K M EMERSON'S SECOND PART. THE NORTH AMERICAN ARITHMETIC. PART SECOND, UNITlf ORAL AND WRITTEN EXERCISES. HUNDRED TENS UNITS m m 138 B O S T O N% JENKS, PALMER ^ CO. SOLD ALSO BY ROE LOCKWOOD, NEW YORK ; HOG AN Ic 1H( ULADEL- PHIA ; GUSHING .\: BROTHER, BALTIMOK . , ,,. ,. , MEECH, ST. LOUIS ; WM. C. TABER & SON, NEW BEDFORD. 55 m EMERSON'S SECOND PART. THE NORTH AMERICAN ARITHMETIC. PART SECOND, UNITING ORAL AND WRITTEN EXERCISES, IN CORRESPONDING CHAPTERS. BY FREDERICK EMERSON, LITE PRINCIPAL IN THE DEPARTMENT OP ARITHMET«0, BOYLSTON SCHOOL, BOSTON. BOSTON: JENKS, PALMER & CO. 1848. Entered, according to Act of Congress, in the year 1832, by Frederick Emerson^ Jn the Clerk's Office of the District Court of the District of Massachusetts. Orders of the School Committee of Boston, At a Meeting of the School Committee, Nov. 18, 1834. Ordered, That Emerson's North American Arithmetic, Second and Third Parts, be substituted in the Writing Schools, for Gol- burn's First Lessons and Sequel.* Ordered, That the Arithmetics now in use be permittea to their present owners ; but that whenever a scholar shall have occasion to purchase a new one, the North American Arithmetic shall be required. ^^^^^^^ g p M'CLEARY, Secretary. * The First Part was already adopted by a previous order. A KEY to this work, containing solutions and answers, [a small book for Teachers only,] is published separately PREFACE. This book is intended for the use of scholars who have been taught in * Part First, ' or by some other means have learned to add, subtract, and multiply numbers as high as 10, mentally. The whole Course of Exercises, of which this is the Sec ond Part, has been divided into three parts, more for the sake of economy and convenience, than on account of any natural division of the subject. The work is not intended to be ai record of the science, — such as might befit the pages of an encyclopedia, — but, a system of induction, through which the scholar may be led to the discovery of arithmet- ical truth, and the proper application of arithmetical ope- rations. Rules, and the technical language necessary to their composition, are avoided in the early part of the course — they are not introduced until the learner is sup- posed prepared, by intellectual improvement from previous lessons, to meet them understandingly. In the arrangement of the exercises in this volume, I have been governed by the natural order of the science; believing, that any deviation from that order, with a view of rendering the work more immediately practical, would render it in reality less practical, as it would necessarily lead the scholar into a habit of performing operations, without comprehending the principles which justify them. The first six chapters consist of oral exercises, and the last six of correspondent written exercises. The work may there- fore be viewed as two entire systems of arithmetic — Oral and Written. Although Part Second does not complete the series of books, entitled * The North American Arithmetic,' still it contains the essential principles, and the common applica- tion of the science. Scholars, therefore, who shall be properly conducted through this volume, will have acquir- ed a knowledge of Arithmetic, adequate to all the pur- poses of common business. Part Third is designed for those, whose continuance at school shall afford oppor- tunity for prosecuting a more extended course of study 4 PREFACE. The mode of teaching arithmetic, and the text-books, used for the purpose, in a great portion of our country, are radically defective. Much of arithmetic is practised at school, but little is learned. The scholar is put to ciphering without adequate mental preparation, and is re- ferred to the direction of rules, whose phraseology and principles are to a learner equally obscure. By a tedious course of practice, perhaps he acquires a certain mechan- ical dexterity in performing operations ; but no sooner does he enter upon the business of life, than he abandons the ru/es of his book, and, in his own way, learns so much of arithmetic as his occupation requires. Whether the following treatise is calculated to afford any remedy for the defects I have alluded to, others will decide. I shall spare myself the task of a prefatory detail of what ^^the author conceives^' to be its advantages, and will only add, that the design and execution of the work, have cost me much time and labor F. Emerson. Boston, January, 1832. NOTE TO TEACHERS. It will be most advantageous for young scholars, to go through with all the Oral Arithmetic before they enter upon the Written Arithmetic. Older scholars. However, after performing the exercises in the first chapter of Oral Arithmetic, may pass immediately to the exercises in the first chapterof Written Arithme- tic; and after concluding this chapter, may take up the two second chapters in the same order ; and thus proceed through the book. Much time has been wasted in some of our schools, by the practice of teach- ing individually f instead of teaching in classes. If this practice has been owing in any degree to the arrangement of text-books, it is hoped the present arrangement will afford a remedy. There can be no more objection to a distinct classification of a school for the purpose of teaching arithmetic, than there is to a like classification for the purpose of teaching orthography: and the advanta- ges of class- instruction in the former branch, are as great as those in the latter. The examples contained in the first six chapters, do not require tlieuseof the elate. The answers, with the process of obtaining them, and the reasons whieb justify the process, are to be given orally. For example, the following question mav be supposed to give rise to the subjoined exercise. jExample. A trader purchased 9 barrels of flour, at 7 dollars a barrel, and Bold the whole for 68 dollars. What did he gain in the trade'? Pupil. * He gained five dollars.' Teacher. * How do you perceive itl' Pupil. ' If one barrel cost seven dollars, nine barrels must have cost nine times seven dollars, which is sixty-three dollars. He must have gained the difference between iixty-three dollars and sixty-eight dollars. 63 from 68 leaves 5.' Learners should not be confined to any form of expression in solutions— their reasoning should be their own. By a little practice, they will acquire an aetoni^ihing acuteness of apprehension, and facilitv of expression. ORAL ARITHMETIC. CHAPTER I. NUMERATIOIV. # Section 1. When we have a large number of articles to count, such as quills, nuts, cents, &c., we may, if we please, count them by tens. Let us suppose we have a quantity of cents before us, and proceed to count them as follows. We first count out ten cents, and lay them in a pile. We then count out ten more, and lay them in another pile; then ten more for another pile; and thus we continue to count out ten at a time, until we have counted ten piles. We put these ten piles together, and they make a large pile containing One Hundred cents. Again we count out ten cents at a time, until we have counted ten small piles, as before. We put these togeth- er, and they make a large pile containing one hundred, like the hundred we first counted. We have now count- ed two hundred cents, and they lie in two large piles. Having learned what is meant by two hundreds^ we pro- ceed to count out one hundred cents more; and after pla- cing them by the side of the two hundreds, the three piles make three hundreds. Four large piles will be four hun- dreds; five piles will be five hundreds; six piles will be six hundreds; seven piles will be seven hundreds; eight piles will be eight hundreds; nine piles will be nine hundreds; and when we have counted out ten of these piles, we put the whole together. They make a pile still larger, and the number of cents contained in it is One Thousand. 6 ORAL ARITHiMiTlC. CHAP I. Examine the arrangement of dots enclosed in ihe lines below, and find how many there are in each en- closure. Observe, that the figures standing over the several enclosures, represent the number of dots con tained therein. 10 100 1000 Example 1 . Which of these numbers is the greatest. One, or Ten^ or One Hundred^ or One Thousand? 2. How many ones are there in a ten ? Sec. 2 NUMERATION. 3. How many tens are there in a hundred ? 4. How many hundreds are there in a thousand ? 5. Ten ones make what number ? Ten tens make what number ? Ten hundreds make what number ? 6. What figures stand to represent the number ten ? 7. What figures stand to represent one hundred ? 8. What figures stand to represent one thousand? Section 2. If one hundred scholars were in school, and one scholar more should come in, the number of scholars would then be one hundred and one; and would be ex- pressed in figures thus; — 101. Again, if you had one hundred books, and you should buy two books more, you would then have one hundred and two books, and their number would be expressed in figures thus; — 102. In Part First, you learned to read figures expressing all numbers, from One to One Hundred, You will now see, m the following columns, how^ the figures stand to express numbers, from One hundred^ to Two hundred. 100 One hundred, 101 one hund. and 102 one hund. and 103 one hund. and 104 one hund. and 105 one hund. and 106 one hund. and 107 one hund. and 108 one hund. and 1 09 one hund. and 110 one hund. and 111 one hund. and 112 one hund. and 113 one hund. and 114 one hund. and 115 one hund. and 116 one hund. and 117 one hund. and 118 one hund. and 119 one hund. and one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, 120 one hund. and twenty, 121 one hund. and twenty-one, 122 one hund. and twenty- two, 123 one hund. and twenty-three 1 30 One hund. and thirty. 140 One hund, and forty. 150 One hund. and fifty. 160 One hund. and sixty. 170 One hund. and seventy. 180 One hund. and eighty. 190 One hund. and ninety. 200 Two hundred. 8 ORAL ARITHMETIC. 1 Edward's mother gave him one hundred walnuts, his sister gave him sixty, and his brother gave him eight; making together, one hundred and sixty-eight. Being required to tell what figures would express the number of his walnuts, Edward looked over the columns of figures on the last page, and discovered, (as you may), that 1 means one hundred^ whenever two figures are standing at the right hand of it; and, that 6 means sixty ^ whenever one figure is standing at the right hand of it. He there- fore said, "1, 6, 8, are the figures." 1. How many tens does the figure 6 represent, when there is one figure standing at the right of it ? 2. What are 6 tens usually called, in reading numbers } 3. How many tens does the figure 4 represent, when there is one other figure standing at the right of it ? 4. What are 4 tens usually called, in reading numbers ^ 5. What number does the figure 1 represent, when there is one other figure standing at the right of it ^ 6. What number does the figure 1 represent, when there are two other figures standing at the right of it.'^ 7. What are 1 hundred and 5 tens usually called ? 8. What are 1 hundred and 9 tens usually called.^ 9. What are 1 hundred and 3 ones usually called ? 10. What are 1 hundred and 8 ones usually called ? 11. What are 8 tens and 2 ones usually called ^ 12. What are 1 hundred, and 7 tens, and 5 ones usu- ally called, in reading numbers ? Note to Teachers, Require the learners to read the numbers expressed id the following columns, without recourse to the preceding columns. 109 172 104 168 113 127 190 110 140 147 145 121 132 122 169 163 143 155 195 183 181 165 176 177 103 118 187 198 159 125 136 154 186 131 158 The comparisons on the next page will show you^ that all the hundreds are expressed in the same manner that one hundred is expressed. X. ADDITION. 9 100 One hundreu. 133 One huiid. & thirty-three. 200 Two hundred. 633 Six hund. & thirty-three. 106 One hund. & six. 149 One hund. & forty-nine. 306 Three hund. & six. 749 Seven hund. & forty-nine. 117 One hund. & seventeen. 154 One hund. & fifty-four. 417 Four hund. & seventeen. 