numbers solution. are too large to be subtracted mentally, 8,397 Minuend. we write the subtrahend below the 3,265 subtrahend. minuend, with the ones under ones, the 5,132 Difference. tens under tens, the hundreds under hundreds^ and the thousands under thousands. Com- menciQg at the right, we take the 5 ones from the 7 ones, and the remainder, 2 ones, we write under the ones. We next take the 6 tens from the 9 tens, and the remainder, 3 tens, we write under the tens. Then 2 hundreds from 3 hundreds leave 1 hundred, which we write under the hundreds; and 3 thousands from 8 thou- sands leave 5 thousands, which we write under the thou- sands. The result, 5,132, is the difference or remainder required. 32. We can subtract apples from apples, dollars from dollars, pens from pens, or hours from hours ; but we can not subtract apples from dollars, nor pens from hours. For 13 apples — 4 dollars = neither 9 apples nor 9 dollars. Again, we can subtract ones from ones, tens from tens, or hundreds from hundreds ; but we can not sub- tract ones from hundreds, nor tens from thousands. For 9 thousands — 4 tens = neither 5 tens nor 5 thousands. Hence, 33t General 'PHnciptes of Subtraction, I. Only numbers expressing the same kind of things can he subtracted the one from the other. n. Only figures occupying the same place in different numbers can be subtracted the onefrmn, the other. (See Manual, page 216.) ^0 INT EGERS. rMOBJLEMS. (1) 62 41 (2) 76 24 (3) 45 34 (4) 57 43 (5) (6) 682 584 850 302 (7) 635 pins 412 pins (8) 3,846 soldiers 2,534 soldiers (9) 7,968 shingles 5,453 shingles (10) 57,908 pounds 43,700 pounds 11. James, having 27 marbles, gave 12 of them to John. How many marbles had he left ? 15. 12. From a piece of muslin containing 39 yards, a merchant sold 13 yards for a dress. How many yards remained in the piece ? 26. 13. Joseph had 46 cents, but he has spent 25 cents for a knife. How many cents has he now ? 14. Ellen attended school 63 days in a term of 75 school- days. How many days was she absent ? 12. 15. A gardener picked 68 boxes of strawberries one fore- noon, and 54 boxes in the afternoon. How many more boxes did he pick in the forenoon than in the afternoon ? 16. Hiram lives 98 rods from the schoolhouse, and Thomas 41 rods. How much farther does Hiram walk in going to school than Thomas ? 57 rods. 17. 435 miles — 314 miles = how many miles ? 121. 18. 6,798 bushels — 2,641 bushels = how many bushels? 19. How many tons are 38,156 tons — 14,044 tons ? 2^,112. 20. A fruit-dealer, having 247 baskets of peaches, sold 125 baskets. How many baskets had he left ? 122. 21. A man whose income is $875 a year, expends $734. How much money does he save ? $lJi'l. 22. A drover bought a lot of cattle for $4,574, and sold them for $5,896. How much did he gain ? $1,322. (23) (24) (25) (26) 57,698 675,004 2,174,943 167,065,149 43,257 245,002 42,301 4,042,136 27. One year a farmer raised 1,898 bushels of oats, and sold 1,427 bushels. How many bushels did he keep for use ? J!i71. SUBTRACTION, OA.SE II. 37 Any figure of the subtrahend greater than the corres- ponding figure of the minuend. FIRST METHOD. 31. Ex. 1. From 16 subtract 9. (See Manual, page 216.) Explanation. — ^We write the mimbers as in solution. Case I ; but as we can not subtract 9 ones 16 from 6 ones, we must unite the 1 ten, wbich _9 equals 10 ones, with the 6 ones, and subtract 7 the 9 from the whole 16 at once. Ex. 2. From 76 subtract 29. Explanation. — ^As we can not subtract 9 ones solution. from 6 ones, we take 1 of the 7 tens and unite — it with the 6 ones, making 16 ones, and sub- o 9 tr acting 9 ones from the 16 ones, we write the t-z remainder, 7, as the ones of the final result. Since we have already used one of the 7 tens, we have now only 6 tens in the minuend, and hence we subtract the two tens from 6 tens, and write the remainder, 4 tens, as the tens of the final result. :pnonjLEM8. 28. In a public school are 45 pupils, and 28 of them are girls. How many are boys ? 17. 29. A groceV sold 35 bars of soap from a box that contained 64 bars. How many bars were left in the box ? 29. 30. One day 92 boats passed Lockport on the Erie Canal, and 47 of them were going east. How many were going west ? 31. A washerwoman had 72 clothes-pins, but she has lost 29 of them. How many has she now ? JiS. 32. A jar filled with butter weighs 52 pounds, and the jar alone weighs 15 pounds. How much does the butter weigh ? 38 INTEGERS. 33. From a barrel of sugar containing 283 pounds, a mer- cliant sold 156 pounds. How many pounds were left in the barrel ? 121. 34. A man bought a village lot for |350, and paid down all but $125. How much did he pay ? 35. A man bought a piano for |475. He paid $267 in cash, and gave his note for the balance. For what sum did he give his note ? $208. 35. Ex. From 853 subtract 467. Explanation. — As we can not subtract 7 solution. ones from 3 ones, we take 1 of the 5 tens ^-^ and unite it with the 3 ones, making 13 ones ; ^ g y and subtracting 7 ones from 13 ones, we write — — — the remainder, 6, as the ones of the final re- sult. Since we have already used 1 of the 5 tens, only 4 tens now remain in the minuend. As we can not subtract the 6 tens from 4 tens, we unite 1 of the 8 hun- dreds with the 4 tens, making 14 tens ; then subtract- ing 6 tens from 14 tens, we write the remainder, 8 tens, as the tens of the final result. Since we have already used 1 of the 8 hundreds, only 7 hundreds now remain in the minuend ; and from this we subtract the 4 hun- dreds, and write the remainder, 3 hundreds, as the hundreds of the final result. FMOBIj EMS. 36. A man who had a farm of 154 acres, gave to his son 65 acres. How much land had he left ? 89 acres. 37. In a certain village school-district are 447 children, of whom only 298 attend school. How many do not attend school ? 1J^9 children. 38. A provision dealer receiving ah order for 525 barrels of beef, has only 354 barrels on hand. How many barrels more will he require to fill the order ? 39. A merchant's sales in January amounted to $1743, and in February to $928. How much did the sales of January ex- ceed those of February ? $815. SUBTRACTION". 39 40. In a certain town 135 men were drafted for tlie army, but 54 of them were rejected by the examining surgeon. How many passed examination ? 41. A regiment entered the service with 1,149 men, and at the close of the war had only 437. How many men had it lost ? ' ^722. 42. A banker's income last year was $12,849, and his ex- penses were $6,768. How much did his income exceed his expenses? $6,081. 43. A market-gardener in one year received $3,730 for fruits and vegetables, and his expenses were $1,850. How much were his profits ? $1,880. 44. One day 724 cattle were received at the Philadelphia Cattle Market, and 648 of them were sold. How many re- mained unsold ? 45. A man having $974 in the bank, drew out $396. How much money had he left on deposit ? $578. 36. Ex. From 3000 subtract 57. Explanation. — ^We can not subtract 7 ones solution. from ones, and as we have in the minuend „ tens to unite with the ones, and hundreds g y to unite with the tens, we must take 1 of the 3 thousands, leaving 2 thousands. This 1 thou- ^^^^ sand = 10 hundreds ; but as we can not subtract ones from hundreds (Prin. 11.), we take 1 of the 10 hun- dreds, leaving 9 hundreds. This 1 hundred = 10 tens ; but as we can not subtract ones from tens, we take 1 of the 10 tens, leaving 9 tens. This 1 ten = 10 ones. We now take the 7 ones from 10 ones, and the remain- der, 3 ones, we write as the ones of the final result. Then 5 tens from 9 tens leave 4 tens, and as there are no hundreds or thousands in the subtrahend, we write the 9 hundreds and the 2 thousands of the minuend, for the hundreds and thousands of the final result. iy^^ZZ^ 40 INTEGERS. JPJROBJLEMS. 46. If I buy a bushel of apples for 65 cents, and give in payment a dollar bill, how much change should I receive ? 47. At a flouring-mill in Baltimore 1000 barrels of flour were made in one week, and 869 barrels of it were sold. How many barrels were unsold ? isi, 48. The Phillips Well on Oil Creek is 460 feet deep, and the Titusville Well 1100 feet deep. How much deeper is the latter well than the former ? SIfifeet. 49. A man divided $7,500 between his son and daughter, giving $4,375 to his son. How much did the daughter re- ceive? $3,225. 50. A shipbuilder received $21,000 for a schooner, which cost him $18,728. How much was his gain ? $2,272. 51. A forwarder had 40,000 bushels of oats in store at Chi- cago. How many bushels had he in store after shipping 25,487 bushels to Buffalo ? U,513 ImsheU. 52. A broker sold stocks for $256,200 which cost him $209,408. Did he gam or lose, and how much ? Gained $46, 792. 53. How many acres are 1,100 acres — 841 acres ? 259. 54. How many cords are 21,610 cords — 19,587 cords ? 55. How many gallons are 110,040 gallons— 90,621 gallons? 56. At the battle of Bunker Hill the Americans lost 449 men, and the British 1054. How much did the British loss exceed the American ? SECOND METHOD. 87. Ex. rrom 7,623 subtract 4,856. Explanation. — In this solution we com- ^^J:^^^' mence at the right, and proceed the same as I'^r^ in the First Method, except that we omit to — — write the partial minuends, 13 ones, 11 tens, 2,767 15 hundreds, and 6 thousands, above the given min- uend. (See Manual, page 216.) 6UBTKAOTION. 41 38. "Rule for Subtraction of Integers, I. Write the subtrahend below the minuend^ placing ones under ones, tens under tens, and so on. n. When the figures of the subtrahend do not ex- ceed in value the corresponding figures of the minuend, 1. Commencing at the right hand, subtract each figure of the subtrahend from the corresponding figure of the minuend, and write the result directly below in the re- mainder. 2. If there are figures in the minuend unthout any cor- responding figures in the subtrahend, write them m the remainder. m. When any figure of the subtrahend exceeds the corresponding figure of the minuend, Add 10 to the figure of the minuend, and from the sum subtract the figure of the subtrahend. In this case, always call the next left-hand figure of the minuend 1 less, or the next left-hand figure of the subtrahend 1 more. (See Manual, page 216.) JPM OBJOEMS, 57. A load of hay with the wagon weighed 2,656 pounds on the hay scales, and the wagon alone weighed 987 pounds. How much did the hay weigh ? 1, 669 pounds. 58. At a certain election 2,649 votes were cast for one can- didate, and 1,975 votes for the other. What majority did the successful candidate receive ? 674 'ootes. (59) (60) (61) (62) 1,000,000 348,794 7,408,215 300,300,333 31,276 127,586 59,826 47,008,296 63. The island of Cuba contains 45,277 square miles, and the State of Ohio 39,904 square miles. How much larger is Cuba than Ohio ? D 4:2 INTEGERS The numbers on this map of Mis- sissippi River show the distances of the different places from the mouth of the river. How many miles is it from 64. Eock Island to Yicksburg ? 65. Memphis to La Crosse ? 1,0 JfS. 66. Cairo to Lake Itasca ? 1,331. 77. St. Paul to Baton Rouge ? 68. The Falls of St. Anthony to the mouth of Red River ? 1, 836. 69. Burlington to Natchez ? i.j^i^. 70. The mouth of Missouri River to Prairie Du Chien ? 538. 71. New Orleans to Quincy ? 1, 399. 73. Quincy to St. Paul ? 632. 73. The mouth of Arkansas River to the mouth of Missouri River ? 64S. 74. Keokuk to New Orleans ? 95. Du Buque to Lake Itasca ? 76. Prairie Du Chien to Natchez? 77. St. Louis to the Falls of St. Anthony ? 798. 78. Memphis to Baton Rouge ? 79. Keokuk to La Crosse? 4I8. 80. Du Buque to Vicksburg ? 81. Cairo to the mouth of Red River ? 850. 83. The mouth of Arkansas River to Lake Itasca ? 1,769. 83. St. Louis to Rock Island ? 84. Burlington to Memphis ? 85. St. Paul to Vicksburg ? 1,630. 86. Quincy to Lake Itasca ? 985, 87. New Orleans to the Falls of St. Anthony? 2,039. 88. Rock Island to Baton Rouge ? 89. St. Louis to New Orleans? SUBTRACTION. 43 90. If I owe $3,496, and I pay $1,748, how much do I then owe? $l,7Ji8. 91. What is the difference between 9,417 and 3,584 ? 5, 833. 93. The greater of two numbers is 11,419, and the less is 7,255. What is their difference ? ^ 16Jf. 93. The running expenses of a machine-shop for a year were $30,456, and the sales amounted to $31,317. What were the net earnings for the year ? $10,761. MEVIJEW PJROBIjEMS. 1. In the year 1860 there were 34 States in the Union, of which 11 seceded. How many States did not secede ? 23. 3. A farmer drew to market seven loads of hay, which weighed 1,577 pounds, 1,891 pounds, 1,648 pounds, 3,154 pounds, 1,736 pounds, 1,954 pounds, and 2,036 pounds. What was the weight of the seven loads ? 3. On board an ocean steamer were 114 cabin passengers, 649 steerage passengers, and a crew of 87 persons. What was the whole number of persons on board ? 850. 4. A father and his two sons earned $1875 in a year, the elder son earning $638, and the younger son $459. How much did the father earn ? $778. 5. A gardener received $318 for cabbages, and $439 for tomatoes. The expense of raising the cabbages was $84, and of the tomatoes $134. What were his profits on the two crops ? $U9. 6. A regiment when it entered the service mustered 1004 men; during the war 37 of these were killed, 48 died, 53 were taken prisoners, and 597 were discharged. How many men served through their term of enlistment ? 269. 7. In five successive weeks 79,747 tons, 84,334 tons, 68,953 tons, 76,081 tons, and 81,168 tons of coal were taken to Phil- adelpliia by the Philadelphia and Reading Eailroad. How many tons were carried over the road in the five weeks ? 8. A contractor furnished 10,000 overcoats for the army, but 1,715 of them were condemned as imperfect. How many of them were accepted ? 44 INTEGERS.. 9. On the first of January an edition of 5,000 copies of a book was published. In January 396 copies were sold, in February 741, in March 1,214, in April 927, in May 643, and in June 584. How many copies remained unsold July 1 ? J,.95. 10. A man was bom in the year 1799, and died at the age of 07 years. In what year did he die ? In the year 18G6. 11. The battle of Lexington was fought in the year 1775, and President Lincoln was assassinated in 1865. How many years between the two events ? 90. 12. In the year 1861 the senior class of a certain college contained 46 students, the junior class 38, the sophomore class 59, and the freshman class 74. Of these, 12 seniors, 14 juniors, 18 sophomores, and 25 freshmen enlisted. How many students enlisted ? 13. How many students remained in each class ? How many remained in college ? I48 reTnained in college. 14. A railroad train left Cincmnati for St. Louis with 435 passengers. On the route 215 passengers left the cars, and 194 went aboard. How many were on the train when it reached St. Louis ? Jf,lJ^. 15. One year a merchant's sales amounted to |37,496. His goods cost $25,267, and his store expenses were $6,485. How much were his profits ? $5, IJ^J^. 16. One day 48,325 letters were received at the New York Post-office. Of these, 21,259 were directed to places m the State of New York, 20,048 to places in other states, and the rest to places in foreign countries. How many were directed to foreign countries ? 7,018. - 17. At the beginning of the year A had property worth $2,350, but he owed $476. During the year he earned $1,156, and expended $879. How much was he worth at the end of the year? $2,151, 18. 1,153 — 967 + 10,000 — 5,308 = how many? 19. What is the difierence between 13,647 + 2,593 — 6,483 4,931 + 5,006 - 7,285 ? 3,033. MULTIPLICATION. 45 SECTION IV. MZrZ TITZIC;A TIOJV. IK-DXJCTION". (See Manual, page 216.) 30« 1. In this picture are 3 stems of a rose-bush, with 3 ro&es on each stem. 3 roses and 3 roses and 3 roses are how many roses ? Then, 3 times 3 roses are how many roses ? 2. On each of the stems are 4 buds. 4 buds and 4 buds and 4 buds are how many buds ? 3 times 4 buds are how many ? 3. On the branches of a cherry-tree we see 5 clusters, with 3 cherries in each cluster. 3 cherries -f 3 cherries + 3 cherries + 3 cherries 4- 3 cherries are how many cherries ? 5 times 3 cherries are how many ? 4. On each of the 5 branches are 4 leaves. 5 times 4 leaves are how many leaves ? 5. A boy at play with his blocks places them in rows on a table. If he counts them one way he has 3 rows, and 7 blocks in each row ; but if he counts them another way he has 7 rows, and 3 blocks in each row. How many blocks has he ? How many are 6. 3 times 7 blocks ? | 7. 7 times 3 blocks ? 8. If a woman can weave 6 yards of rag carpet in 1 day, how many yards can she weave in 4 days ? 46 INTEGERS. 9. A shoemaker can make 5 pairs of children's shoes in 1 day. How many pairs can he make in 8 days ? 10. How much will 5 oranges cost, at 8 cents a piece ? The sum of 7 + 7 + 7 + 7 + 7 is 35, and 5 times 7 are 35. By each of these methods we have fotuid the sum of five 7's, or of as many 7's as there are ones in 5 ; but the second method is shorter than the first. 40. The process of finding, by a method shorter than Addition, the sum of as many times one number as there are ones in another, is Multiplication. 41. The result thus found is the Product, and 42. The numbers themselves are Factors. 43. The factor which is to be taken any certain num- ber of times is the Multiplicand, and 44. The factor which shows how many times the mul- tiplicand is to be taken, is the Multiplier. The number of ones in the product must equal the number of ones of the multiplicand taken as many times as there are ones in the multipher. 11. What is the product of 6 times 8 loaves of bread ? 12. What is the product of the factors 10 and 7 ? 13. 7 times 8 are 56. Which of these numbers is the mul- tiplicand ? Which is the multiplier ? Which is the product ? Which are the factors ? 45. This sign x, written between two numbers, signi- fies that they are to be multiplied together. It is called the Sign of Multiplication, and is read times, or multiplied hy. Thus, 6 X 9 = 54 may be read 6 times 9 equal 54, 6 times 9 are 54, 6 multiplied by 9 equal 54, or 6 multiplied by 9 are 64. 14. Read 7 x 12 = 84. 16. Read 3x8 = 4x6. 15. Read 4 x 25 = 100. 17. Read 3 x 4 x 5 = 60. 18. 6 X 7 plums are how many plums? MULTIPLICATION. 47 19. 10 X 4 apples = how many apples ? 20. 3 X 2 X 5 balls = how many balls ? ,46. Numbers applied to objects or things are Concrete Numbers; as, 4 apples, 19 men, 237 books. 47f Numbers not applied to objects or things are Abstract Numbers; as, 4, 19, 237. (See Manual, page 216.) 48. MULTIPLICATION TABLE. 1]? 1 2 1 1 3 4 5 6 7 8 9 1111111 oJO 123456789 0|6 666666666 1 2 3 4 5 6 7 8 9 6 12 18 34 30 36 42 48 54 o(0 1 2 2 2 3 4 5 6 7 8 9 2 2 2 2 2 2 2 „jO 123456789 7|7 777777777 2 4 6 8 10 12 14 16 18 7 14 21 28 35 43 49 56 63 3\t 1 2 3 3 3 4 5 6 7 8 9 3 3 3 3 3 3 3 QJO 123456789 0I8 888888888 3 6 9 12 15 18 21 24 27 8 16 24 33 40 48 56 64 72 412 1 2 4 4 3 4 5 6 7 8 9 4 4 4 4 4 4 4 ftjO 123456789 9]9 999999999 4 8 13 16 20 34 28 32 36 9 18 27 36 45 54 63 73 81 123456789 555555555 5 10 15 20 35 30 35 40 45 irt( 0133456789 lUjio 10 10 10 10 10 10 10 10 10 10 20 30 40 50 60 70 80 90 OUAIj XJXJSJtCISES. 1. — 1. Count by 3's to 20, in this manner : One 2 is 2, or once 2 is 2 ; two 3'8 are 4, or 3 times 2 are 4 ; three 2's are 6, or 3 times 2 are 6 ; four 3^s are 8, or 4 times 3 are 8; and so on. 3. Multiply from times 3 to 10 times 3, thus ; times 2 is 0, once 2 is 3, 3 times 3 are 4, 3 times 3 are 6, 4 times 3 are 8, and so on. 3. Multiply from 10 times 2 to times 2, thus ; 10 times 2 are 20, 9 times 3 are 18, 8 times 3 are 16, 7 times 3 are 14, and so on. 2. — 1. Count by 3's to 30, in this manner : One 3 is 3, or once 3 is 3 ; two 3's are 6, or 3 times 3 are 6; three 3's are 9, or 3 times 3 are 9; four 3's are 13, or 4 times 3 are 12, and so on. 2. Multiply from times 3 to 10 times 3, thus ; times 3 is 0, once 3 is 3, 2 times 3 are 6, 3 times 3 are 9, and so on. 3. Multiply from 10 times 3 to times 3, thus ; 10 times 3 are 30, 9 times 3 are 27, 8 times 3 are 24, and so on. 3. — 1. Count by 4's to 40, thus : One 4 is 4, or once 4 is 4; two 4's are 8, or 2 times 4 are 8 ; three 4's are 13, or 3 times 4 are 13, and so on. 48 INTEGERS. 2. Multiply from times 4 to 10 times 4, thus ; times 4 is once 4 is 4, 2 times 4 are 8, 3 times 4 are 12, and so on. 3. Multiply from 10 times 4 to times 4, thus ; 10 times 4 are 40, 9 times 4 are 36, 8 times 4 are 32, and so on. 4. — 1. Count by 5'8 to 50. (See Manual, page 216.) 2. Multiply from times 5 to 10 times 5. 3. Multiply from 10 times 5 to times 5. 6.— 1. Count by 6's to 60. 2. Multiply from times 6 to 10 times 6. 3. Multiply from 10 times 6 to times 6. 6.— 1. Count by 7's to 70. 2. Multiply from times 7 to 10 times 7. 3. Multiply from 10 times 7 to times 7. 7.— 1. Count by 8's to 80. 2. Multiply from times 8 to 10 times 8. 3. Multiply from 10 times 8 to times 8. 8.— 1. Count by 9's to 90. 2. Multiply from times 9 to 10 times 9. 3. Multiply from 10 times 9 to times 9. 9.— 1. Count by lO's to 100. 2. Multiply from times 10 to 10 times 10. 3. Multiply from 10 times 10 to times 10. C u^ S E I . The Multiplier One Figure. FIRST METHOD. 49. Ex. 1. 37 + 37 + 37 + 37, or 4 times 37, are how Ulciliy . FIRST SOLUTION. Explanation. — ^In adding these numbers, s^ Addition. we first find the sum of 7 ones taken 4 37 times, which is 28. We write the 8 ones 37 of this sum as the ones of the final result, ^|^ and the 2 tens we write in tens' place be- — tween the two parallel lines. We next find _2_ the sum of 3 tens taken 4 times, which 148 is 12 tens ; and adding to this sum the 2 tens of the first result, we write the 14 tens, which equal 4 tens and 1 hundred, as the tens and hundreds of the final result. MULTIPLICATION. 49 In the second solution we write the 37 «=^<^^ solution. _ T J 1 • 1 • J 1 J3u Multiplication. only once, and as this number is to be 07 taken 4 times, we write 4 under the ^ right-hand figure. 7 ones + 7 ones + 7 —z- ones + 7 ones, or 4 times 7 ones, are 28 ones. We write the 8 ones of this sum ^^^ or product as the ones of the final result ; and the 2 tens, which are to be added to the sum or product of the tens, we write in tens' place between the two paral- lel Hues. Then, 3 tens + 3 tens + 3 tens + 3 tens, or 4 times 3 tens, are 12 tens ; and adding to this sum or product the two tens of the first result, we write the 14 tens as the tens and hundreds of the final result. (See Manual, page 216.) Ex. 2. What is the product of 3,794 multipHed by 6 ? Explanation. — ^We write the mul- solution. tipher under the multipHcand, and ^^94 Multiplicand. commence at the right to multiply. ^ ^^ ^^^' 6x4 ones = 24 ones, or 4 ones and 452 2 tens. We write the 4 ones for the 22764 Product ones of the final product, and the 2 tens in tens' place between the parallel Hues. 6x9 tens = 54 tens, and 54 tens + 2 tens = 56 tens, or 6 tens and 5 hundreds. We write the 6 tens for the tens of the final product, and the 5 hundreds in hundreds' place between the parallel lines. 6x7 hundreds = 42 hundreds, and 42 hundreds + 5 hundreds = 47 hun- dreds, or 7 hundreds and 4 thousands. We write the 7 hundreds for the hundreds of the final product, and the 4 thousands in thousands' place between the paral- lel lines. 6x3 thousands = 18 thousands, and 18 thousands + 4 thousands = 22 thousands, or 2 thou- sands and 2 ten-thousands, and these we write for the thousands and ten-thousands of the final product. E 50 INTEGERS. PJROBIjJEMS, 1. How much will 3 tons of hay cost, at $12 a ton ? $36. 2. In 1 day there are 24 hours. How many hours in 4 ? 96. 3. How much will 4 pounds of raisins cost, at 22 cents a pound ? 88 cents. 4. If a canal-boat goes 42 miles in a day, how many miles will it go iQ 6 days ? 252. 5. A farm laborer worked 5 months for $16 a month. How much did his wages amount to ? $80. 6. What is the product of 7 x 74 ? 518. 7. The multiplicand is 96, and the multiplier is 8. What is the product ? 768. 8. A livery-man bought 3 horses, at $125 apiece. How much did they cost him ? 9. How much will it cost for a party of 6 persons to go from New York to Liverpool on an ocean steamer, if the fare is $137 ? $822. 10. A music dealer sold 7 pianos, at $325 each. How much did he receive for all of them ? $2,275. 11. How many quarts of milk will be used in a hotel in a week, if 36 quarts are used each day ? 12. The factors are 2,147 and 5. What is the product ? 13. A cooper sent to the mill 9 loads of flour-barrels, and each load contained 146 barrels. How many barrels did he send? 14. In one mile there are 5,280 feet. How many feet in 8 miles ? 15. The Pennsylvania Central Kailroad Company bought 6 locomotives, at $28,675 each. How much did they all cost ? $172,050. 16. How much will 7 bushels of potatoes cost, at 56 cents a bushel ? S92 cents. 17. A farmer raised 8 acres of wheat, and each acre pro- duced 25 bushels. How many bushels of wheat did he raise ? MULTIPLICATION. 61 SECOND METHOD. 50. Ex. Multiply 473 by 9. Explanation. — ^We multiply each figure of solution. tlie multiplicand by the multiplier, as in the 473 First Method. 9x3 ones are 27 ones, or 7 9 ones and 2 tens. We write the 7 ones as the 4,257 ones of the product, but instead of writing down the tens' figure, 2, we reserve it in the mind, to be added to the product of the tens. 9x7 tens are 63 tens, and 63 tens + 2 tens = 65 tens, or 5 tens and 6 hundreds. We write the 5 tens as the tens of the prod- uct, and reserve the 6 hundreds in the mind, to be added to the product of the hundreds. 9x4 hundreds are 36 hundreds, and 36 hundreds + 6 hundreds = 42 hundreds, or 2 hundreds and 4 thousands, and these we write as the hundreds and thousands of the product. This is the method generally used. PM OBLEMS. 18. A druggist sold 8 gallons of kerosene, at 85 cents a gal- lon. How much did he receive for it ? 680 cents, 19. The distance from New York to Washington is 226 miles. How many miles does a man travel, who goes from New York to Washington and back ? JiS2. 20. The factors are 9 and 147. What is the product ? 21. How many gallons in 6 hogsheads, each containing 124 gallons ? 7U' 22. A market gardener bought 5 acres of land, at $635 per acre. How much did the land cost him ? 23. Nine cars of a freight train are loaded with flour, and each car contains 96 barrels. How many barrels of flour on the train ? 86 Jf. 24. How many pounds in 6 barrels of Onondaga salt, each barrel containing 256 pounds ? 25. How much will 122 pairs of boots cost, at $8 a pair ? 52 INTEGERS. 51* In the picture on page 45 we see that 7 times 3 blocks are the same as 3 times 7 blocks ; or that the product is the same, whichever of the two numbers is taken for the multipUcand. In problem 25, $8 is the true multiplicand, because, if one pair of boots costs $8, 122 pairs will cost 122 times $8 ; but since 122 times 8 is the same as 8 times 122, in solving the problem we may place 122 as the multiplicand, and use 8 as the multipHer. 52* General Principles of Multiplication. I. In the solution of problems, either factor may he used as the multiplicand. n. The true multiplicand is that factor which would be used in solving the problem by Addition. m. The multiplicand may be either an abstract or a concrete number. rV. The multiplier must always be an abstract number. Y. The product is always of the same kind as the true multiplicand. (See Manual, page 216.) JPMOBIjJEMS. 