854 Eight hund. & fifty-four. 121 One hund. & twenty-one. 199 One hund. & ninety-nine. 621 Five hund. & twenty-one. 999 Nine hund. & ninety-nine. Note to Teachers. The learners may be required to read tlie several numbers expressed in the following columns of figures. 814 293 552 466 851 372 947 444 664 767 528 381 786 391 579 451 619 369 940 296 CHAP. 11. ADDITIOIV. Section 1. 1. The Humane Society gave Charles a premium of 6 dollars, for saving a boy from drowning, and a lady gave him 5 dollars more. How much did he receive } Solution, 6 dollars and 5 dollars are 11 dollars. 2. A merchant sold 7 barrels of flour to one man, and 5 to another. How many barrels did he sell ? 3. If you should pay 9 cents for a book, and 4 cents for a pencil, how much would you pay for both ? 4. A farmer paid 10 dollars for a plough, and 9 dol- lars for a harrow. How much did he pay for both ? 5. A baker bought 8 barrels of flour of a merchant, and 8 more of a miller. How many did he buy } 6. Thomas gave 9 cents for a purse, and had 7 cents left to put in it How many cents had he at first ? 7. A farmer sold 5 cows, and then had 6 cows left How many cows had he at first? 10 ORAL ARITHMETIC. II. 8. If you should receive 9 dollars from one man, and 5 from another, how many dollars would you receive ? Section 2. 1. Two little boys went into a shop to be weighed. The oldest of them weighed 40 pounds, and the youngest, 30 pounds. How many pounds would they weigh both together ? Solution, 40 is the same as 4 tens^ and 30 is the same as 3 tens. Then 4 tens and 3 tens are 7 tens; — and 7 tens are the same as 70. 2. There were 40 oranges in one basket, and 20 in another. How many were there in both baskets.'* 3. What is the whole number of scholars in a school, that consists of 20 boys and 30 girls ? 4. A baker paid 50 dollars for a horse, and 30 dollars for a cart. -How many dollars did he pay for both ? 5. If I read 50 pages of history, and 40 pages of poetry, how many pages do I read of both ? 6. If a man has lived 20 years in the city, and 10 years in the country, how old must he be ? 7. James paid 60 cents for his Reader, and 40 for his Arithmetic. How many cents did they both cost ? 8. Suppose you should buy 60 quills at one store, and 50 at another; how many quills would you have ? Solution, 60 is 6 tens, and 50 is 5 tens. 6 tens and 5 tens are 11 tens. 11 tens are 1 hundred and 1 ten; — that is, 110. 9. Suppose 70 books are upon my table, and I put on 50 more; how many will then be on the table ? 10. If a gold watch cost 90 dollars, and the chain 40 dollars; how many dollars do they both cost.'* 11. In a certain orchard, there are 80 pear trees and 60 peach trees. How many trees in the orchard ? 12. If 90 persons should enter a hall at one door, and 60 at another; how many would there be in the hall.'* 13. If I purchase 80 barrels of flour from one man, and 80 from another; how many barrels shall I have ^ 14. A miller had 90 bags of wheat on hand, and re* ceived 80 bags more. How many bags had he then ? 9. 3. ADDITION. 11 15. If a horse cost 90 dollars^ and a gig 90 dollars, how much do the horse and gig both cost ? 16. How many pounds of honey in two jars; — there being 70 pounds in one jar, and 60 in the other ? Section 3. 1. A gardener called three boys to the garden gate to give them some grapes. To the first boy he gave 40 grapes, and to the second 40; but the third boy attempted to push the others aside, and the gardener seeing it, gave him only 6. How many did he give them all ? Solution. 40 grapes and 40 grapes are 80 grapes. Then 80 grapes and 6 grapes are 86 grapes. 2. John, James, and Henry went a fishing. John caught 30 fishes, and James caught 40; bu^t Henry caught only 9. How many did they all catch ? 3. How many are 30 and 40 and 9 ? 4. A traveller gave 70 dollars for his horse, 20 dollars for his saddle, and 5 dollars for his bridle. How many dollars did he give for the whole ? 5. How many are 70 and 20 and 5 ? 6. A farmer kept 50 sheep m one pasture, 30 in another, and 7 in another. If he had kept them all in one pasture, how many would there have been together ? 7. How many are 50 and 30 and 7 ? 8. How many cents will it take to buy a seal, a blank-book, and a pencil; supposing the seal to cost 60 cents, the blank-book 20 cents, and the pencil 8 cents ? 9. How many are 60 and 20 and 8 ? 10. An escort went out to meet Gen. Layfayette: 40 men rode on horseback, 30 rode in gigs, and 10 rode in coaches. Of how many did the escort consist? 11. How many are 40 and 30 and 107 12. How many are 40 and 30 and 3 r 13. How many are 60 and 20 and 5 i 14. How many are 30 and 30 and 7 i * 15. How many are 50 and 40 and 9 i 16. How many are 50 and 50 and 8 i 17. How many are 60 and 40 and 4.? 18. How manv are 70 and 30 and 6 f 12 ORAL ARITHMETIC. XL Section 4. 1. A certain class consists of 11 studious boys, and 2 idle boys. How many are there in the class ? 2. Hovvmanyarelland2? 11 and 3? 11 and 4? Hand 5? llandG? Hand?? llandS? Iland9? llandlO? 3 Alfred paid 11 cents for a pen-knife, and 10 cents for a writing-book. How much did he pay for both ? 4. If you should, pay 12 cents for a slate, and 3 cents for an orange, how many cents would they both cost ? 5. How many are 12 and 2? 12and3? 12and4? 12 and 5.^ 12and6? 12and7? 12and8? 12and9? 12andl0? 6. If 12 boys play at foot-ball on one side, and 8 boys on the other, how many are there in the play ? 7. A certain class consisted of 13 small boys, and 4 large boys. How many were there in the class ? 8. How many are 13 and 2? 13and3? 13and4? 13and 5? 13and6? 13and7? 13and8? 13and9? 13andl0? 9. A number of sheep are in a fold; — 13 are lying down, and 6 are standing up. How many are there ? 10. There were 14 hats hanging up, and 5 more *ying down. How many hats were there in all ? 11. How many are 14and2? 14and3? 14and4? 14and 5? 14and6? 14and7? 14and8? 14and9? 14andl0r 12. If you give 14 cents for a bow, and 4 cents for an arrow, how much do the bow and arrow cost? 13. A wagoner drove 15 miles in the forenoon, and 6 in the afternoon. How many miles in the day ? 14. How many are 15and2.'^ 15and3? 15and4? 15and 5? 15and6? 15and7.^ 15and8.^ 15and9? 15andlO.? 15. If a cow be worth 15 dollars, and a sheep 2 dol- lars, what are the cow and sheep together worth ? 16. David wrote 16 fines in the forenoon, and 7 in the afternoon. How many lines did he write in the day.'* 17. How many are 16and2? 16and3? 16and4? 16and 6.? 16and6? 16and7? 16and8? 16and9.? 16andl0.? 18. A trooper gave 16 dollars for his saddle, and 9 dollars for his bridle. How much did he pay for both ? 4. 5. ADDITION. It 19. A man lost S doJars, and still had 17 dollars left. How many dollars had he before he lost any ? 20. Howmanyarel7and2? 17and3? 17and4? 17and 5? 17and6? 17and7? llandS? 17and9? 17andl0.? 21. If a time-piece cost 17 dollars and a looking-glas«? 7 dollars, how many dollars do they both cost ? 22. While 18 doves were upon a roof, 9 doves more lit among them. How many were then upon the roof? 23. Howmanyarel8and2? 18and3? 18and4? 18and 5? 18and6? 18and7? 18and8? 18and9? 18and'10? 24. A man rolled 18 barrels of flour out of a mill and a boy rolled out 5 more. How many did both roll out ? 25. A young man began studying law at the age of 19 vears, and studied 3 years. At what age did he finish ? 26. Howmanvarel9and2? 19and3? 19and4? 19and 5? 19and6? 19and7? 19and8? 19and9? 19andlO? 27. A farmer mixed 19 bushels of oats with 10 of corn. How'many bushels were there of the mixture ? Note to Teachers. Tlie following combinations may be embraced in Beparate questions by the teacher ; thus, — How many are 19 and 4 ? 19 and 4 14 and 2 19 and 9 15 and 9 16 and 3 19 and 6 16 and 8 13 and 7 18 and 5 16 and 6 13 and 5 14 and 4 12 and 2 15 and 4 18 and 8 14 and 8 17 and 7 18 and 2 14 and 6 12 and 5 19 and 7 17 and 6 12 and 9 14 and 7 15 and 5 19 and 3 13 and 3 17 and 8 19 and 8 16 and 5 12 and 7 12 and 6 17 and 9 15 and 8 16 and 6 16 and 9 12 and 8 16 and 4 17 and 5 1 3 and 8 Section 5. 1. Charles had 25 books in his library, and his falhe gave him 8 more. How many had he then ? Suggestion. You will easily perceive how many .25. and 8 are, since you already know that 5 and 8 are 13, and that 15 and 8 are 23. 14 ORAL ARITHMETIC. II. 2. A father said to his son, ' You are 7 years old, and I am 47 — How old shall we each of us become, In 9 years from this time?' What should have been the answer? 3. James bought a small book for 6 cents, and David bought a large book for 56 cents. For how many cents must each boy sell his book, in order to get 4 cents more than he gave ? 4. Julia was returning from a walk in the garden, with 8 red roses, and 63 white roses. She met her brother, who gave her 6 more red roses, and 6 white ones. How many of each kind had she then? 5. William has 9 cents, and John has 79 cents. If they should each of them get 10 cents more, how many would each boy then have ? Note to Teachers. The following combinations maybe embraced inques- tions by the teacher; thus, — How many are 3 and 9 ? 3 and 9 54 and 6 41 and 10 38 and 8 13 and 9 67 and 8 53 and 9 41 and 7 7 and* 6 72 and 9 65 and 6 53 and 9 27 and 6 85 and 7 77 and 3 65 and 2 8 and 8 90 and 5 89 and 5 77 and 10 38 and 8 IS and 4 92 and 7 89 and 6 1 and 7 26 and 7 14 and 10 92 and 4 41 and 7 39 and 2 26 and 7 14 and 8 Section 6. 1. A trader paid 29 dollars for a chest of tea, 4 dollars for a box of lemons, and 5 dollars for a box of raisins. What did he- pay for the whole ? 2. How many are 29 and 4 and 5 ? 3. If I pay 38 dollars to one man, 6 to another, and 3 to another, how many dollars do I pay out ? 4. How many are 38 and 6 and 3 ? 5. Stephen had 47 books; he bought 5 more, and then his uncle gave him 6 more. How many had he at last ? 6. How many are 47 and 5 and 6? 7. On a certain day, a passenger travelled 56 miles in the stage, 4 miles in a wagon, and 7 miles on foot. How many miles did he travel on that day? 8. How many are 56 and 4 and 7-^ 6 7. ADDITION. 15 9. If a yoke of oxen be worth 65 dollars, a sheep 8 dol- lars, and a lamb 2 dollars, how much are they all worth? 10. How many are 65 and 8 and 2? 11. A school boy paid 74 cents for a reading book, 7 cents for a writing book, and 9 cents for some quills. How many cents did he pay for the whole ^ 12. How many are 74 and 7 and 9 ? 13. A meeting was held. in a country village, to which 83 persons walked, 9 rode on horseback, and 8 rode in gigs. How many attended the meeting ? 14. How many are 83 and 9 and 8? 15. A market man received 92 dollars for butter, 9 dollars for cheese, and 5 dollars for poultry. How many dollars did he receive for the whole ? 16. How many are 92 and 9 and 5 } Section 7. 1. The captain of a steam-boat received the following fassengers; — 45 gentlemen, 20 ladies, and 8 children, low many passengers were there in all ? Solution. 45 and 20 are 65; then 65 and 8 are 73. Answer^ 73 passengers. 2. If a quire of paper cost 23 cents, a book 30 centS; and a pencil 9 cents, what do they all cost } 3. How many are 23 and 30 and 9? 4. Alfred paid 25 cents for his penknife, and 20 cents for his wallet, and then had 5 cents left. How many cents had he at first ? 5. How many are 25 and 20 and 5 } 6. A lady gave 57 cents for a fan, 30 cents for a work bag, and 4 cents for some needles. How m^ny cents did she lay out ? 7. How many are 57 and 30 and 4 ? 8. A fowler went out one morning to shoot birds; — he shot 46 plovers, 50 snipes, and 6 quails. How many birds did he shoot ? 9. How many are 46 and 50 and 6 ? 10. If a cart cost 26 dollars, a plough 10 dollars, and i chain 5 dollars, what do they all cost ? 11. How many are 26 and 10 and 5 } 16 ORAL ARITHMETIC. III. 12. A farmer sold a horse for 75 dollars, a cow for 30 dollars, and a sheep for 5 dollars. How many dollars did he get for the whole ? 13. How many are 75 and 3C and 5 ? 14. Wilham gave 64 cents for a handkerchief and 40 cents for a pair of gloves, and then had 9 cents left. How many cents had he at first ? 15. How many are 64 and 40 and 9 ^ 16. How many are 5 and 9 and 2 and 8 and 6 and 4? 17. How many are 8 and 3 and 7 and 6 and 5 and 3.^ 18. How many are 6 and 8 and 4 and 9 and 7 and 5.'* 19. How many are 9 and 7 and 2 and 8 and 3 and 6.'' 20. How many are 17 and 5 and and 9 and 6 and 8.'^ 21. How many are 23 and 8 and 1 and and 9 and 7? 22. How many are 48 and 6 and 7 and 4 and and 2? 23. How many are 71 and 3 and 9 and and 6 and 9? CHAP. III. SUBTRACTION* Section 1. 1. There were 9 passengers in a stage; 3 of them got out to walk: how many remained in the stage ? Solution. 3 from 9 leaves 6. Answer. 6 passengers. 2. A boy having 10 cents, paid 6 cents for a kite, and *ost the remainder. How much did he lose ? 3. Ann has 12 books andJuHahas7. Howmanymore must Julia have, to make her number equal to Ann's ? 4. Andrew has 11 cents, and James has only 5 cents. How many cents has Andrew more than James ? 5. Stephen has 8 cents, and wishes to buy a knife worth 16 cents. How many more cents does he want? 6. A lady went to buy goods, carrying 13 dollars; she returned with 9 dollars. How much did she spend ? 7. A merchant bought a box of goods for 10 dollars, and sold it for 14 dollars. How much did he gain ? 1. 2. SUBTRACTIOIN. 17 8 Jonathan is 7 years old, and his brother is 11 years old. What is ^he difference in their ages ? 9. Henry bought a book and a pencil for 19 cents: he gave 10 cents for the book; what did the pencil cost? 10. A man who owed a debt of 12 dollars, paid 5 dol- lars of it. How many dollars remained unpaid.'* 11. John sold a knife for 18 cents, which was 9 cents more than he gave for it. How much did he give for it ? 12. A farmer agreed to give 17 dollars for a cow: and he paid S dollars down. How much did he still owe ? Section 2. 1 . A sloop of war went out with a crew of 70 men, and fell into an engagement, in which 30 of her men were killed. How many of the crew were still living ? Suggestion. Consider the numbers to be, 7 tens^ and 3 tens; — you may then take 30 from 70 as easily as you can take 3 from 7. 2. A market woman had 60 oranges, and sold 20 of them. How many had she remaining ? 3. 20 from 60 leaves how many ? How many are 20 anrf 40 ? 4. A certain school consists of 50 scholars, 30 of whom are girls. How^ many boys are there ? 5. 30 from 50 leaves how many ? Hoic many are 30 and 20 ? 6. A baker had 80 dollars to lay out for a horse and cart. After having paid 50 dollars for a horse, how many dollars had he left to purchase the cart ? 7 . 50 from 80 leaves how many ? Hoio many are 50 and 30 ? 8. If your lesson for the whole day be 40 questions in ihis book, and you answer 20 questions in the forenoon, how many are there left for the afternoon } 9. 20 from 40 leaves how many ? How many are 20 am/ 20 ? 10. I have read 40 pages, in a book which contains 90 pages. How many pages remain to be read ^ 11. 40 from 90 leaves how many.'* Hoiv many are 40 and 50? 12. James had 70 cents, and paid 40 of them for a school-book. How many cents ha'd he left ^ 1 3. 40 from 70 leav es how ma'hy ? How many are 40 arw? 30? 18 ORAL ARITHMETIC. HI Section 3. 1. A man received 12 dollars for work, and paid 2 dollars for his board. How many dollars did he save ? 2. How many will remain, if we take 2 from 12 ? 2 from 13? 2froml4? 2from 15 ? 2 from 16 ? 2 from 17? 2 from 18? 2 from 19? 2 from 20 ? o. A stable keeper owned 14 fine horses. After sell- ing off 3 of them, how many had he remaining ? 4. How many will remain, if we take 3 from 13 ? 3 from 14? 3 from 15? 3 from 16? 3 from 17? 3 from 18 ? 3 from 19 ? 3 from 20 ? 3 from 21 ? 5. 16 boys were dismissed, but 4 of them were called back for being noisy. How many were allowed to go? 6. How many will remain, if we take 4 from 14 ? 4 from 15? 4 from 16? 4 from 17? 4 from 18? 4 from 19 ? 4 from 20 ? 4 from 21 ? 4 from 22 ? 7. A man, who had 18 dollars, paid 5 dollars for a pair of boots. How many dollars had he remaining? 8. How many will remain, if we take 5 from 15 ? 5 from 16 ? 5 from 17? 5 from 18? 5 from 19? 5 from 20 ? 5 from 21 ? 5 from 22 ? 5 from 23 ? 9. If you had just 20 cents, and you should lose 6 cents, how many cents would you then have ? 10. How many will remain, if we take 6 from 16? 6 from 17? 6 from 18? 6 from 19? 6 from 20 ? 6 from 21 ? 6 from 22 ? 6 from 23 ? 6 from 24 ? 11. A man who had 22 dollars on hand, lent 7 dollars to his neighbour. How many dollars had he remaining ? 12. How many will remain, if we take 7 from 17? 7 from 18 ? 7 from 19 ? 7 from 20 ? 7 from 21 ? 7 from 22 ? 7 from 23 ? 7 from 24 ? 7 from 25 ? 13. 24 peaches grew upon a young peach tree, and the owner took off 8 of them. How many remained on? 14. How many will remain if we take 8 from 18? 8 from 19? 8 from 20 ? 8 from 21? 8 from 22 ? 8 from 23 ? 8 from 24 ? 8 from 25 ? 8 from 26 ? 3.4. CORRESPONDENT EXAMPLES. 19 15. Suppose you had 26 cents, and paid 9 of them for a dozen of quills; how manj cents have you left.^ 16. How many will remain, if we take 9 from 19 .'^ 9 from 20? 9 from 21? 9 from 22? 9 from 23? 9 from 24 ? 9 from 25 ? 9 from 26 ? 9 from 27 ? 17. James received 28 cents, and Charles received 10 cents less than James. How many did Charles receive ? 18. Howmany will remain, if we take 10 from 20? 10 from21? 10from22? 10from23? 10from24? lOfrom 25? 10 from 26? 10 from 27? 10 from 28? Note to Teachers. The following combinations of numbers may be em- braced in quesions by the teacher, thus, — 2 from 17 leaves how many? 2 from 17 8 from 19 7 from 24 5 from 23 7 from 21 3 from 20 9 from 21 6 from. 24 9 from 24 9 from 26 3 from 18 8 from 23 3 from 16 7 from 18 8 from 22 3 from 21 8 from 26 2 from 15 6 from 17 9 from 27 6 from 19 6 from 22 5 from 18 7 from 25 5 from 22 5 from 19 4 from 16 2 from 20 4 from 20 4 from 15 2 from 19 4 from 22 10 from 25 10 from 27 10 from 26 10 from 29 Section 4. CORRESPONDENT EXAMPLES. 1. A farmer, who had 19 dollars on hand, received 5 dollars for a sheep. How many dollars had he then ? 2. A butcher, who had 24 dollars on hand, paid out 5 dollars for a sheep. How many dollars had he left? 3. A jeweller gave 17 dollars for a silver watch, and sold it for 6 dollars more than he gave for it. For how many dollars did he sell it ? 4. A young man gave 23 dollars for a watch, and was obHged to sell it for 6 dollars less than he gave. For how much did he sell it ? 5. It was 4 years ago, that Samuel left the academy, and he was then 15 years o.d. How old is he now ? 6. Sarah is 19 years old; her father died when she was 15. How manv years is it, since her father died ? 20 ORAL ARITHMETIC. III. 7. A farmer who had 26 sheep, purchased 8 more. How many sheep had he then ? 8. A farmer who had 34 sheep, sold 8 of his flock. How many had he remaining ? 9. A merchant who had 9 dollars on hand, received 27 dollars more, for a quantity of goods. How many dol- lars had he then ? 10. A merchant who had 36 dollars in his pocket, paid a small debt, and then had 27 dollars left. How many dollars did he pay ? 11. A wagon passed along, carrying 38 empty barrels and 7 full ones. How many barrels in all } 12. In a store-room there were 45 barrels, only 7 of which w^ere filled. How many were empty .^ 13. Edward paid 46 cents for a book, and then had 9 cents left. How many cents had Edward before he purchased the book ^ 14. Joseph's father gave him 55 cents, to buy a book, but he obtained the book for 46 cents. How many cents did Joseph save ? 15. How many are 57 and 5.*^ Then if we take 5 from 62, how many remain } 16. How many are 64 and 6.^ Then if we take 6 from 70, how many remain ^ 17. How many are 79 and 8.^ Then if we take 8 from 87, how many remain } 18. How many are 86 and 6 } Then if we take 6 from 92, how many remain } 19. How many are 48 and 9.'^ Then if we take 9 from 57, how many remain ^ 20. How many are 75 and 7 ^ Then if we take 7 from 82, how many remain ^ 21. How many are 36 and 5 } — how many are 5 and 36 i Then 5 from 41 leaves how many } — 36 from 41 leaves how m.any? 22. How many are 43 and 9 } — how man}^ are 9 and 43 ? Then 9 from 52 leaves how many? — 43 from 52 .feaves how many } 5. MISCELLANEOUS EXAMPLES. 21 23. How many are 54 and 6 ? — how many are 6 and 54 ? Then 6 from 60 leaves how many ? — 54 from 60 leaves how many? 24. How many are 68 and 4? — how many are 4 and 68 ^ Then 4 from 72 leaves how many? — 68 from 72 leaves how many? 25. How many are 79 and 8? — how many are 8 and 79? Then 8 from 87 leaves how many? — 79 from 87 leaves how many? 26. How many are 87 and 5? — how many are 5 and 87? Then 5 from 92 leaves how many.'* — 87 from 92 leaves how many? Section 5. MISCELLANEOUS EXAMPLES. 1. George and David went out to gather lilies; George fot 56, and David 49. On the way home, George gave )avid 8. How many had each boy then ? Solution, At first, George had 56; he gave away 8; 8 from 56 leaves 48 At first, David had 49; he re- ceived 8 more; 49 and 8 are 57. 2. A clerk went out to collect some money. He re- ceived 60 dollars from one man, 9 dollars from another, and 20 from another ; and he paid a debt of 7 dollars. How many dollars had he to bring in ? Solution. 60 and 9 are 69, and 20 are 89; — this is the number of dollars he collected. He then paid 7 dollars. 7 from 89 leaves 82. 3. Harriet answered 23 questions in arithmetic, and Mary answered 7 more than Harriet. How many ques- tions did they both answer ? 4. Edward answered 36 questions in arithmetic, and Stephen answered 6 less tb^a Edward. How many questions did they both ansv. er ? 5. A blacksmith who had 100 dollars, laid his money out as follows — For iron 6Q dollars, for steel 30 dollars, and the remainder for coal. How many dollars did he pay for coal '^ 22 ORAL ARITHMETIC. Ill 6. A carpenter paid 31 dollars for boards, 10 dollars for shingles, 6 dollars for nails, and 5 dollars for screws. How many dollars did he spend ? 7. A trader gave 4S dollars for a chest of tea, and 3 dollars for getting it home. For how much must he sell the tea, in order to gain 8 dollars ? 8. If I have 70 dollars on hand, and pay out 4 dollars to one man, 20 to another, and 30 to another, how many dollars shall I have remaining ? 9. A gentleman travelled 8 miles before breakfast, 30 more before dinner, and 40 more after dinner. How many miles did he travel during the day ? 10. A merchant, who had 80 barrels of flour, sold to one man 52 barrels, to another 6 barrels, and to another 5 barrels. How many barrels had he left .^ 11. Ohver's penknife is worth 33 cents, and Edwin's is worth only 19 cents. Now if they exchange pen- knives, how many cents must Edwin give Oliver ? 12. Arthur's penknife was worth 25 cents, and Wal- ter's was worth only 18 cents: still, A. gave W. 6 cents to exchange. How much did A. lose ? 13. On the Fourth of July, Robert had 50 cents given him. He spent 6 cents for fruit, 12 cents for confection- ary, 20 cents for a picture of Gen. Washington, and gave away 5 cents. How many cents had he left? 14. Leonard has 32 cents, and Albert has 49. How many cents has Albert more than Leonard ? 15. Francis being asked how old he was, answered, that in 14 years more, he should be 25 years old. How old was he, at the time he was asked ? 16. If a cow be worth 22 dollars, and a calf 5 dollars, how much more is the cow worth than the calf? 17. A jockey gave 85 dollars for a horse, and sold him for 68 dollars. How much did he lose ? 18. A trader gave 83 dollars for a hogshead of sugar, and sold it for 96 dollars. How much did he gain ? 3 9. A farmer gave 24 dollars for a cart, and 12 dollars for a, plough. How many dollars did both cost? 20. If a gold watch be worth 64 dollars, and a gold chain 18 dollars, how much are they both worth? 5. MISCELLANEOUS EXAMPLES. 23 21. A man gave 13 dollars for the improvenienl of a piece of ground, paid 36 dollars for having it cultivated, and then sold the produce for 48 dollars. How many dollars did he lose ? 22. A market-man bought some butter for S dollars, some cheese for 15 dollars, and some poultry for 12 dollars; and then sold the whole for 39 dollars. Did he gain or lose ? — and how much ? 23. John bought a penknife for 25 cents; he exchanged It for a better one, paying 16 cents, and then sold the better one for 40 cents. Did he gain or lose ? — and how much ? 