26. Thomas attended public school 76 days, and his tuition was 3 cents a day. How much was his school bill ? 228 cts. 37. In one week a n.ewsboy sold 246 papers, at 5 cents each. How much did he receive for them ? 28. A man's family expenses are $4 a day. How much are they for a year, or 365 days ? $1,460. 29. How much will 2,755 army blankets cost, at $2 apiece ? 30. How many pounds in 6 bales of cotton, each weighing 478 pounds? 2,868. 31. What will be the cost of building a horse railroad 4 miles long, at $12,678 a mile ? $50 J 12. 32. How many gallons are 8 times 27,645 gallons ? 33. How many pounds are 5 times 32,051 pounds ? 34. What is the product of 6 times 1,026,348 ? 6, 158,088, MULTIPLICATION. 53 C^SE II. The Multiplier any number of Tens, Hundreds, Thou- sands, and so on. 53. Ex. Multiply 254 by 10. Explanation. — We write the numbers as soltition. shown in the solution, and multiply each 254 figure of the multiplicand by the multiplier, 10 as in Case I. 10 x 4 ones = 40 ones ; 10 x 5 2540 tens =50 tens, and 50 tens + 4 tens = 54 tens ; 10 X 2 hundreds = 20 hundreds, and 20 hundreds + 5 hundreds = 25 hundreds. The figures of the product are the same as those of the multiplicand, with a cipher on the right. Hence, 54* Annexing a cipher to any number multiplies it by 10. 55* Annexing a second cipher multiplies by 10 again ; that is, annexing two ciphers to any number multiplies it by 10 times 10, or 100. 56t Annexing three ciphers to a number multiplies it by 10 times 100, or 1,000. 57. Annexing four ciphers multiplies by 10,000 ; annex- ing five ciphers, by 100,000 ; and annexing six ciphers, by 1,000,000. 35. How much will 10 barrels of mess pork cost, at |23 a barrel? $230. 36. In constructing a telegraph line 100 miles long, how many poles will be required, allowing 16 poles to the mile ? 37. How many panes of 8 by 12 glass in 1,000 boxes, each box containing 75 panes ? 75, 000. 38. In one barrel of flour there are 196 pounds. How many pounds in 10,000 barrels ? 1, 960, 000. 39. Multiply 5,675 yards by 100,000. 567,500,000 yards. 40. What is the product of 393 pounds multiplied by 1,000,000 ? 54 INTEGERS. 10 SECOND SOLUTION. 58. Ex. Multiply 254 by 30. Explanation. — 30, or 3 tens = 3 times ^^^^ solution. 10 ones, or 10 times 3 ones. Hence, 30 ^^^ times 254 are 10 times 3 times 254. We - may therefore multiply by 3, as in Case ^^^^ I., and the product thus obtained by 10. The final result, 7,620, is 10 times 3 times 7620 254, or 30 times 254. In the second solution, after multiply- 254 ing 254 by 3, we have annexed a cipher to 30 the result. rjQ^Q 59. To multiply by 300, we multiply by 3, and annex two ciphers to the product ; to multiply by 3,000, we mul- tiply by 3, and annex three ciphers, and so on. PUOBI.EMS, 41. At $65 a hogshead, how much will 300 hogsheads of molasses cost ? $19,500. 43. How many bushels of oats- will be required to keep 800 cavalry horses a year, if 187 bushels are required for each horse ? 43. How many sheets of paper in 3,000 copies of Worcester's Dictionary, there being 113 sheets in each copy ? 336,000. 44. A man bought a farm of 70 acres, at $135 an acre. How much did the farm cost him ? $8, 750. 45. How many pounds of beef in 814 barrels, each barrel containing 300 pounds ? J^2,800. 46. A railroad company bought 9,000 cords of wood, at $5 a cord. How much did the wood cost them ? 47. A vessel at New York took on a cargo of 7,000 barrels of kerosene. If each barrel contained 43 gallons, how many gallons of kerosene in the cargo ? 301, 000. 48. A wholesale grocer bought 500 chests of tea, each con- taining 56 pounds. How many pounds did all the chests con- tain ? 28,000. MULTIPLICATION 55 49. In one hour there are 60 minutes. How many minutes in 1 day, or 24 hours ? ljJi40. 50. How many pounds of cotton in 40,000 bales, each weighing 394 pounds ? 15, 760, 000. 51. An Illinois farmer had 80 acres of com, which yielded 94 bushels an acre. How many bushels had he in the whole crop? 7,520. 53. Multiply 349 by 4,000,000. 996,000,000. 53. What is the product of 600,000 times 972 ? 54. The factors are 90,000 and 3,165. What is the product ? C^SE III. The Multiplier more than One Figure. FIRST METHOD. Multiply 563 by 34. FIB8T SOLUTION. Multiplying ly 4. 563 4 2252 Multiplying by 30. 563 30 16890 Adding partial products. 2252 16890 19142 60. Ex. Explanation. — ^As 34 consists of 4 ones and 3 tens, or of 4 and 30 ; and as we can not multiply 563 by the whole 34 at once, we first multiply it by 4 and then by 30, and after- ward add the two results or partial products. 19,142, the result thus obtained, is the sum of 4 x 563 plus 30 X 563, or 34 x 563. In the First Solution, each of the second solittion. three steps stands by itself; in the Second Solution they are placed to- gether. In multiplying by 3 tens, or 30, we may first write a cipher in ones' place, and then to the left of it write the product obtained by multiplying by 3. solve the following problems. 563 34 2252] Partial 16890 ) P^«•■ j i -' i I ! 11 ...J-.. — .-J..J.J...JJ...J... i i M i 1 — j— .-- i i i 1 i r ■ _ 1 ' 1 I 1 ; io n e|T|e!njt ihl 96 DECIMALS. "When 1 tenth of any thing or number is divided into 10 equal parts, each part will be 1 hundreth of the whole thing or number. 114. Since the value of a figure in any place is 1 tenth of the value of a Hke figure in the next place at ihe left, a figure written at the right of tenths must be Jiundreths. Thus, .11 is one tenth and 1 hundredth. .83 is 8 tenths and 3 hundredths. 3.27 is 3 ones, 3 tenths, and 7 hundredths. .05 is tenths and 5 hundredths. 115. .35 is 3 tenths and 5 hundredths ; but 3 tenths = 30 hundredths, and 30 hundredths + 5 hundredths = 35 hundredths. Tenths and hundredths are read together as hundredths. .52 is 5 tenths and 2 hundredths, and is read 52 hun- dredths. .96 is 96 hundredths. .03 is 3 hundredths. 5.24 is 5 and 24 hundredths. 298.05 is 298 and 5 hundredths. EXEJtCISES, 9. Read 11.18, 10.24, 81.6. 10. Read 40.93, 128.52, 50.07. 11. Read 7.08, 217.01, 3000.02. 12. Write 7 tenths and 3 hundredths, or 73 hundredths. 13. Write 5 tenths and 1 hundredth, or 51 hundredths. 14. Write 27 hundredths ; 3 hundredths. 15. Write 4 and 15 hundredths ; 4 and 5 hundredths. 16. Write 800 and 21 hundredths. 17. Write 18000 and 1 hundredth. 116. A figure at the right of hundredths is thou- sandths ; and tenths, hundredths, and thousandths are read together as thousandths. NOTATION AND NUMERATION. 97 .456 is 4 tenths, 5 hundredths, and 6 thousandths, and is read 456 thousandths. .209 is 2 tenths, hundredths, and 9 thousandths, and is read 209 thousandths. .063 is read 63 thousandths. .004 is read 4 thousandths. 3.528 is read 3 and 528 thousandths. 80.082 is read 80 and 82 thousandths. 117. Numbers expressed by ones, tens, hundreds, etc., are Integers or Whole Numbers. 118. Numbers expressed by tenths, hundredths, thousandths, etc., are Decimals. 119. A number consisting of an integer and a deci- mal is a Mixed Number. 120. In writing decimals and mixed numbers the decimal point must always be used. exeh CIS es. 18. Read .275, 7.463, 32.416. 19. Read 86285.419, .507, 700.256. 20. Read 11.092, .048, .002. 21. Read 214.005, .001, 217.908. 22. "Write 5 and 376 thousandths. 23. Write 3250 and 615 thousandths. 24. "Write 43 thousandths ; 81 thousandths. 25. "Write 87 and 87 thousandths. 26. Write 401 thousandths ; 7 thousandths. 27. Write 9000 and 9 thousandths. 28. Write 18 and 305 thousandths. 29. Write 101000 and 101 thousandths. 121. A figure at the right of thousandths is ten-thou- mndths; and a decimal containing tenths, hundredths, thousandths, and ten-thousandths is read as ten-thou- sandths. .2574 is 2 tenths, 5 hundredths, 7 thousandths, and 4 ten- thousandths, and is read 2574 ten-thousandths. 98 DECIMALS. .0452 is 452 ten- thousandths. 6.0048 is 6 and 48 ten-thousandths, 59.0006 is 59 and 6 ten-thousandths. 6000.3001 is 6000 and 3001 ten- thousandths. 204.0809 is 204 and 809 ten-thousandths. 122. A figure at the right of ten-thousandths is hundred-thousandths. Thus .32516 is 3 tenths, 2 hun- dredths, 5 thousandths, 1 ten-thousandth, and 6 hun- dred-thousandths, and is read 32516 hundred-thou- sandths. A figure at the right of hundred-thousandths is millionths. Thus .259361 is 259361 mmionths. A figure at the right of millionths is ten-millionthSy and a figure at the right of ten-miUionths is hundred- millionths. .63015 is 63015 hundred-thousandths. .40003 is 40003 hundred -thousandths. .029304 is 29304 millionths. .000007 is 7 millionths. .2367592 is 2367592 ten-millionths. .0000024 is 24 ten-millionths. .59642108 is 59642108 hundred-millionths. .00000009 is 9 hundred-millionths. >* TABLE OF VALUES OF DECIMAL NUMBERS. One decimal figure expresses tenths. Two decimal figures express liundredtJis. Three u " thousandths. Four u " ten-thousandths. Five u " hundred-thousandths. Six (( " millionths. Seven u " ten-millionths. Eight u u " hundred-millionths. (See Manual, page 218.) NOTATION AND NUMERATION. 99 124. Figures standing in places at equal distances to the right and left of ones have naines that correspond to each other, as shown in the following DIAGRAM OF DECIMAL NOTATION. 765432 1. 234567 8 125( The place which any decimal figure occupies in a number determines the value expressed by it in that number. 126. The relative values of the different places in decimals are shown by the following DECIMAL NOTATION AND NUMERATION TABLE 1 one 1 tenth 1 hundredth 1 thousandth 1 ten-thousandth 1 hundred-thousandth 1 millionth 1 ten-millionth 10 tenths 10 himdredths 10 thousandths 10 ten-thousandths 10 hundred-thousandths 10 millionths 10 ten-milHonths 10 hundred-millionths is 10 tenths. " 10 hundredths. " 10 thousandths. " 10 ten-thousandths. "10 hundred-thousandths. " 10 millionths. " 10 ten-millionths. " 10 hundred-millionths. are 1 one. " 1 tenth. ". 1 hundredth. " 1 thousandth. 1 ten-thousandth. 1 hundred-thousandth. 1 millionth. 1 ten-millionth. (See Manual, page 218.) 100 DECIMALS. 127. Since the value of every figure in a decimal is determined by the place it occupies, and since ciphers on the right of a decimal do not change the places of the other figures in the number, it follows that I. Ciphers may he annexed to any decimal, or decimal ciphers to any integer, without changing its value ; and II. Ciphers may be omitted from the right of any deci- mal, or decimal ciphers from the right of any integer, with- out changing its value. (See Manual, page 218.) JEXEMCISES. „^ ^ , . (See Manual, page 218.) 30. Read .523, 4.376, .009, .027, .209. 31. Read 9.018, 100.003, 435.125. 32. Read .2987, 4.0232, 18.0901, .1805, 14.0029. 33. Read 365.0007, 60.1273, 400.5017. 34. Read .7025, .7005, .0005, 30.6008. 35. Read .24731, .09671, .10006, .00008. 36. Read .00055, .70438, 8.52804. 37. Read .90052, 250.46031, 349.30116, 1000.20084. 38. Read .256153, 709.400365, .4366576. 39. Read .1413948, 32876.2850041, 25.530016. 40. Read 217.1800624, 59.00654387, 3054.26405746. 41. Read 32957251.5283563, 172000650.50040036. 42. Write sixteen ten-thousandths. 43. Write five hundred seventeen and three thousand six hundred forty-seven ten-thousandths. 44. Write thirty-six thousand two hundred seventy-three hundred-thousandths. 45. Write fourteen hundred-thousandths. 46. Write five thousand eighteen hundred-thousandths. 47. Write two hundred seventeen thousand four hundred fifty-six millionths. 48. Write six hundred thousand two hundred eighty-four millionths. 49. Write one hundred ninety-three millionths. ADDITION. 101 50. "Write sixteen million three hundred fifty-eight thousand seven hundred twenty-four ten-millionths. 51. Write forty-six million two hundred seventy-four thou- sand five hundred eight hundred-millionths. 52. Write 78,319 ten-millionths. 53. Write 6 and 49 hundi'ed-millionths. 54. Write 106,204 hundred-millionths. 55. Write 7,017 and 4 millionths. 56. Write 4 and 68,001 ten-millionths. 57. Write 44 and 44 hundred-thousandths. 58. Write 975 million 206 thousand 410 and 3 tenths. 59. Write 50 million and 50,512 hundred-millionths. 60. Write 101 million 101 thousand 101 and 1,001,001 hun- dred-millionths. SECTION II. A. S) ^ I T I O JV. 128. Ex. What is the sum of 45.75, 29.36, and 442? Explanation. — ^We write tens under tens, soLirrioN, ones under ones, tenths under tenths, and 45.75 hundredths under hundredths. The decimal 29.36 points then stand in a column. We commence ^-^^ at the right, and add as in integers. The sum 79.53 of the column of hundredths is hundredths, and the sum of the column of tenths is tenths. We must therefore place the decimal point at the left of the 5 in the sum. (See Manual, page 219.) 129. The decimal point in the sum must always he placed directly below the decimal points in the parts. 102 DECIMALS. JPMOB LEMS. 1. A farmer had 98.7 acres of land, and bought 15.5 acres more. How many acres had he then ? IIJ^.2. 2. A lady bought two carpets, the first contaming 27.5 yards, and the second 23.5 yards. How many yards of carpeting did she buy ? 51 yards. 3. A farmer who had three meadows cut from the first 18.68 tons of hay, from the second 15.27 tons, and from the third 13.54 tons. How much hay did he cut from the three meadows ? 4. A silversmith used 3.65 ounces of silver in making a set of tea-spoons, 8.72 ounces for a set of table-spoons, and 11.63 ounces for a set of forks. How much silver did he use for all ? , ^Jf. ounces. 5. A gentleman has three village lots, one of which contains .8 of an acre, another .5, and the third 3.4 acres. How many acres has he in the three lots ? 6. A housekeeper burned .728 of a ton of coal in December, .835 of a ton in January, .694 of a ton in February, and .532 of a ton in March. How many tons did she burn in the four months ? 2.789. 7. At a Pennsylvania iron mine 216.845 tons of iron were mined on Monday, 204.376 tons on Tuesday, 198.275 tons on Wednesday, 220.615 tons on Thursday, 187.945 tons on Friday, and 206.004 tons on Saturday. How many tons were mined in the week ? 1234.06. 8. A wood dealer has on hand 57.75 cords of hickory wood, 139.75 cords of maple wood, 67.65 cords of beech wood, and 78.26 cords of hemlock wood. How many cords of wood has he ? 34S.4I. 9. A farmer has four meadows. The first contains 10.15 acres, the second 9.76 acres, the third 12.25 acres, and the fourth 7.82 acres. How much land has he in the four meadows ? S9.98 acres. 10. The cargo of a coal barge consisted of 45.75 tons of chestnut coal, 69.54 tons of stove coal, 36.94 tons of egg coal, and 51.25 tons of lump coal. What was the whole amount of coal in the cargo ? 203 4S tons. ADDITION. 103 11. A man traveled 125.75 miles by stage, 313.75 miles by railroad, and 89.45 miles by steamboat. How many miles did lie travel in all ? 428.95. 13. A tavern keeper bouglit four loads of hay, the first con- taining 1.156 tons, the second 1.328 tons, the third .987 tons, and the fourth 1.048 tons. How much hay did he buy ? 13. What is the sum of 1.325 + .865 + .655 ? S.745. 130. Ex. AMiat is the sum of 7.4675, 836.5, 85.275, and 973 ? Explanation. — After writing the num- solution. . bers with the decimal points in a column, 7.4675 thus bringing ones under ones, tenths q^ 97^0 under tenths, etc., we annex ciphers to 973 0000 the second, third and fourth numbers ■ ' until each contains as many decimal fig- 1902.^4^5 ures as the first number. (See 127.) We then add as in integers, and place the decimal point in the sum directly under the decimal points in the parts. JPU OBLJEMS. 14. A tailor bought two pieces of cassimere, one containing 08.5 yards, and the other 41.35 yards. How many yards of cloth in both pieces ? 79.75. 15. One hour a railroad train ran 33.6 miles, the next hour 38.83 miles, and the third hour 33.37 miles. How far did it mn in the three hours ? 7^.69 miles. 16. In four successive weeks an ice dealer sold 18.363 tons, 15 967 tons, 17.4 tons, and 16.48 tons of ice. How much ice •lid he sell in the four weeks ? 68.11 tons. 17. How many tons of straw are there in five stacks, the first of which contains 3.7 tons, the second 4.1375 tons, the third 3.875 tons, the fourth 5.3 tons, and the fifth 7.65 tons ? 23.5525. 18. A fruit dealer bought five lots of cranberries. The first lot contained 5.75 bushels, the second lot 8.5 bushels, the third lot 6.635 bushels, the fourth lot 9 bushels, and the fifth lot 4.35 bushels. How many bushels were there in the five lots ? 104 DECIMALS. 131« llute for Addition of decimals. I. Write the numbers so that the decimal points shall stand in a column, II. Add in the same manner as in integers^ and place the decimal point in the sum directly under the decimal points in the parts. PJROBLEMS. . 19. What is the sum of 5.0084 + 641.385 + 9.00843 + 21.000001 + 5.064 ? 681465831. 20. A merchant sold 1.75 bushels of clover seed to one farmer, 3 bushels to another, 2.5 bushels to a third, 4.225 bushels to a fourth, and 3.25 bushels to a fifth. How many bushels did he sell to the five farmers ? U.725. 21. What is the sum of five and five tenths, five and five hundredths, eight and seventy-five thousandths, twenty-one and three thousand six ten-thousandths, and five thousand nine and six hundred forty thousand seventeen millionths ? 50J^9.565617. 22. A gardener sold 4.5 bushels of beans to one grocer, 7 bushels to another, 3.25 bushels to a third, 1.625 bushels to a fourth, and 2.125 bushels to a fifth. How many bushels did he sell to all ? 18.5. 23. What is the sum of three hundred and three hundredths, one thousand seven and two hundred thousand six millionths, one hundred seventeen thousand seven hundred nine and six hundred four ten-thousandths, and eight and fifty-two mil- lionths ? 1190^.290458. (24) (25) (26) (27) 47.25 .967 .125 8000.1 5.00695 .00054 1.25 96.2006 193.9 953.5 12.5 504.40307 5.876 7.375 .0125 2046.25 94.376964 1000.0001 .00125 9.0004 9.00005 6.75 7.3 28.4 290.050063 8.80808 .0827 167.283 483 .000006 5.5008 SUBTKACTION. 105 SECTION III. S ZrS T^A. C TIOJV, 132. Ex. From 38.25 subtract 16.78. Explanation. — ^We write the numbers so solution. that the decimal point of the subtrahend shall ?5*Ho stand directly under that of the minuend, and — 'J— subtract as in integers. Since hundredths 21.47 subtracted from hundredths leave hundredths, and tenths subtracted from tenths leave tenths, there are hundreths and tenths in the remainder ; we there- fore place the decimal point before the 4. 133* The decimal point in the remainder must always he directly under the decimM point in the minuend and subtrahend. PMOBIiEMS. 1. A merchant sold 16.7 yards of calico from a piece that contained 33.4 yards. How many yards remained in the piece ? 16.7. 2. A farmer raised 33.25 bushels of clover seed, and sold 24.75 bushels. How many bushels had he left ? 7.5. 3. A druggist bought a cask of wine containing 36.5 gal- lons. After selling 19.5 gallons, how much remained in the cask ? 17 gallons. 4. At night the ice on the river was 7.37 inches thick, and in the morning it was 9.03 inches thick. How much ice had formed during the night ? 1.66 inches thick. 5. A wood dealer bought 398.65 cords of wood, and sold 276.25 cords. How many cords remained unsold ? 122.^. 6. Of a railroad, which is to be 156.325 miles long when finished, 83.875 miles are built. How many miles remain unfinished ? 72.J^5. 7. Mr. West's farm is 116.36 rods in length and 29.28 rods less in width. What is its width ? 87.08 rods. I 106 DECIMALS. 134. Ex. From 130.5 subtract 93.1875. Explanation. — Since cipliers may be an- solution. nexed to any decimal number without 130.5000 changing its value, (see 127), we annex 93.18 75 ciphers to the minuend until it contains as 37.3125 many decimal figures as the subtrahend, and then subtract, and place the decimal point as be- fore explained. PM OBIjJEMS. 8. From a cask containing 31.5 gallons of vinegar 20.75 gallons were drawn. How many gallons remained in the cask ? 10.75. 9. A man put into his wood-house 24.5 cords of wood for his year's supply, and at the end of the year he had 2.875 cords left. How many cords had he used ? 21.625. 10. Two men built 134 rods of stone fence, one of them building 65.87 rods. How many rods did the other build ? 11. From a hogshead of molasses containing 135.5 gallons, 1.175 gallons leaked out. How many gallons were then in the hogshead ? ^ 134.325. 12. A farmer having 217.625 acres of land, sold 87.0375 acres. How much land had he left ? 130.5875 acres. 135. ^tcle for Subtraction of decimals, I. Write the numbers with the decimal point of the sub- trahend directly under that of the minuend. n. Subtract in the same manner as in integers, and place the decimal point in the remainder directly under the decimal point in the subtrahend. PMOJil^ EMS. 13. From a box that contains 10 pounds of indigo a grocer sold 7 0625 pounds. How many pounds remained in the box? ^'^^'^• 14. From ninety-five and forty-four thousandths take ten and eight thousand five ten-thousandths. SUBTRACTION. 107 15. A liberty pole 83.5 feet long was set in the ground 8.75 feet. How many feet from the ground to the top of the pole? 7Jt.75 16. A cubic inch of silver weighs 6.061 ounces, and a cubic inch of marble 1.641 ounces. How much heavier is the silver than the marble ? 44^ ounces. 17. 923.85 miles — 385.275 miles = how many miles ? 18. A silver dollar which weighs 412.5 grains, contains 41.25 grains of copper. How much pure silver does it con- tain ? 371.25 grains. 19. A lady used 19 yards of silk in making dresses for her two daughters, using 9.875 yards for one dress. How much silk did the other dress contain ? 9.125 yards. 20. 48.2175 acres — 39.5 acres are how many acres? 21. At a certain point near Sandy Hook, N. J., the water at low tide is 5.649 feet deep, and at high tide it is 11.249 feet deep. How much does the tide rise and fall at that point ? 22. The walls of a school room measure 120 square yards, but the openings (windows and doors) measure 17.75 square yards. How many yards for plastering are there on the walls ? 23. The ceiling of the same room measures 51.5 square yards. How many yards less of plastering in the ceiling than in the walls ? 50.75. (34) (25) (26) 87.006 1.03045 20000.2875 9.84 .0009 482.52006- 27. From eight hundred sixty and four hundredths take nmeteen and nine thousand eighty-four hundred-thousandths. Bemainder^ 840.94916. 28. A cubic foot of gold weighs 1203.625 pounds, and a cubic foot of iron 450.4375 pounds. How much more does the gold weigh than the iron ? 753.1875 pounds. 29. A vessel sailed from Boston for Havana with' a cargo of 438.275 tons of ice, but 156.895 tons melted on the voyp- How much of the ice reached Havana ? 281.38 ' 80. How much more ice reached Havana than rr voyage ? 108 DECIMALS. SECTION IV. MUZ TI^ZICA. TIOJV. .3 .9 .03 _3 .09 .003 3 .009 ca.se I, One Factor an Integer. 136t 3 times 3 apples are 9 apples, 3 times 3 ones are 9 ones, and 3 times 3 tens are 9 tens. So, also, 3 times 3 tenths are 9 tenths, 3 times 3 hundredths are 9 hundredths, and and 3 times 3 thousandths are 9 thousandths. (See 52, V.) Ex. 1. Multiply 43.21 by 4. Explanation. — We write the factors and mul- tiply as in integers. Since 4 times 1 hundredth are 4 hundredths, and 4 times 2 tenths are 8 tenths, the 4 in the product is hundredths, and the 8 is tenths. We must therefore place the decimal point before the 8. Ex. 2. Multiply .12815 by 7. Explanation. — Since the multiphcand is hundred-thousandths, 7 times this multipli- cand must also be hundred-thousandths ; and since hundred-thousandths are expressed by five decimal figures, (see 123), there must be five decimal figures in this product. Hence, When the multiplier is an integer , there are as many decimal figures in the product as in the multiplicand. PR OBIjEMS. 1. In one rod there are 16.5 feet. How many feet across a street that is 3 rods wide ? Ji9.5. 2. How many acres in 8 fields, each containing 12.84 acres ? 172.84 SOLUTION. .12815 7 .89705 MULTIPLICATION. 109 3. How many bushels of oats are there in 4 bins, solution of i • • .^/^ «^ 1 1 1 ,. PROBLEM 3. each containing 30.75 bushels ? ^ 4. How many miles will a man travel in 8 days, 4 if he travels 44.635 miles each day ? 357. co^ 5. K .085 of a pound of butter be made from one quart of milk, how much butter can be made from 9 quarts ? .765 of a -pound. 6. A farmer put 8 loads of hay in a stack, and each load weighed .9375 of a ton. How many tons of hay in the stack ? (7) (8) (9) (10) 210.735 634.04 19.125 250.375 9 6 8 6 11. What is the product of 16.24 multiplied by solution of 14 ^ PKOBLEM 11. 12. If a man builds 2.5 rods of stone-wall in one \^ day, how many rods can he build in 18 days ? Ji5. tjtt 13. If a mason can plaster 145.75 square yards ig24 of wall in a day, how many yards can he plaster in ^^„ „^ 26 days? S789.5. 14. A merchant bought 39 pieces of sheeting, each piece containing 40.25 yards. How many yards did he buy ? 15. What is the weight of 47 reairfs of printing-paper, each ream weighing 38.125 pounds ? 1791.875 'pounds. 16. How many rods of ditch will a laborer dig in 64 days, if he digs 7.38 rods each day ? 1^72.32. 17. A jeweler made 85 finger rings, using .0225 of an ounce of gold for each ring. How much gold did he use ? 1.9125 ounces. 18. A farmer sheared 113 sheep, and the fleeces averaged 4.0625 pounds. What was the amount of his wool clip ? 459.0625 pounds. 19. If a gardener can raise 405 bushels of onions on one acre of ground, how many bushels can he raise on 3.276 acres ? 20. How much is 7 million times 7 millionths ? 49. 21. If 94.3 tons of iron are required for one mile of railroad, how many tons will be required for a road 164.35 miles long ? 110 DECIMALS, — U..j....;....i — i — j. — ji — ; — I..... CASE II. Each Factor a Decimal or a Mixed Number. 137. If from a board 1 foot square a part be taken, 7 tenths of a foot long and 5 tenths of a foot wide, the area of the part may be found by multiplying its length by its breadth. By the diagram it will be seen that the square foot contains 10 times 10, or 100 small squares ; and hence, each of these small squares is 1 hundredth of the whole board. The part taken contains 5 times 7, or 35 of these small squares, or 35 hundredths '7 of the whole board. Hence, 7 tenths multiphed _:r by 5 tenths must produce 35 hundredths. .35 138i The product of tenths multiplied by tenths is hun- dredths. Ex. Multiply 24.3 by .3. Explanation. — ^We first multiply as in in- tegers. Then, since the product of tenths multiplied by tenths is hundredths, and since hundredths are expressed by two decimal figures, we place the decimal point before the 2 in the product. PBOB LUJMS. 23. Multiply .8 by .3. . Product, .U, 23. What is the product of .9 multiplied by .7 ? 24. What is the product of 16.7 multipUed by .5 ? 8.35. 25. Multiply 38.3 by 8. Product, 30.64. 26. If a man can chop 2.5 cords of wood in a day, how many cords can he chop in .7 of a day ? 1.75. 27. How many square feet are there in a board 16.5 feet long and .9 of a foot wide ? 14.85. SOLTTTIOIf. 24.3 .3 7.29 MULTIPLICATION, 111 139. If from a cubic foot a part be taken, 7 tenths of a foot long, 4 tenths of a foot wide, and 3 tenths of a foot thick, the capacity of the part may be found by multiplying its length, width, and thickness together. By the first figure it will be seen that the whole cubic foot contains 10 times 10 times 10, or 1,000 small cubes ; and hence each small cube is 1 thousandth of the cubic foot. By the second figure it will be seen that the jpart taken contains .7 times 4 times 3, or 84 small cubes, which are 84 thousandths of the cubic foot. Hence, the product of 7 tenths, 4 tenths, and 3 tenths is 84 thousandths. To obtain the product of three factors, we multiply the product of the first two by the third. The product of 7 tenths and 4 -tenths is 28 hundredths. Then, 28 hundredths multi- plied by 3 tenths must produce 84 thousandths. .7 ^4 .28 .3 .084 140. The product of hundredths multiplied by tenths is thousandths. When tenths are multiplied by tenths, there is 1 decimal figure in each factor and 2 iu the product. When hundredths are multiphed by tenths, there are 2 decimal figures in one factor, 1 in the other, and 3 112 DECIMALS. in the product. In each case there are as many deci- mal figures in the product as in both factors. 141. The product must always contain as solution. many decimal figures as both factors. 44.76 Ex. Multiply 44.76 by 8.23. — '— Explanation. — Since there are four deci- oqko mal figures in both factors, there must be 35808 four decimal figures in the product. qfifi q74.« 142. ^ule for MuUipUcation of decimals, I. Write the numbers, and multiply as in integers. n. Place the decimal point in the product so that it shall contain as many decimal figures as both factors. m OBLEMS. 28. What is the product of .43 multiplied by .4 ? .172. 29. Multiply .84 by .6. Product,, .504. 30. If one ton of iron ore yields .685 of a ton of iron, how many tons of iron will 893.056 tons of ore yield ? 611.7 Jf336. 31. How many tons of broom-corn can be raised from .85 of an acre of land, if 1.8764 tons can be raised from one acre ? l.BOJfOJ^. 32. How many gallons of linseed-oil can be obtained from 249.5 bushels of flaxseed, if 3.15 gallons of oil can be obtained fi-om one bushel of seed ? 785.925. 33. If 3.75 gallons of cider can be made from one bushel of apples, how much cider, can be made from 38.5 bushels ? UJf.375 gallons. 34. If one yard of cassimere can be made from 1.625 pounds of wool, how many pounds of wool will be required for 54.25 yards ? 88.15625. 35. In one square rod there are 272.25 square feet. How many square feet in 108.4 square rods ? 29511.9. 36. If 22.73 pounds of starch be made from one bushel of com, how many pounds can be made from 83.25 bushels ? DIVISION. 113 37. Multiply .0854 by 0.33. Since there are seven decimal figures in both solution of factors, and in the product but five, we must pre- ^g^^ fix two ciphers to the product, and place the * Qg^ decimal point before the first one. v^ 38. Multiply .084 by .07. Product, .00588. 2552 39. What is the product of .00393 multiplied 0027328 by .006? .00002358. 40. What is the product of .057 and .00049 ? .00002793. 41. Multiply .06052 by .066. Product, SECTION V. ^irisiojsr. CA.SE I The Divisor an Integer. 143. One fourth, of 8 apples is 2 apples, one fourth of 8 feet is 2 feet, one fourth of 8 ones is 2 ones, and one fourth of 8 tens is is 2 tens. So also one ^ [4 ^08 [4 ^008 [4 fourth of 8 tenths is 2 .2 .02 .002 tenths, one fourth of 8 hundredths is 2 hundredths, one fourth of 8 thou- sandths is 2 thousandths, etc. Ex. 1. Divide .9275 by 7. Explanation. — ^We write the numbers solution. and divide as in integers. Since one j??I? I ^ seventh of 9 tenths is 1 tenth with a re- .1325 mainder, the first quotient figure 1 is tenths, and the decimal point must therefore be placed before it. 114 DECIMALS. Ex. 2. Divide 1047.15 by 45. Explanation. — Di- partial solutiox, viding 1047 by 45 we 10^7.15 | 45 obtain a quotient of 23 -^- [ 23 and a remainder of 12, 147 as shown in the Par- 1^^ SOLTJTIOX. 1047.15 I 45 90 147 135 [ 23.27 tial Solution. Since 12 121 the three of the quo- 90 tient is ones, we must place the deci- 325 mal point after it, before writing the 315 next quotient figure. We then con- tinue the division until all the figures of the dividend have been used. 144. When a decimal or a mixed number is divided by an integer, there mill be as many decimal figures in the quotient as in the dividend. PMOBJLEMS. 1. A physician fed .675 of a ton of hay to his horse in 5 weeks. How much hay did he feed each week ? .135 of a ton. 3. If .804 of a ton of coal will last a family 6 weeks, how much do they bum in a week ? .13 J}, of a tan. 3. A bell-founder cast 8 bells of equal size, and they weighed together .984 of a ton. What was the weight of each bell ? .123 of a ton. 4. A lady put 16.24 pounds of grapes into 8 fruit cans. How many pounds did she put into each can ? 2.03. 5. A father divided 217.5 acres of land equally among his 5 sons. How many acres did each son receive ? Jf3.5. 6. If 2.1875 barrels of flour will last a family 7 months, how much flour will they use in one month ? .3125 of larrel. 145. Ex.1. Divide 4 by 8. Explanation. — Since 8 is not contained in 4. n I « 4, we annex a decimal cipher to the four ones — ^ before dividing. (See 127.) -^ DIVISION. 115 Ex. 2. How many times is 32 contained in 164 ? Explanation. — The quotient of 164 solution. divided by 32 is 5 with a remainder of 1^^ 4, and since the quotient figure 5 is [ 5.125 ones, we place the decimal point at the 4000 right of it. We then annex decimal ^^ ciphers to the remainder 4 ones (see 80 127), and continue the division until ^^ there is no remainder. 160 160 PROBLEMS. 7. A silver-ware manufacturer made 25 sets of table-spoons which weighed 331 ounces. How much did one set weigh ? 9.2Ji, ounces. 8. An Iowa farmer raised 3045 bushels of corn from 56 acres. How much was the yield to the acre ? 54.375 Imshds. 9. If a man can chop 44.635 cords of wood in 31 days, how many cords can he chop in one day ? 2.125. 146. Ex. What is the quotient of .099 di\ided by 36 ? Explanation. — Since one thirty-sixth solution of 99 thousandths is 2 thousandths with a remainder, we write the 2 m the quotient as thousandths by pre- fixing two decimal ciphers, and then continue the division until there is no 180 remainder. - °J; PMOBLEMS. 10. The dividend is .897 and the divisor 39. What is the quotient ? .023. 11. If .7505 of an ounce of gold-leaf will cover 79 square feet, how much gold-leaf will be required to gild one square foot ? .0095 of an ounce. 13. A manufacturer put up 33 gallons of lemon extract in 736 bottles. How much did eacl? bottle contain ? .099 36 V2 270 252 . .00275 116 DECIMALS. 13. A dairyman made 7.2 pounds of butter from 128 quarts of milk. How much butter was that from one quart of milk ? 14. Divide .7 by 112. Quotient, .00625. 15. If one bushel of Onondaga salt can be made from 35 gallons of brine, how many bushels can be made from 618.625 gallons ? 17.675. C^SE II. The Divisor 147. The quo- tient of 15 divid- ed by 5 is 3. If we multiply both dividend and di- visor by 4, and divide the new dividend by the new divisor, the quotient will be 3, the same as before. If we multiply both terms by 23 and divide, we obtain the same quotient. 148« ^ the dividend and divisor are both multiplied by the same number, the quotient remains unchanged. Ex. 1. Divide 91 by .7. soltttion. _, _,_ _ Dividend. 91 I .7 Divisor. Explanation. — ^We mul- iq iq tiply both terms by 10, -— — to make the new divisor New Dividend. 910 | 7 New Divisor. a Decimal or a Mixed Number. 15 [5 Divisor. 15 Dividend. 3 Quotient. 23 45 5 Divisor. 15 5 Divisor. 30 23 4 4 345 115NewDiv 60 20 New Divisor. 345 3 Quotieni 60 3 Quotient. a whole number, and di- vide as in integers. Ex. 2. Divide 32.5 by .13. Explanation. — We multi- ply both terms by 100, to make the new divisor a whole number, and divide as in in- tegers. 130 Quotient. SOLUTION. 32.5 [ .13 Divisor. 100 100 3250 26 65 65 13 New Divisor. 250 Quotient. DIVISION. 117 Ex. 3. Divide 6.95835 by .423. Explanation ^We mul- tiply both terms by 1,000 by removing the deci- mal point three places to the right, and divide as in Case I. Ex. 4. Divide .27 by 4.32. Explanation. — We multi- ply both terms by 100 by removing the decimal point two places to the right, and divide as in Case I. 80LTTTI0N. 6.95835 [ .423 Divisor. 6958.35 423 2728 2538 1903 1692 423 New Divisor. • 16.45 Quotient. 2115 2115 .27 [ 4.32 Divisor. 27.00 I 432 New Divisor. 2592 [ -^ .0625 Quotient. 1080 864 From these examples we 2160 see that when the divisor 2160 contains one decimal figure, we multiply both terms by 10 ; when it contains two decimal figures, by 100 ; when three decimal figures, by 1,000, and so on. That is, 149i Before dividing, both terms must be multiplied by a number composed of 1 with as many ciphers annexed as the divisor contains decimal figures. 16. If a family use .75 of a barrel of flour each month, how long will 9 barrels last them ? 12 months. 17. A farmer carried 15 bushels of wheat to mill, and re- ceived one sack of flour for every 1.25 bushels. How many sacks of flour did he get ? 12. 18. If 2.125 yards of linen can be made from .85 of a pound of flax, how much flax will be required for one yar^ of linen ? .^ofa pouTid. 118 DECIMALS, 19. If a woman can weave 6.35 yards of rag carpet in one day, how long will it take her to weave 40 yards ? 64 days. 20. At how many loads can a teamster draw 780.7 cubic yards of gravel, if he draws .925 of a yard at each load? 84J^. 21. A druggist put up .875 of gallon of sweet-oil in bottles, each containing .0625 of a gallon. How many bottles did he fill? U. 22. A merchant tailor sold 21 yards of silk, in vest patterns of .875 of a yard each. How many patterns did he sell ? 2Jf. 23. If a farm hand can plow 13.5 acres of land in one week, how long will it take him to plow 47.25 acres ? 3.5 weeks. 24. What is the length of a lane which contains 21.12 square rods and is .8 of a rod wide ? 26.J!i, rods. 25. What is the quotient of 32.625 divided by 43.5 ? .75. 26. If a miller makes one barrel of flour from 4.5 bushels of wheat, how many barrels will he make from 49.5 bushels ? 27. A lady put up 58 quarts of strawberries in cans, putting 1.8125 quarts in each. How many cans did she fill ? 82. 28. If one suit of clothes can be made from 5.75 yards of cloth, how many suits can be made from 109.25 yards ? 19. 150. (Rule for division of S)eci77iats* I. When the divisor is an integer. 1. If necessary, annex decimal ciphers to the dividend till the figures of the dividend will contain the divisor, 2. Divide as in whole numbers. 3. Place the decimal point in the quotient so that it shall contain as many decimal figures as the dividend. II. When the divisor is a decimal or a mixed number. 1. Omit the decimal point from the divisor, and remove the decimal point in the dividend as many places to the right as the original divisor contains decimal figures. 2. Divide and place the decimal point in the quotient as before. DIVISION 119 I^BOBLJEMS. 29. Divide 7 by 43.75. OvMimt, .16, 30. Divide 525 by 9.375. Quotient^ 56. 31. If .625 of a yard of cloth be made from one pound of wool, how many pounds of wool will be required for 12 yards of cloth? 19.2. 32. A farmer cut 639 cords of wood from 11.25 acres of woodland. How many cords was that to the acre ? 56.8. 33. In how many hours can you empty a cistern that con- tains 204 barrels of wafer, if you pump out 18.75 barrels each hour ? 34. If a steamboat runs 156.25 miles in a day, in how long a time will it run 80 miles ? .512 of a day. 35. How much land will be required to raise 21 bushels of corn, if the yield is at the rate of 65.625 bushels per acre ? 36. If 112.59 pounds of maple sugar are made from 625.5 gallons of sap, how much sugar can be made from one gallon of sap ? .18 of a pound. 37. The dividend is 28 and the divisor .64. What is the quotient ? Jf3.75. 38. What is the quotient of 85 -^ .272 ? 312.5. 39. Divide 267.66 by 11.896. Quotient, 22.5. 40. A perfumer put up 73 gallons of cologne in bottles, putting .0625 of a gallon into each. How many bottles did he fill ? 1168. 41. How many cars will be required to transport 80410.5 tons of freight, allowing each car to carry 6.2625 tons ? 12840. 42. How many blocks of marble each weighing .9376 of a ton will together weigh 15.9392 tons ? 17. 43. If one gallon of alcohol can be made from .38 of a bushel of corn, how many gallons can be made from 15.39 bushels ? 40.5. 4:4:. In a fence .9 of a mile long how many lengths are there, each length being .00225 of a mile long? 4OO. 45. If .0196 of a cord of wood will make one bushel of char- coal, how many bushels will .833 of a cord make ? 4^.5. 120 DECIMALS. CASE III. True Remainders. 151. Ex. How many barrels each holding 3.5 bushels can be filled from 237 bushels of apples, and how many apples will be left ? Explanation. — Since the divisor solution. contains one decimal figure, we mul- ^^ ' ^-^ tiply both terms by 10 before com- ^370 I 35 mencing to divide. But since 2,370, ^-^^ I 67 barrels. the dividend used, is 10 times as 270 great as the given dividend, the ^^ remainder 25, which is a part of 25 Remainder, this 2,370, is ten times as ffreat as fT^^ . , . mi o -(5. True remainder. the true remainder. Thereiore, to find the true remainder, we divide the 25 by 10, which we do by placing a decimal point before the 5. Hence, 67 barrels and 2.5 bushels over is the result required. 152. When the quotient is an integer, the true remainder must always contain as many decimal figures as there are in the given term having the more decimal figures. PROBJOEMS, 46. A farmer has 494 gallons of cider. How many barrels can he fill putting 31.5 gallons into each ? 15 f and heme 21.5 gallons left. 47. If 9.5 tons of freight make 1 car load, how many car loads are 124.3 tons ? .1! of a ton more than 13 car loads. 48. A farmer who has 134 bushels of wheat, wishes to ex- change it for sheep. If he gives 3.5 bushels for 1 sheep, how many sheep can he buy ? 55, and haw 1.5 lushels of wheat left. 49. Into how many building lots each containing 1.35 acres can I divide 9.5 acres ? Into 7, and .75 of an acre remaining. 50. A quartermaster has 834.35 pounds of coffee. If he dis- tributes 71.35 pounds to his regiment daily, how many days' rations of coffee has he, and how much over ? 11 days'* rations^ and 50.5 pounds over. UNITED STATES MONEY. 121 SECTION VI. ujsriT^:^ STATBS MOj\rBr. 153* United States Money — also called Federal Money — consists of dollars, cents, and mills. 10 mills are 1 cent. I 1 dollar is 100 cents. 100 cents are 1 dollar. I 1 cent is 10 mills. 154. Since 100 cents are 1 dollar, 1 cent is 1 hun- dredth of a dollar, and is written $.01. And 155. Since 10 mills are 1 cent, 1 mill is 1 tenth of a cent or 1 thousandth of a dollar, and is written $.001. The divisions of a dollar into cents and mills corres- pond to the decimal divisions of a doUar into hun- dredths and thousandths. Hence, 156. Gents may always he written as hundredths, and mills as thousandths, of a dollar. 85 cents are written $.35. 8 cents are written $.08. | 6 mills are written $.006. 10 cents 5 mills are written $.105. 7 dollars 93 cents are written $7.93. 5 dollars 56 cents 8 mills are written $5,568. 20 dollars 30 cents 1 mill are written $20,201. EXEIt CIS ES. 1. Eead $.15, $.60, $318.75, $14.06, $5.94, $8.01. 2. Eead $.255, $.004, $300,567, $12,108, $575.10, $900.25. 3. Write 37 cents, 80 cents, 6 cents. 4. Write 13 dollars 4 cents, 75 dollars 50 cents. 5. Write 8 mills, 15 cents 6 mills. 6. Write 83 dollars 12 cents 5 mills. 7. Write 400 dollars 8 cents 1 mill. 157. Decimal parts of a dollar less than mills or thou- sandths are read as decimals of a mill. $.0006 is 6 tenths of a mill ; $.0085 is 8 and 5 tenths mills. $.2943 is 29 cents 4 and 3 tenths mills. $15.65425 is 15 dollars 65 cents i and 25 hundredths mills. K 122 DECIMALS oom:i>ut.a.xions of xj. s. :m:one;y 158. Ex. 1. What is the sum of $108.50, $10,875, and $.458? Explanation. — We write the numbers with dollars under dollars, cents under cents, and mills under mills ; and then add the parts, and place the decimal point in the sum, as in Addition of Decimals. Ex. 2. From $45.25 subtract $17,625. Explanation. — We write the numbers with dollars under dollars, cents under cents, and mills under mills ; and then subtract, and place the decimal point in the remainder, as in Subtraction of Decimals. SOLUTION. $108.50 10.875 .458 1119.833 SOLUTION. 145.250 17.625 $27,625 Ex. 3. Multiply $8,126 by 2.7. Explanation. — We write the multipHer under the multipHcand ; and then multiply, and place the decimal point in the product, as in Multiphcation of Decimals. i SOLUTION. $8,125 2.7 56875 16250 $21.9375 Ex. 4. Divide $436.72 by 53. Explanation. — We write the divisor at the right of the dividend ; and then divide, and place the decimal point in the quotient, as in Division of De- cimals. Ex. 5. How many times are 127.50 contained in $1168.75? 159. In business, when the mills in any final result are 5 or more, they are regarded as 1 cent, and when less than 5, they are rejected. $436.72 424 127 106 53 $8.24 212 212 SOLUTION, $1 168.75 116875 11000 6875 5500 13750 13750 UNITED STATES MONEY. 123 PJROB IjEMS. Find the sum of the several amounts of money in the first four problems. (1) (3) (3) (4) $121.10 $ 7.28 $.58 $2000 38.47 241.09 .145 5.75 92.86 .42 .0275 48.01 ?82.79 .96 .5625 .495 810.04 44.52 .095 859.17 5. One day a toll-gate keeper received $17.56, and the next day $28.25. How much toll did he receive in the two days 1 6. A furniture dealer sold a wash-stand for $6.50, a bureau for $11.63, and a rocking-chair for $8.25. For how much did he sell all of them ? $26.38. 7. On Saturday night a laborer paid $1,625 for flour, $.85 for tea, $.75 for sugar, and $.375 for butter. How much money did he pay out ? $3.60. (8) (9) (10) (11) From $250.35 $.104 $100,000 $1,000 take 187.50 .087 5.875 .065 12. I owe $167.45. If I pay $94.50, how much will I then owe ? $72.95. 13. A lawyer having $256.56 in the bank, drew out $98.75. How much money had he left in the bank ? $157.81. 14. One week a laborer earned $12.50, and expended $8.38. How much of his earnings had he left ? $Ji,.12. (15) (16) (17) (18) Multiply $194.17 $310.75 $50.44 $249.60 by 8 36 7.5 ^ 19. How much will 7 bushels of wheat cost at $1,125 a bushel ? - $7,875. 20. If an ounce of indigo costs $.15, how much will 9 ounces cost ? $1.35. 21. A builder bought 37 thousand feet of pine lumber at $28.25 a thousand. How much did it cost him ? $10^5.25. 124 DECIMALS. 22. How much will 8.5 gallons of kerosene cost at $.625 a gallon ? $5.3125. Find the quotient in problems 23, 24, 25, and 26. (23) (24) (25) (26) $51.75 1 9 $156.06 I 8^ $405.65 I $ 21.35 $7.82 I $.085 27. If 7 pounds of sugar cost $.91, what is the price of a pound ? $.13. 28. At what price per head must I sell 105 sheep, to receive $564,375 for them ? $5,375. 29. A grocer paid $35 for a barrel of sugar at $.125 a pound. How many pounds did he buy ? 280. 30. A chair maker received $172.50 for chairs at $7.50 a set. How many sets did he sell ? 23. 31. If 31.5 gallons of linseed-oil cost $59.0625, what is the price of a gallon ? $1,875. 160« ^ule for Computations of ZTnited States JKoney, Write the numbers, and add, subtract, multiply, divide, and place the decimal points in the results, as in Decimals. Pit om^EMS. 32. A man bought a village lot for $325, and after paying $22.63 for taxes, he sold it so as to gain $72.37. For how much did he sell it ? $420. 33. I bought a hat for $4,875, a coat for $28, and a pair of boots for $7.50. How much did they cost me ? $40,375. 34. A farmer sold a jar of butter to a merchant for $10.37, receiving in payment groceries to the amount of $4.63, and the balance in money ? How much money did he receive ? $5.74. 35. How much will 125 pounds of nails cost at $.06 a pound ? $7.50. 36. What will be the cost of .84 of a ton of plaster at $4.25 a ton ? $3.57. REVIEW PROBLEMS. 125 37. At $2.50 a bushel, how many plums can be bought for $1,875? .75ofalushel. 38. How much will 29 rolls of wall-paper cost at $.44 a roll ? $12.76. 39. How much muslin can be bought for $24.36, at $.56 a yard ? JfS.B yards. 40. I paid $.78 for a piece of fresh beef at $.12 a pound. How much did it weigh ? 6.5 pounds. 41. A mechanic earned $56.25 in January, $45.63 in Feb- ruary, $67.50 in March, $65,875 in April, and $75 in May. How much did he earn in the five months ? $310,255. 42. Mr. Stevens bought a watch for $32.25, and sold it to Mr. Adams for $27.75. How much did he lose by the trans- action ? $^.50. 43. Mr. Adams afterward sold it for $30,625. How much did he gain ? $2,875. 44. One season a nurseryman sold 2840 young apple-trees at $.375 apiece. How much did he receive for them ? $1065. 45. A stationer paid $114 for pocket-knives at $.95 apiece. How many did he buy ? 120. 46. How much must be paid for transporting .456 of a ton of freight from New York to Toledo by railroad, at $28.60 a ton ? $13.0416. 47. A fruit dealer sold 686 baskets of peaches for $1543.50. What was the price per basket ? $2.25. SECTION VII. TliOSZBMS IJ\r ^£JClMjiZS, 1. A merchant deposited $59.17 in the bank on Monday, $62.86 on Tuesday, $48.12 on Wednesday, $75.48 on Thurs- day, $88.57 on Friday, and $110.72 on Saturday. What was the amount of his deposits for the week ? $Jf.Jf,Jf..92. 2. What vrill be the cost of 15,000 bushels of wheat at $2.0625 a bushel ? $30937.50. 126 DECIMALS. 3. When the price of rice is $.0625 a pound, how many- pounds can be bought for $3.50 ? 56. 4. If wheat is worth $1.4375 per bushel in Chicago, and $2,125 in New York, how much is added to its value by transportation ? $.6875 per bushel. 5. How much will it cost to build 18.4 rods of picket-fence at $3,125 a rod ? $57.50. 6. A fruit dealer paid $14.25 for 95 quarts of strawberries. How much did he pay a quart for them ? $.15. 7. I cut .912 of a ton of hay from my door yard, and the yield was at the rate of 1.92 tons to the acre. How much land is there in my door yard ? .^75 of an acre. 8. When cranberries are worth $5.00 a bushel, what part of a bushel can be bought for $.625 ? .1^5. 9. A shoemaker paid $385.40 for sole-leather, $216.94 for upper-leather, $104.05 for linings, $24.28 for thread and silk, and $12.75 for pegs. How much did he pay out for stock ? 10. What is the quotient of .016 -^ .512 ? .03125. 11. When the dividend is .01 and the divisor 12.8, what is the quotient ? .00078125, 12. Mr. Butler had^ a silver watch worth $18.75, which he exchanged for a gold watch worth $80, paying the balance in money. How much did he pay to boot ? $61.25, 13. If you pay $24 for 7.5 reams of letter-paper, how much does it cost you a ream ? $3.20. 14. How many packages each containing .875 of a pound can be filled from a chest which contains 55 pounds of tea, and how much tea will be left ? 62 packages ; .75 of a pound left. 15. A man on a journey paid $32.17 for railroad fare, $12.44 for steamboat fare, $37.25 for hotel bills, and $7.32 for other expenses. What were his total expenses ? $89.18. 16. How much hemlock bark at $8.25 a pord will pay a biU of $57.75 for boots and shoes ? ' 7 cords. 17. At $4,375 a head, how much will 1000 sheep cost ? 18. A merchant's sales in September were $2174.15, in October $1416.24, in November $1765.93, and in December $2443.76. How much did his sales average per month ? KEVIEW PROBLEMS. 127 19. A man who owed $250, paid at one time $65.48, at another time $47.81, and at another $93.37. How much did he then owe ? $UM' 20. What is the quotient of .315 -=- .3125 ? 1.008. 21. If my expenses for five consecutive weeks are $12.25, $13.61, $14.09, $11.52, and $13.78, how much are my average weekly expenses ? $13.05. 22. A fruit dealer paid $15.45 for oranges, $20.34 for lemons, $27.59 for pine-apples, and $16.72 for cocoa-nuts. How much did the fruit cost him ? $80.10. 23. If I buy goods to the amount of $8.45, and in paying for them give a 10-dollar bill, how much change ought I to receive ? $1.55. 24. A merchant tailor sold a piece of damaged cloth at a loss of $19.88, and the cloth cost him $87.50. For how much did he sell it ? $67.62. 55. A dealer in agricultural implements paid $199.50 for plows, at $7,125 each. How many plows did he buy ? 28. 26. What will be the cost of 32.5 yards of tapestry carpet at $2.75 a yard ? $89,375. 27. If 1 coat can be made from 3.125 yards of cloth, h'ow many can be made from 52.5 yards ? 16, with a remnant of 2.5 yards. 28. A spice dealer put up 280 pounds of ground cinnamon in boxes, each holding .25 of a pound. How many boxes did he fill ? 1120. 29. From a piece of cloth containing 44.5 yards, a tailor made as many suits of clothes, each containing 8.375 yards, as he could. How many yards were left in the piece ? 2.625. 30. How many pounds of metal will it take to cast one church bell weighing 2,765 pounds, and 7 factory bells, each weighing 325 pounds ? 5,0J^0. 31. From a cask which contained 37.175 gallons of alcohol, a druggist drew, at different times, .125 of a gallon, 1.5 gal- lons, .25 of a gallon, 2.75 gallons, .625 of a gallon, .0625 of a gallon, and .75 of a gallon. How many gallons were then left in the cask ? 31.1125. 128 DECIMALS. 32. How much delaine at $.5635 a yard can he bought for $9 ? 16 yards. 33. A butcher paid $58.60 for an ox, and after killing it, he retailed the meat for $59.76, sold the tallow for $4.18, and the hide for $7.88. How much were his profits ? $ 13.22. 34. A provision dealer bought pork at $.125 a pound, and sold it at $.13. How much did he gain a pound ? 35. If a family use .785 of a barrel of flour in one month, .825 of a barrel the next month, .73 of a barrel the third, .8 of a barrel the fourth, and .76 of a barrel the fifth, what is the average amount used per month ? .78 of a Mrrd. 36. A farmer harvested 713.5 bushels of oats from a field of 13 acres, and 576.25 bushels from a field of 9 acres. What was the average yield per acre of the two fields ? 58.625 dushels. 37. 11 miles of a certain railroad cost $13875.30 per mile, 14 miles cost $15251.64 per mile, and 28 miles cost $14588.45 per mile. What was the length of the road, and what the average cost per mile ? Cost per mile, $1^615.62. 38. George Wells bought 5 yards of casimere at $1,875, 1 yard of alpaca for $.875, 13 yards of calico at $.25, and 14 yards of muslin at $.35. What was the cost of the whole ? (See Manual, page 219-) ^ouj,/^ ojf f. B. ^hams S^ (^a, S/a/tei ^oe^, /^«^y//4; /^-^ '(/ tJaymen/, c^ 3. ^c/a^ f^o. CHAPTER III. COMPOUND NUMBERS. SECTION I. J\rOTA.TIOJV :dJV^ "E^D ZfC TIOJV. 161* We find the amount or quantity of articles bought and sold, by measuring or ^Yeighing them. Some articles are sold by the quart or gallon ; some by the peck or bushel ; some by the foot or yard ; some by the acre ; some by the cord, and some by the pound or ton. 162. The names applied to particular amounts or quantities of articles are Denominations ; as the gallon, bushel, foot, mile, pound, and dozen. 163. Numbers applied to denominations are Denomi- nate Numbers ; as 4 yards, 9 bushels. 164. A number expressed in two or more denomina- tions is a Compound Number ; as 4 hours 30 minutes, 3 yards 2 feet 6 inches. A denominate number may be an integer, as 3 bush- els ; a decimal, as .5 of a mile ; a mixed number, as 6.75 tons, or a compound number, as 5 pounds 6 ounces. In writing compound numbers, the denominations are generally abbreviated, as shown in the tables. (See Manual, page 219.) 