24. A gentleman gave 32 dollars for a piece of cloth, and 13 dollars for having it made into a suit of clothes. How much did the suit cost ? 25. A tailor gave 54 dollars for some cloth and trim- mings; he made the whole into clothes, which .he sold for 72 dollars. How much did he gain by the work? 26. If a man, having 50 dollars, should buy a barrel of sugar for 24 dollars, and a barrel of molasses for 13 dollars, how many dollars would he have left ? 27. A scholar gave 55 cents for a geography, and 42 cents for an arithmetic. What did he give for both .^ 28. Ellen had 30 cents; her father gave 16 more, and her mother 10 more; she then bought a book for 45 cents. How many cents had she remaining ? 29. James had 38 cents, and his father gave him 14 more — William had 33 cents, and his father gave hhn 19 more. Which boy then had the most money ? 30. Lucy has 75 cents, and she intends buying a book, that will cost 63 cents. How much money has she more than the book will cost ? 31. A grocer purchased some oranges for 18 dollars, some lemons for 8 dollars, some raisins for 4 dollars, and some figs for 6 dollars; and then sold the whole for 44 dollars. Did he gain or lose ? — and how much ? 32. A company marched 82 miles in three days. It marched 25 miles the first day, 36 miles the second day, . and the remainder of the distance the third day. How many miles did it march the third dav ? M ORAL ARITHMETIC. IV. CHAP. IV. MULTIPLICATIOir. Note to Teachers, In tlie second and third sections of this chapter, the learners are required to find the products of factors as high as 10 and 20. There are few scholars, who will easily comuj it these products to memory; and it will, theiefore, be necessary to adopt a mental process, by which they may leadiiy be found. 'J'he following example will show the process. Question. How many are 8 times 16 1 Solution. 16 is made up of 10 and 6. Eigli* times tea are eighty; eight times six are forty-eight. 80 and 48 ar€ 128. Section 1. 1. On the Fourth of July, George and Richard went to the Celebration, and whenever George spent 1 cent, Richard spent 7. In the course of the day, George spent 6 cents. How many did Richard spend ? Solution, Richard must have spent six times seven cents. 6 times 7 are 42. 2. If 4 yards of cloth are required to make 1 cloak, how many yards are required to make 5 cloaks .'^ 3. If 1 boat will carry 5 men across the river, how many men will 3 boats of the same size carry ? 4. If a man can earn 8 dollars in one week, how many dollars can he earn in 3 weeks ^ 5. If a traveller ride 6 miles in one hour, how many miles can he ride in 9 hours ? G. In a garden, there are 5 rows of plum trees; 8 trees in each row. How many trees are there in the garden } 7. In a field there are 8 rows of apple trees; 5 trees in each row. How many trees are there m the field ? 8. There were 7 boys, who gave a poor man, 4 cents apiece. How many cents did the man receive ? 9. How many merit-marks will Susan get in 6 days, provided she gets 4 every day ? 10 How many errors will Jane make in 8 days, pro- vided she makes 3 errors every day ? 11. If the price of 1 quart of nuts b® 7 cents, for how many cents can you buy 7 quarts .'' 12. If you should read 9 pages every day, how many pages would you read in 8 days ? 1. 2. MULTIPLICATION. 25 13. How many dollars must I pay for 9 yards of cloth, that is worth 6 dollars a yard ? 14. How many dollars must I pay for 9 barrels of flour, when the price is 7 dollars a barrel ? 15. If 4 bushels of wheat are required for 1 barrel of flour, how many bushels are required for 8 barrels ? 16. What is the cost of 7 reams of letter paper, that is sold at 6 dollars per ream ? 17. How much will a market-man get for 10 melons, if he sell them at 9 cents apiece ? 18. If a man can earn 9 dollars in one month, how many dollars can he earn in 6 months ? Section 2. 1. John and Henry together, caught 13 fishes every morning; and it always happened, that John caught 10 of them, and Henry 3. Now how many did each boy catch in 6 mornings ? — Then how many did they both catch in 6 mornings ? 2. 10 and what number make 13 ? How many are 6 times 10? 6 times 3? How many are 60 and 18? — Then 6 times 13 are how many ? 3. If the diamond in a ring cost 10 dollars, and the ring 4 dollars, what does the diamond-ring cost ? What would 5 diamonds cost ? What would 5 rings cost ? — Then what would 5 diamond-rings cost ? 4. 10 and what number make 14 ? How many are 5 times 10? 5 times 4? How many are 50 and 20.^—^ Then 5 times 14 are how many ? 5. A Northern hunter received a bounty of 10 dollars from the state, and 5 from the county, for killing a wolf: how much did he receive from both ? How much would he receive from the state for killing 7 wolves ? How much from the county for killing 7 ? — Then how much from state and county both for killing 7 ? 6. 10 and what number make 15 ? How many are 7 times 10 ? 7 times 5 ? How many are 70 and 35 ? — Then 7 tim.es 15 are how many ? 7. A man hired a horse and gig, to pay 10 cents a mile for the horse, and 6 cents a mile for the gig. How c 26 ORAL ARITHMETIC. IV. much a mile did he pay for both ? How much for the horse 8 miles ? How much for the gig 8 miles ? — Then how much for the horse and gig, both 8 miles ? 8. 10 and what number make 16 ? Haw many are 8 times 10 ? 8 times 6 ? How many are 80 and 48 ? — Then 8 times 16 are how many ? 9. If a man gather 10 barrels of apples, and a boy 7 barrels, in a day, how many barrels do they both gather? How many barrels can the man gather in 9 days ? How many barrels can the boy gather in 9 days ? — Then how many barrels can they both gather in 9 days ? 10. 10 and what number make 17 ? How many are 9 times 10 ? 9 times 7 ? How many are 90 and 63 ? — Then 9 times 17 are how many ? 11. If a man eat 10 ounces of meat in a day, and his wife eat 8 ounces, how many ounces do they both eat in a day ? How many ounces will the man eat in 4 days ? How many ounces will the wife eat in 4 days ? — Then how many ounces will they both eat in 4 days ? 12. 10 and what number make 18 .'^ How many are 4 times 10 .^ 4 times 8 ? How many are 40 and 32 ? — Then 4 times 18 are how many? 13. If a company of soldiers march 10 miles in the forenoon, and 9 miles in the afternoon, how many miles do they march in a day ? How many miles would they march in 6 forenoons ? How many in six afternoons ? — Then how many miles in 6 days ? ' 14. 10 and what number make 19? How many are 6 times 10 ? 6 times 9 ? How many are 60 and 54 ? — Then 6 times 19 are how many? 15. How many 'are 6 times 10? 6 times 2? — Then how many are 6 times 12 ? 16. How many are 7 times 10? 7 times 3.^ — Then how many are 7 times 13 ? 17. How many are 8 times 10? 8 times 4? — Then how many are 8 times 14 ? 18. How many are 9 times 10? 9 times d? — Then how many are 9 times 15 ? 19. How many are 10 times 10 ? 10 times 6 f — Then how many are 10 times 16 ? 2. 3. MULTIPLICATION. 27 ■Then -Then —Then -Then 20. How many are 3 times 10? 3 limes 7 how many are 3 times 17 .'^ 21. How many are 2 times 10? 2 times 8 how many are 2 times IS ? 22. How many are 5 times 10? 5 times 9 how m_any are 5 times 19 ? 23. How many are 4 times 10? 4 times 9 how many are 4 times 19 ? Section 3. 1 . Albert spends 1 1 cents every month, for stationary. How many cents will he spend in 3 months ? 2. How many are 2 times 11? 3 times 11? 4 times 11? 5 times 11? 6 times 11? 7 times 11? 8 times 11? 9 times 11? 10 times 11? 3. If you should write 11 copy-lines every day, how many copy-lines would you write in 10 days ? 4. If a pound of Malaga raisins cost 12 cents, how many cents will 4 pounds of raisins cost ? 5. How many are 2 times 12? 3 times 12? 4 times 12? 5 times 12? 6 times 12? 7 times 12? 8 times 12? 9 times 12? 10 times 12? 6. If you should read 12 verses every morning, how many verses would you read in 9 mornings ? 7. Suppose a st^am boat will go 13 miles in an hour;— how many miles will it go in 5 hours ? 8. Howmanyare2timesl3r 3timesl3? 4timesl3? 5 times 13? 6 times 13? 7 times 13? 8 times 13? 9 times 13? 10 times 13? 9. If a labourer can earn 13 dollars in a month, how many dollars can he earn in 8 months ? 10. If it take 14 men to navigate one ship, how many men will it take to navigate 6 ships ? 11. How many are 2 times 14? 3timesl4? 4timesl4? 5 times 14? 6 times 14? 7 times 14? 8 times 14? 9 times 14? 10 times 14? 12. If a pound of honey be worth 14 cents, Jhow many cents are 7 pounds of honey worth? 28 ORAL ARITHMETIC. IV 13. If a carpenter can make 15 hat boxes in a day, how many hat boxes can he make in 2 days ? 14. Howmanyare2timesl5? 3timesl5? 4timesl5? 5 times 15? 6 times 15? 7 times 15? 8 times 15? 9 times 15? 10 times 15? 15. If a shoemaker make 15 pairs of shoes in a week, how many pairs will he make in 10 weeks ? 16. How much would a man earn in 3 months, pro- vided his wages were 16 dollars a month? 17. Howmanyare2timesl6? 3timesl6? 4timesl6? 5 times 16? 6 times 16? 7 times 16? 8 times 16? 9 times 16? 10 times 16? 18. How much would a man spend in 9 months, if his expenses were 16 dollars a month? 19. If 17 barrels of flour can be carried on one wagon, how many barrels may be carried on 4 wagons ? 20. How many are 2 times 17? 3timesl7? 4timesl7? 5 times 17? 6 times 17? 7 times 17? 8 times 17? 9 times 17? 10 times 17? 21. If one hogshead of molasses be worth 17 dollars, what is the value of 8 hogsheads of molasses ? 22. If 5 men should pay me 18 dollars apiece, how many dollars should I receive from them all ? 23. How many are 2 times 18? 3timesl8? 4timesl8? 5 times 18? 6 times 18? 7 times 18? 8 times 18? 9 times 18? 10 times 18? 24. If I should pay to 7 men, 18 dollars apiece, how many dollars should I pay to all of them? 25. How much would a farmer get for 3 cows, if he should sell them for 19 dollars apiece ? 26. Howmanyare2timesl9? 3timesl9? 4timesl9? 5 times 19? 6 times 19? 7 times 19? 8 times 19? 9 times 19? 10 times 19? 27. If you answer 19 questions at every recitation, how many would you answer in reciting 6 times? 28. Paper is generally packed in reams, of 20 quires each. How many quires are there in 4 reams? 4. MULTIPLICATION. 29 29. How many are 2 times 20? 3 times 20? 4 times 20? 6 times 20 ? 6 times 20 ? 7 times 20 ? 8 times 20 ? 9 times 20? 10 times 20? 30. If a trader make 20 cents on every pound of tea he sells, how much will he make on 5 pounds. Section 4. 1. An ounce is a small weight, 16 of which make a pound. How many ounces of tea are there in 3 pounds and 9 ounces of tea? Solution. In 1 pound there are 16 ounces, and in 3 pounds there are 3 times 16 ounces. 3 times 16 ounces are 48 ounces. 48 ounces and 9 ounces are 57 ounces. 2. How many ounces in S- pounds and 4 ounces ? 3. 20 penny-weights of gold make 1 ounce. How m.any penny-weights in 4 ounces and 13 penny- weights ? 4. How many penny- weights are there in 6 ounces and 9 penny-w^eights ? 5. The brewer sells his beer by the firkin, and a fir- kin holds as much as 9 gallon measures. How many gallons are there in 10 firkins and 6 gallons ? 6. In 7 firkins and 3 gallons, how many gallons ? 7. 40 rods, measured by a surveyor's chain, make 1 furlong. How many rods in 3 furlongs and 17 rods ? 8. In 4 furlongs and 8 rods, how many rods ? 9. There are 12 months in a year. How many months are there in 6 years and 6 months ? 10. In 9 years and 10 months, how many months ? 11. 4 pecks of oats, peas, beans, or any other dry commodity, make 1 bushel. How many pecks of wheat are there in 6 bushels and 3 pecks ? 12. In 10 bushels and 1 peck, how many pecks ? 13. 12 pence, in English money, make 1 shilling. How many pence are .there in 8 shillings and 6 pence? 14. In 10 shillings and 9 pence, how many pence ? 15. 10 cents, in Federal money, make 1 dime. How Hiany cents are there in 7 dimes and 6 cents ? 16. In 4 dimes and 8 cents, how many cents ? 