165. Those denominations in a compound number which express the greater amount are Higher Denomi- nations : and L 130 COMPOUND NUMBERS. 166t Those which express the less amount are Lower Denominations. Thus, a quart is a higher denomina- tion than a pint, and a lower denomination than a gallon. 167. Changing numbers from one denomination to another without changing their value is Reduction. 168. Keducing numbers from higher to lower de- nominations is Eeduction Descending ; and 169. Keducing Numbers from lower to higher de- nominations is Reduction Ascending. 170. Table I,— Liquid Pleasure. The denominations gallons, quarts, pints, and gills constitute Liquid Measure. They are used in measur- ing oil, molasses, wines, milk, and other liquids. 4 gi. (gills) are 1 pt. (pint). 3 pt. " 1 qt. (quart). 4 qt. " 1 gal. (gallon). 1 gal. is 4 qt. 1 qt. " 2 pt. 1 pt. " 4 gi. NOTATION AND REDUCTION. 131 EXBJl CISJES. 1. Read 5 gal. 3 qt. 1 pt. 1 gi. ; 14 gal. 3 qt. 1 pt. 2. Read 11 gal. 1 pi. 2 gi. ; 7 gal. 1 pt. ; 3 qt. 1 gi. 3. Write fisre gallons one quart one pint tvvo gills. 4. Write seventeen gallons two quarts three gills. REDXJCTIOIN" DESCENDING^. SOLTTTION, 171. Ex. 1. How many pints are equal to 3 gaUons ? ^ 9^^- Explanation. — Since 3 gal. are 3 times — 1 gal., and 1 gal. is 4 qt., 3 gal. are 3 times ^ ^^' 4 qt., or 12 qt. ; and since 12 qt. are 12 — times 1 qt., and 1 qt. is 2 pt., 12 qt. are 12 24^^. times 2 pt., or 24 pt. Hence, 3 gal. = Ex. 2. How many pints are equal to 2 gal. 3 qt. 1 pt.? Explanation. — Since 2 gal. are 2 times 1 gal., solution. and 1 gal. is 4 qt, 2 gal. 2 gal. 3 qt. 1 pt. are 2 times 4 qt., or 8 qt., - and 8 qt. + 3 qt. are 11 8 + 3 = 11 qt. qt. Since 11 qt. are 11 _2 times 1 qt., and 1 qt. is 2 22 + 1 = 23 pt. pt., 11 qt. are 11 times 2 pt, or 22 pt., and 22 pt. + Hence, 2 gal. 3 qt. 1 pt. = 23 pt. 1 pt. are 23 pt. PMOBIjEMS. 1. Reduce 11 gallons to quarts. (See Ex. 1.) J^ qt. 2. How many gills are there in 4 quarts ? 32. 3. A wholesale druggist put ten gallons of sweet-oil into bottles which held 1 gill each. How many bottles did he fill ? 320. 4. How many pint bottles will be required to hold 7 gal- lons of currant wine ? 56. 5. In 2 gal. 3. qt. 1 pt. 3 gi., how many gills ? (See Ex. 2.) 95. 132 COMPOUND NUMBERS, 6. A druggist lias 11 gal. 2 qt. of alcohol. How long will it last him, if he sells 1 qt. each day ? JjB days. 7. How many pint bottles will 4 gal. 3 qt. 1 pt. of catchup fill ? 8. A grocer bought a barrel containing 31.5 gallons of vine- gar, which he sold by the quart. How many quarts did he sell ? 126, 9. Reduce 5 gal. 3 qt. 1 pt. 3 gi. to gills. RKJDXJCTlOlSr ASCENDING-. 172. Ex. 1. How many gallons are 48 pints ? Explanation. — Since solution. every 2 pt. are 1 qt., and 2 pt. are contained in 48 pt. 24 times, 48 pt. are 24 qt. And since every 4 qt. are 1 gal., and 4 qt. are contained in 24 qt. 6 times, 24 qt. are 6 gal. Ex. 2. How many gallons in 79 pints ? Explanation. — Since every 2 pt. are 1 qt., and 2 pt. are contained in 79 pt. 39 times with a re- mainder of 1 pt., 79 pt. are 39 qt. 1 pt. And since every 4 qt. are 1 gal., and 4 qt. are contained in 39 qt. 9 times with a remainder of 3 qt., 39 qt. are 9 gal. 3 qt. In the first, or EuU So- lution, we have written all the numbers mentioned in the explanation, both 4^pL ["Ipt. 24 times. MqL [4 qt 6 times. 24 qt. = 6 gal Hence, 4S pt = 6 FULL SOLUTION. Idpt. [2pt. 39 times and 1 pt. rem. 7dpt. = 3dqt. Ipt. ddqt. [4 qt. 9 times and 3 qt. rem. 39 qt. = 9 gal 3 qt. Hence, 79 pt. = 9 gal 3 qt. 1 pt, COMMON SOLUTION. Idpt. [2pt. 39 qt. Ipt [4 qt 9 gal 3 qt Hence, 79 pt = 9 gal 3 qt. 1 pt. NOTATION AND REDUCTION. 133 abstract and concrete ; but in tlie second, or Common Solution, we have omitted the abstract quotients, and written only the denominate numbers. rnoBJj^EMS. 10. How many gallons are 356 quarts ? (See Ex. 1.) 11. How many gallons of cider will it take to fill 104 pint bottles ? 13. 13. Reduce 160 gills to gallons. 5 gal 13. Reduce 140 gills to higher denominations. (See Ex. 3.) 4 gal. 1 qt. 1 pt. 14. A woman buys of a milkman 1 pt. of milk a day. How much does she buy in a year or 365 days ? JjS gal. 2 qt. 1 pt. 15. In one week a grocer sold 190 quart cans of oysters. How many gallons of oysters did he sell ? Ji7.5. 16. Reduce 655 gills to higher denominations. 17. One morning a farm hand, in watering cattle, pumped 879 strokes, and at each stroke the pump discharged 1 pint of water. How much water did he pump ? 109 gal. 3 qt. 1 pt. We have now learned that gallons are reduced to quarts, quarts to pints, and pints to gills, by multiply- ing ; and that gills are reduced to pints, pints to quarts, and quarts to gallons, by dividing. Since the reduction of gallons to gills is from a higher to a lower denomination, and the reduction of gills to gallons is fi'om a lower to a higher denomination, the explanations already given are sufficient to establish the following 173. General Principles of deduction, I. A denominate number is reduced to lower denominor tions by multiplication. n. A denominate number is reduced to higher denomi- nations by division. 134 COMPOUND NUMBERS /^ ^^ ^^fe. /--^ 1' ^ \ /^ ^a^iN Ml ^^ x^ ^MM ^^/-\ )p^ ^ I'm ^R r^^^^^^^F ^«H T^ JKlALF l|| ji^^ ^^^^SsSSj^^BtB^mr sN^HHI Hi'/ '^Hb .^&^i wOsHEd i'^ QiPPsHiH ii^hBIHh^K/'*? ^HHB |f| 10 aHpj ■H^ft' ' ii^Mii[H pusHEiyr liH \\'^' *» T^T^^^^ H^^ -^— "--", ." dticK^ HaMi* iF i J^^SB^BPS*^^^ :£ES==:. — -~-E^ n4i Table II*— Dry Measure, The denominations bushels, pecks, quarts, and pints constitute Dry Measure, They are used in measuring grain, seeds, fruits, berries, several kinds of vegetables, lime, charcoal, and some other articles. In measuring grain, seeds, peas, beans, and small fruits, the measure must be ex^en full. But in measuring large fruits, coarse vegetables, and other bulky articles, the measure must be heaping fall. 4 heaped measures must equal 5 even measures. 2 pt. are 1 qt. 1 bu. is 4 pk. 8 qt. " 1 pk. (peck.) 1 pk. " 8 qt. . 4 pk. " 1 bu. (bushel.) 1 qt. " 2 pt. The quart and pint of Dry Measure are larger than the quart and pint of Liquid Measure. 6 quarts Dry Measure are equal to nearly 7 quarts Liquid Measure. EXEMCISJES. 1. Eead 17 bu. 1 pk. 2 qt. 1 pt. ; 2 pk. 1 pt. 2. Read 19 bu. 5 qt. ; 3 pk. 4 qt. 1 pt. 3. Write one bushel two pecks four quarts one pint. 4. Write five bushels six quarts one pint. 6. Write twenty-eight bushels three pecks. NOTATION AND REDUCTION. 135 175. Ex. 1. Keduce 13 bu. solution. 3 qt. to quarts. 1| ^^- ^ P^' ^ $^- Explanation. — Since there — are no pecks, while there are ^^ P'^' denominations both higher and lower than pecks, we ^1^ + 3 = 419 qt. write a in the place of Hence, 13 Im. 3 qt. = 419 qt. pecks in the Solution. Ex. 2. Eeduce 195 qt. to bu. solution. Explanation. — In reduc- 195 qt. \ 8 qt. ing 24 pecks to bushels, 24 pk. Z qt. [4 pk, we have pecks remaining. ~^ , There are no pecks, there- fore, in the final result. ^ence, 195 qt. = 6lu.3qt. 176. 'Rules for deduction. I. For Reduction Descending. 1. Multiply the highest denomination given by that num- ber of the next lower denomination which equals 1 of this higher, and to the product add the given lower denomination. 2. In the same manner, reduce this result to the next lower denomination ; and so continue until the given num- ber is reduced to the required denomination. n. For Reduction Ascending. 1. Divide the given denomination by that number of this denomination which equals 1 of the next higher, writing the quotient as so many of the higher denomination, and the remainder as so many of the denomination divided. 2. In the same manner, reduce this quotient to the next higher denomination ; and so continue until the given num- ber is reduced to the required denomination. 3. Write the last quotient and the several remainders in their order, from the highest denomination to the lowed, for the required result. 136 COMPOUND NUMBERS jPH OB Jj JEMS. 18. How many quarts are there in 18 bu. 2 pk. 3 qt. ? 595. 19. How long will 3 bu. 3 pk. 4 qt. of com last my liens, if 1 feed them 1 pt. each day ? 18 4 days. 20. How many pint papers of seed-corn are equal to 7 bu. 8 pk. 5 qt. 1 pt. ? 21. Reduce 553 pints to higher denominations. 22. A farmer's boy fed to his colt 1 pint, of oats each day for 150 days. How many oats did he feed ? 21m. 1 ph. 3 qt. 23. A blackberry girl sold 10 quarts of blackberries each day for 18 days. How many berries did she sell ? 5 lu. 2pk. J^ qt. 24. Reduce 275 bu. 7 qt. to quarts. 8,807 quarts. 25. A gardener put 4 bu. 5 qt. of strawberries into quart boxes. How many boxes did he fill ? 133. 26. How many times can a pint measure be filled from 3 bu. 2 pk. 1 pt. of chestnuts ? 225. 27. Reduce 261 quarts to higher denominations. 8 hu. 5 qt. 28. A dealer in garden seeds put up 353 pint papers of marrowfat peas. How many peas did he put up ? 51m. 2 ph Ipt. 29. If 1 peck of clover seed will seed one acre of land, how much land can a farmer seed with 5.5 bushels ? 22 acres. 177t Table III,— Linear Measure, The denominations miles, rods, yards, feet, and inches constitute Linear or Line Measure. They are used in measuring distances, and also the length, width, thick- ness, height, and depth of things. This line i.a.__^_i...ii_ is one inch long. 12 in. (inches) are 1 ft. (foot.) 3 ft. "1 yd. (yard.) 5.5 yd. " 1 rd. (rod.) 320 rd. "1 mi. (mile.) 1 mi. is 320 rd. 1 rd. " 5.5 yd. 1 yd. « 3 ft. 1 ft. " 12 in. NOTATION AND REDUCTION, 137 In this picture tlie gateway is represented as 1 rod wide ; the board fence as 40 rods, or 1 eighth of a mile long ; the large tree as 80 rods, or 1 fourth of a mile from the comers ; the corners as 160 rods, or 1 half mile from the gate ; and the house as 1 mile from the gate. EXEIt CIS ES. 1. Read 4 mi. 114 rd. 4 yd. 1 ft. 10 in. ; 17 mi. 46 rd. 3.5 ft. 2. Write eight miles twenty-six rods two yards two feet six inches. 3. Write three hundred nineteen miles sixty-seven rods three and seventy-five hundredths yards. PR OB JL JEMS. 30. How many feet are 63 rd. 4 yd. 2 ft. ? 31. How many blocks each 1 inch long can be cut from a board 15 feet long? 180. 32. Reduce 5 mi. 187 rd. 2 yd. 1 ft. 9 in. to inches. 353,919 inches. SOLUTION OF PROBLEM 63 rd. 4 yd. %ft. 5.5 315 315 346.5 -t- 4 = 350.5 yd. 3^ 1051.5 + 2 = 1053.5 /i. Hence, 63 rd. 4 yd. 2 ft. — 1053.5 ft. 138 COMPOUND NUMBERS. 33. How many feet are 200 mi. 4 yd.? 1,056,012. 34. How many rods of fence will it take to inclose a farm winch, is 1 mile long and .5 of SOLUTION OF PROBLEM 36. 398 yd. 1 5.5 yd. 3980 [55 385 ]J^rd. 130 110 3.0 yd. Hence, S98 yd. = 72 rd. 2 yd. a mile wide ? 960. 35. Reduce 398 yards to rods. 36. In a bundle of lath there are 100 pieces, each 4 feet long. K laid lengthwise in a row upon the ground, how far would they reach ? 2JtTd.lyd. 1ft. 37. Reduce 1,530 inches to higher denominations. 7 rd. 4 yd. 38. How many tiles each 1 foot long will be required for 1 mi. 68 rd. 2 yd. of tile-drain ? 6,408. 39. How many miles are there in the fences that inclose the farm shown in the map on page 25 ? 2 mi. 242 rd. 178« Table IV.— Square Measure. The denominations square miles, acres, square rods, square yards, square feet, and square inches consti- tute Square Measure. They are used in measuring land, flooring, plastering, and other surfaces. A square foot is 1 foot or 12 inches long, and 1 foot or 12 inches wide. Hence, it contains 12 times 12, or 144 square inches. 144 sq. in. (square in.) are 1 sq. ft. 9 sq. ft. " 1 sq. yd. 30.25 sq. yd. " 1 sq. rd. 160 sq. rd. " 1 A. (acre.) 640 A. "1 sq. mi. w.. g & zn ^ ^ (itt 12 inebLes wide 1 sq. mi. is 640 A. 1 A. " 160 sq. rd. Isq.rd. " 30.25 sq. yd. 1 sq. yd. " 9 sq. ft. 1 sq. ft. " 144 sq. in. NOTATION AND BEDUCTION. 139 EXEMCISES. 1. Bead 14 sq. mi. 84 A. 28 sq. rd. 2. Read 25 sq. rd. 16 sq. yd. 84 sq. in. 3. Write two hundred nine square miles eighty six acres one hundred seven square rods. 4. Write five square yards eight square feet thirty-six rquare inches. PMOBJOEMS. 40. Reduce 34 square miles to square rods. 3,431,600 sq. rd. 41. A farmer planted one hill of com upon every square yard of a 10-acre lot. How many hills did he plant ? 48,400. 42. Reduce 84 sq. rd. 4 sq. ft. to square feet. 22,873 sq.ft. 43. In 25.3 square miles how many acres ? 16,192 A. 44. How many square miles are there in 312,000 square rods ? 45. A fruit grower has an orchard containing 6,386 peach trees, and each tree occupies 1 square rod of land. How much land is there in the orchard ? 39 A. I46 sq. rd. 46. In covering a roof a tinsmith used 1,152 sheets of tin, each covering 14 by 20 inches. How many square feet were in the roof? 2,240. 47. Reduce 334,976 square inches to higher denomina- tions. . 8 sq. rd. 16 sq. yd. 4 sq.ft. 32 sq. in. 179. Table V,— Cubic Measure. The denominations cu- bic yards, cubic feet, and cubic inches constitute Cubic Measure. They are used in measuring earth, timber, stone, and many other articles, and in esti- mating the capacity of bins, boxes, etc. A cubic foot is 1 foot or 12 inches long, 1 foot or 12 inches wide, and 1 foot or 12 inches 140 COMPOUND NUMBERS. thick, and lience it contains 12 times 12 times 12, or 1728, cubic inches. A cubic yard is 3 feet long, 3 feet wide, and 3 feet tliick, and contains 27 cubic feet. 1738 cu. in. (cubic incli) are 1 cu. ft. l 1 cii. yd. is 27 cu. ft. 27 cu. ft. " 1 cu. yd. | 1 cu. ft. " 1728 cu. iu. EXEB, CISE8. 1. Read 30 cu. yd. 10 cu. ft. 1008 cu. in. 2. Read 215 cu. yd. 49 cu. in. 3. Write fourteen cubic yards twenty-four cubic feet. 4. Write one hundred nine cubic yards ninety-two cubic inches. PItOnLE3IS. 48. In 3 cu. yd. 17 cu. ft. 112 cu. in., how many cubic inches ? 49. Reduce 846,296 cubic inches to higher denominations. 18 cu. yd. 3 cu. ft. 130 Jf, cu. in. 50. Reduce 5 cu. yd. 948 cu. in. to cubic inches. 23Jf,228. 51. How many cubical blocks, each containing 1 cubic inch, will be required to make a pile that shall contain 16.5 cubic yards? 769,824. 52. A brick is 8 inches long, 4 inches wide, and 2 inches thick. If 100,000 bricks are piled together, how many cubic yards will there be in the pile ? 137 cu. yd. 4 cu. ft. 1216 cu. in. 53. Reduce 4,713,256 cubic inches to higher denominations. 101 cu. yd. 1000 cu. in. 180. Table VI,— Wood 3Ieasure, The denominations cords, cord feet, and cubic feet constitute Wood Measure. They are chiefly used in measuring wood. Kough stone is also commonly sold by the cord. A pile of wood 8 feet long, 4 feet wide, and 4 feet high contains 1 cord ; and 1 foot in length of such a pile contains 16 cubic feet, and, therefore, is 1 cord foot. NOTATION AND REDUCTION. 141 16 cu. ft. are 1 cd, ft. (cord foot.) 8 cd. ft. " 1 cd. (cord.) 128 cu. ft. " 1 cd. 1 cd. is 8 cd. ft. 1 cd. '^ 138 cu. ft. 1 cd. ft. " 16 cu. ft. :EXEItCISES. 1. Eead 7 cd. 5 cd. ft. 8 cu. ft. ; 19 cd. 4 cd. ft. 2. Write twenty cords seven cord feet six cubic feet. 3. Write two hundred fifty-one cords two cord feet. PMOBLEMS. 54. How many cubic feet are 13 cd. 5 cd. ft. 12 cu. ft. ? 1,756. 55. In 2,240 cubic feet of cobble stone, how many cords ? 142 COMPOUND NUMBERS. 56. How many cords of wood in a pile 76 feet long, 4 feet wide, and 4 feet high ? 9.5. 57. How much wood is there in a pile 120 feet long, 4 feet wide, and 6 feet high ? 22.5 cords. 58. How much wood will a teamster draw at 10 loads, if each load is 12 feet long, 4 feet wide, and 3 feet high ? 59. If a pile of wood is 4 feet wide and 4 feet high, how long must it be to contain 56.25 cords ? 450 feet. 181. Table VII.— Weight, Most kinds of produce, provisions, and groceries, also iron and other metals, coal and many other arti- cles, are bought and sold by Weight. The denomina- tions in common use are tons, hundred-weight, pounds, and ounces. 16 oz. (ounces) are 1 lb. (pound.) 1 T. is 20 cwt. 100 lb. " 1 cwt. (hundred-weight.) 1 c^i;. " 100 lbs. 20 cwt. " 1 T. (ton.) 1 lb. " 16 oz. 200 lb. of pork, beef, or fish are 1 bbl. (barrel.) 196 1b. of flour are 1 bbl. NOTATION AND REDUCTION. 143 EXER CISES. 1. Read 5 T. 17 cwt. 34 lb. 4 oz. ; 39 T. 94 lb. 2. Write seven tons thirteen hundred-weight fifty-nine pounds fourteen ounces. 3. Write one ton nine hundred forty-eight pounds five ounces. PMOBIjEMS. 60. How many ounces are there in 9 tons ? 288,000. 61. Reduce 4 T. 16 cwt. 83 lb. to pounds. 9,683. 63. One month a manufacturer put up 5 T. 4 cwt. 39 lb. of saleratus in pound packages. How many packages did he put up ? 10,429. 63. How many pound bars of lead will weigh 3 T. 54 lb. ? 64. How much will 6 barrels of mackerel cost at $.085 a pound ? $102. 65. A grocer retailed 13 barrels of flour at $.055 a pound. How much did he receive for it ? $140.14. 66. How many tons are 34,000 pounds ? 17 T. 67. If 1 oz. of lead is used in making 1 rifle ball, how much lead will be required to make 60,000 balls ? 1 T. 17 cwt. 50 lb. 68. How many barrels of flour are 6,860 pounds ? 35. 69. Reduce 64,015 oz. to higher denominations. 2 T. 15 oz. 182. Table VIII.— Counting. In counting articles for market purposes, the de- nominations dozen and gross are used. 13 ones are 1 doz. (dozen.) I 1 gro. is 13 doz. 13 doz. " 1 gro. (gross.) I 1 doz. " 13 ones. EXEItCISES. 1. Read 24 gro. 5 doz. ; 29 gro. 7 doz. 2. Write seventeen gross six dozen. 3. Write fifty gross nine dozen. 144 .COMPOUNDNUMBERS. FJROBJLEMS. 70. If a cook uses 6 eggs each day, in how many days will she use 9 doz. eggs ? 18. 71. How many steel pens are there in 7 boxes each con- taining 1 gro. ? 1,008. 72. How many clothes-pins in 47 doz. ? 561^. 73. How many dozens are 132 brooms ? 11. 74. In one year a tailor used 23 gro. 9 doz. buttons. What number of buttons did he use ? 3,420. 75. If 18 crayons are used every week in a certain school, how many gross will be used in 40 weeks ? 5. 76.~ In 1,844 screws, how many gross ? 12 gro. 9 doz. 8. 183. Table IX.— Paper. Paper is bought and sold by the ream, quire, and sheet. 24 sheets are 1 quire. I 1 rm. is 20 quires. 20 quires " 1 rm. (ream.) I 1 quire " 24 sheets. EXJEIt CIS ES. 1. Bead 4 rm. 8 quires 12 sheets ; 11 rm. 10 quires. 2. Write thirteen reams fifteen quires four sheets. PR OBLEMS. 77. How many sheets of paper are there in a ream of fools- cap ? 480- 78. If 1 sheet of printing paper will make 4 handbills, how many bills will 5 rm. 8 quires make ? 10,368. 79. How many letters, each requiring 1 sheet, can be written on 11 rm. 12 quires of commercial note paper ? 6.,568. 80. In 384 sheets of letter paper, how many quires ? 16. 81. If a lawyer uses 18 sheets of legal cap paper every day, how many reams will he use in 320 days ? 12. 82. If a merchant uses 3 quires of wrapping paper every day of the 313 week-days of the year, how much paper does he use in the year ? 46 rm. 19 quires. NOTATION AND REDUCTION. 145 184. Table X,—Time, The denominations centuries, years, months, weeks, days, hours, minutes, and seconds are used in express- ing different portions of time. The day and the year are the natural divisions of time, the other denomina- tions, except centuries, being parts of these. 60 sec. (seconds) are 1 min. (minute.) 60 min. 24 h. 7 da. 52 wk. 1 da. or 365 da. 52 wk. 2 da. or 366 da. 100 yr. are 1 h. (hour.) " 1 da. (day.) " 1 wk. (week.) 1 common yr. (year.) (" 1 leap-yr. 1 century. 1 century 1 leap-year 1 common yr. " 1 da. " Ih. " 1 min. " is 100 yr. u \ 52 wk. 2 da. I or 366 da. 52 wk. 1 da. or 365 da. 24 h. 60 min. 60 sec. Every fourth year from the beginning of a century is a leap-year. EXEIt CIS ES. 1. Read 3 yr. 6 mo. ; 12 h. 30 min. 15 sec. 3. Read 9 wk. 3 da. 10 h. ; 4 da. 15 h. 45 min. 3. Write five hours fifteen minutes thirty seconds. 4. Write fourteen weeks six days four hours. 6. Write twenty-eight years nine months. M 146 COMPOUND NUMBERS. Seasons. Months. Bays. Winter. ( 1st a I 2d ' 10. January, Jan. 31 ' February, Feb. 28 ( 3d ' ' March, Mar. 31 Spring. ■| 4th ' ' April, Apr. 30 ( 5th ' May, May. 31 C 6th ' ' June, June. 30 Summer. ] 7th ' July, July. 31 i 8th * ' August, Aug. 31 ( 9th ' ' September, Sept. 30 Autiimn. KlOth ' ' October, Oct. 31 Cllth ' ' November, Nov. 30 Winter. 12th ' ' December, Dec. 31 February iias 28 days in a common year, and 29 in a leap-year. (See Manual, page 219.) 83. How many minutes in the month of January ? 4.4,640. 84. How many seconds are there in a common year ? 81,536,000. 85. Reduce 3 wk. 20 min, to minutes. 80,260 min. 86. Reduce 50,400 minutes to weeks. 5 weeks. 87. How many seconds in the three summer months ? 7,948,800. 88. How long will it take a clock to tick 1,000,000 times, if it ticks once every second ? 1 wk. 4 da. 13 h. 46 min. 40 sec. 89. How many days were there from the beginning of the year 1857 to the end of the year 1866, two of the years, 1860 and 1864, bemg leap-years ? 3, 652. 90. In a leap-year, how many hours ? 8,784. 91. Reduce 875,665 sec. to higher denominations. 10 da. 3 Ti. 14 min. 25 sec. 92. After the 9th day of October, how many hours remain in the month ? 528. 93. If you can count 75 every minute, how much time would you spend in counting 27,000,000 ? 35 wk. 5 da. NOTATION AND REDUCTION. 147 185. The Metric System of Weights and Measures. In the year 1866, the Congress of the United States passed a bill authorizing the use of a new system of weights and measures. In this system the principal denomination is the Metre, from which all the other denominations in all the tables are . derived. Hence, this system is called the Metric System. The principal denomination for the Measure of Sur- face is the Are; for the Measure of Capacity, the Litre ; and for Weight, the Gram. (See Manual, page 219.) The lower denominations in each table are tenths, hundredths, or thousandths of these ; and their names are formed by prefixing deci, centi, or mUli to the name of the principal denomination. The higher denominations are 10, 100, 1,000, or 10,000 times the principal denomination of any table ; and their names are formed by prefixing deJca, hecto, lilOy or myria to the name of that principal denomina- tion. TABLE OF DENOMINATIONS AND THEIB EELATIVE VALUES. PREFIXES FOR LOWEB DENOMINATIONS. MilU (mill-ee) .001 of Centi (sent-ee) .01 of Ded (des-ee) .1 of NAMES OP PKINCIPAL DENOMINATIONS. Metre (mee-ter) Are (are) Litre (li-ter) Gram PREFIXES FOR HIGHER DENOMINATIONS. BeJca (dek-a) 10 Eecto (hec-to) 100 Kilo (kill-o) 1,000 Myria (mir-e-a) 10,000 The weights and measures of this system have not yet come into use in this country. They are in general use in France, Belgium, Spain, and Portugal ; and their use has been legaHzed by Great Britain, Italy, Norway, Sweden, Greece, Mexico, and most of the South American governments. 148 COMPOUND NUMBERS, UEES OF 10 millimetres are 10 centimetres " 10 decimetres " 10 metres " 10 dekametres " 10 hectometres" 10 kilometres " 1 centimetre 1 decimetre 1 metre 1 dekametre 1 hectometre 1 kilometre 1 myriametre LENGTH. millimetre centimetre decimetre METRE dekametre hectometre kilometre myriametre y-oV(T metre jh metre yV metre S9.37 inches. 10 metres 100 metres 1,000 metres 10,000 metres MEASURES OF SURFACE. 100 centares are 1 are 100 ares " 1 hectare 1 centai'e 1 ARE 1 hectare is y^o are {( j 100 sq. metres, ( 119.6 sq. yd. " 100 ares MEASURES OF CAPACITY. 10 millilitres are 10 centilitres " 10 decilitres '' 10 litres 1 centilitre 1 decilitre 1 litre 1 dekalitre 10 dekalitres 10 hectolitres 1 hectolitre r millilitre is y^jVo ^^^^^ litre litre 1 centilitre " jli 1 decilitre " j\ fl cu. decimetre 1 LITRE "I .908 qt. dry meas. [ 1.0567 qt.liq.meas. 1 dekalitre " 10 litres kilolitre, or 1 hectolitre " 100 litres \^^IGHT. 10 milligrams are 1 centigram decigram gram dekagram hectoOTam 10 centigrams " 10 decigrams " 10 grams " 10 dekagrams " 10 hectograms " 10 kilograms j. ^^ or kilos ) 10 myriagrams" 10 quintals " kilogram myriagram quintal millier or tonneau 1 milligram 1 centigram 1 decigram 1 GRAM 1 dekagram 1 hectogram 1 kilogram) or 1 kilo) 1 myriagram 1 quintal 1 millier f oVo gram tIo gram TO gram 15.432 grains 10 grams 100 grams 1000 grams, or 2.2046 pounds 10 kilos 100 kilo^ 1,000 kilos ADDITION. 149 SECTION II. 186. Ex. What is the sum of 5 rd. 4 yd. 2 ft. 3 in., 7 rd. 1 yd. 1 ft. 9 in., 2 yd. 1 ft., 2 ft. 11 in., and 12 rd. 5 yd. 5 in. ? Explanation. — We write the solution. numbers so that like denomina- 5 ^^^ 4 r^.^ 2 ft. 3 in. tions stand in the same columns. 7 119 Commencing with the lowest 2 10 denomination, we add ; 5 in. 2 11 -t- 11 in. + in. + 9 in. + 3 in. 1? 5_ 9__? =: 28 in. But 28 in. =: 2 ft. 4 26 rd. 3 yd. 2 ft. 4 in. in. ; we therefore write the 4 in. as the inches of the sum, and the 2 ft. we add with the feet of the given numbers. 2 ft. + ft. + 2 ft. + 1 ft. + 1 ft. + 2 ft. = 8 ft. But8ft. =r 2 yd.2ft ; we there- fore write the 2 ft. in the sum, and add the 2 yd. with the column of yards. 2 yd. + 5 yd. + 2 yd. + 1 yd. + 4 yd. = 14 yd. But since 14 yd. = 2 rd. 3 yd., we write the 3 yd. in the sum, and add the 2 rd. with the column of rods. 2 rd. 4- 12 rd. + 7 rd. + 5 rd. = 26 rd., which we write as the rods of the sum. The result, 28 rd. 3 yd. 2 ft. 4 in., is the sum required. PBOBZEMS. Find the sum of the compound numbers in problems 1, 2, 3. (1) (3) (3) 9 A. 96 sq. rd. 3 rm. 5 quires 16 sheets 34 gal. 2 qt. jJt. 11 44 4 8 35 1 1 8 108 3 20 6 33 3 1 10 56 2 18 14 36 1 4. A painter used 5 gal. 3 qt. 1 pt. of linseed-oil one week, and 3 gal. 2 qt. 1 pt. the next week. How much did he use in the two weeks ? 9 gal. 2 qt. 150 COMPOUND NUMBERS. 5. A farmer used 3 bu. 1 pk. 6 qt. of clover seed in seed- ing one field, and 3 bu. 3 pk. 4 qt. in seeding another. How much did he use upon the two fields ? 6 bu. 1 ph 2 qt. 6. How much wood is there in three piles, the first of which contains 5 cd. 3 cd. ft. 13 cu. ft., the second 6 cd. 6 cd. ft., and the third 9 cd. 4 cd. ft. 4 cu. ft. ? 21cd.6 cd.ft. 7. A father is 28 yr. 164 da. older than his son, and the son is 17 yr. 235 da. old. How old is the father ? 46 yr. 24 da. 187. !%igU for A.ddltion of Compound JVumbers, I. Write the parts with like denominations in the same column. n. Add each denomination separately, beginning with the lowest ; and when the sum is less than 1 of the next higher denominatioUj write it under the denomination added. m. When the sum of any denomination is equal to or more than 1 of the next higher denomination, reduce it to that higher denomination, write the remainder under the denomination added, and add the quotient with the next higher denomination. PM OBIjEMS. 8. A stationer sold 5 gro. 3 doz. 8 pens of one kind, and 2 gro. 6 doz. 6 pens of another. How many pens did he sell ? 7 gro. 10 doz. 2. 9. A livery-man bought 3 loads of hay, the first weighing 1 T. 2 cwt. 17 lb., the second 1 T. 3 cwt. 96 lb., and the third 19 cwt. 49 lb. How much hay did he buy ? ST. 5 cwt. 62 lb. 10. A housekeeper made 2 gal. 1 qt. 1 pt. of currant wine, 1 gal. 3 qt. of blackberry wine, and 4 gal. 2 qt. 1 pt. of grape wine. How much wine did she make ? 8 gal. 3 qt. 11. In five successive days, a fruit dealer sold 3 pk. 3 qt. 1 pt. of cherries, 1 bu. 1 pk. 5 qt., 1 bu. 3 pk. 7 qt., 1 bu. 2 qt. 1 pt., and 3 pk. 1 pt. How many cherries did he sell ? 12. What is the sum of 5 rm. 14 quires 12 sheets, 7 rm. 11 quires 9 sheets, and 9 rm. 15 quires 9 sheets ? ADDITION. 151 13. A railroad train runs from Detroit to Ann Arbor in 1 h. 45 min. ; to Jackson in 1 b. 40 min. more ; to Marshall in 1 h. 25 min. more ; to Kalamazoo in 1 h. 35 min. more ; to Mies in 3 h. 15 min. more ; and to Chicago in 4 h. 30 min. more. What is the running time of the train from Detroit to Chicago ? 13 h. 10 min. 14. In an ax factory are six large grindstones, which weigh 2 T. 1 cwt. 18 lb., 1 T. 16 cwt. 24 lb., 2 T. 3 cwt. 7 lb., 2 T. 2 cwt. 7 lb., 1 T. 18 cwt. 87 lb., and 1 T, 19 cwt. 69 lb. What is their total weight ? 12 T. 1 cwt. 12 lb. 15. A telegraph company put up 1 mi. 14 rd. 3 yd. of wire one day, 318 rd. 5 yd. the second day, 1 mi. 39 rd. 4 yd. the third day, and 1 mi. 67 rd. the fourth day. How much wire did they put up in the four days ? 4 mi. 120 rd. 1 yd. 16. What is the sum of 9 cu. yd. 20 cu. ft. 388 cu. in., 218 cu. yd. 14 cu. ft. 524 cu. in , and 145 cu. yd., 11 cu. ft. 1415 cu. in. ? 373 cu. yd. 19 cu.ft. 599 cu. in. 17. There are 35 sq. yd. 5 sq. ft. of plastering in the ceiling of a room, 22 sq. yd. 2 sq. ft. in each of the two side walls, and 17 sq. yd 7 sq. ft. in each of the two end walls. How much plastering in the room ? 115 sq. yd. 5 sq.ft. 18. Add 4 yd. 2 ft. 4 in., 3 yd. 1 ft. 8 in., 5 yd. 2 ft. 6 in. 19. Add 28 wk. 4 da. 14 h. 45 min. 45 sec, 11 wk. 3 da. 10 h. 30 min. 15 sec, and 6 wk. 6 da. 3 h. 25 min. 30 sec. SECTION III. S VS T^A. C TIOJV. 188. Ex. 1. From 8 bu. 1 pk. 7 qt. subtract 4 bu. 3 pk. 2 qt. Explanation. — We write the de- nominations of the subtrahend un- solution. der the Hke denominations of the ?^^- \^^' I minuend. Commencing with the 4 3 2 lowest denomination, we subtract; ^^^- ^P^- ^9^- 152 COMPOUND XUMBEKS. 2 qt. from 7 qt. leave 5 qt., wliicli we write as the quarts of the remainder. Since we can not subtract 3 pk. from 1 pk., we take 1 bu. (=4 pk.) of the 8 bu. in the minuend, add it to the 1 pk., and from the 5 pk. thus obtained, we subtract the 3 i)k., writing the dif- ference, 2 pk., in the remainder. Finally, we subtract 4 bu. from the 7 bu. now left in the minuend, and write the difference, 3 bu., in the remainder. The compound number, 3 bu. 2 pk. 5 qt., is the remainder required. Ex. 2. From 20 gal. subtract 5 gal. 2 qt. 1 pt. Explanation. — Since we can not subtract 1 pt. from pt., and there solution. are no quarts in the minuend to i^ ? ? reduce to pints, we take 1 of the ^^ ^al' ^ t 1 t 20 gal., leaving 19 gal. From this — — l^^JPjL. 1 gal. (or 4 qt.) we take 1 qt., leav- 14 (/aZ. 1 qt. Ipt. 3 qt. ; and this 1 qt. == 2 pt. The form of the minuend is now changed from 20 gal. to 19 gal. 3 qt. 2 pt., and from this we subtract 5 gal. 2 qt. 1 pt., obtaining a remainder of 14 gal. 1 qt. 1 pt. /^N PJtOBZEMS. /2) From 7 da. 3 h. 20 min. 5 mi. 220 rd. 4 yd. 2ft. 5 in. Subtract 3 9 15 2 264 3 2_^_ 3. From a cask that contained 33 gal. 2 qt. of vinegar, a erocer drew 17 gal. 3 qt. How much vinegar was left in the Lk? 15 gal 3 qt. 4. A farmer raised 614 bu. 1 pk. of oats, and sold all but 133 bu. 3 pk. How many oats did he sell ? 4^0 hi. 2 ph 5. A physician bought a load of hay which weighed, with the wagon, 1 T. 8 cv/t. 21 lb. The wagon alone weighed 12 cwt. 43 lb. What was the weight of the hay ? 15 cwt. 78 lb. 6. A merchant tailor bought 32 gro. 6 doz. rubber buttons, and sold 24 gro. 8 doz. 6. How many buttons had he left ? SUBTRACTION. 153 7. The walls of a room measure 68 sq. yd. 4 sq. ft., and the windows and doors 18 sq. yd. 7 sq. ft. How many square yards of plastering on the walls ? ^9 sq. yd. 6 sq.ft. 8. A farmer exchanged a farm of 200 acres for another con- taining 113 A. 38 sq. rd. How much more land was there in the first farm than in the second ? 86 A. 132 sq. rd. 9. A bookseller having 23 rm. 13 quires of letter-paper, sold 13 rm. 16 quires 13 sheets. How much paper had he then ? 8 rm. 15 quires 12 sheets. 10. A grocer bought a crock of butter which weighed 44 lb. 6 oz. The crock alone weighed 7 lb. 10 oz. How much did the butter weigh ? 86 lb. 12 oz. 11. A druggist put 5 gal. 3 qt. 1 pt. of alcohol into a can which would hold 30 gal. How much more alcohol would the can have held ? IJf, gal. 1 qt. 1 pt. 12. A laborer agreed to dig a cellar 18 ft. long, 16 ft. wide, and 5 ft. deep. After digging 44 cu. yd. of earth, how much more has he to remove ? 9 cu. yd. 9 cu.ft. 13. William is 16 yr. 28 da. old, and Edward is 11 yr. 284 da. old. How much older is William than Edward ? 4 yr. 109 da. 14. A farmer contracts to deliver at a railroad station 1,000 cords of wood. He has 384 cd. 5 cd. ft. already cut. How much more wood must he chop ? 615 cd. S cd. ft. 15. From 90 cu. yd. subtract 39 cu. yd. 18 cu. ft. 966 cu. in. 50 cu. yd. 8 cu. ft. 762 cu. in. Ex. 3. How many years, months, and days from April 15, 1864, to July 4, 1867 ? Explanation. — Since the solution. later of two dates is expressed ISQT yr. 7 mo. 4: da. by a greater compound num- ber than the earlier, we write 3 yr. 2 mo. 19 da. the later date for the minuend, and the earlier for the subtrahend, writing the number of the year, month, and day in order. We then sub- tract as in other compound numbers, calling 30 days a month when the number of days in the subtrahend is greater than that in the minuend. 164 COMPOUNDNUMBERS. 189. "Rule for Subtraction of Compound JVumbers, . I. Write the denominations of the subtrahend under the like denominations of the minuend. II. Subtract each denomination of the subtrahend from the like denomination of the minuend, and write the result as the same denomination in the remainder. III. WJien any denomination of the subtrahend exceeds that in the minuend, before subtracting, reduce 1 of the next higher denomination of the minuend to this lower denomi- nation, and add it to the number of this denomination given in the minuend. rV. In the last case, in subtracting the next higher de- nomination, we may either call the number in the minuend 1 less, or that in the subtrahend 1 more. jpm oblems. 16. Benjamin Franklin was born Jan. 17, 1706, and died Apr. 17, 1790. How old was he when lie died ? SJj. yr. 3 mo. 17. George "Washington was born Feb. 32, 1732, and died Dec. 14, 1799. At what age did he die ? 67 yr. 9 mo. 22 da. 18. A note dated June 7, 1863, was paid Apr. 4, 1865. How long did it remain unpaid ? 1 yr. 9 mo. 27 da. 19. A note was given Sept. 10, 1867, payable Feb. 4, 1868. How long had it to run ? J^ mo. 2^ da. 20. Robert was bom Oct. 9, 1858. How old was he, May 11, 1867 ? 8yr.7 mx).2 da. 21. Washington Irving died Nov. 28, 1859, aged 76 yr. 7 mo. 25 4a. What was the date of his birth ? J._p7*. 3, 1783. 22. A farmer in a country district agrees to furnish 10 cd. 4 cd. ft. of wood for the winter term of school. After drawing 4 cd. 7 cd. ft., how much has he yet to draw ? 5 cd. 5 cd.ft. 23. From 11 mi. 84 rd. 4 yd. 1 ft. take 5 mi. 186 rd. 2 yd. 3 ft. 5 mi. 218 rd. 1 yd. 2 ft. 24. In a storehouse there is a bin which will hold 240 bu., and in the bin are 183 bu. 3 pk. of wheat. How much more wheat will the bin hold ? 56 Jyii. 1 ph. MULTIPLICATION. 155 SECTION IV. MZrZ TITZIC^A TIOJV. 190. Ex. Multiply 6 wk. 2 da. 8 h. by 7. Explanation. — We write the mul- solution. tiplier under the lowest denomina- q y^j^^ 2 da. 8 h. tion of the multiplicand. Then, 7 commencing at the right, we mul- 44 wk. 2 da. 8 h. tiply ; 7 times 8 h. = 56 h. But 56 h. = 2 da. 8 h. ; we therefore write the 8 h. in the product, and reserve the 2 da. to be added with the days of the product. 7 times 2 da. = 14 da., and 14 da. +2 da. = 16 da. But since 16 da. = 2 wk. 2 da., we write the 2 da. in the product, reserving the 2 wk. to be added with the weeks in the product. 7 times 6 wk. == 42 wk., and 42 wk. + 2 wk. = 44 wk., which we write in the product. The result, 44 wk. 2 da. 8 h., is the product required. /■^N l^n OBLEMS. ,<^\ Multiply ^ T. 4: cwt. \^ lb. 1 cu. yd. 14 cu.ft. 356 cu. in. by 6 l_ 3. Multiply 4 bu. 2 pk. 7 qt. by 9. 4. If a painter uses 3 gal. 3 qt. 1 pt. of linseed-oil in paint- ing 1 lumber wagon, how much will he use in painting 5 wagons ? lJi> gal. 1 qt. 1 pt. 5. If a man can cradle an acre of wheat in 3 h. 20 min., how long will he be in cutting 7 acres ? 23 h. 20 min. 6. If the rate of speed of a railroad train is 28 mi. 216 rd. an hour, how far will it run in 14 hours ? JfOl mi. IJ^Jf. rd. 7. What is the weight of 50 bales of cotton, each weighing 4 cwt. 96 lb. ? 12 T. 8 cwt. 8. How many bushels of wheat will a field of 9 acres yield, at an average of 14 bu. 3 pk. 4 qt. an acre ? 133 hi. 3 ph 4 ^• 9. How much wood can a team draw at 18 loads, if they draw 1 cd. 2 cd. ft. 12 cu. ft. at each load ? 24, al 1 cd.ft. 8 cu.ft. 156 COMPOUND NUMBERS. 191 • Hule for Muttipllcation ofConipou7idJVunibers. I. Write the multiplier under the lowest denomination of the multiplicand. II. Multiply each denomination of the multiplicand, in order, by the multiplier, as in integers ; and when the prod- uct is less than 1 of the next higher denomination, write it under the denomination multiplied. in. When any product is equal to, or greater than, 1 of the next higher denomination, reduce it to that higher denomination, write the 7^emainder under the denomina- tion multiplied, and add the quotient with the next higher denomination in the final result. Pit OBJLJEMS. 10. Multiply 12 A. 84 sq. rd. by 27. 338 A. 28 sq. rd. 11. If a farmer uses 1 bu. 3 pk. 2 qt. of seed- wheat to the acre, how much will he use in seeding 15 acres ? 27 bu. 6 qt. 12. How much cider will it take to fill 8 demijohns, each holding 3 gal. 2 qt. 1 pt. ? 29 gal. 13. A publisher uses 2 rm 7 quires 12 sheets for each num- ber of a weekly newspaper. How much paper does he use in a year ? 123 rm. 10 quires. 14. If the water of a river flows at the rate of 3 mi. 280 rd. an hour, how far will a log float in 219 hours ? SJiS mi. 200 rd. 15. If 33 cd. 7 cd. ft. of wood make 1 canal-boat load, how much wood will make 19 loads ? 6JiS cd. 5 cd.ft. 16. A farm hand can plow an acre of corn in 4 h. 15 min. How long will it take him to plow 25 acres, if he works 10 hours a day ? 10 da. 6 h. 15 min. 17. How much land is there in 24 village lots, each 5 rods front and 7 rods deep ? 5 A. Ji,0 sq. rd. 18. A teamster drew 32 loads of freight, each load weighing 1 T. 2 cwt. 25 lb. How much freight did he draw ? 35 T. 12 cwt. 19. If a manufacturer makes 15 gro. 4 doz. 9 clothes pins each day, how many does he make in the 308 working-days of the year? J^,7Jfl gro. 11 doz. DIVISION. 157 SECTION V. DITISIOjY, 192i Ex. Divide 16 rm. 9 quires 14 sheets of paper into 5 equal parts. Explanation.— We write solution. the dividend and divisor, 16 rm. 9 qu i res 14 wheel s \ 5 and commence at the left 3 rm. 5 quires 22 sheets of the dividend to divide, as in integers. One fifth of 16 rm. is 3 rm. mth a re- mainder of 1 rm. Writing the 3 rm. in the quotient, we reduce the 1 rm. remainder to quires, and to it add the 9 quires, making 29 quires. One fifth of 29 quires is 5 quires with a remainder of 4 quires. Writing the 5 quires in the quotient, we reduce the 4 quires remain- der to sheets, and to it add the 14 sheets, making 110 sheets. One fifth of 110 sheets is 22 sheets, which we write in the quotient. The result, 3 rm. 5 quires 22 sheets, is the quotient required. FROBL EMS. Find the quotient in problems 1, 3, 3. (1) (2) (3) 53 ha. 3 pi. 4 ^. I 9 300 gro. 4 doz. j 16 401 cd. 5 cd.ft. I 17 4. A ship sailed 59 mi. 30 rd. in 7 hours. What was her average hourly distance ? 8 mi. lJf.0 rd. 5. A farmer put 385 gal. 3 qt. 1 pt. of cider into 9 casks. How much cider did he put into each ? Jf2 gal. 3 qt. 1 pt. 6. If a glazier can set the glass for 8 windows in 10 h. 40 min., how long will it take him to set the glass for 1 window ? 7. A teamster fed to his horses 67 bu. 3 pk. of oats in 30 days. How many oats did he feed each day ? 21m. 1 pic. 8. If a locomotive bums 13 cd. 6 cd. ft. 4 cu. ft. of wood in making 13 trips, how much does it bum in making 1 trip ? 158 COMPOUND NUMBERS. 193. ^ule for division of Compound JVumbers. L Write the dividend and divisor, as in integers. n. Divide the highest given denomination, as in in- tegers, and write the result as the corresponding denomina- tion in the quotient. ni. Eeduce the remainder to the next lower denomina- tion, add to the result the number given of this lower denomination, and divide the same as before. IV. Proceed in the same manner until all the denomina- tions of the dividend are divided. PM OB JO JEMS. 9. From 15 acres of meadow a farmer cut 28 T. 13 cwt. 75 lb. of hay. What was the yield per acre ? 1 T. 18 cwt. 25 lb. 10. He harvested 376 bu. 3 pk. 4 qt. of barley from 12 acres. What was the yield of barley per acre ? 311m. 1 ph 5 qt. 11. A workman laid 64 rd. 3 yd. 1 ft. of stone-wall in 26 days. How much did he lay each day ? 2 rd. 2 yd. 2 ft. 12. How long will it take a cooper to make 1 flour barrel, if he can make 8 in 10 hours ? 1 h. 15 min. 13. A bridge pier containing 448 cu. yd. of stone was built in 36 days. What was the average amount of stone laid daily ? 12 cu. yd. 12 cu.ft. 14. Divide 69 T. 3 cwt. 29 lb. 8 oz., by 19. Quotient^ 3 T. 12 cwt. 80 lb. 8 oz. 15. A cook uses 12 doz. eggs in 18 days. How many eggs does she use each day ? 8. 16. An ink manufacturer put up 3 gal. 2 qt. 1 pt. 2 gi. of ink in 59 bottles. How much did each bottle contain ? 2 gi. 17. If 1 rm. 10 quires of paper are used in making 10 doz. writing-books, how many sheets are used in making 1 doz. books ? How many in making 1 book ? 6 sheets in 1 booh 18. If 12 men can chop 132 cords of wood in 4 days, how many cords can 1 man chop in 1 day ? 2 cd. 6 cd. ft. 19. If 7 men can mow 26 A. 40 sq. rd. of grass in 10 hours, how much can 1 man mow in 1 hour ? 60 sq. rd. REVIEW PROBLEMS. 159 SECTION VI. "P^RO^SLBMS IJ\r COM'POU'J\r^ JVZrM:SB'RS. 1. How many bales, each weighing 250 lb., will 7 T. 5 cwt. of hay make ? ^^• 2. At $3 a rod, how much will it cost to build a fence around a lot 8 rods long and 5 rods wide ? $'78. 3. A man traveled by stage 78 miles in 16 hours. How far did he travel in an hour ? Jt mi. 280 rd. 4. How much will 3 T. 5 cwt. 56 lb. of iron castings cost, at 7 cents a pound ? $J^58.92. 5. How many cu. yd. of stone are there in an abutment 56 ft. long, 8 ft. wide, and 15 ft. high ? 2J^8 cu. yd. 2J^ cu.ft. 6. How many gallons of wine will 25 doz. quart bottles hold? 75. 7 Five wood-cutters worked together during the winter. The first chopped 118 cd. 4 cd. ft., the second 109 cd. 2 cd. ft. 8 cu. ft., the third 106 cd. 5 cd. ft, the fourth 98 cd. 3 cd. ft. 8 cu. ft., and the fifth 91 cd. 7 cd. ft. How much wood did they chop ? 52 J^ cd. 6 cd.ft. 8. From Dec. 30 to Aug 1 of the following year, how many months and days ? 1 mo. 2 da. 9. How many minutes from Aug. 18, at noon, to Oct. 9, at noon? 7J!i,,880 min. 10. How much cider can be made from 100 bu. of apples, if 3 gal. 1 qt. 1 pt. can be made from 1 bu. ? 337 gal. 2 qt. 11. From a barrel which contained 42 gal. 2 qt. of syrup, 7 gal. 2 qt. were drawn one day, 3 gal. 1 qt. 1 pt. the next, and 4 gal. 3 qt. the third ? How much syrup remained in the barrel ? 26 gal. 3 qt. 1 pt. 12. A railroad company has a pile of wood 576 ft. long, 25 ft. wide, and 18 ft. high. How much wood is there in the pile? 2,025 cd. 13. In 17 gal. 1 pt., how many gills ? 5J^8. 14. At $.625 a cu. yd., how much will it cost to dig a cellar 27 ft. long, 19 ft. wide, and 7 ft., deep ? $83,125. 160 COMPOUND NUMBERS. 15. A way-freight car was loaded with 3 T 3 cwt. 48 lb. of groceries, 3 T. 19 cwt. 40 lb. of hardware, 1 T. 1 cwt. 94 lb, of furniture, and 18 cwt. 64 lb. of dry goods. How much freight was in the car ? 9 T. 3 cwt. lf,6 Tb. 16. A druggist bought 9 casks, each containing 20 gal. 3 qt. 1 pt. of brandy. How much did they all contain ? 17. How much seed-corn will be required for 5,000 pint papers ? 75 lu. 4 qt. 18. How many cu. in. are 3 cu. yd. 18 cu. ft. 334 cu. in. ? 19. If 2 show-bills can be printed on 1 sheet, how much paper will be required for 2,400 bills ? 2 rm. 10 quires. 20. At $35 an acre, what will be the cost of a piece of land 140 rd. long, and 112 rd. wide ? $3,JjS0. 21. How many 3-pint bottles will 32 gal. 2 qt. 1 pt. of cider fill ? 87. 22. If it is 18 feet around the hind wheel of a carriage, how many times will the wheel turn over in runnmg 6 mi. 174 rd. 3 yd.? 1,920. 23. A certain room is 22 ft. long, 18 ft. wide, and 12 ft. high. How many sq. yd. are there in the ceiling ? W 24. How many square yards are there in the four sides of the same room ? 106 sq. yd. 6 sq.ft. 25. A grocer bought 6 doz. brooms for $19.34, and retailed them at 40 cents apiece. How much did he gain ? 26. In making a road 250 rods long, 5 cu yd. 3 cu. ft. of gravel were used to the rod. How much gravel was used ? 1,277 cu. yd. 21 cu.ft. 27. If a manufacturer makes 1,000 lead-pencils in a day, how many gross will he make in 26 days ? 180 gro. 6 doz. 8. 28. A man bought 7 acres of land, at $450 an acre, and sold it in building lots, each 10 rd. long and 4 rd. wide, at $150 apiece. How much did he gain ? $1,050. 29. In 1 liquid gallon there are 231 cu. in. What is the capacity in gallons of a cistern 7 ft. long, 5.5 ft. wide, and 9 ft.deep? ^ ^^^^^• 30. How long will 3 cwt. of sugar last a family, if they use lib. a day? 42 wh 6 da. REVIEW PROBLEMS. 161 31. What will be the cost of a pile of stone 30 ft. long, 8 ft wide, and 4 ft. high, at $6 a cord ? $45. 32. How many miles will a locomotive run in 4 hours, run ning at the rate of 124 rods in a minute ? 93. 33. Last year, I sold from my garden 5 bu. 1 qt. of cherries, dried 1 bu. 3 pk. 1 qt., put up in cans 3 pk. 5 qt., and 3 pk. 2 qt. were eaten in my family. How many cherries grew in my garden ? 8 hi. 2 ph. 1 qi. 34. These cherries grew upon 7 trees. What was the aver- age yield per tree ? l'bu.7qt.' 35. How many tons of hay will a span of horses eat in 15 weeks, if they eat 45 lb. a day ? 2 T. 7 cwt. 25 lb. 36. How many oats will it take to last them the same time, if they eat 24 qt. a day ? 78 du. 3 pi. 37. If a family use 3 bu. 1 pk. of potatoes each month, how much will a year's supply cost them, at $.5625 a bushel ? 38. If a housekeeper uses a half-pint of molasses each day, how long will 20 gallons last her ? Jf5 wTc. 5 da. 39. A plumber has a coil of lead pipe 34 ft. long, which weighs 189 lb. 2 oz. How much does 1 ft. of the jDipe weigh ? 40. How much will 15 ft. of the same pipe weigh ? 41. If I deposit $4.50 in a savings-bank eveiy week, and draw out $12.50 each month, how much will I have on deposit at the end of the year ? $8Jf. 42. How many cords of stone will it take to build a stone fence 76 ft. long, 4 ft. high, and 2 ft. thick ? ^.75 cd, 43. One day a carman drew 14 T. 18 cwt. 52 lb. of freight, at 17 equal loads. How much did he draw at each load ? 44. A grocer bought four hogsheads of molasses, which con- tamed 118 gal. 3 qt., 123 gal. 2 qt., 109 gal., and 122 gal. 1 qt. How much molasses did he buy ? 45. From the first hogshead he sold 49 gal., from the sec- ond 68 gal. 3 qt., from the third 39 gal. 1 qt., and from the fourth 54 gal. 2 qt. How much molasses did he sell ? 46. How much molasses was left in each hogshead ? In M and 3d, 69 gal. 3 qt. ; in 2d, 54 gal. 3 qt. ; in 4th, 67 gal. 3 qt. 47. How much molasses had he on hand ? 262 gal. N CHAPTER IV. FRACTIONS. SECTION I. ij\ri>iTCTioj\r Ajv^ j\roTATioj\r, 194i When an apple is divided into 2 equal parts, 1 of the parts is 1 half. When a pear or any thing is divided into 3 equal parts, 1 of the parts is 1 third, and 2 of the parts are 2 thirds. When a thing is divided into 4 equal parts, the parts are fourths. 1 half is written \ ; 1 third, \ ; 2 thirds, § ; 1 fourth, \ ; 2 fourths, I; 3 fourths, |. When a thing or a number is divided into 5 equal parts, the parts are fifths ; when into 6 equal parts, they are sixths ; when into 7 equal parts, they are sevenths ; and when into 8 equal parts, they are eighths. Fifths, sixths, sevenths, and eighths are writ- ten thus : 1 fifth, \, 2 sixths, 1, 1 seventh, ^, 4 eighths, f, 2 fifths, 1, 4 sixths, f , 3 sevenths, 5^ 5 eighths, |, 4 fifths, |. 5 sixths, %. 6 sevenths, %. 7 eighths, \, INDUCTION AND NOTATION. 1G3 195. A number that represents one or more of the equal parts into which a thing is divided is a Fraction. 196. The two numbers that are used in writing a fraction are the Terms. Thus the terms of the fraction § are 5 and 6. The 6 shows that a whole thing is divided into 6 equal parts, and the 5 represents 5 of these parts. 197. The term that expresses the number of equal parts into which a whole one is divided is written below a horizontal line, and is the Denominator ; and 198. The term that represents the number of these equal parts is written above the Hne, and is the Numerator. Thus, in the fraction ^, 4 and 7 are the terms ; 4 is the numerator, and 7 is the denominator. 199. When the numerator is less than the denom- inator, the fraction is less than 1 ; as, |, f , |, j^, -f\. When the numerator and denominator are equal, the value of the fraction is 1 ; as, f = 1, | = 1, j| = 1, 15 _ 1 24 _ 1 15 — ^} 24 — •*•• When the numerator is greater than the denomina- tor, the value of the fraction is greater than 1 ; as, |, |, 12 JJl 15 2 7 ~o'> 9 » «d2' 2 0* 200. A fraction whose value is less than 1 is a Proper Fraction ; as, |, |, j\, Jf , 3^. 201. A fraction whose value is equal to or greater than 1 is an Improper Fraction ; as, |, |, f , -y-, J/-, if. 202. Fractions whose denominators are alike are Similar Fractions ; as, |, |, and | ; |, |, J, and |. 203. Fractions whose denominators are unlike are Dissimilar Fractions ; as, |, |. 204. A number composed of an integer and a frac- tion is a Mixed Number; as, 4f, 31|. (See 119.) 164 FRACTIONS. 3x| f, and 4 X I = I. 205. Two of the 8 equal parts of this cake, or |, are 2 times as much as 1 of the parts, or I; and 1 are 2 times as much as f . Hence, 2 x | = I, and 2 X f = |. So, also, 3 X I = If we divide § of the cake into 2 equal parts, i part will be I ; if we divide | of it into 2 equal parts, each of the 2 equal parts will be f . So, also, |^2 = |, f-^2 — 7» 3 • ^ — 3> TT • ^ — TT* Jj-cnce, I. A fraction may he multiplied by multiplying its numerator. II. A fraction may he divided hy dividing its numerator, 206. If we divide a whole cake into 2 equal parts, each part wiU be | ; if we divide ^ of it into 2 equal parts, each part will be | ; and if we divide | of it into 2 equal parts, each part will be |. That is, 1 -^ 2 = ^, I — 2 = I, and ^ -^ 2 = i. Again, 2 of the eighths put together are |, 2 of the fourths together are ^, and the 2 halves are the whole X 1 = 4, 2 X i = i, and 2 X A = 1. caJce, or 1. That is, 2 I. A fraction may be divided hy multiplying its denom- inator. II. A fraction may he multiplied hy dividing its denom- inator. 207. If we divide ^ of a melon into 2 equal parts, we shall have | of a melon ; and if we divide | of a melon INDUCTION AND NOTATION 16i into 2 equal parts, we shall have § of a mel- on. That is, ^ and I But the ^ may be changed to f , and the i to f , by multiply- ing both terms of each frac- tion by 2. Thus, 1X2 — 2 3 X 2 — 4> and 1 X 2 4X2 Again, | of the melon are together equal to | of it? and I of a melon are together equal to ^ of it. That is, 1 = 1, and j = J. But the f may be changed to I, and the f to |, by dividing both terms of each frac- tion by 2. Thus, § i i = |, and 1 1: 1 = i- Hence, I. The value of a fraction is not changed by multiplying both terms by the same number. n. The value of the fraction is not changed by dividing both terms by the same number. 208. All operations in fractions are based upon the following General Principles of JFraclions, I. Afrajction is multiplied, 1. By multiplying its numerator ; or, 2. By dividing its denominator. n. A fraction is divided, 1. By dividing its numerator ; or, 2. By multiplying its denominator. m. TJie value of a fraction is not changed, 1. By multiplying both terms by the same number; or, 2. By dividing both terms by the same number. (See Manual, page 219.) 166 FRACTIONS. SECTION II. ^B^ ZTC TlOJSr. C-A.se I, Fractions to Lo^vest Terms. 209. When tlie terms of a fraction can not both be exactly divided by any integer greater than 1, the frac- tion is in its lowest terms. Thus, f is in its lowest terms, because no integer greater than 1 will exactly divide both 5 and 7. 210. To reduce a fraction to its lowest terms is to change its numerator and denominator to the smallest numbers possible, without changing the value of the fraction. Thus, ^ — \=^\. Ex. Keduce jf to its lowest terms. Explanation. — We reduce || to lower first solution. terms by dividing both numerator and i| = | = | denominator by 2, (jf = -|) ; and the fraction, f , thus obtained, we reduce second solution. to still lower terms by dividing both i| = | terms by 3, (f = |,) as shown in the First Solution. (See Prin. HE., 2.) Or we can reduce jf to its lowest terms at one operation by dividing both terms by 6, as shown in the Second Solution. 1. Reduce the fraction \ to its lowest terms. ^. 2. Reduce /^ mi. to its lowest terms. |- mi. 3. To what lower terms can || be reduced? J<^^ ^, or f. 4. Reduce ^-^^\\, and /^ to their lowest terms. |-^ |, j.. 5. In what lower terms can lyV^ be expressed ? REDUCTION. 167 6. Reduce f |, \^, and || to their lowest terms. 7. Redu6e the fractions f f and \y to their lowest teims. h V-. 8. What are the lowest terms of the fractions {■q%, j%, t^, 1^, rVa, and iff ? ff, |, f, |, Iv, f ." C-A.se II. Fractions to Given Denominators. 211. "We have already seen that the value of a frac- tion is not changed by multiplying both terms by the same integer. Thus, |^ | == J, f >^ f _ _6^^ | x 4 _, ^2^ I X i = If, and so on. Ex. Eeduce | to thirty-sixths. Explanation. — ^To reduce 2 ninths to solution. thirty-sixths, we must multiply both terms 36 [ 9 by such an integer as will give 36 for the 4 new denominartor. We find this integer by dividing 36 by 9. Multiplying both I x 4 = 3% terms of | by 4, the integer thus found, we have /g. Hence, f = /g^. JPJtOBZJEMS. 9. Reduce f to twelfths. Z^-. 10. Reduce f to twenty-firsts. 11. Reduce | to eighteenths, and | to twenty-fourths, -f^, /j. 13. Reduce | to sixths, to ninths, and to fifteenths. A 1^ and if 13. Reduce f to forty-fifths, and j\ to thirty-thirds. 14. Reduce f to fourteenths and to twenty-firsts. 15. Reduce f to fortieths and | to fortieths. ||^ If. 16. Reduce f and f to twenty-eighths. 168 FBACTIONS. CASE III. Dissimilar Fractions to Similar Fractions. 212i Ex. 1. Beduce | and J to similar fractions. Explanation. — Thirds can not be solution. reduced to halves, nor halves to 2 x 2 _ 4 thirds. But since 2 times 3 = 6, we reduce | to sixths by multiply- ^- ^ | = | ing both terms by 2 ; and since 3 times 2 = 6, we reduce ^ to sixths Hence, §, ^=§, §, by multiplying both terms by 3. Ex. 2. Keduce f , 1, and f to similar fractions. Explanation.— Since 3 x 4 x 7 = 84, solution. we reduce these fractions to 84ths ix4x7 = if by multiplying both terms of the | ^ § x 7 _ 2 1 first by 4 and 7, both terms of the 5X3X4_6o second by 3 and 7, and both terms ^ >< s x 4 - 84 of the third by 3 and 4. Hence, f , |, f = f f, f |. f f . 213. Hence, to reduce dissimilar to siii^ilar fractions, we Multiply both terms of each fraction by the denominators of all the other fractions. 