17. In 1 month there are 30 days. How many days are there in 3 months and 15 days ^ so 55^>C«C00<>OOC<>00OC>0<.:<>C:-0-: o 00 ^ CO (^^ GO CO c^ g^ QO CI QD lO GO CO CO CO CO o CO CI a GO o o GO o ^0 00 o CO CO QO o CI GO CI lO ? VO o >0 o lO lO o CO o CO lO Ci o Ci ^ wo o QD GO c^ o CO o CO QO c^ o CO o CO c^ CO CO Oi c^ o GO o 00 o CO CO CO CI CO 00 CI d CO CO QO CI o CI CO CO o QO QO o QO CI CO CO 10 00 o CI CO CO QO GO o o CI CO CO »iO CO CO CI 00 6 S 2 2 ot^ c S5 6 5oS ^. o 00 - i « « V ^ £ S o- •6 C ^ a o -':.> bjD c: ^ ^ i) 5 « S fl ^ a O (V I C»0<>00<>C>OOOC-<>C'-C«.>C>OOC<:<:'0<>^ »-i -S '*^ ** o .s C3 ^ w 0) W v^ OT ^ "^ (ij It- er ^^ ^ !> C g p. 1 DIVISION. 31 CHAP. V. DIVISIOJV. Section 1. 1. A lady divided 15 peaches among some little girls, giving 3 to each girl. How many girls were there ? Solution. As many times as 3 peaches are contained in 15 peaches, so many girls there were. 2. If you had 16 cents to lay out in pencils, and the price of the pencils were 4 cents apiece, how many could you buy for all the money ? 3. How many times is 4 contained in 16 ? 4. If 4 horses are required to draw one wagon, how many wagons might be drawn by 20 horses ? 5. How many times 4 in 20 ? How many are 5 times i? 6. If a man can travel 4 miles in one hour, how many hours will it take him to travel 12 miles ? 7. How many times 4 in 12 ? Hotv many are 3 times 4? 8. How many yards of broadcloth, that is sold at 7 dollars a yard, can be purchased for 14 dollars ? 9. How many times 7 in 14 ? How muny are 2 times 7? 10. How many lead pencils could you buy for 18 cents, if they were sold at 6 cents apiece ? 11. How many times 6 in 18 ? Hoiv many are S times 6? 12. In an orchard there are 35 trees, standing in rows, 7 trees in a row. How many rows are there ? 13. How many times 7 in 35? Howmanyare 5times7? 14. A man bought sheep at 4 dollars apiece, and paid for them all, 24 dollars. How many did he buy ? 15. How many times 4 in 24 ? How many are Q times A7 16. A gardener set out 30 peach trees, in rows, put- ting 5 trees in a row. How many rows were there } 17 How many times 5 in 30.^ How m^ny are 6 times 5? 18. A farmer got 36 dollars for some sheep, that he Bold at 6 dollars apiece. How many were there ? 19 How many times 6 in 36 } How many are 6 times 6? 32 ORAL ARITHMETIC. V\ 20. A trader wishes to pack 56 hats in boxes, putting 8 hats in a box: — how many boxes are wanted ? 21. How many times 8 in 56 ? How many are 7 times S? 22. How many dozen of eggs can you buy for 63 cents, when they are sold at 9 cents a dozen? 23. How many times 9 in 63 .'^ How many artV times 9? •24. If an orange be worth 6 cents, and a hme 1 cent, then how many oranges are 60 limes worth ? 25. How many times 6 in 60 ? How many ar elO times G? 26. If one silk bonnet be Worth 7 handkerchiefs, how many bonnets ar-e 56 handkerchiefs worth ? 27. How many times 7 in 56 ? How many are S times 7? 28. If I give a barrel of flour for 4 bushels of wheat, how many barrels must I give for 36 bushels ? 29. How many times 4 in 36.^ How many are 9 times ^'t 30. If a man can build S^rods of fence in a day, how many days will it take him to build 72 rods ? 31. How many times 8 in 72 ? Eoicf many ar e 9 times S? 32. If 5 bushels of wheat will pay for a yard of broad cloth, how many yards will 45.bus]iels pay for? 33. How many times 5 in 45? How many are 9 times ^? .34. Lafayette was 42 days on iiis passa ge from Toulon to New York. How many weeks !was hi, ^ passage ? 35. How many times 7 in 42?' Howma\ iyare6times7? Section 2. 1. Suppose 2 men have 8 biscuit to diviv ^^ equally between them; — how many must each.m^n tak ^^ ^ Observution, There are 2 men to share the biscuit, and if there were only 2 biscuit to her divided, ^ ^ man would take 1 biscuit. Therefore,! man wii ^ ^^^^ take 1 biscuit, of every 2 bisctiit. Solution, As many times as 2 is contained in t '•> ^^ many biscuit must each man take. 2 is contained it ■ ^^ 4 jtimes. ^. Jf 12 dollars be divided equally betwe.en '2 mei h \y many dollars does each man receive? \ Jf 18 chestnuts should be divii^ed equally between ^g, how many would each jbw ^^give ? 2. DIVISION 33 4. A tenant cultivated a piece of corn, agreeing to give the owner of the land 1 bushel of every 2 bushels that he might raise. He raised 22 bushels. How many- bushels should the owner of the land receive ? 5. Suppose 3 boys have 12 oranges to divide equally between them; — how many must each boy take } Observation, If the 3 boys had only 3 oranges to di- vide, each boy would take I orange; — if they had 2 times 3 oranges, each boy would take 2 times 1 orange: — and thus, each boy will take as many times 1 orange, as there are threes in the number to be divided. Solution. As many times as 3 is contained in 12, so many oranges must each boy take. 3 in 12, 4 times, 6. If 15 biscuit be divided equally between 3 men, how many biscuit does 1 man receive.^ 7. 3 careless boys must pay 24 cents for breaking a square of glass. What must each boy pay.'^ 8. A man set out 27 trees, in 3 rows; an equal num- ber in each row. How many were there in one row ? 9. 4 boys have 12 oranges to divide equally between them. How many will each boy receive ? Observation. If the 4 boys had only 4 oranges to di- vide, then each boy would receive 1 orange. Therefore each boy must receive 1 orange of every 4 oranges that there are in the number to be divided. Solution. As many times as 4 is contained in 12, so many oranges will each boy receive. 4 in 12, 3 times. 10. I have 32 minutes to spend on 4 lessons. How many minutes can I, spend on each lesson? 11. Suppose I wish to give* 28 quills to 4 boys; — how many must I give to each boy? 12. 4 men received 20 dollars for doing a piece of work. How much was each man's share ? 13. A fisherman hired a boat, agreeing to give the owner, 1 fish of every 5, that he might catch: he caught 20. How many should he give the owner ? 14. If 35 pounds of beef be divided among 5 soldiers, how many pounds does each soldier receive ? 15. 5 men have agreed to pay equal shares of 50 dol- lars How many dollars must one man g^y? 34 ORAL ARITHMETIC. V. 16. Charles was one of 6 boys, who owned together 42 books. They divided the books, and Charles receiv- ed 1 book of every 6 books. How many did he receive.'* 17. If 24 books should be divided equally* among 6 boys, how many would each boy receive ? 18. 6 men have agreed to pay 36 dollars in equal shares. How many dollars must each man pay.^ 19. Edward is one of 7 boys, who are to have 23 peaches divided equally among them. How many will Edward receive for his share ? 20. If 7 writing-books be made of 42 sheets of paper how many sheets are there in each book ? 21. 8 boys owned together 72 quills; and, in order to share them equally, each boy took 1 quill from every 8 quills in the number. How many did each boy take ? 22. I have 48 dollars to divide among 8 men. How many dollars must I give to one man .'' « 23. 9 men shared 45 bushels of corn among them, each man taking for his share, 1 bushel of every 9 bush- els. How many bushels did each man take ? 24. 9 persons have agreed to make up a purse of 72 dollars. How many dollars must each one put in ? 25. 10 sailors are to receive 90 dollars for retaking their ship. How much will each sailor receive ? Section 3. 1. Charles has 25 cents, which he has engaged to appropriate as follows. Whenever he blots his writing- book, he is to lay it aside, and pay 6 cents for a new one. Now how many books can he pay for; and how many cents will he have remaining, after his number becomes too small to buy another book. Observation. If his whole number of cents were 30, he could then pay for five books ; because, 6 cents are con- tained in 30 cents, 5 times. Again, if his whole number of cents were only 24, he could then pay for 4 books; because, 6 cents are contained in 24 cents, 4 times. 2. Suppose it takes 8 buttons to trim a vest: — how many vests can the tailor, who has only 34 buttons, trim* and what number of buttons will he have remaining ? 3. DIVISION. 35 3. How many times 8 in 34; and how many over ? 4. How many glass tumblers, at 10 cents apiece, can a woman that has only 53 cents, buy; and how many cents will she have remaining ? Solution, As many times as 10 is contained in 53, so many tumblers she can buy. 10 is contained in 53, 5 times, and there is three over. 5. How many kegs, that will hold 7 gallons apiece, may be filled from a cask of wine containing 46 gallons; and how many gallons will remain in the cask ? 6. How many times 7 in 46; and how many over.'* 7. A hat-maker has 53 hats finisiied; and, in order to send them to market, he must pack them in boxes, that will hold 8 hats apiece. How many full boxes can he send; and how many hats will remain on hand ? 8. How many times 8 in 53; and how many over ? 9. A trader has 69 dollars with which he wishes to purchase hats. If he should pay 7 dollars apiece for the hats, how many could he purchase; and how many dol- lars would he have remaining ? 10. How many times 7 in 69; and how many over -* 11. If 4 yards of cloth will make 1 cloak, how many cloaks can be made from a piece of cloth containing 38 yards; and how many yards will there be over .'^ 12. How many times 4 in 38; and hew many over .'^ 13. How many times is 4 contained in 29; and how many over r How many are 7 times 4, and 1 more? 14. How many times is 6 contained in 53; and how many over } How many are 8 times 6, and 5 more ? 15. How many times is 9 contained in 57; and how many over } How many are 6 times 9, and 3 more ? 16. How many times is 7 contained in 68; and how many over } How many m-e 9 times 7, and 5 more ? 17. How many times is 5 contained in 49; and how many over : How many are 9 times 5, and 4 mx)re ? 18. How many times is 3 contained in 2Q'., and Jbow many over } How many are 8 times 3, and 2 more ? 19. In 15, how many times 4; and how many ovqy? In 17 ? In 26 ? In 33 ? In 27 } In 42 .? 36 ORAL ARITHMETIC. V. 20. In 27, how many times 5; and how many over? In 29? In 36? In 32 ? In 44 ? In 48 ? 21. In 28, how many times 6; and how many over? In 37 ? In 38 ? In 46 ? In 49 ? In 22 ? 22. In 30, how many times 7; and how many over ? In 36? In 43? In 48 ? In 51 ? In 59 ? 23. In 28, how many times 8; and how many over? Tn 35 ? In 46 ? In 52 ? In 61 ? In 75 ? 24. In 31, how many times 9; and how many over ? tn 34 ? In 42 ? In 50 ? In 67 ? In 70 ? 25. A gallon measure, used for jneasuring wine, beer, milk, &c. will contain as much as 4 quart measures. Suppose I have 15 quart measures full of water; — how many gallon measures can I fill from them; and how many quarts will there be over ? 26. How many gallons are there in 34 quarts ? 27. 3 feet, measured on a line, are the same as 1 yard. How many yards are there in 29 feet ? 28. How many yards are there in 17 feet ? 29. 60 minutes, by the clock, make 1 hour. How many hours are there in 128 minutes ? 30. 12 inches, on the carpenter's rule, make 1 foot. How many feet long is a board, that is 65 inches long ? 31. How many feet are there in 38 inches ? 32. 8 drams of medicine, weighed by the apothecary, are the same as 1 ounce of medicine. How many ounces are there in 46 drams ? 33. How many ounces are there in 30 drams ? 34. 9 square feet, measured upon the floor, make 1 square yard. How many square yards are there in 20 square feet? 35. How many square yards in 51 square feet? 36 7 days are 1 week. How many weeks in 33 days ? 37. How many weeks are there in 52 days ? 38. 12pence, in Enghsh money, make Ishilhng. How .^any shillings are there in 69 pence ? 39. How many shillings are there in 46 pence ? 40. 10 cents, in Federal money, make 1 dime. How many dimes are there in 48 cents ? 41. How many dimes are there in 95 cents ? 4. MULTIPLICATION AND DIVISION. 37 Sectiojv 4. CORRESPONDENT EXAMPLES. 1. How much will 5 pounds of dates cost, when the price of them is 7 cents a pound ? Solution, If one pound cost 7 cents, 5 pounds will cost 5 times 7 cents. 5 times 7 are 35. 2. How many pounds of dates can you buy for 35 cents; the price being 7 cents a pound ? Solution. I can buy as many pounds of dates as 7 is contained times in 35. 7 in 35, 5 times. 