214. Fractions having like denominators are said to have a common denominator ; and reducing dissimilar to similar fractions is sometimes called reducing them to equivalent fractions having a common denominator. JPjR oblems. 17. Reduce f and ^ to similar fractions. ^§, ^V* 18. Reduce ^ and f to similar fractions. 19. Reduce f and ^ to similar fractions. §^, 5^. 20. What similar fractions are equal to | and I ? 21. What similar fractions are equal to | and ^g ? 22. What similar fractions are equal to jf and I ? REDUCTION. 169 23. Reduce | and i\ to similar fractions. §§, ^|. 24. Reduc-j i, f, and f to similar fractions. ^^ ||_, |^. 25. Reduce f , ^, and f to similar fractions, ff^^ f-^-^^ ±^^. 26. Reduce |, |, ^, and f to equivalent fractions having a common denominator. //^, //^, //^, //g.. 37. Reduce |, |, f , and f to similar fractions. 28. Reduce |, j^2, and f to similar fractions. 29. What similar fractions are equal to |, ^, and | ? 30. Reduce i\, ^, f , and y*j to similar fractions. 31. Reduce ^, |, and f to equivalent fractions having a com- mon denominator. -^^, ||-_, ^^, O-AlSE IV. Improper Fractions to Integers or Mixed Numbers. 15( Ex. 1. In y- how many ones ? Explanation. — Since every 4 solution. fourths are 1, the number of 12 fourths [ 4: fourths times 12 fourths contains 4 ~^ fourths is the number of I's in 12 fourths, and 12 fourths con- Hence, ^ = S. tains 4 fourths 3 times. Ex. 2. In J3O- how many ones ? solittion. Tji^^^ .^,. „ -in +i.,-^^« «^^ 10 thirds I 3 ^/iWs ILxPLANATiON. — 10 tmros con- — ^ tains 3 thirds 3 times, with a ^i remainder of 1 third. (See Manual, p. 219.) Hence, ^ = S{'. 216. Hence, to reduce an improper fraction to an integer or a mixed number, we Divide the numerator by the denominator, PJROBZEMS, 32. In -3/- how many I's ? 13. 83. ?f ^ are how many I's ? 34. -Y" apples are how many apples ? 35. Y- melons are how many melons ? <^4. O 170 FEACTIONS. 36. Reduce V to a mixed number. IS^, or 13^, 37. Reduce Y? V) ^^^ V/ *^ mixed numbers. 38. \%^ gallons are liow many gallons ? <^||. 39. How much hay in 53 bales, each containing ^ T. ? 40. How many bushels of peaches in 176 baskets, each con^ taining | of a bushel ? S8§. 41. Reduce Yi^ 'IP? ^^^ W^ to integers or mixed num- bers. 7^, 2SS^§-, 23. :]Lntegers or Mixed Numbers to Improper Fractions. 217i Ex. 1. Keduce 5 to fourths. Explanation. — Since 1 is 4 fourths, 5 are 5 times 4 fourths, or 20 fourths. BOLUTION. 4 fourths _5 20 fourths Hence, 5 ~ Ex. 2. Reduce 5| to an improper fraction. Explanation. — Since 1 is 4 fourths, 5 are 5 times 4 fourths, or 20 fourths, and 20 fourths + 3 fourths are 23 fourths. FULL SOLUTION. 4:fourths 5 20fourths d fourths 2'Sfourths Hence, 5| COMMON BOLUTION. f 20 + 3 = 23 (See Manual, page 219.) 21 8t Hence, to reduce an integer or a mixed number to an improper fraction, we* First multiply the integer by the denominator, and, if there be a numerator, add it to the product ; then write this result for the numerator, and the given denominator for the denominator of the required fraction. PMOBI.JSM8. 42. In 15 how many thirds ? 43. In 24| how many fourths ? 44. What improper fraction is equal to 17| ? ADDITION. 171 28A 45. Reduce 31^ to an improper fraction. 46. Reduce 37 to sixths, and 11^*5 to fifteenths. 47. What improper fraction'is equal to 4:-^^ ? 48. How many fifty-seconds in 1\\ ? 49. 12,^3 equals how many thirteenths ? 50. 41 = how many thirty-firsts ? ' HV- 51. Reduce 24^, and 22^3 to improper fractions. 52. Reduce 5||, 214|, and IIS^V, to improper fractions. SECTION III. • CASE I. All the Parts Fractions. 219. Ex. 1. What is the sum of tV+t3 + t5 ai^d t-j? Explanation. — Since the parts boltttion. in these fractions are all of the J^ + ^3^ 4. _5^ j^ _2^ — j j same kind or denomination, (twelfths), and since the numerators express the num- bers of the parts, we add the fractions by adding their numerators. 1 + 3-1-5 + 2 = 11; and since the parts are twelfths, we write the denominator 12 under the 11. Ex. 2. What is the sum of |, j, and | ? Explanation. — ^^^ solhtiox. Fifths, thirds, and J + ^l = iS + IB + l§=W-=l|| fourths do not ex- press the same second solution. kind of things or l + KI = ''^^0^'' = W = 1|« parts, and hence they can not be directly added (See 20, 1). But they can all be reduced to similar fractions, (sixtieths), and these similar parts can be added, as shown in Ex. 1. 172 FRACTIONS. In reducing the dissimilar fractions to similar ones, the common denominator need be written but once, and the several numerators may be written above it, as shown in the Second Solution. (See Manual, page 219.) 220. Hence, to add fractions, we Reduce all dissimilar to similar fractions, add the nu- merators, and under the sum write the common denom- inator. JPM OBJ0EM8. 1. What is the sum of f and ^ ? §f, 2. Add I and I 3. Add I and |. Add f% and j-%. f|; /^. 4. If a family bum ^ T. of coal one month, and | T. the next, how much do they bum in the two montks ? 5. What is the sum of f , ^, and | ? ±f§, or 1//^. 6. A merchant sold | bu. of clover seed to one farmer, i bu. to another, and | bu. to a third. How much clover seed did he sell ? ^//-, or i| hu. 7. A teamster drew in three loads, | cd. of wood, f| cd., and \l cd. How many cords of wood did he draw in the three loads ? f^f§, or ^|f cd. 8. What is the sum of | and f ? 9. Add -y,, f, and f. f§j, or 2/^. 10. A market gardener has f of an acre of blackberries, ^ of an acre of raspberries, and f of an acre of strawberries. How many acres of berries has he ? 11. Keduce ^, f, and f to twelfths, and find their sum. 12. Reduce ^ da. ^ da. | da. and | da. to twenty-fourths of a day, and add them. §^, or l^j dwys. 13. Reduce %\, $f , $,-\, and IgV to himdredths of a dollar, and find their sum. $l-f^Q. 14. A sewing girl paid %\ for a thimble, $yV for needles, %% for silk braid, IfW ^^^ sewing silk, and %^-^ for a spool of cotton. How many himdredths of a dollar did each article cost ? How much money did she pay out ? ^l^U' % ADDITION. 173 CA.SE II. All or Some of the Parts Mixed Numbers. 221. Ex. What is the sum of 2|, 5^, 7, and ^ ? Explanation. — We write the parts with boltjtion. the integers in the same column, and re- 2 1 = 2f ^ duce the dissimilar fractions to similar 5| = 5Jg ones. Fifths, halves, and thirds can be 7=7 reduced to thirtieths. Since | = |4, 2| 4| = 4f g must equal 2f ^. So, also, 5 J = 5|f, and ^^'9 4f = 4§g. Adding the fractions, we have ^^ |§, or If J. We write the f § in the result, and add the 1 with the given integers. The sum of aU the integers, 19, written before the f §, gives 19§§, the required sum. PROBLEMS, 15. Add 2f and 4f . ^ 7/y. 16. What is the sum of ^ and 3f ? * 17. A copper-smith used 5f bu. of charcoal one month, and 6| bu. the next. How many bushels did he use in the two months ? (Fourths may be reduced to eighths.) 12 f du. 18. A farmer has 85| A. of cleared land, and 47f A. of wood- land. How many acres are in his farm ? 19. What is the sum of 241 j%, 4f , and 1§ ? 24-7^. 20. Add 8|, 6^, and 27f . 21. Mr. Wood's farm is If mi. long, and f mi. wide. What is the distance around it ? 3^§ mi. 22. A teamster drew two loads of straw, one weighing l^'V T., and the other H T. How much straw in both loads ? 23. What is the sum of $4f , $61, $23, and $if ? 24. What is the sum of l/o, 3^, 5f , and lOf ? 21^§i. 25. A fruit dealer bought l^f bu. of walnuts of one boy, and 2 ^ bu. of another. How many walnuts did he buy ? S§§ bu. 26. Add 3i, 14, 5t^3, I 41. 27^- 27. Add 19f , j\, I, 651, and 23. 109//^. 174 FRACTIONS. SECTION IV. S U:B T^A C TIOJV. OA.SE I. Both Numbers Fractions. 222. Ex. 1. From \l subtract j^. Explanation. — Since tlie fractions solution. are similar, we subtract tbe numerator j J — j^^ — -^^ of the less fraction from that of the greater. 11 — 7 = 4 ; and since the parts are all of the same kind or denomination, (fifteenths), we write the denominator, 15, under the 4. Ex. 2. From i subtract |. Explanation. — Since fifths first solution. and thirds do not express the 5~3==T5"~T5— To same kind of things or parts, g^^oxD solution, (See 33, I), we reduce the 4 _ 2 _. 12 - ro _ _2_ given fractions to similar 53 lo lo fractions, (fifteenths), and then subtract the less from the greater in the same manner as shown in Ex! 1. In reducing the dissimilar fractions to similar ones, the common denominator need be written but once. 223. Hence, to subtract fractions, we Reduce all dissimilar to similar fractions, subtract the less numerator from the greater, and under the difference write the common denominator. PB OBZEM8, 1. From i subtract f^g. -^j^. 2. From | subtract f. ^J. 3. What is the diflference between }f and \ ? 4. What is the diflference between $| and $/g ? ${-, SUBTRACTION. 175 5. From a jug that contained | of a gallon of boiled cider, a woman used 5 of a gallon. How much cider was left in the jug? §gal 6. One day A worked -{'^ of the day, and B ^ of the day. Which worked the longer ? How much the longer ? 7. A housekeeper bought i pk. of cranberries, and used I pk. the same day. How many had she left ? ^^ ph 8. From f subtract |. From ^ subtract ^^ 9. Charles lives f| mi. from the schoolhouse, and John ? mi. Which lives the greater distance from the schoolhouse ? How much the greater ? Charles, /^ mi. 10. The snow was -^^ ft. deep one night, and the next morn- ing it was I ft. deep. What depth of snow had fallen during the night? ^ft. 11. From y\ subtract f. 12. From y\ subtract ^. ^^. 13. If I pour y\ qt. of wine from a bottle containing I qt., hpw much wine will be left in the bottle ? -/g- qt. 14. From f | cd. of wood a teamster took {'^ cd. How much wood remained ? 15. A farmer bought ^l T. of plaster, and sowed yV T. on his clover lot. What part of a ton had he left ? §§ T. 16. What is the difference between f§ mi. and f mi. ? /s- mi. C-A.se II. The Minuend a Mixed Number or an Integer. 224. Ex. 1. From 5| subtract 2^. Explanation. — ^We write the subtrahend solution. under the minuend, and reduce the frac- 5| = S/j tional parts to similar fractions. We then 2| == 2j% subtract the fractional part of the subtra- ^5 hend from that of the minuend, and the *' integer of the subtrahend from the integer of the min uend. The result, Sjh, is the required difference. 1 -2 176 FRACTIONS. Ex. 2. From 7| subtract 4|. Explanation. — After reducing the boltttion. fractional parts of the given numbers 7| = 7^| = 6f g to similar fractions, we find that the 4| = 4:|§ = 4|g fraction of the subtrahend is greater W^ than that of the minuend. "We ^^ therefore take 1 of the 7, and unite its value (|J) with the |§, thus changing the minuend to 6f §. From this we subtract 4|§, in the manner shown in Ex. 1. Ex. 3. From 15 subtract 4f . solution. Explanation. — ^Before subtracting, we 15 = 14| reduce 1 of the 15 to sevenths, thus 4f = 4^ changing the minuend to 14|. j^T PItOBIj:EM8. 17. From 7| subtract 3|. 4^, 18. From 7^ subtract 3f. S§§, 19. From 8f subtract fi. 7§§.M 20. A merchant bought a cheese which weighed BOf lb., and sold 23| lb. of it. How much cheese had he left ? 57§- lb. 21. A piece of cloth that measured 43^ yd. before it was dressed, shrank 2f yd in fulling. How many yards did it then contain ? ^^H- 22. From 19y«y subtract lOf. 23. From 5| subtract 3^. 24. From 7 subtract dj%. 3/g, 25. A grocer bought 123^ lb. of butter. After selling 52 1 lb. of it, how much had he left ? 70§ lb. 26. A farmer cut 24f T. of hay from two meadows, cutting 9|| T. from one of them. How much hay did he cut from the other? HUT. 27. From 19f subtract f§. l^jh 28. If I have 5|^ acres of land, and I sell f of an acre, how much land have I left ? J^,^-^ acres. 29. From a bin which contained 4| bu. of potatoes, a house- keeper used 2| bu. How many potatoes were left in the bin ? MULTIPLICATION. 177 30. A merchant buys boots at $5} a pair, and sells them at $7 a pail'. What are his profits on each pair ? 31. A bin that will hold 190 bu., contains 104| bu. of wheat. How many bushels more will the bin hold ? SS^. 32. From 171 subtract ^. 33. From 103 subtract 40j\. 62j%: 34. Wishing to pay my butcher $8|, I hand him a lO-doUar bill. How much change ought I to receive ? ^1^, SECTION V. CASE I. The Multiplicand a Fraction. 225. Ex. Multiply | by 4. Explanation. — Since a fraction is multi- boltttion. plied by multiplying its numerator (See 2x4=| 208, 1), we multiply 2, the numerator of the given fraction, by 4, and under the product write the denominator. mOBJLEMS. 1. Multiply 2% by 5. ^. 2. What is the product of /^ multiplied by 7 ? 3. How much is 3 times f ? ^^ or f. 4. If a man and team can plow j\ of an acre in an hour, how much land can they plow in 4 hours ? ^ A. 5. How much is 7 times f ? ^^-, or 2f. 6. How much will 8 bushels of oats cost, at $| a bushel ? 7. How much will 5 cloth caps cost, at $| a piece ? 8. How far will a locomotive run in 34 minutes, at the rate of I of a mile a minute ? 15 mi. 9. If Y^o of an acre will pasture 1 cow through the summer, how many acres will pasture 18 cows ? 16^ A, 178 FRACTIONS. 10. A carpenter built 15 lengths of board fence, and each length was f | of a rod long. How long was the fence ? 11. Multiply f by 11, ii by 9, and j\ by 15. 12. How much is 6 times ff lb. ? 8 times j\ doz. ? 4^^ lb. ; 4-3 do^- C^SE II. The Multiplier a Fraction. 226. Ex. 1. Multiply 15 by f ; that is, find § of 15. Explanation. — | is equal to 2 times I, and | is the result of solution. dividing 1 by 3. Hence, to get | of 15 -^ 3 = 5 15, we first divide it by 3 to find j 5 x 2 = 10 of it, and then multiply the result, _ • 5, by 2, to find f of it. S^^^^' ^^ ^ f =^^- Ex. 2. Multiply 15 by j% ; or,' find j% of 15. Explanation. — ^We first di- solution. ^ vide 15 by 12 to find J^ of 15 -^ 12 = H 15, and then multiply the 15 y^^—T_5 — q_3 _.gi result, i|, by 5, to find {^ of '^ '^ '^ ' 15. Hence, 15 x /^ = 6j^. 227. Hence, to multiply any number by a fraction, we Divide the multiplicand by the denominator j and multi- ply the result hy the numerator. PMOJBZJSMS. 13. Multiply 18 by I ; that is, find f of 18. IS. 14. What is the product of 45 multiplied by § ? 15. How much is ^ of 43 yards of ribbon ? 24- yd. 16. Multiply 7 by j\. j-f, or Sj\. 17. I bought 300 lb. of nails, and used \ of them in building a bam. How many nails did I use ? 262^ lb. 18. A fat ox weighed 1,173 lb., and, when killed, the beef weighed \\ as much. How much did the beef weigh ? MULTIPLICATION. 179 19. Last year I gathered 13 bushels of plums from my gar- den, and f of them were damsons. How many damson plums had I ? 20. Multiply 57 by j'V, and 23 by i§. 26§, 2^. 21. A and B bought a mowing machine for $145, A paying i^g of the cost, and B {'^. How much did each man pay ? C-A.se III. Both Factors Fractions. 228. Ex. Multiply § by f ; or, find | of |. Explanation. — ^We first divide | by 3 to bolutiok. find^ of |. This we do by multiplying 4 _ _4^ the denominator by 3. (See 208, n.) We 4 x2 _ _8^ then multiply the result, /_, by 2, to find | ^^ ~ '^ of |. Hence, ixf,or§off = -,%. 229. Hence, to multiply a fraction by a fraction, we Multiply the numerators together for a new numerator, and the denominators together for a new denominator. The word of between fractions signifies multiplica- tion ; thus, I of J = I X |. PROBLEMS. 22. Multiply I by f. ^. 23. WhaHs the product of | x | ? ^, 24. Multiply ^ by I ; ^, by ^. 25. How much will f gal. of syrup cost, at $f a gal. ? $^. 26. How much will | yd. of brown linen cost, at $| a yd. ? 27. How much is f of \\ lb. ? |^ ^. 28. How much is | of ^ of an apple ? §■ of an apple. 29. How much wood is f of | of a cord ? 30. What part of a melon is f of f of a melon ? 31. Three men own a factory in equal shares. How much of the factory does each man own ? If one man sells \ of his share, what part of the factory does he sell ? i-. 180 FRACTIONS. 33. A man who owned { of a sMp sold f of his share. How much of the ship did he sell ? -fj^, 33. I of y'^o = what fraction ? 34. What is the product of f x f x f ? We multiply all the numerators together for the numerator of the product, and all the denominators together for the denominator of the product. 35. What is the product of | x f x y3_ ? ^^o_^ ^ ^^. 36. ixfx4xf = how many ? /-j-. 37. ^ of f is what part of 1 ? /y. 38. I of i of J is what part of 1 ? /g-. 39. What is the product of |, ^, and ^^ ? 0-A.SE3 I"Vr. One or both Factors Mixed Numbers. 230. Ex. Multiply 3^ by 2|. Explanation. — ^We first reduce solxttion. the mixed numbers to improper 3} = ? and 2? = § fractions, and then multiply, as 7x-8 = 56_92__.9i in Casein. ^ ^ 2 J n! Hence, S^x 2§=^ 9^. PBOBZEMS. 40. What is the product of 6 times 3f ? 22 j-. 41. How much will 9 barrels of flour cost, at $10| a barrel ? 43. If a man builds 4| rd. of stone fence in 1 day, how much can he build in 13 days ? 60§ rd. 43. Multiply 8 by 4|. 37^. 44. How much will 3| lb. of opium cost, at $9 a lb. ? 45. How much will | yd. of vesting cost, at $3| a yd. ? 46. Multiply 3,--^^ by 2f. 8§§. 47. What is the product of 4| x 3/y ? 17^. 48. Multiply 9f by ^^ ; 4^ by 11|. 5|| ; W- 49. How many sq. rd. in a field 36| rd. long, and 21 f rd. wide? 791 fi. DIVISION. 181 SECTION VI. ^irisioj\r. CASE I. The Divisor an Integer. 231. Ex. Divide f by 4. Explanation. — To divide I by 4, is to find | of |. To ^ ^ joLtrxioN^ _ e - 2 do this, we write \ of f , and I "^ ^ — I o^ I = 3% = I multiply, as in Case HI., tt o / _ o MultipHcation. (See 230.) ■^^^^®' ^ -^ ■♦ - #• 232* Hence, when the divisor is an integer, we Write it as the denominator of a fraction with 1 for a numerator t and multiply the given fraction by the fraction thusformed. FROBZJEMS. 1. Divide f by 4. /^, or ^. 2. Divide j\ by 5. Divide |f by 16. /j. ^. 3. If 4 lb. of sugar cost $||, how much does 1 lb. cost? 4. Six boys gathered || of a bushel of chestnuts, and shared them equally. How many chestnuts in 1 boy's share ? 5. A butcher packed f of a ton of pork in 8 barrels. How much did he put in each barrel ? ^^ T. 6. A seamstress used | of a yard of linen in making 9 col- lars. How much linen did she use for each collar ? 7. If 4 oz. of iudigo cost $f , what is the price of 1 oz. ? $^y, 8. Divide 2f by 8. Before dividing, reduce the mixed number to an im- proper fraction. 9. What is the quotient of 3| divided by 4 ? ^. 10. If a teamster draws 3| cords of stone at 15 loads, how much does he draw at each load ? ^ cd. 182 FRACTIONS. 11. What is the quotient of 15| divided by 18 ? |-||. 12. If 13 boxes of strawberries cost $3^, how much does 1 box cost ? 13. If 7 men can bind 22}f acres of wheat in one day, how much can 1 man bind ? S^ A. 14. How many times is 9 contained in 414 ^ -4#f • 15. Divide 400^^ by 23. 11 j\. C^SE II. The Divisor a Fraction. 5-^| Ex. 1. Divide 5 by | Explanation. — Since the quotient is not changed by multiplying both dividend and divisor by the same num- ber (See 208, HI), we multi- ply them both by 3, and thus obtain 15 for a new divi- dend, and 2 for a new divi- sor. Then, 15 -^ 2 = ^- = T^, the required quotient. Ex. 2. Divide | by |. Explanation. — We first multiply both dividend and divisor by 5, the denomina- tor of the divisor, (See 148), and then divide the new divi- dend, J^^-, by the new divisor, 2, as in Case I. (See Manual, page 220.) FIRST SOLTTTION. 5 X 3 = 15 I X 3= 2 15 -- 2 = -y = Hence, 5 ~ % = 7i. SECOND SOLTTTIOII. = 15-^2 Hence, 5 • f = n- Jj4-^2 FIE8T SOLUTION. I X 5 = -'/- |x5= 2 — 1 ^f 1 5_ _ _ 5 Ol -4- — Hence, | -^ SECOND SOLUTION. Hence, f -=- # = 1? =1? DIVISION. 183 234. From these examples it will be seen that, to divide by a fraction, we Multiply the dividend by the denominator of the divisor, and divide the result by the numerator. FMOBIjEMS 16. Divide 3 by f . 7|. 17. Divide 7 by ^, and 6 by f. 21; 8, 18. At $1 a bushel, how many bushels of apples can be bought for $5 ? ^I hi. 19. The dividend is 9, and the divisor is f. What is the quotient ? Jfi^. 20. What is the quotient of f^ divided by f ? 1^. 21. How many times is f contained in ff ? 22. How much ribbon, at %^^ a yard, can be bought for $f ? 23. K a horse walks a mile in y*j of an hour, how far will he walk in 8 hours ? 30 miles. 24. At $1 a pound, how much candy can be bought for $| ? 25. Divide 3} by 4|. Before dividing, reduce the mixed numbers to im- proper fractions. Thus, Si-J- 4| = -L«- -t- ^- = 4f = U- 26. What is the quotient of 8 -j- 2| ? 2^, 27. Divide \l by 3|. /^. 28. The dividend is 5f, and the divisor is f. What is the quotient? i^|.. 29. A shoe-dealer paid $150 for a case of boots, at $6| a pair. How many pairs of boots were in the case I 30. A man whose daily wages were $2 j, received at the end of the week $12f . How many days had he worked ? 5f . 31. I paid %ll for 5i lb. of rice. What was the price per lb. ? 32. How many rolls of wall-paper, each containing 4^ sq. yd., will be required to cover 58^ sq. yd. of wall ? 13, 33. What is the quotient of f divided by f ? 34. Divide 1 by i| ; h^ by ^ 3.5. . ^_5_. 35. What is the quotient of 15| -r- 24^ ? 144. 184: FRACTIONS. SECTION VII. CAJ\rCBZ ZiATIOJSr. ca.se I. In Multiplication. 235. What is the product of f , 4, and \\ ? Explanation. — ^In the First Solution we re- ^ ^ 2TL'''''T''l 10 duce the product, ^^^^, 5X7X22- tVs^ - suf = W to its lowest terms, by bbcond boltttion. dividing both dividend 13 and divisor by 4, and ^ x s^ x ^f = J| both terms of the result thus obtained by 7. But, in the Second Solution, we di- vide the numerator, 4, and the denominator, 8, by 4, and the numerator, 21, and the denominator, 7, by 7. We then multiply the remaining numbers in the numer- ators together for the numerator of the product, and the remaining numbers in the denominators together for the denominator of the product. The results in the two solutions are the same. (See Manual, page 220.) 236. The process of dividing a numerator and a denominator by any number, either in the same frac- tion, or in fractions which are to be multiplied together, is Cancellation. In multiJ)Ucation of fractions, whenever a numerator and a denominator contain the same factor, the process can be shortened by Cancellation. In this case, the product will always be in its lowest terms. PJROBIjEMS. 1. Multiply 4 by I ; I by /o ; and f by |f. /^,|, and J^. 3. What is the product of f , |, and f ? ^. 3. Multiply I, I, §, and | together. Division. 185 4. How much is | of | of | of i\ ? 5. f X 2| X 1| X I = what fraction? f. 6. j\ of I of I of f is what part of 1 ? ^, 7. How much must I pay for Q^\ ft. of gas-pipe, at $f a ft. ? 8. How many sq. ft. of oil-cloth will be required to cover an office table 3f ft. long, and 2| ft. wide ? 10. 9. How much will If^ yd. of linen cost, at $if a yd. ? 10. How much must I pay for cutting f of ^ cd. of wood, at |of$i|acord? $f C^SE II. In Division. 237. Ex. 1. Divide | by |. Explanation. — In this Solution we have ^'^^'^ s^^^™^- multiplied the nu- ^ • ^ — '^' • ^ — s oi -^ — 2^ merator of the divi- dend by the denominator of the divisor (5x4 = 20), and the denominator of the dividend by the numerator of the divisor (9 x 3 = 27). If we change the places of the second solution. terms of the divisor, and multi- f-^4=f^3=i? ply the dividend by |, the frac- tion thus formed, we shall multiply the same numbers together as in the First Solution. This is shown in the Second Solution. Hence, 238. In dividing by a fraction, we may Change the places of the terms of the divisor , and mul- tiply the dividend by the fraction thus formed. Ex. 2. Divide f^ by j|. Explanation. — ^We first in- solution. vert the divisor — that is, _5 _i. is _ J \a _ 4 change the places of the terms ^ ^ * * ^ ~ ^J^ ^~ -and then proceed as in Case I. P 186 FRACTIONS. PMOBIjEMS. 11. Divide f by |, and f by f . /^, and f . 12. How many times is ^^ contained m^^^'i 2^-. 13. I paid $58^ for building 3| rods of door-yard fence. How much did it cost me a rod ? $15^. 14. Divide 11 by 4^? 15. How many brooms can be made from 22i lb. of broom- corn, if y5^ lb. is used for 1 broom ? 6 doz. 16. A hardware dealer paid $45 for sheep shears, at %^-^ a pair. How many pairs did he buy ? 17. How many boxes, each 5| in, long, 41 in. wide, and 3 in. deep, will be required to hold as much as one box 8| in. long, 8^ in. wide, and 6 in. deep ? 6. 18. Divide i?_ of f by i of I of %. /^. SECTION VIII. ^HO'SZBMS IJ\r FHACTIOJVS, 1. A provision dealer sold 7 barrels of pork for $113|. What was the price per barrel ? $16{, 2. How many cubic feet are there in a block of marble 4^ ft. long. If ft. wide, and \ ft. thick? ^if. 3. How much will 1 pound of sugar cost, if 4| pounds cost $H? $2%' 4. An errand boy earned $f on Monday, %\ on Tuesday, and II on Wednesday. How much did he earn in the three days ? 5. Reduce |f to its lowest terms. 6. In 17y^, how many elevenths ? 7. How much can a man earn in f of a day, at $1| a day ? 8. A grocer buys cheese at %^§-^ a pound, and sells it at %\ a pound. How much does he gain per pound ? 9. If you pronounce 63 words each minute in reading aloud, how long vdll it take you to read a chapter containuig 5,796 words ? 111.82 min. REVIEW PROBLEMS. 187 10. The sum of two numbers is 83§, and one of them is 47^. What is the other ? 36^^^. 11. The difference of two numbers is 2f, and the greater number is 61 ^q. What is the less number ? ^^/^. 13. The difference of two numbers is j-^^, and the less num- ber is ^\. What is the greater number ? i-||-. 13. How many lights of glass, each containing | of a square foot, are there in a box of glass which contains 50 square feet ? 14. Reduce 3/y to forty-fourths. 15. Reduce f , f , and f to similar fractions. 16. Reduce ^%\^-2 to a mixed number ? 17. If llf doz. eggs cost %'^j%, how much will 13f doz. cost ? 18. How many pieces of stone flagging, each 23 ft. square, will it take to make a sidewalk 148^ ft. long and 5^ fb. wide ? 19. What part of 19^ is 18yL ? i.^. .20. One Saturday morning, Frank caught five fish, the first of which weighed 3|\ lb., the second 3| lb., the third 2| lb., and the fourth and fifth each 2| lb. What was the weight of all? 21. A farmer sold f of his land, and afterward bought 37f acres. He then had 1122^^ acres. How much land had he at first? ^ 12JfA. 22. What is the difference between f^ of 18f and | of 17| ? 23. How much will a turkey weighing 8}^ pounds cost, at $/;japound? ^%VV 24. If you sleep 1-^^ hours each day, how much time do you spend in sleep in 1 year ? 16 wh IQ-ji^ h. 25. A grocer bought a barrel of sugar, and after selling y\ of it, ^ of it, and -^% of it, he had 119 pounds left. How many pounds were in the barrel at first ? 272. 26. Reduce ^Vq ^^ its lowest terms. 27. A laborer laid 189 ,V rods of tile-drain in 13| days. How much did he lay each day ? ISf rods- 28. I paid 12 cents a pound for llf pounds of beef, but -^ of it was bone. As the bone was worth nothing, how much a pound did the meat cost me ? 16§^ cents. 188 FRACTIONS. 29. If a man can earn $17.50 in 26 days, when he works 10 hours a day, how much can he earn in 19 days, when he works 12 hours a day ? $62.70. 30. How much will 375 pounds of bone-dust cost, at $38 a ton? $7,125. 31. William is ^-^ as old as his father, his father is ^ as old as his grandfather, and his grandfather is 72 years old. How old is William ? 12 years. 32. Two men dug a well in five days, digging 11| ft. the first day, 6| ft. the second, 5i ft. the third, 3| ft. the fourth, and 3/2 ft. the fifth. How deep was the well ? 33. If Andrew can run 968f rods in 25 minutes, and Rich- ard can run 963 rods in 24 minutes, which can run the faster ? 34. If they start together, and run in the same direction for 18^ minutes, how far will they be apart ? ^-^/g- '"'o^^- 35. The minuend is 201 f, and the subtrahend \^. Wha* is the remainder ? 36. The dividend is {^ of If, and the divisor is | of 87|. What is the quotient ? j^j. 37. The sum of three numbers is 357 j^^, and two of them are 265 1 and \\. What is the third number? 90^^. 38. A sewing girl earns $| a day, and pays $2| a week for her board. How much money will she have at the end of 13 weeks, after paying for her board ? $16^. 39. How many square feet are there in a board 16 ft. long and I ft. wide ? 40. Four boys spent one Saturday in gathering walnuts. On dividing them, Albert received \ of all they gathered, Eobert |-\, Thomas f^^, and David 17 1 quarts. How many walnuts did each of the other boys have ? 41. How many walnuts did they gather in all ? 21m. 1 pk. S qt. 42. If 12| acres of wheat can be cut with a reaper in 1 day, how many acres can be cut in 3,4 days ? Jj^js' CHAPTER Y. PERCENTAGE. SECTION I. 239« In business transactions, hundredths of any thing or number are commonly called Fer Cent. Thus, .04 is 4 per ceni., .09 is 9 per cent., .48 is 48 per cent. Hence, I. Any per cent, of a thing or number is so many hun- dredths of that thing or number ; and, II. Per cent, may always be written as a decimal. 240. This character, ^, placed after a number, is the Commercial Sign of per cent. Thus, 15% signifies 15 hundredths, and is read 15 per cent. EXER CIS ES. 1. Write, decimally, 8 per cent,, 3 per cent., 5 per cent. 2. Write 14 per cent, 17 per cent, 28 per cent, 33 per cent., 40 per cent., 65 per cent., 10 per cent 3. Read the following decimals as per cent. : .07, .19, .30 .42, .50, .69, .06, .99, .75. 4. Eirst read, and then write, decimally, 12^, 29^, 63^, 90^. 5. Read, and then write, 1^ 9^, 10^, 56^, 47^. 6. Write, with the sign of per cent., the decimals in Exer- cise 3. 241. 125^ is written, decimally, 1.25, 308^ is written 3.08. VALUES. OF FRACTIONS IN DECIMALS. ^ = .5, and ^^ = .005 I = .25, and Ifo = .0025 f = .75, and |^ = .0075 I —. .2, and 4^ = .002 I = .125, and Ifo = .00125 f = .375, and Ifo = .00375 f = .625, and |^ = .00625 I = .875, and l^ = .00875 190 PERCENTAGE. 12^^ is written .12^ or .125, i% or .25% is written .001 or .0025. Hence, I. To express per cent, decimolly, two decimal figures are always required. II. To express more than 99 per cent.y an integer or a mixed decimal number is required. III. To express parts of 1 per cent., decimal figures or fractions at the right of hundredths are required. JEXEMCISES. 7. Write, decimally, 114^, 159^, 237^, 475^ 108^, 200^. 8. Read .038, .0425, .165, .43f, .311, .05^. 9. Write 6,% 16]^, 10^^, 18.7^, 22.5^, 31.25^ {^, Ifc. SECTION II. GBJSTB'RAZ dTTZIC^ATIOJSrS. 242» Per cent, may be applied to any number, great or small, concrete or abstract. 243. The process of finding any per cent, of a num- ber is Percentage. The result of the computation is also called Percentage. 244. The number expressing the per cent, is the Bate, or Bale Per Gent. 245. The number on which the percentage is com- puted is the Base. Ex. How much is 15^ of 125 ? '°''''"''''' Explanation.— Since 15^ of a ^"f^ ^'^ number is .15 of that number, -^ we multiply 125 by .15, as in -1^25 MultipHcation of Decimals. -^^^^ lo. f O Percentage. 246. ^ule for 'Percentaffe, Multiply the base by the rate expressed decimally. GENERAL APPLICATIONS. 191 PBOBI^EMS* 1. Of a flock of 125 sheep, 4^ were killed by dogs. How many sheep were killed ? 5. 3. At a school of 125 pupils, the average daily attendance is 88^ of the whole number. What is the daily attendance ? 3. From a cask of 44 gallons of oil, 5^ leaked out. How much oil leaked out ? 2.2 gallom. 4. Crystal Lake covers 87^ of a square mile. How many acres does it cover ? 556.8. 5. Of a cargo of 14,865 bushels of wheat, 12^ was injured by water. How many bushels were injured ? 1783.8. 6. A piece of cotton cloth containing 42 yards, shrank 6^ in bleaching. How many yards did it then contain ? 100^ of a number is the whole of it. 100^ — 6^ = 94^; and 94^ of 42 yd. = S9.4S yd. 7. A fruit-grower set out 250 peach-trees, but 18^ per cent. of them died. How many trees lived ? 205. 8. A steamboat company bought 5,280 cords of wood, of which 21^ was hickory, 33^ was beach, and the balance was maple. How many cords of each kind did they buy ? 9. A man husked 284 bushels of com, and- received 12^^ of it for his labor. How much com did he receive ? SECTION III. COMMISSIOJV. 247. A person who buys and sells goods for another, receiving for his services a certain 122.5U Interest (the principal), = $472.50, the 35Q p^,„,jp,, ^^OU-Di' 1472:50 Amount. JPJ2 OBZJEMS. 1. What is the interest of $25 for 1 year, at 6^ ? $1.50. 2. Find the interest of $132 for 1 year, at 5fo. $6.60. 3. Find the interest of $76.50 for 1 year, at 7^. $5,355. 4. What is the interest of $216.25 for 3 years, at 8fo ? $51.90. 5. A man paid a debt of $188.65, with interest at 10^, 1 year after it was due. What amount did he pay ? $207,515. 6. What is the amount of $3,750 for 3 years, at 5^ ? 7. February 11, 1864, I borrowed $250. How much did the debt amount to, February 11, 1867, interest at 6^? 8. What is the interest of $560.10 for 2 years, at 6i^ ? C^SE II. Interest for Months. 262. Ex. What is the interest of $84.18 for 7 months, at 6^? Explanation. — The interest for soltttion. 1 year is $5.0508 ; and the inter- $84.18 est for 7 months, or ^^ of a year, '^^ is y\ of $5.0508, which is $2.9463. $5.0508 mt. for 1 yr. To find ^2 of $5.0508, the in- I terest for 1 year, we multiply it $35.3556 [ 12 by the numerator, 7, and divide $2.9463 intfor^vyr. the product by the denominator, 12 (see 227) ; but 7 is the given number of months, and 12 is the number of months in a year. Hence, INTEEEST. 197 To find the interest for any number of months. Multiply the interest for 1 year by the given number of months, and divide the product by 12. (See Manual, page 220.) The answers to the remaining problems in this Chap- ter are given in accordance with the principle stated in Art. 159. PHOBZEMS. 9. What is the interest of $153.17 for 3 months, at 6^ ? 10. What is the interest of $18.72 for 5 months, at 4^ ? 11. Find the mterest of $584.34 for 1 yr. 4 mo., at 7^. (1 yr. 4 mo. = 16 mo.) $54.53. 13. If I have $1876.50 at interest for 3 yr. 5 mo., at 6^, how much interest will I receive ? $272.09. 13. What is the amomit of $394.25 for 6 mo., at 5^ ? 14. K I give my note for $275, Jan. 11, 1867, and pay it Feb. 11, 1868, with 7^ interest, how much do I pay ? 15. If I borrow $782, at 12^ interest, and pay the debt in 1 month, how much do I pay ? $739.32. CA.SE III. Interest for Days. 263. In computing interest, 30 days are called a month. Hence, Every 3 days are j^^, or .1 of a month, and Every 1 day is \ of j^^, or .0| of a month. 264. Ex. 1. What is the interest of $675 bolittion. for 18 days, at 7^? ^g^g* ^ Explanation. — Since every 3 days are .1 .07 of a month, 18 days are .6 of a month. $47^ We therefore multiply $47.25, the interest .6 for 1 year, by .6, and divide the product $28 850 1 J^^S^ by 12, as in Case II. $2i3625~^ "^^ 03r ^ « 198 PERCENTAG SOLUTION. $169.44 .06 Ex. 2. What is tlie interest of $169.44 for 1 yr. 3 mo. 17 da., at 6^? Explanation. — Since 1 yr, 3 mo. |10 1664 are 15 mo., and 17 da. are .5§ of a 15.5^ mo., the whole time is 15.5f mo. SS888~ We therefore multiply $10.1664, the 38888 interest for 1 year, by 15. 5 1, and di- 508320 vide the product by 12, as in Case 508320 n., and the result, $13.19, is the in- 101664 terest required. In multiplying by $158.25696 [12 I, we take | of the multiplicand twice. $13,188 TearSy months, and days can be expressed as months and tenths of a month, 16. What is the interest of $116.25 for 24 days, at 6^? $.JiB5. 17. If I borrow $819 for 20 days, at 8^, how much interest must I pay ? $3.64" . 18. How much interest will I have to pay, at 7^, on a loan of $1296, for 9 mo. 15 da. ? $1!1.82. 19. What is the interest of $936 for 3 yr. 2 mo. 29 da., at 10^ ? $sos,H- 20. Find the interest of $718 for 1 yr. 14 da., at 6^. 21. What is the interest of $48, from Nov. 23, 1866, to Dec. 8, 1867, at 7^ ? ^'^•^^• General ^ule for Interest. I. For 1 year, Multiply the principal by the rate. n. For 2 or more years. Multiply the interest for 1 year by the number of years. III. For any other time. Multiply the interest for 1 year by the time expressed in months and tenths of a month, and divide the product by 12. IV. For the amount. Add the interest to the principal. INTEREST. 199 PHOBJLEMS. 27. How much interest must I pay for the use of $756.50 for 5 years, at 7^ ? $261^.775. 28. A note of $1834.75, dated Oct. 9, 1867, was paid Oct. 9, 1868, with interest at 6^. What was the amount paid ? 29. There is a mortgage on my house and lot for $1244, with interest at 7^. How much interest is due annually ? 80. A teacher in St. Louis bought a house and lot for $3750, and paid for it at the end of 3 years, with interest at 6^. What amount did he pay ? ^^4^5. 31. What is the interest of $752.50 for 8 months, at % ? 32. What is the interest of $87.36, at 10^, from Feb. 10, 1866, to Oct. 10, 1867 ? $U.56. 33. A young man bought a watch for $85, and paid for it in 9 months, with interest at Ifc. How much did he pay ? SECTION VII. 1. On opening a box of 128 panes of glass, a glazier found ^\^0SZBMS; Mmhracing all the Principles and MetJwds of Computation in the Preceding Cliapters, (See Manual, page 220.) 1. A drover paid $35 each for 32 head of cattle, and $4'} each for 23 head. How many cattle did he buy, and how much did they cost him ? They cost him $2,086. 2. A farmer sold a farm of 96.23 acres at $85 an acre, and afterward bought another of 123.47 acres for $52.50 an acre. Which fann came to the more money ? How much the more ? The first farm; $1697.37^. 3. A dealer in garden seeds put up 8 bu. 2 pk. 5 qt. of peas in papers holding 1 pt. each. How many papers did he have ? 654" 4. How many years, months, and days old are you to-day ? 5. How long a time has elapsed since the Declaration ol American Independence, which was made July 4, 1776 ? 6. How many bushels of lime can be bought for $7.35, at $.20 a bushel ? 36.75. 7. A lady paid $52.50 for 28 yards of carpetmg. What was the price per yard ? 8. At what price per lb. must I sell 84:5 lb. of raisins, to receive $37.18 for them ? $.44. 9. If 1 lb. of cheese can be made from 3f qt. of milk, hov/ many lb. can be made from 35 1 qt. ? 9|. 10. A gardener has 12 hives of bees, and last summer he obtained S\ lb. of honey from each hive. How much honey did he get in all ? 11. A hop grower bought 3,875 hop-poles, at $22 a thou- sand. How much did be pay for them ? Since 1,000 hop-poles cost $22, 1 hop-pole must cost y^^oo (or .001) of $22, which is $.022 ; and 3,875 hop- poles, at $.022 apiece, must cost 3,875 times $.022, which is ^85.25. Q 202 MISCELLANEOUS PROBLEMS. 13. A nursery-man sold 144 apple-trees, at $12 a hundrecl. How much did he receive for them ? $17.28. 13. In building a house, I used 21,375 feet of clapboards, for which I paid $35 per thousand. How much did they cost me? $7Jj.8.12^. 14. What will be the cost of 7 pieces of sheeting, each con- taining 39 yards, at $.37^ a yard ? $102.37^, 15. How many cubic feet in a plank 16 ft. long, 1 ft. wide, and .25 ft. thick ? 16. A roadway 500 ft. long, and 16 ft. wide, is to be cov- ered 1.5 ft. deep with gravel. How many cu. yd. of gravel will be required ? J^J^Jf. cu. yd. Jf, cu.ft. 17. Two men start from the same place at the same time, and travel, one 43.82 miles, and the other 34.57 miles a day. How far apart will they be at the end of 7.5 days, if they travel in opposite directions ? 587.925 miles. 18. How far apart will they be in the same time, if they travel in the same direction ? 69.375 miles. 19. How many pieces of wall-paper, each 9 yd. long and \ yd. wide, will be required to paper 81 sq. yd. of wall ? 18. 20. A regiment was mustered into the army with 938 men. During the term of service, 93 men were killed, 76 died of sickness, 214 were mustered out, 295 were taken prisoners, 183 deserted, and 349 new recruits joined the regiment. How many men belonged to the regiment when its term of service expired ? J,,26. 21. If 1.25 lb. of rags are required for 1 yd. of rag carpet, how many yards of carpet can be made from 30 lb. of rags ? 22. How many miles of telegraph line can be constructed with 6,116 poles, if 22 poles are set to the mile ? 23. At the beginning of the year, the population of a cer- tain city was 31,675. During the year the number of deaths was 764, and the number of births 803 ; 1,236 people removed from the city, and 1,394 removed to it. What was the popu- lation at the end of the year ? 31, 872. MISCELLANEOUS PROBLEMS. 203 24. A freight train of 13 cars is loaded with 96 barrels of flour to each car. How many barrels of flour on the train ? How many tons ? 122.301^ tons. 25. An ice dealer delivered to his customers 1,296 lb. of ice daily for 27 days, 1,794 lb. daily for 26 days, 2,146 lb. daily for 56 days, 1,834 lb. daily for 24 days, and 1,310 lb. daily for 21 days. How many lbs. of ice did he deliver ? 26. How many tons of ice did he deliver ? 136.669. 27. How many pounds was his average daily delivery ? 28. In 13,427 pints of beans, are how many bushels ? 29. Into how many building lots can 12 acres of land be divided, each lot being 4 rods front and 10 rods deep ? 30. How many bricks can be put into a pile 12 ft. long, 6 ft. wide, and 4 ft. high, each bnck being 8 in. long, 4 in. wide, and 2 in. thick ? (See Manual, page 220.) 7, 776. 31. Reduce 204,080 cu. in. to higher denominations. ^ cu. yd. 10 cu.ft. 176 cu. in. 32. Reduce 11 mi. 4 yd. 2 ft. to feet. 68, 09 J^ ft. 33. If a load of hay with the wagon weighs 3,165 pounds, and the wagon alone weighs 1,249 pounds, how much is the hay worth, at ^11 a ton ? $10,538. 34. A grocer paid $60.75 for oranges, at $6.75 a box. How many boxes did he buy ? 35. A paper manufacturer paid $231.40 for rags, at $65 a ton. How many rags did he buy ? 3.56 tons. 36. A barrel inspector examined 400 barrels, and condemned ^\fo of them. How many barrels bore inspection ? 390. 37. A provision dealer bought pork at $12.50 a hundred, and sold it at an advance of 20^. At what price did he sell it ? $15 a hundred. 38. If I sell railroad stock which cost me $2,500, at a loss of 8|^, how much do I receive for it ? $2287.50. 39. How much will it cost to insure a factory for $28,000, at 2\% premium ? 204 MISCELLANEOUS PROBLEMS. 40. The owners of the brig, Ivanhoe, paid 1|^ for an insurance of $17,750 for a single voyage to the "West Indies. How much did the insurance cost them ? $199.68^. 41. How much must be paid for an insurance of $8,650 on a cargo of wheat from Milwaukee to Oswego, at |^ ? $54-06 j^. 42. A dairyman sent 4,320 lb. of butter to a commission- merchant, whose rate of commission was 4^ on his sales. He sold the butter for $.37^ a pound. How much did the dairy- man realize ? $1555.20. 43. An agent receives 40^ commission for selling maps. How much will his commission be on sales amounting to $3,280, and how much will the map publisher receive ? $1,312; $1,968. 44. What is the interest of $341.08 for 3 yr. 10 mo., at 5i^ ? $71.91. 45. To how much will $74.18 amount in 2 yr. 2 mo., at dfc ? 88.64^. 46. Find the amount of $250 for 5 yr. 7 mo., at 6^ ? 47. The owners of a vessel that was overdue from Liver- pool, fearing that she was lost, paid 18^ for an insurance of $32,000 upon her. How much did the insurance cost them ? 48. A gentleman whose house cost him $18,000, had it insured for $14,000, at |^ premium. Should the house bum down, what would be his entire loss ? $4122.50. 49. How many cubic feet in a pile of 20 planks, each 12 ft. long, 10 in. wide, and 2 in. thick ? 33^. 50. A farmer raised 8.5 acres of flax, which yielded 850 pounds to the acre. How much was the crop worth, at $.06 a pound ? $433.50. 51. How many acres are there in 100 miles of a road 4 rods wide ? 800. 52. In a box containing 50 sq. ft. of window-glass how maAy panes are there, each pane being 10 in. long and 8 in. wide ? 90. 53. If 5 cows eat 2y^^ T. of hay in 5 wk., how much hay will 1 cow eat in 1 wk. ? 175 lb. MISCELLANEOUS PROBLEMS. 205 54. How many yards of carpeting will it take to cover the floor of a parlor 6| yd. long and 5^- yd. wide? 35§r, 55. Keduce V/ ^^ a mixed number. 56. A wood-chopper cut three trees into cordwood. The first tree made 3| cd., the second 4j%, and the third 5^ cd. How much wood did the three trees make ? 57. What will be the cost of 850 handbills, at the rate of $3.50 for the first hundred, and $1.25 for each succeeding hun- dred? $12.87^, 58. Bought 18,280 ft. of pine flooring, at $24 a thousand. How much did it cost ? $J,S8.72. 59. A bushel contains 2150.42 cubic inches. How many cubic inches in 87.5 bushels ? 188161.75. 60. How many bushels in 135476.46 cubic inches ? 61. A farmer has a bin 7 ft. long, 6 ft. wide, and 4 ft. deep. How many bushels will it hold ? 135 hi., nearly. 62. A gallon liquid measure contains 231 cubic inches. How many cubic inches in 275 gallons ? 63. How many gallons in 13051.5 cubic inches ? 56.5. 64. A box 11 in. long, 7 in. wide, and 3 in. deep, will hold how many gallons ? 65. What is the capacity in gallons of a cistern 8 ft. long, 8 ft. wide, and 7 ft. deep ? 3351j\. 66. A tanner has a vat that will hold 1075.21 gallons. How many bushels will it hold ? 117.5. 67. At 2.^^ commission, how much will a commission-mer- chant receive for selling 750 barrels of pork, at $21 a barrel ? 68. A traveling agent sold 1,600 young apple-trees, at $16 a hundred. How much did his commission amoimt to, at 25^? $64. 69. How much will $290 amount to in 3 years, at 8^ in- terest ? 70. What is the interest of $2,750 for 8 mo., at 7.3^ ? 71. At 7^, how much interest must I pay for the use of $195.75, from May 13 to November 8 following ? $6.66. 206 MISCELLANEOUS PKOBLEMS. 72. A seamstress bought a sewing-machine for $75, paying $i5 down, and the balance in 4 months, with interest at 7^. What was the amount of the last payment ? $61.^0. 73. A note for $417.13, dated Jan. 10, 1866, was paid Dec. 14, 1867, with Qfo interest. What was the amount paid ? 74. A music dealer sold a piano, which cost him $324, for 33^^ above cost. How much did he get for it ? $4S2. 75. If 23 lb. of starch can be made from 1 bu. of corn, how much corn will be required for 63,365 lb. of starch ? 76. A railroad company having 343 A. of woodland, cut from 28.5 A. 58.75 cd. of wood per acre, from 93.3 A. 52.5 cd. per acre, and from 112.7 A. 48.25 cd. per acre. How much wood was cut, and how many acres of wood remained standing ? 120104 cd.; 108.5 A. 77. A farmer sowed 13^ bu. of wheat on 10^ acres of land, and the yield was 14^ bu. per acre. How much wheat did he get above his seed ? 139^ iu. 78. How many tons will 500 barrels of flour weigh ? 49. 79. 23 teams were employed 47 days in drawing earth for a railroad embankment, each team averaging 15 cu. yd. 24 cu. ft. a day. How much earth was in the embankment ? 17175 cu. yd. ^ cu.ft. 80. How much can I save in a year, if I earn $100 a month for 10 months, and spend $68.63 every month ? $176.^. 81. How many feet in a board 16 ft. long and 9 in. wide ? 82. A manufacturer finds that merino wool loses .42 of its weight in cleaning. How much weight will 2345.5 pounds lose ? 985.11 'pounds. 83. Find the interest of $25 for 7 yr. 3 mo. 24 da., at 10^. -■$18.29. 84. What is the amount of $110.62 for 3 yr. 7 mo. 28 da., at 7^? $138.97. 85. What sum must be paid to cancel a debt of $219.16, which has been due 1 yr. 6 mo. 14 da., at the rate of interest in this State ? 86. I bought a house for $1,850, and afterward sold it at an advance of 30^. How much was my gain, and for how much did I sell it ? Qain^ $555 ; Belling price^ $^,405. MISCELLANEOUS PROBLEMS. 207 87. The sum of three parts is 100 mi., and two of the parts are 33 mi. 225 rcL 2 yd. 2 ft., and 17 mi. 90 rd. 3 yd. What is the third part ? 88. 131 + I + 5^ + f 4- what number = 4rt ,% ? 27^h 89. After traveling 21^ and 18^^ of a journey of 425 miles, what i of the journey had I yet to travel ? How many miles had 1 yet to travel ? ^^9.25 miles. 90. How much will |^ of a ton of hay cost, at $7.50 a ton ? 91. Brass is composed of copper and zinc. In a quantity of brass that weighed f ^ of a ton, the copper weighed ^\ of a ton. What was the weight of the zinc ? ^^ T. 92. If a family burn f| of a cord of wood in 30 days, how much wood will they burn in 1 day? tIs ^* 93. From Tvhat number must I subtract 984,006 to obtain a remainder of 9,276,985 ? 10,260,991. 94. The subtrahend is six thousand and twenty-four ten- thoqsanths, and the remainder four thousand ninety-six hun- dred-thousandths. What is the minuend ? 95. The minuend is 17 cu. yd., and the remainder 16 cu. yd. 1,596 cu. in. What is the subtrahend ? 26 cu.ft. 132 cu. in. 96. If 16 shoemakers make 5^0 pairs of shoes in 16.25 days, how many pairs can 1 workman make in 1 day ? 2. 97. A farmer exchanged 85 lb. of butter at $.21 a pound, for flannel at $.35 a yard. How many yards did he receive ? 98. If a young man smokes 1,284 cigars in a year, how much will his year's supply cost him, at $28.50 a thousand ? $36.59. 99. What will be the cost of 17,890 feet of hemlock lumber, at $11 a thousand ? 100. What number, multiplied by 7,296, will produce 292,518,528 ? 101. The multiplicand is 19^, and the product 319?. What is the multiplier ? 16^, 102. The product of three factors is 29.2923, and two of the factoi-s are 11.4 and 285.5. What is the third factor ? 103. How many square inches in a mirror 5 ft. 3 in. high and 26 in. wide ? 208 MISCELLANEOUS PROBLEMS. 104. In a village lot 66 ft. front by 133 ft. deep, are how many sq. rd. ? 32. 105. K 16 men in 17.25 days can mine 529.92 tons of iron ore, how miicli ore can 1 man mine in 1 day ? 1.92 T. 106. A clergyman had his household furniture insured for $850, and his library for $650, at |^. What premium did he pay annually ? $11.25. 107. The dividend is 9, and the quotient 576. What is the divisor ? .015625. 108. What number divided by 19^ "3 will give a quotient of ••■1 21 * 109. A farmer cures his hams by the following recipe : For every 100 lb. of meat, 9 lb. of salt, 5 oz. of saltpeter, 4 oz. of ground pepper, and 1 qt. of molasses. What quantity of each ingredient must he use for 675 pounds of meat ? Salt^ 60.75 lb. ; saltpeter, 83.75 oz. ; ground pepper, 27 os. ; molasses^ 6.75 qt. 110. In salting beef, the same farmer uses the following recipe: For every 100 lb. of meat, 6 qt. of salt, 1 qt. of molasses, and 4 oz. of saltpeter. What quantity of each in- gredient must he use for 380 lb. of meat ? ^ Salt, 22.8 qt. ; molasses, 3.8 qt. ; saltpeter, 15.2 oz. 111. Which is the more advantageous, to borrow $175 at 7^ interest to pay house-rent m advance, or to pay $200 rent at the end of the year ? How much the more advantageous ? To lorrow, hy $12.75. 112. A man can hire a farm of 97 acres for $500 per annum, or he can buy it for $70 an acre. If money is worth 6^, which is the cheaper course, and how much the cheaper ? To luy the farm, Iry $92.60 per annum. 113. A farmer bought 80 sheep, at $4.20 a head, giving his note, payable in 6 mo., at 7^. At the end of the 6 mo. he sold the sheep at $5.25 a head cash, and paid the note. How much did he get for the keeping of the sheep ? $72.24. 114. A man had his life insured for $5,000 at the age of 29 years, and died at the age of 47. His yearly premiums aver- aged $71.85. How much more did his family receive than he had paid ? MANUAL OP METHODS AND SUGGESTIONS. A Word with Teachers. — This Manual is intended to give you hints and suggestions rather than detailed methods of instruction ; and to call your attention to those points which require your special efforts, if you would secure that thoroughness in your pupils which is essential to real progress. It is not intended to lay down pre- scribed forms for you to follow, hut to give you such hints and sug- gestions as will enable you to work out details of methods of in- struction, and to adapt them to your classes, in accordance with your methods of thought. Use of Objects. — Children gain ideas more readily by perception than by reflection. You should, therefore, illustrate the subjects of lessons by Visible Objects whenever this is practicable. For example : In Notation. — A child may be aided in comprehending the idea of a ten and of a hundred by the use of counters. Take a quantity of beans, or other suitable objects, make a pile of ten of them for a ten, and ten such piles near together for ten tens or a hundred. In Addition.— Illustrate the addition of two numbers, as 37 and 48, thus : 3 piles of tens and 7 counters or ones may represent 37 ; and 4 piles of tens and 8 counters or ones may represent 48. Then put- ting the 7 counters and 8 counters together, there will be enough counters for 1 ten and 5 counters or ones more : The 1 ten, 3 tens, and 4 tens, together, are 8 tens, and the 8 tens and 5 ones, are 85. In Subtraction. — The process may be illustrated in a similar manner. In Compound Numbers. — Let the pupils see and handle the various measures, and they will get clear ideas of a quart, a bushel, a foot, a pound, or any other denomination. Let them measure water, and see that 2 pints are 1 quart, and 4 quarts 1 gallon. Let them exercise their judgment upon the capacity of vessels, by estimating how much a pail, a pan, a pitcher, a cup, or a bottle, will hold ; and then require them to measure the vessel to correct their judgment. In the same manner let them measure sand, or com, to become familiar with the denominations of dry measure. Also, let them by trial see if a quart liquid measure is the same as a quart dry measure, etc. To make them familiar with the denominations of distance, let them draw lines, an inch, a foot, and a yard long, upon the black- hoard : judge of the width, length, and height of the room, or the 210 MANUAL FOK TEACHERS. house ; and then measure the various distances to correct their judg- ment. Again, for greater distances, let them measure the distance of a rod along the fence, then 40 rods or 1 eighth of a mile on the road. Let them estimate the distances between different objects in the room, about the yard, and along the road, and afterward measure them. If no other measures can be obtained, cut from a lath, or any other straight stick, a piece 1 inch long ; another 1 foot long, and mark it off into inches with a knife or pencil ; and another 1 yard long, and mark it off into feet. A cord or a piece of rope may be used for the measure of a rod. In square measure, have a square inch, a square foot, and a square yard. You can show that these might be used to measure with, but that it is more convenient to take dimensions with linear measures. The measure of a cubic inch may be made by cutting out a piece of pasteboard in this form, cutting about one half through the thickness where the lines cross it, folding it together, and fastening the joining edges. Larger cubes may be made in the same manner. A box 1 foot long, 1 foot wide, and 1 foot high, is a measure of a cubic foot. The surface should be marked off into square inches. If your schoolroom is not supplied with scales and weights, you can prepare a few weights by making packages or bags of shot, sand, or pebbles, weighing 1 oz., 1 fourth lb., 1 half lb., 1 lb., 5 lb., and 10 lb. "With these and other objects you can exercise pupils until they can judge of weight with considerable accuracy. Original Illustrations. — In many cases you will interest your pupils, as well as assist them to comprehend a problem, by drawing figures or diagrams upon the blackboard. Tables of Combinations. — Require the pupils to construct the Tables of Addition, Subtraction, Multiplication, and Division, before committing them to memory. They can do this by using counttrs to form the combinations. In this way they will more fully compre- hend the meaning of tlie tables, and also prove them to be correct. Only a small part of any table should be assigned for one lesson, and upon that the pupils should be made thorough before an advanced lesson is assigned. Let them from memory write that part of the table learned upon their slates or paper, and upon the blackboard. Also yourself write upon the blackboard portions of the table, not in regular order, and without the results, and require the pupils to give the results without hesitation. For example, take the 4's in the Ad- dition Table, and writing them as directed, 44:44:4:44:444 4 8 3 9 5 7 MANUAL FOR TEACHERS. 