3. How many quarts of wine are there in 10 gallons; there being 4 quarts in one gallon ? 4. How many gallons of wine are there in 40 quarts; every 4 quarts making one gallon ? 5. How many pence are there in 5 shillings; there being 12 pence in 1 shiUing ? 6. How many shillings are there in 60 pence; there being 12 pence in 1 shiihng ? 7. In 1 penny there are 4 farthings. How many far- things in 9 pence and 3 farthings ? 8. How many pence are there in 39 farthings; — and how many farthings are there over ? 9. A gentleman went on a journey of 9 days, and paid for his expenses, 4 dollars per day. How much were his expenses during the whole journey ? 10. A gentleman, who had been away on a journey for 9 days, found on his return, that he had spent 36 dol- lars. How much did he spend a day ? Suggestion. 9 dollars would allov/ him 1 dollar a day. 11. If 6 bushels of onions grow upon 1 square rod of ground, how many bushels will grow upon 10 rods ? 12. A man raised 60 bushels of onions upon 10 rods of ground. How many bushels grew upon 1 rod } 13. If a ship sail 7 miles an hour, how many miles will she sail in 7 hours ? 14. If a ship sail 49 miles in 7 hours, how many mileg does she sail in 1 hour ':' 38 ORAL ARITHMETIC. V. Section 5. CONNECTED OPERATIONS. 1. A market man sold 10 pounds of cheese at 8 cents a pound, and received his pay in sugar at 10 cents a pound. How many pounds of sugar did he receive ? Solution. The price of 1 pound of cheese being 8 cents, the price of 10 pounds is 10 times 8 cents, or 80 cents. 80 cents will pay for as many pounds of sugar, as there are times 10 in SO. 10 in 80, 8 times. 2. In 10 times 8, how many times 10? Solution. 10 times 8 are 80. 10 in 80, 8 times. 3. 4 coaches went from Baltimore to Washington, each carrying 6 boys: the same boys returned, riding 8 in a coach. In how many coaches did they return ^ 4. In 4 times 6, how many times 8 ? 5. How maoy boxes of raisins at 6 dollars a box, will pay for 4 kegs of tobacco at 9 dollars a keg ? 6. In 4 times 9, how many times 6 ? 7. How many sheep worth 4 dollars a head, must be given for 6 tons of hay, worth 8 dollars a ton ? 8. In 6 times 8, how many times 4 ? ' 9. How many reams of paper at 3 dollars a ream, will pay for 5 dozen of books at 6 dollars a dozen ? 10. In 5 times 6, how many times 3 ? 11. A hunter, in Michigan, sold 7 pelts at 5 dollars a pelt, agreeing to take his pay in muskets at S dollars apiece The purchaser counted out as many muskets as the pelts would pay for, and finding there was still a balance due to the hunter, he paid this in money. How many mus- kets and how much money did the hunter receive ? 12. In 7 times 5 how many times 8; how many over 13. In 3 times 4 how many times 5; how many over 14. In 8 times 5 how many times 7; how many over 15. In 7 times 7 how many times 5; how many over 16. In 5 times 6 how many times 4; how many over 17. In 9 times 3 how many times 6; how many over 18. In 4 times 8 how many times 9; how many over 19. In 6 times 9 how many times 8; how many over 5. 6. MISCELLANEOUS EXAMPLES. 39 Note to Teachers. In performing oral solutions which involve several of»- erations, it will be found convenient, although not absolutely necessary, to use the terms j)/ttsand minus. These terms, however, should not be adopted until they are perfectly understood; and the duty of explaining them is here confided to the Teacher. It will not suffice, merely to tell the learner, that "plus means more J and minus means less,'' for the idiom of our language does not allow him to use more and less, as he is called upon to use plus and minus. The ex- pressions, " 7 more 5 is 12," and, ** 10 less 4 is 6,'* although true, are not very likely to meet the understanding of a young scholar. Perhaps it will be found necessary to resort to illustrations like the following. Place 8 books in a pile before you, and say, — " Here is a pile of 8 books, and I shall make the number of books in the pile 3 more." I'hen, placing 3 additional books upon the pile, say, — " 8 books plus 3 books are 11 books." Again proceed to say,—" I shall now make the number 2 less." Then, taking off 2 books, say, — " 11 books, minus 2 books are 9 books." Section 6. MISCELLANEOUS EXAMPLES. 1. How many days in the three Summer months; — ' there being 30 in June, 31 in July, and 31 in August ? Solution, 30 days plus 31 days are 61 days; 61 days plus 31 days are 92 days. 2. There were 42 gallons of wine in a cask; but, the cask not being tight, 7 gallons have leaked out. How many gallons still remain in the cask ? Solution. 42 minus 7 is 35. Answer. 35 gallons. 3. A boy that had 97 cents, paid 62 cents for a book, 18 cents for a morocco wallet, and 6 cents for a pencil. How many cents had he remaining ? Solution. 62 cents (for book), plus IS cents (for wallet), is 80 cents; 80 cents plus 6 cents (for pencil), is 86 cents. 97 cents minus 86 cents is 11 cents. 4. A black-smith bought 9 tons of coal, at 4 dollars per ton, and gave 3 dollars for having it drawn to his shop. How much did the coal cost him ? Solution, If the price of 1 ton was 4 dollars, the price of 9 tons was 9 times 4 dollars, or 36 dollars; 36 dollars plus 3 dollars (for having it drawn) is 39 dollars. 5. A schoolmaster laid out 96 cents in writing-books, at 8 cents apiece, and then gave away 5 of them. How many books had he remaining ? Solution. He bought as many books, as 8 is contained times in 96, which is 12. Then, 12 books minus 5 books, (that he gave away), are 7 books. 40 ORAL ARITHMETIC. V 6. A trader gave 16 dollars for a keg of tobacco, and after selling it, found he had gained 6 dollars. For how much did he sell it ? 7. A rich farmer in Vermont had a flock of 100 sheep; they went upon a mountain, and the wolves destroyed IB of them. How many sheep had he remaining ? 8. If a man spend 4 dollars in a week, how many dol lars will he spend in 9 weeks ? 9. How many dozen of eggs can you buy for 64 cents, when the price of them is 8 cents per dozen ? 10. A trader gave 48 dollars for 7 barrels of flour, and sold it for 6 dollars a barrel. What did he lose ? 1 1 . How many weeks are there in 35 days ? 12. Four men made up a purse of 40 dollars, for a charitable purpose. The first man put in 9 dollars, the second 12 dollars, and the third 7 dollars; — how much did the fourth man put in ^ 13. A sum of money was divided equally among 8 sailors; and Jack, who was one of the number, received for his share, 15 dollars. What was the sum divided? 14. How many days are there in 13 weeks and 5 days.^ 15. A trader bought 3 reams of paper, at 5 dollars per ream, and 7 maps, at 5 dollars apiece. How much did he give for the whole ? 16. If 49 bushels of corn should be divided equally among 7 men, how much would one man receive ? Solution. As- many times as 7 is contained in 49, so many bushels would one man receive. 17. If 45 dollars be divided equally between 5 men, how many dollars does each maa receive ? 18. A man bought a turkey weighing 10 pounds, for 8 cents a pound, and then sold it for 3 cents a pound more than he gave. For how much did he sell it ? 19. Charles had 25 cents; his father gave him 4 mel- ons, which he sold for 6 cents apiece; he then paid 12 cents for a book. How many cents had he left ? 20. If a man earn 6 dollars a week, how many weeks will it take him to earn 48 dollars ? 21. Geographers consider the United States in four classes. There are 6 Eastern states, 4 Middle states, 6. MISCELLANEOUS EXAMPLES. 41 8 Southern states, and 6 Western states. How many states are there in the Union ? 22. A merchant paid 43 dollars for some iron, and sold it for 35 dollars. How many dollars did he lose ? 23. A man paid 78 dollars for a piece of land, and 16 dollars for*having it fenced; and he then sold it for 100 dollars. Did he gain or lose; — and how much ? 24. A cabinet-maker sold 6 tables, at 14 dollars apiece. How many dollars did he receive ? 25. If I bay 10 yards of cloth, at 7 dollars a yard, how many five-dollar bills must I pay for it ? 26. How many boxes of strawberries can you buy for 36 cents, when they are sold at 9 cents a box ? 27. Suppose a trader, who has 12 barrels of flour on hand, should lay out 35 dollars in buying more flour, at 5 dollars a barrel; how many barrels would he have ? 28. If I pay 19 dollars to one man, 13 to another, and 31 to another, how many dollars do I pay out ? 29. If a laborer can earn 7 dollars in a week, how many weeks will he be in earning 42 dollars ? 30. How many hats, that are sold at 6 dollars apiece, can a man who has 50 dollars pay for; — and how many dollars will he have remaining ? 31. If you should perform 19 examples in arithmetic, every day, how many would you perform in 6 days ^ 32. Samuel Moderate earns 7 dollars a month, and John Smart earns 15 dollars a month. How much more will John earn than Samuel, in 6 months ? 33. If 1 man do 1 day's work in 1 day, how many men will it take to perform 7 days' work, in 1 day ? 34. If 4 men will perform 4 days' work in 1 day, how many days' work will 4 men perform in 9 days ^ 35. How many days will it take 4 men to dig a cellar, that 1 man would be 36 days in digging ? 36. How many days will it take 7 men to clear a piece of wood-land, that 28 men can clear in one day ? 37. How many men will it take to perform as much work in 1 day, as 1 1 men can perform in 6 days ? 38. How many days will it take 4 men to perform the same work, that 12 men can perform in 3 days ? 42 ORAL ARITHMETIC. V 39. A trader has three bundles of bank notes; — 23 dollars in one bundle, 15 dollars in another, and 34 dol- lars in another; but in one of the bundles there is a note of 5 dollars, which is counterfeit. How many dol- lars of good money has he ? 40. Stephen has lost 30 cents, and has found 10 cents; he now has 18 cents. How much had he at first ? 41. A farmer went to the city with 8 barrels of cider, which he sold at 4 dollars a barrel. He then purchased 3 hogsheads of salt, at 3 dollars per hogshead, and paid an old debt of 12 dollars. How many dollars had he to carry home ? 42. If I pay 3 dollars apiece for 7 umbrellas, and 6 dollars apiece for 6 hats, for how many dollars must I sell the whole, in order to gain 7 dollars ? 43. A man borrowed 75 dollars, and the next day paid all but 14 dollars of it. How much did he pay ? 44. A. spent 5 dollars as often as B. spent 3 dollars. How much did A. spend, while B. spent 27 dollars ? Solution, As many times as 3 dolkrs are contained in 27 dollars, so many times 5 dollars did A. spend. 3 dollars are contained in 27 dollars 9 times; therefore A. spent 9 times 5 dollars. 45. A man and a boy were gathering corn; — the man gathered 7 rows, in the same time that the boy gathered 4 rows. How many rows would the man gather, while the boy was gathering 32 rows ? 46. A man and a boy were digging potatoes; — the man dug 11 bushels in the same time that the boy dug G bushels. How many bushels would the boy dig, while the man was digging 55 bushels ? 47. Suppose butter to be worth 12 cents a pound, and tea 42 cents a pound; — how many pounds of butter must be given for 2 pounds of tea ? '48. A farmer sold 2 cows at 23 dollars apiece, and 9 sheep at 5 dollars apiece; he received in payment, 3 ploughs at 8 dollars apiece, and the rest in money. How much money did he receive ? 49. 4 boys found a purse containing 29 dollars. They paid 2 dollars for advertising it; and, as no owner ap- (5. MISCELLANEOUS EXAMPLES. 43 peared, they agreed to take 6 dollars apiece to themselves, and give the remainder to a poor woman. How much was there remaining for the woman ? 50. What sum of money must be divided among 16 men, in order that one man shall receive 4 dollars ? 51. Two classes are studying arithmetic. The first class is 81 examples in advance of the second; the sec- ond performs 40 examples in a day, and the first, 31. In how many days will the second overtake the first ? 52. A lady paid 6 dollars for silk, 9 dollars for cam- bric, 7 dollars for linen, and then had 13 dollars remain- ing. How many dollars had she at first ? 