211 point to the numbers forming any combination, ai:!d require instant answers. Tliis may be done first with the whole class, and then with each pupil separately. Pursue a similar course with review exercises in all the tables. Also, give numerous problems to lae class, and require instantaneous answers ; and let pupils give original problems to each other for mental solution. Inductive and Oral Exercises. — '* Make haste slowly" in pass- ing over these portions of the book. Exercise pupils upon the definitions and signs until they are perfectly familiar with them. The Oral Exercises should be assigned to the class in connection with the Tables of Combinations. For example, when the class have learned the addition of 3's from the table, assign the oral exercise in counting by 3's. The forms given for the first two or three exercises under each number should be continued through all the exercises. Give your pupils frequent practice upon these exercises. Let the school to- gether recite from them 5 minutes at a time, or when they are passing into and out of the room, or when classes are moving. All the possible combinations of the numbers used are given in the oral exercises. If pupils are ready in these, they will be ready and accurate in computations. Conducting Recitations. — In conducting recitations, never for- get that the ends to be accomplished are fourfold, viz. : 1st. To impart new and valitable instruction, adapted in kind and amount to the condition of the minds of your pupils ; 2d. To teach pupils to think, by so guiding their inquiries that they shall discover truths for themselves ; 3d. To make them thorough, by always requiring accurate recitations and explanations ; and, 4:th. To keep them interested in their studies. The following order in conducting recitations has been found to secure these results : 1st. Hear as many of the class recite the lesson assigned as time will permit, requiring them to go through the recitation without inter- ruption from other members of the class, and with as little prompting and as few questions as possible from you. Throw no stumbling- blocks in their way at this time ; for pupils who recite a new lesson well, do all you have a right to ask of them at first. 2d. After this, test their knowledge of the lesson, by fair but criti- cal questions. In this way you will find what instruction they need. Sd. Impart the needed instruction and no more, always observing this rule : ^^ Never tell a child anything you wish Mm to remember, with- out requiring him to tell it to you again.'''' 4:th. Make practical applications of the lesson. 5th. Review such portions of previous lessons as you deem im- portant. 212 MANUAL FOB TEACHERS. To express in words what we have learned, is to make the knowl- edge our own, to make it more clear to our understanding, and to fix it in our memory. Therefore, require pupils to write out upon their slates or paper a full explanation of one or more of the problems in the lesson, before coming to recitation, and also upon the black- board at recitation. The explanations of examples solved in this book may be taken as models ; but as many of these, especially in integers, are more minute than you should require for the solution of the problems, a few hints are here given that may aid you in properly directing your pupils in their explanations. Addition. — Suppose you have assigned a solution and explanation of problem 81, page 29, to be written out by a pupil. His explanation may be as follows:. i%e farmer raised as many 'bushels of grain as the sum of 687 hushels^ 1,229 bushels, 643 bushels, 184 bushels, 259 bushels, and 296 bushels. Since only figures occupying the same place in different numbers can be added, I wrote the given numbers (or the given parts) with ones undfr ones, tens under tens, and so on. I then commenced with the ones, and added each column separately, writing the right-hand figure of each sum under the column added, and adding the left-hand figure with the next column. The sum, 3,198, is the total number of bushels of grain which the farmer raised. Subtraction. — For example, take problem 57, page 41. The pupil's explanation may be this : The hay weighed as much as the difference between 2,656 pounds and ^^^ pounds. Since only figures occupying the sam£ place in different numbers cam. be subtracted the one from the other, I wrote the subtrahend below the min/uend, placing ones under ones, tens under tens, and so on. I then commenced with the ones, and subtracted each figure of the subtra- hend from the corresponding figure of the minuend, and wrote the result directly below in the remainder. The remainder, 1,669, is the number of pounds of hay in the had. You may require the pupil to embrace, in his explanation, the pro- cess when any figure of the subtrahend exceeds the corresponding figure of the minuend, until you are sure he is familiar with it ; after- ward it may be omitted. Multiplication. — Take problem 107, page 61. The explanation of the solution may be as follows : The cost of manufacturing SGO pianos was 360 times as much as the cost of manufacturing 1 piano. I therefore multiplied $270, the cost of manufacturing 1 piano, by 360, and I obtained $97,200, the total cost. You may require the pupil to explain the process — writing the fac- tors, multiplying, adding partial products, and annexing ciphers for MANUAL FOR TEACHERS. 213 the final product— until he is familiar with all the stepg ; after which an explanation like the above is sufficient. Division. — First Form: both terms like denominations. — Take problem 37, page 74. Since $6 was the price of 1 cord of wood, for $6,828 as many cords were as the number of tiwM $6 is contained in $6,828, which is 1,138 Second Form : the divisor an abstract number. — Take problem 51, page 76. He received 1 twelfth as much for building 1 rod offence as he did for building 12 rods. I therefore divided $156, the cost of building 12 rods, by 12, and the result, $13, is the cost of building 1 rod. You may require the pupil to explain all the steps in the process of Division untU he is familiar with them ; after which the above form of explanation is sufficiently full. The above five explanations may be modified to meet the conditions of any problem in integers, decimals, compound numbers, percentage, and for most problems in Fractions : Fractions. — The two following explanations may be of some aid in Multiplication and Division of Fractions. Problem 17, page 176. — Seven eighths of any number is 7 times as much as one eighth of the number; and one eighth of any number is found by dividing it by 8. I therefore divided 300 pounds by 8 to find 1 eighth of them, and then mul- tiplied the quotient, 37^, by 7, to find 7 times 1 eighth, or 7 eighths of them. The residt, 262^, is the number of pounds used. Problem 22, page 181. — Since $^ is the price of 1 yard of ribbon, for %i as many yards can be bought as the number of times $^ is contained in $f . $To is contained in $| as many times as 10 times $^, or $3, is contained m 10 times $|, or $\° ; and $3 is contained in $\°, ^J" or 2§ times. Therefore, 2i yards is the required result. Combinations. — Rapidity in computation is an acquirement much to be desired by all. Stimulate pupils to pronounce results rapidly, and without naming the numbers combined. Thus, 9 In Addition— Insiedu^ of allowing them to say, 7 and 5 are 13, ^ 12 and 6 are 18, 18 and 9 are 27, they should be taught to pro- 7 nounce the partial results orally, as they point to each figure ~ added; thus, 7, 12, 18, 27. In Subtraction — The usual manner is this : 2 from 6 leave S406 4 ; 3 from 10 leave 7 ; 5 and 1 are 6, and 6 from 14 leave 8 ; 1 ^^? and 1 are 2, and 2 from 3 leave 1. Teach pupils to perform ■^*^* all the combinations mentally, and to pronounce the partial results orally ; thus, looking at the 6 and 2, the pupil says 4 ; looking at the and 3, he says 7 ; looking at the 4 and 5, he says 8 ; and looking at the 3 and 1, he says 1. 214 MANUAL FOR TEACHERS. In Multiplication — Teach him as each figure of the mul- 834:5 tiplicand is reached, to pronounce the product and the "^ sum; thus, 35 ; 28, 31 ; 21, 24; 56, 58. 584:15 In Short Division— Ld him name only the quotient 11364(4 figure and the remainder ; thus, 2 and 3 over, 8 and 1 2841 over, 4, 1. Require pupils to study the solutions and explanations, and to state the principles upon which any given method is based. No rule should be assigned as a lesson until the principles involved are clearly understood. Pupils should commit to memory the definitions, principles, and other matter in Italics, but in all cases they should be required to show that they understand what they have memorized. If you find that pupils ' work to get the answers' given in the book, rather than to understand and apply principles and methods, change one or more figures in the problems, when you assign a lesson, and the printed answers will be of no help to the pupils. Many of the Review Problems in each chapter may be solved in more than one way. Call out the diflferent methods from the class, and thus stimulate them to think : and after the solutions have been presented, exercise their judgment by requiring them to state which of the solutions presented is the best, and why. We will now pass to the suggestions to which references are made in various parts of the book. Page 10.— Exercise pupils in writing and reading all the numbers to 100. Require them to write the numbers given in the Exercises, in columns, neatly, placing ones under ones, and tens under tens, and to bring their work to recitation for inspection. Page 11. — Drill the class in writing and reading numbers of three figures until they make no mistakes. Then require them to write the exercises given on page 12, first upon their slates, and afterward upon the blackboard. See that their work is neatly done, the figures well made, and ones written under ones, tens under tens, etc. Page 13.— Practice pupils upon the two periods, until the places and their names are familiar to them. You will aid them in this, by writing upon the blackboard two periods of ciphers, and under these require the pupils to write the exercises, and also to name the place occupied by each cipher. Page 15. — To enable pupils to write and read numbers readily, they should be able to tell promptly how many figures are required to express any given number less than 1,000,000,000. To secure this ability on their part, you can frequently question them in this man- ner : How many figures express ones ? Thousands ? Millions ? etc. MANUAL FOR TEACHERS. 215 How many figures are required to express 50 thousand ? Ho-w many to express 50 thousand 7 hundred ? To express 50 thousand 38 ? etc. How great a number can be expressed by four figures ? By seven figures ? By five figures ? etc. How many figures are required to express ten-thousands ? How many to express hundreds ? Hundred-millions ? etc. Require pupils to point off all whole numbers iuto periods of three figures each, commencing at the right. Explain to them that, in reading numbers, they must always com- mence at the left, and read each period by itself as a distinct number, pronouncing after it the name of the period. Before assigning the Review Exercises on page 16, drill the class both in reading and writing numbers : first, upon numbers containing no ciphers ; next, upon numbers containing one cipher, — the cipher occupying a different place in each number ; then upon numbers con- taining two ciphers in all possible places, both together and separate ; then upon numbers containing three ciphers, and so on. The Heview Uzercises (page 16) are test exercises ; and pupils should be able to write all of them before passing to the next section. Page 17. — See suggestions on Inductive and Oral Exercises, page 211. Page 19.— See suggestions on TaUes of Combinations, page 210. Page 20. — See suggestions on Tables of Combinations, page 210. Page 22. — Explain to pupils that, since we can only add figures occupying the same place in different numbers, and since it is more convenient to have the figures to be added stand in a column, we write the parts, for convenience, ones under ones, tens under tens, etc. Also explain that we commence at the right to add, not from necessity, — for we may commence at any other place, — ^but because it is more convenient. In order that your pupils may do their work methodically, require them uniformly to commence either at the top or at the bottom of columns of figures to add them. Page 24. — ^Explain the terms parallel and Tiorizontal, bo that your pupils will clearly understand their meaning. Page 25. — The signs for wood-land, meadow, fences, streams, etc., on this map are the conventional signs used by topographers and surveyors. They always mean the same thing whenever found upon a properly drawn map. Problem 41. — Be sure that the pupil discovers that there are two fences, each 47 rods long, and two others each 34 rods long. Page 26. — ^If more problems are required, a large number may be formed from this table of railroad distances. Thus, assign for a lesson 216 MANUAL FOE TEACHERS. to find the distance from Worcester to each of the other places named in the table. Page 37.— Require the pupil to add an example, first by the method given on page 34, and then by the method here given. Then require him to compare the results, and thus to discover that the latter method is only an abridgment of the former. Page 31. — See suggestions on Inductive and Oral Exercises^ page 211. Page 33. — See suggestions on Tables of Combinations^ page 210. Page 33.— See suggestions on Oral Exercises^ page 211. Page 35. — If Principle II. is not understood by your pupils, lead them to see the truth of it, by questions. Thus : What is the dificrence between $7 and $3 ? Between 7 boys and 3 boys ? Between 7 tens and 3 tens ? Between 7 thousands and 3 thousands ? etc. Page 37.— The method of adding 10 to both minuend and subtra- hend before subtracting, when any figure in the subtrahend exceeds the corresponding figure in the minuend, is thought to be too difii- cult for quite young beginners, and hence it is not introduced into this book. Those teachers who prefer to use it, will, of course, do so. By the aid of objects show the class that 1 ten and 6 ones are 16 ones, and that 7 tens and 6 ones are 6 tens and 16 ones. Page 40.— Require the pupil to solve this example first by the method given on page 37, and then by the method here given. He will thus see that the latter method is only a abridged form of the former. Page 41.— Require your pupils uniformly to call the next left- hand figure of the minuend 1 less, or the next left-hand figure of the subtrahend 1 more. Page 45. — See suggestions on Indtcctive and Oral Exercises, page 211. Page 47. — See suggestions on Tables of Combinations, page 210. Page 48. — See suggestions on Indicctive and Oral Exercises, page 211. Page 49. — Very few learners fully comprehend the fact that every problem in multiplication can be solved by addition. In order to fix this fact firmly in the minds of your pupils, require them to solve the first 10 problems on page 50, both by addition and multiplication. Page 53, — Require pupils to speak of the true multiplicand only as the multiplicand. For example, in finding 345 times 7, they may multiply 345 by 7 ; but in the explanation of their solution, require them to say 345 times 7. 67000 2100 67000 21 6700O 134000 67 134 140700000 1407000 lOO MANUAL FOR TEACHERS. 217 Page 61,— If this explana- tion is not fully understood, solve the example in diiferent ways, and compare the results. Thus, you may use either of the foUowing solutions : 140700000 But, in practice, never permit pupils to write the factors for multi- plication as shown in the first of these solutions. Page 62.— First Reference.— Be sure to give your pupils clear ideas of the terms /actors and parts. Thus, 7, 5, and 2 are factors of 70, but they are parts of 7 + 5 + 2, or 14, and also of 752. Second Reference.- Round trip, out and back again, or twice over the road, once each way. This is also called doubling the road. Third Reference. — See suggestions on Conducting JRecitations, page 211. Page 64.— See suggestions on Inductive Exercises^ page 211. Page 65. — Illustrate, by the use of objects, fractional division, or the division of a given number of things into a certain number of equal parts. For example, with a quantity of beans or com, or a large number of other counters before the class, divide the number of objects into 2, 3, 4, 5, etc., equal parts, and then require each pupil to do the same. Also, ask numerous questions like these : How can you find 1 half of these objects ? How 1 third of them ? How 1 fifth of them ? 1 fifteenth ? 1 fortieth ? etc. How can you find 1 third of any number? How 1 eighth of it ? How 1 twenty-fourth ? 1 seventy- first ? 1 ninetieth ? etc. Page 66, — See suggestions on TaUes of Comiinations, page 210. Page 67.— See suggestions on Oral Exercises, page 211. Page 73. — Require the class now to solve the problems on pages 70, 71, 72, and 73, by short division. Page 75.— Do not permit pupils to leave these 12684 l 28 principles until they thoroughly comprehend them. 1 4413 You can test their understanding by placing erro- ^^o neons processes upon the blackboard, and requiring -^g them to point out the errors, and to tell why they 28 are wrong. Thus, place upon the blackboard this 84 example, and require pupils to correct it, and to ^ give reasons for the correction : Page 98. — If your pupils are slow to learn the notation of deci- mals, you may find the following direction of value to them : To Wrue any Decimal Number, Write it first as an integer, and then place ihe deeimdl poirU according to the table of values of decimal numbers. R 218 MANUAL FOR TEACHERS. Exercise your pupils until they are familiar with this table. Ask such questions as the following, until they can give correct answers promptly : How many decimal figures express thousandths ? How many hun- dred-thousandths ? Hundredths? Ten-millionths ? etc. "What decimal is expressed by four decimal figures ? By three figures ? By eight figures ? etc. Page 99. — Pupils should be thoroughly grounded in this table, as a knowledge of it will enable them to master the theory of computa- tions in decimals with increased facility. For a test of their knowl- edge, write a line of I's on the blackboard, 111111111.11111111, and, commencing at the right, require your pupils to give the law of increase toward the left, first to the decimal point, and afterward to hundred-millions ; thus, 10 hundred-millionths are 1 ten-millionth, 10 ten-millionths are 1 millionth, 10 millionths are 1 hundred-thou- sandth, and so on, to 10 ten-millions are 1 hundred-million. Then, commencing at the left-hand figure of the integer, require them to give the law of decrease to the right ; thus : 1 hundred-mil- lion is 10 ten-millions, 1 ten-million is 10 millions, 1 million is 10 hun- dred-thousands, 1 one is 10 tenths, 1 tenth is 10 hundredths, 1 hundredth is 10 thousandths, 1 ten-millionth is 10 hundred-mil- lionths. After the class can run through this number with facility, write lines of 4's, 7's, 3's, and so on, and practice them upon the numbers thus formed in the manner above directed. Page 100.— First Reference.— Illustrate these principles by numerous examples like these; show that .5 = .50 = .500 = .5000; that .75 = .750 = .75000, and so on ; also, that annexing a cipher changes the values of the figures of an integer, by removing ones to tens, tens to hundreds, etc. Second Reference.— In reading numbers, the word and should be used only between the integral and decimal parts of mixed numbers. If 441 be read four hundred and forty-one, certain entirely difier- cnt numbers would be read exactly alike. Thus 400.041 and .441 would both be read four hundred and forty-one thousandths. The following unlike numbers would be read alike : 15000,00008 and .15008; 5000.0024 and .5024; 964000.000072 and .964072. Ten-thousand, hundred-thousand, ten-thousandths, hundred-thou- Bandths, etc., should be written as compound words ; otherwise, three hundred ten thousandths might mean .310 or .0300. The exercises on page 100 will test the ability of pupils to write and read decimals. For additional practice, write decimal numbers upon the blackboard for your pupils to read, and require them to write numbers from dictation, until they can write and read decimals with the same facility as integers. MANUAL FOK TEACHERS. 219 Page 101, — ^Explain the difference between Addition of Decimals and Addition of Integers. Do the same with Subtraction, Multipli- cation, and Division. Page 129. — Be sure that pupils always write abbreviations cor- rectly, and in their proper places, i. e.y that they write the proper ab- breviated form, and place a period after it ; and that the abbreviation be placed after the number, except the sign of dollars (^), which should always be placed before the number. Before your pupils take up the tables in Compound Numbers, read carefully the suggestions on Use of Objects, page 209, Page 146. — Make your pupils so familiar with this table, that they can tell at once the number of any month, and the number of days in it. Page 147. — In the table of weight (see 185), the denomination grain is not named, as it is not used in commercial weight. Explain to your pupils that if a pound is divided into 7000 equal jmrts, 1 of these parts is 1 grain : and hence, a grain is 1 seven-thousandth of a pound; a gram is 15432 seven-miUionths of a pound; and a pound contaitis a little more than 453.6 gram^. Page 165.— Thorough drills upon these principles are indispens- able to the further intelligent progress of the pupil. Therefore, be sure that he understands them before passing to Section II. Page 169. — The common solution is similar to Reduction Ascend- ing, and it may be explained in the same manner, by regarding the numerator as a number to be reduced to a higher denomination, and the denominator as that number of the given denomination that equals one of the required higher denomination. Thus, the process of reducing 17 fourths to a mixed number is the same as that of reducing 17 pecks to bushels, or 17 quarts to gallons. Page 170.— The common solution is similar to Reduction De- scending, and may be explained in the same manner, by regarding the integer as the higher denomination, and-the numerator of the fraction as the number of the next lower denomination. Thus, the process of reducing 5| to fourths is the same as that of reducing 5 bu, 3 pk. to pk., or 5 gal, 3 qt, to qt. Page 173.— Fix in the minds of pupils the fact that numerators only are added or subtracted, and denominators are written to give name or denomination, the same as in compound numbers. Give numerous illustrations, such as 4 men + 3 men + 5 men ; $4 + $3 + $5; 4 parts + 3 parts + 5 parts ; 4 ninths + 3 ninths + 5 ninths, or £i^s •*• ^!ths + ^s' 1^ P^°^-S P^^«5 S13-$8; 13 parts- 13 8 8 parts ; 13 ninths — 8 ninths, or —. — , etc. ninths ninths 220 MANUAL FOR TEACHERS. Page 182.— The method "Invert the divisor and proceed as in multiplication," is introduced in Cancellation, p. 184. Therefore pupils should be required to solve and explain these problems in the manner here shown, as this course wiU make them familiar with the reasons for the process, which is not likely to be the case if they at once adopt the mechanical convenience of inverting the divisor. Page 184.— Be sure that pupils understand that a factor in either term will cancel only one like factor in the other term. Also alwavs require them to write 1 in the place of a canceled term, as pupils are liable to think that when a term is canceled, a belongs in its place. Test their knowledge upon this point, by questions and ex- amples. Page 19T.-Exp]ain the fact that any number of months forms the numerator of a fraction, of which 12 (the number of months in a year) is the denominator; and that this multiplying fraction can often be reduced to lower terms before multiplying. Thus, 2 mo. = /- = i yr., 3^ mo. = xV - i yr., 4 mo. = ^\ = \ yr., 6 mo. = r% = i yr., 8 mo". - /a - i yr., 9 mo. = t\ = f yr., 10 mo. = ia = s yj.^ ^ familiarity with this fact will often enable pupils to abridge their work by multiply- ing the interest for 1 year by the time expressed in years and frac tions or decimals of a year. Page 201. — When pupils pass over these General Review Prob- lems, exercise their ingenuity in producing diflferent methods for solving the same problem. Thus, problem 30, page 203, may be solved in three ways : 1st Find the contents of the pile in cubic inches, and then divide them by the number of cubic inches in a brick. 2d. Find the cubic contents of the pile in feet, and multiply them by the number of bricks in a foot ; and, M. Find how many bricks long, wide, and high, the pile is, and then multiply these numbers togetjier. Again, Problem 50, page 104, may be solved in two ways : Is^. Find the whole number of pounds raised, by multiplying the number of pounds per acre by 8.5, the number of acres, and multiply $.06, the price of 1 pound, by this product ; and, 2d. Find what he received for 1 acre of flax, by multiplying $.06, the price per pound, by 850, the number of pounds raised to the acre ; and then multiply this result by 8.5, the number of acres raised. This course will teach pupils that, in many cases, there is more than one right way to solve a problem, and it will also teach them to think methodically. ^ ^ YB 17389 ^ UNIVERSITY OF CALIFORNIA LIBRARY i^himmw^',-:" Wi-^^ ^^f^: SUPERIOR SCHOOL BOOKS PUBLI£HEr> JY HARPER & E OTHERS, Franklin Square, New York. I. FRENCH'S ARITHMilTICS. 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