53. How many barrels of flour, at 6 dollars per barrel, can the baker who has 45 dollars, purchase; and how many dollars will he have remaining ? 54. A trader, that has 48 dollars, wishes to buy all the boots he can pay for, at 5 dollars a pair, and then lay out the remainder of his money in shoes, at 1 dollar a pair. How many pairs of boots, and of shoes, must he buy ? 55. What sum of money must be divided among IS men, in order that one man shall receive 7 dollars ? 56. Three men made up a purse of 40 dollars. The first man put in 6 dollars, and the second 3 times as much as the first. How much did the third put in ^ 57. Eliza gave a poor woman 4 cents, Augusta gave her 3 times as much as EHza, and Lucy 3 times as much as Augusta. How much did the woman receive ? 58. If a man dig 30 bushels of potatoes in a day, and a boy 13 bushels, how many bushels will they both dig in a day ? How many will they both dig in 3 days ? 59. A farmer purchased 15 sheep; — he sold 8 of them at 4 dollars apiece, and the remainder at 3 dollars apiece; and ihen found that he had gained 7 dollars. How much did be give for the sheep ? 44 ORAL ARITHMETIC VI CHAP. VI. FRACTIOI^^S. Section 1. Note to Teachers. The subsequent progress of the learner, will depend much on a proper conception of the division of unity, and a correct application of the nomenclature of fractions. Therefore, this section, however simple it may appear, should not be slighted. It should be recited with the books closed. The picture of a board. This board, as it is presented above, is a whole thing. The same board appears hereafter divided into parts; and the parts are named according to their number and size Divided now into 2 equal parts, ^m One of these parts is one-half, g 1 . How many halves are there in the whole of any thing? 2. Suppose I can write a letter on 1-haif of a sheet of paper; how much paper shall I use, in writing 2 letters ^ 3. How much is 1-half and 1-half, added together? Divided now into 8 equal parts. One of these parts is one-third. \ 4. How many thirds are there in the whole of any thing? 5. If a carpenter can make 3 door-panels of 1 board, what part of one board will he use, in making 1 panel ? 6. Which is the greater part, 1-half, or 1 -third? Divided now into 4 equal parts. I One of these parts is one-fourth, 7. How many fourths are there in the whole of 1 thing? 8. I gave 1 -fourth of an orange to John, and 2-fourths to Frances. How much of the orange did I give a|ray ? 9. Which is the greater part, 1 -third, or 1 -fourth? Divided now into 5 equal parts. One of these parts is one-ffth. 10. How many fifths are there in the whole of any thing? 11. Charles divided a melon, equally among 5 boys. What part of the melon, [how many fifths,] had 2 boys t 12. Which is the smaller part, 1 -fourth, or 1 -fifth ? I FRACTIONS. 4S Divided now into 6 equal parts. _^^ One of these parts is one-sixth. '^^ 13. Howmany sixths are there in the wholeof any thing.^ 14. If 3 girls and 2 boys should each of them-eat 1-sixth of a pie, what part of the whole pie would they all eat ? 15. Which is the greater part, l-fifth, or 1-sixth.^ Divided now into 7 equal parts, ^m^m 16. How many sevenths are there in the whole of 1 thing.'* 17. John broke ofF2-sevenths of a new pencil, and cut off 1-seventh more. How much of it was then wasted ^ 18. Which is the smaller part, 1-sixth, or 1-seventh? Divided now into 8 equal parts. ^s=^^^^^^^^^s^^^g 19. How many eighths are there in the whole of 1 thing .^ 20. If a boy earn 3 eighths of one dollar, and find 4- eighths mora, what part of one dollar will he then have ^ 21. Which is the smaller, 1-seventh, or 1 -eighth .? Divided now into 9 equal parts. ^^ ^J ^p^M^^^^^^s^P^^ One of these parts is one-ninth. ^^^g ^B^ p^j^s^g^s^^ 22. How many ninths are there in the whole of 1 thing.'* 23. Stephen paid 3-ninths of all his money for a slate, and 6-ninths for a blank-book. How much had he left.'* 24. Which is the greater part, 1-eighth, or 1-ninth? Divided now into 10 equal parts, ^g^ One of these parts is one-tenth, ^g^ 25. Howmany tenths are there in the whole of 1 thing.** 26. If a book cost 5-tenths of a dollar, and a penknife cost 4-tenths, what part of 1 dollar will they both cost .'* 27. Which is the greater part, 1-ninth, or 1-tenth ? Remark 1st. It appears from the examples above, that, ONE-HALF of any thing, is one of two equal parts of the thing; — ONE-THIRD of any thing, is one of three equal parts of the thing; — ONE-FOURTH of any thing, is one ot four equal parts of the thing; and so on. Remark 2nd. The greater the number of parts is, into which any thing is divided, the smaller the parts are. 46 ORAL ARITHMETIC. Section 2. VI Note to Teachers. One object in this section is, to lead the pupil to apply rxjrrectly the terms expressing fractional parts. Every answer, therefore, must oe given in a vulgar fraction, unreduced. For example, two fourths is the answer which must be given to the 3d question. The books to be closed dur ing tlie recitation of this section. 1. If we divide any thing into 2 equal parts, and take away 1 of the parts, how much of the thing is left } 2, If we divide any thing into 3 equal parts, and take away 2 of the parts, how much of the thing is left .'* 3. If we divide any thing into 4 equal parts, and take away 2 of the parts, how much of the thing is left ? 4. If we divide any thing into 5 equal parts, and take away 3 of the parts, how much of the thing is left ? 5. If we divide, any thing into 6 equal parts, "and take away 3 of the parts, how much of the thing Is left ? 6. If we divide any thing into 7 equal parts, and take away 2 of the parts, how much of the thing is left ^ 7. If we divide any thing into 8 equal parts, and take away 5 of the parts, how much of the thing is left ? 8. If we divide any thing into 9 equal parts, and take away 3 of the parts, how much of the thing is left ? 9. If we divide any thing into 10 equal parts, and take away 4 of the parts, how much of the thino; is left ? 10. Into how many parts must any thing be divided^ so that 1 part shall be 1-eleventh ? — Into how many, so that 1 part shall be 1 -twelfth ? — 1 -thirteenth ^ 2 3. FRACTIONS. 47 Section 3. Note to Teachers. The learners may be referred to Remark Ist. under Uie^first section of examples in this chapter, for a correct form of expression to be adopted in answering the 1st., 4th., 7lh., and other similar questions .n this section. Books to be closed during tiie recitation of this section. 1. What is meant by one-half oi any thing } 2. Suppose you have 1-half of 1 dollar; — what part of a dollar more must you get, to make up 1 dollar ? 3. How many halves are equal to a whole one ^ 4. What is meant by one-third of any thing .'' 5. If I should cut 1 orange into thirds, and give you 2-thirds of it, what part of an orange would you still want, to make up 1 orange by joining the parts together ? 6. How many thirds are equal to a whole one ? 7. What is meant by one-fourth of any thing ? 8. Suppose you have 1-fourth of 1 dollar, — what part of a dollar must you get, to make up 1 dollar ? 9. How many fourths are equal to a whole one ? 10. What is meant by one-fifth of any thing } 11. If I should cut 1 apple into fifths, and give you • 4-fifths of it, what part of an apple would you still want, to make up 1 apple by joining the parts together } 12. How many fifths are equal to a whole one } 13. What is meant by one-sixth of any thing ? 14. If I own 2-sixths of 1 acre of land, and I wish to own 1 acre, what part of 1 acre must I buy } 15. How many sixths are equal to a whole one .'^ 16. What is meant by one-seventh of any thing .^ 17. A man bought 4-sevenths of a pound of tea at one shop, and enough more at another shop to make 1 pound. What part of 1 pound did he buy at the last shop ? 18. How many sevenths are equal to a whole one } 19. What is meant by one-eighth of any thing .-^ 20. James had 5-eighths of a dollar given him, and he earned 3-eighths more. How much money had he then -^ 21. How many eighths are equal to a whole one? 22. What is meant by one-ninth of any thing }' 23. If I have 7-ninths of 1 acre of land, and I wish to own 1 acre, what part of 1 acre must I buy } 24. How many ninths are equal to a whole one ? 48 ORAL ARITHMETIC. VI. 25. What is meant by one-tenth of any thing ? 26. Suppose you have 8-tenths of 1 dollar, — what part of a dollar must you get, to make up 1 dollar ? 27. How many tenths are equal to a whole one ? RELATIONS OF NUMBERS. Section 4. Note to Teachers. The object of this section is, to show the correspondence of the division of a unit, with the division of a collection of units. The ques- tions that inquire, what part of one number is another number, must be answered ii5 the terms of vulgar fractions. For instance, — the answers required in the 9th example are, — 1 is 1-fourth of 4; 2 is 2- fourths of 4; 3 is 3-fourths of 4. A collection of units is now to be viewed as a single thing; therefore the Verb singular Avill be used thus — 3 times 4 is 12. 1. If 1-half of a sheet of paper be worth 1 cent, what is a whole sheet worth ? 2. Suppose 2 cents are lying upon the desk before us; — v/hat part of the 2 cents is 1 cent ? 3. What part of 2 is 1 ? 4. If 1 -third of a loaf of bread be worth 1 cent, what is 2-thirds of it worth ^ What is a whole loaf worth ? 5. Suppose 3 cents are in a pile before us; — what part of the pile is 1 cent ? What part of the pile is 2 cents ? 6. What part of 3 is 1 ? What part of 3 is 2 ? 7. If 1-fourth of a yard of ribbon cost 1 cent, what will 2Tfourths of a yard cost ? What will 3-fourths of a yard tost } What will a whole yard cost ? 8. Suppose 4 cents are in a pile before us; — what part of the pile is 1 cent ? is 2 cents? is 3 cents ? 9. What part of 4 is 1 : is 2 ? is 3 .^ 10. If 1 -fifth of a barrel of flour be worth 1 dollar, what is 2-fifths of a barrel worth ? 3-fifths of a barrel ? 4'"fifths of a barrel ? What is 1 barrel worth ? 1 1 . What part of 5 is 1 ? is 2 ? is 3 ? is. 4 ? 12. If 1-sixth of a yard of ribbon cost 1 cent, what will 2-sixths of a yard cost ? 3-sixths of a yard ? 5-sixths of a yard ? What will 1 yard cost ? 13. What part of 6 is 1 ? is 2 ? is 3 f is 4? is 5 ? 14. If a horse trot 1 mile in 1 -seventh of an hour, how many miles will he trot in 2-sevenths of an hour ? in 6-serenths of an hour ? How many miles in 1 hour r 15. What part of 7 is 1 ? is 2 ? is 3 .^ is 4 .? is 6 .^ 4. 5. RELATIONS OF NUMBERS. 49 16. If 1 -eighth of a bar of silver be worth 1 dollar, what is 3-eighths of the bar worth ? What is 5-eighths of the bar worth ? What is the whole bar worth ? 17. What part of 8 is 1 ? is 2 ? is 3 ? is 5 ? is 7 ? IS. If 1-ninth of a pound of sugar cost 1 cent, what will 2-ninths of a pound cost ? What will 8-ninths of a pound ? What will 1 pound cost ? 19. What part of 9 is 1 ? is 2 ? is 4 ? is 6 ? is 8 ? ^ 20. If a man can build 1 rod of fence in 1-tenth of a day, how many rods can he build in 4-tenths of a day ? in (3-tenths of a day ? How many rods in 1 day ? 21. What part of 10 is 1.? is 2 ? is 3 .^^ is 4.^ is 6 .^ Section 6. 1. A carpenter having sawed a board into halves, finds by measuring, that 1-haIf of the board is 2 feet long. How long was the whole board ? 2. Suppose 2 is 1-half of some number,- — what is the whole of the number ? 3. If 1-haIf of a pound of rice be worth 3 cents, what is a whole pound worth ? 4. 3 is 1-half of what number ? 4 is 1-half of what number? 7 is 1-half of what number ? 5. How many times 1-half of any number will nfiake the whole number ? 6. There is just room for 2 boys to sit upon 1 -third of a certain board. What number of boys could sit up- on the whole of that board ? 7. Suppose 2 is 1-third of some number, — what is the whole of the number ? 8. If 1 -third of a box of raisins be worth 3 dollars, what is the whole box worth ? 9. 3 is 1-third of what number ? 4 is 1-third of what number ? G is 1-third of what number ? 10. How many times l-third of any number will make the whole number ? 11. If 1 -fourth of a yard of broad-cloth cost 2 dollars, what will a yard cost ? 12. 2 is 1-fourth of what number ? Solution, 2 is 1-fourth of 4 times 2. 4 times 2 is 8 E OO ORAL ARITHMETIC. VL 13. If a ship sail 3 miles in l-foiirth of an hour, how many miles will she sail in an hour ? 14. 3 is 1 -fourth of what number? 4 is 1-fourth of what number ? 10 is 1-fourth of what number ? 15. How many times 1 -fourth of any number will make the whole number ? 16. 3 men reaped 1 -fifth of a field of wheat in a day. What number of men would have reaped the whole field ? Sohition. If 3 men reaped l-fifth, 5 times 3 men' would have reaped 5-fifths, or the whole. 17. If l-fifth of a pound of loaf sugar be worth 4 cents, what is 1 pound worth ? 13. 4 is l-fifdi of what number ? 5 is l-fifth of what number ? 8 is l-fifth of what number ? 19. How many times l-fifth of any number will make the whole number .'' 20. 4 sheets of paper is 1-sixth of a quire. How many sheets are there in a quire f .21. 4 is 1-sixth of what number ? 2 is 1-sixth of what number ? 3 is 1-sixth of what number ? 22. How many times 1-sixtli of any number will make the whole number ? 23. If 1 -seventh of a ton of hay be worth 2 dollars, what IS a whole ton worth .^ 24. 2 is 1-seventh of what number.^ 3 is 1-seventh of what number ? 9 is 1-seventh of what number ? 25. 5 is 1-eighth of what number ? 3 is 1-ninth of what number ? 6 is 1 -tenth of what number ? SECttON 6. 1 . If a yard of cloth be worth 2 dollars, what is 1-half of a yard worth ? What is 1-half of 2 ? Observation. Smce you know that l-half of 2 is 1, you will perceive, that l-half of any other number must be as many times 1 as there are ticos in the number. 2. What is 1-half of 4 ? of 12 ? of 18 ? . 3. If a bar of silver weigh 3 pounds, what will 1 -third of it weigh ? What is 1 -third of 3 ? Obsertation. Here notice, that 1 -third of any nu mbei is as many times 1, as there are threes in the number. 6. RELATIONS OF NUMBERS. 51 4. If a peck of oats cost 9 cents what will 1 -third of a peck cost ? What is 1 -third of 9 ? 5. What is 1 -third of 6 ? of 15 ? of 24 ? 6. If a man earn 4 shilhngs in 1 day, how much does he earn in 1-fourth of a day ? What is 1-fourth of 4 ? Observation. Here notice that 1-fourth of any number is as many times 1, as there are fours in the number? 7. If a pound of raisins cost 20 cents, what will 1-fourth of a pound cost ? What is 1 -fourth of 20 ? 8. What is 1-fourth of 8 ? of 28 ? of 64 ? 9. If a yard of cambric cost 5 dimes, what will 1 -fifth of a yard cost ? What is 1 -fifth of 5 ? 10. If 3,5 drums of figs w^ill pay for a hogshead of su- gar, how many drums will pay for 1 -fifth of a hogshead ? 1 1 . What is 1-fifth of 35 ? of 25 ? of 40 ? 12. If a man earn 6 dollars in a week, how much does he earn in 1 -sixth of a week ? What is 1-sixth of 6 ? 13. Suppose writing paper costs 24 cents a quire; — what will be the price of 1-sixth of a quire ? Solution, If a quire cost 24 cents, 1-sixth of a quire will cost 1-sixth of 24 cents. 1-sixth of 24 is 4. 14. What is 1-sixth of 12 ? of 36 .? of 48 ? 15. What number of days is 1 -seventh of a week ? 16. If you eat 21 meals in a week, how many do you eat in 1 -seventh of a week, or 1 day ? 17. What is 1-seventh of 14 ? of 21 ? of 5.6 .? IS. If a melon weigh 8 pounds, what is the weight of 1-eighth of it ^ What is 1-eighth of 8 ? 19. If a yard of silver wire cost 32 cents, what will 1-eighth of a yard cost } 20. What is 1-eighth of 16 .? of 32 ? of 64 } 21. If a drum of figs weigh 9 pounds, what is the weight of 1-ninth of a drum ^ What is 1-ninth of 9 } 22. Suppose a watch-chain consists of 18 links; — how many links are there in 1-ninth of the chain ? 23. What is 1-ninth of 18 ? of 36 ? of 72 ? 24. W'hat number of cents is 1-tenth of a dime ? 25. If a chest of Souchong tea be worth 30 dollars, what is 1-tenth of a chest worth ^ 26. What is 1-tenth of 30 } of 50 } of 100 ? 53 ORAL ARITHMETIC. VI. Section 7. Note to Teachers. In solving the following questions, the learner should nrst state what proportional part of the number to be divided will be the answer, and tlience proceed to find the answer in tlie denomination of tlie dividend.— See solution under example 2d. 1 . If any number of oranges should be divided 'equally between 2 boys, what part of the number would 1 boy receive ? What part would 1 boy receive, if the oranges were divided among 3 boys } 4 boys } 5 boys } 6 boys } 7boys.^ 8boys.^ Sboys.^ lOboys ? 2. 5 sailors received 40 dollars, which they divided equally among them. What did 1 sailor receive ? Solution, If 5 sailors received 40 dollars, 1 sailor must have received 1 -fifth of 40 dollars. 1 -fifth of 40 dollars is 8 dollars ? 3. 2 fishermen caught 24 fishes, which they shared equally. How many was each man's share ? 4. If a traveller spend 23 dollars in travelling a week, how much does he spend a day ^ 5. A farmer can keep 9 cows on 36 acres of land: — . how many acres would it take to keep 1 cow } 6. 3 men have a bill of 30 dollars to pay: — how much must each man pay ? 7. If a stage run 42 miles in 6 hours, what distance does it run in 1 hour ? 8. Suppose 8 dozen of biscuit to be worth 72 cents; — what is the value of 1 dozen ^ 9. A tailor made 10 cloaks of 40 yards of cloth. How many yards did he put into each cloak ? • Section 8. Note to Teachers. Require the learner, as in tlie last section, to com- mence every solution by stating what proportional part of the given number is to be taken for the answer. 1 . If a man can travel 35 miles in a day, what distance can he travel in 1 -seventh of a day ? What distance in 4-sevenths of a day ^ Solution. If he can travel 35 miles in a day, he can travel 1 -seventh of 35 miles in 1 -seventh of a day; 1 -sev- enth of 35 miles is 5 miles. He can travel 4 times 5 miles in 4-sevenths of a day; 4 times 5 miles are 20 miles 7. 8 RELATIONS OF NUMBERS. 53 2. There are 24 sheets of paper in a quire. How many sheets are there in 1 -eighth of a quire ? How many sheets in 3-eighths of a quire ? . 3. What is 1 -eighth of 24 ? 3-eighths of 24 ? Solution, 1 -eighth of 24 is 3; 3-eighths is 3 times 3. 4. If a pound of coffee be worth 15 cents, what is 1-third of a pound worth ? 2-thirds of a pound ? 5. What is 1-third of 15 ? 2-thirds of 15 ? . 6. If a bushel of barley cost 50 cents, what does 1-fifth of a bushel cost ? 4-fifths of a bushel ? 7.. What is 1-fiflh of 50 ? 4-fifths of 50 ? 6. If a yard of ribbon cost 28 cents, what will 1-fourth of a yard cost ? 3-fourths of a yard ? 9. What is 1-fourth of 28 ? 3-fourths of 28 ? 10. If an acre of land will produce 30 bushels of rye, how much will 2-sixths of an acre produce ? Direction. First get what 1-sixth of an acre will yield. 11. What is 5-sixths of 30 ? Direction. First get 1-sixth of 30; thence 5-sixths. 12. If a stage run 81 miles in a day, what number of miles will it run in 7-ninths of a day? 13. What is 7-ninths of 81.? 14. There are 100 cents in a dollar. What number of cents are there in 8-tenths of a dollar ? 15. What is 8-ienths of 100? 16. Albert's kite line was 32 yards long, and he cut off 2-eighths of it for a fish hne. What was the length of the fish line ? 17. Suppose a boy having 42 quills, should give away 3-sevenths of them; — how many would he give away? — - and how many vi juld he have left ? 18. A man having 40 dollars, paid away 3-eighths of his money for a Ion of hay. What was the price of the hay? — and how many dollars had he left ? 19. A boy, who had 45 cents, paid away 3-fifths of hU money for a (juire of paper. What was the price of th« paper ? — and how many cents had he left ? 20. If 75 men can build a mile of fence in a'day, whai number of men njust be employed, to build 2-thirds of a mile in the same time ? 54 ORAL ARITHMETIC. VI Section 9. 1 . If 48 dollars should be divided equally among 8 men, what part of the money, — and what number of dollars, would 3 men receive ? Solution. 3 men would receive 3-eighths of the money. 1 -eighth of 48 dollars is 6 dollars; 3-eighths is 3 times 6 dollars, or 18 dollars. 2. A tierce, holding 42 gallons of molasses, has been emptied into 6 kegs, of equal size. What part of the molasses, — and what number of gallons in 5 kegs ? 3. What part of 6 is 5 ? What is 5-sixths of 42 ? 4. If a piece of broad-cloth containing 30 yards will make ten suits of clothes, what part of the piece, — and what number of yards, will make 6 suits ? 5. What part of 10 is 6 ? What is 6-tenths of 30 ? 6. 5 girls had 15 oranges, which they shared equally: 2 of the girls gave their shares to a sick woman. What part of 15 oranges did the woman receive? — and what number of oranges did she receive ? 7. W^hat part of 5 is 2 ? What is 2-fifths of 15 ? 8. 7 men owned 56 sheep in company, and 3 of the men took out their shares. What part of the flock, — and what number of sheep did they take out ? 9. What part of 7 is 3 ? What is 3-sevenths of 56 ? 10. If 4 men eat 28 biscuit in a day, what part of 28 biscuit, — and what number of biscuit will 2 men eat ? 1 1 . What part of 4 is 2 ? What is 2-fourths of 28 ? 12. 3 brothers owned 60 acres of land together, and the 2 younger sold their shares to the oldest. What part of the land, — and how many acres did they sell ? 13. What part of 3 is 2 ? What is 2-thirds of 60 .? Section 10. 1. If 3 men can fell 18 trees in a day, how many trees can 4 men fell in the same time ? Solution. If 3 men can fell 18 trees in a day, 1 man can fell 1-third of 18 trees, or 6 trees; 4 men can fell 4 times 6 ti*^es, or 24 trees. 2. What is 4 times 1-third of 18 ? Solution. 1 third of IS is 6: then 4 times 6 is 24. ' 9. 10. 11. RELATIONS OF NUMBERS. 55 3. If 5 men will cut 20 cords of wood in a day, how many cords will 3 men cut in the same time ? 4. What is 3 times 1 -fifth of 20 ? 5. If 4 barrels of flour cost 24 dollars, how much will 7 barrels cost, at the same price per barrel ? 6. What is 7 times 1-fourth of 24 ? 7. If 2 boats will carry 16 passengers across the river, how many passengers will 5 boats carry? 8. What is 5 times 1-half of 16 ? 9. Suppose a cooper can make 27 barrels in 9 days; — how many barrels can he make in 5 days ? 10. What is 5 times 1 -ninth of 27 ? 1 1 . Suppose 6 kegs will hold 36 gallons of molasses; — what number of gallons will 4 kegs hold ? 12. What is 4 times 1 -sixth of 36 ? 13. If 8 soldiers eat 56 pounds of beef in a week, how many pounds will 9 soldiers eat in a week ? 14. What is 9 times 1-eighth of 56 ? 15. If a workman can earn 49 dollars in 7 weeks, how many dollars can he earn in 6 weeks ? 16. What is 6 times 1-seventh of 49 ? 17. If 10 casks of claret wine cost 80 dollars, what would be the price of 8 casks of the same wine ? Section 11. 1. Suppose there are 10 links in 2-thirds of a watch chain; — how many links are there in 1 -third of the chain.'* How many links in the whole chain ? Solution. If there be 10 Imks in 2-thirds of the chain, there is 1-half of 10 links in 1-third of it: 1-half of 10 is 5. If 5 links be 1-third of the chain, there are 3 times 5 hnks in the chain: 3 times 5 is 15. 2. 10 is 2-thirds of what number ? 3. If 3-fourths of a pound of honey cost 1 5 cents, what will 1-fourth cost? What will a pound cost? 4. 15 is 3-fourths of what number? Solution. Since 15 is 3-fourths of the required num- ber, 1-third of 15 must be 1-fourth of the number: 1-third of 15 is 5. If 5 be 1-fourth of the number, 4 times 5, o) 20, is the whole number. 56 ORAL ARITHMETIC. VI. 5. If 2-fifths of a bushel of oats cost 18 cents, what will 1 -fifth cost ? What will a bushel cost ? 6. IS is 2-fifths of what number ? 7. If 4-sevenths of a kite line be 36 yards long, how long is 1-seventh ? How long is the whole line ? 8. 36 is 4-sevenths of what number ? 9. If 3-sixths of a chest of tea cost 21 dollars, what will 1 -sixth cost ? What will a whole chest cost ? 10. 21 is 3-sixths of what number? 11. If 5-eighths of a pipe of wine be worth 30 dollars, what is the value of the whole pipe? Direction. First find what 1-eighth is worth. 12. 30 is 5-eighths of what number? 13. If a man can earn 40 cents by working 4-sevenths of a day, how much can he earn by working a whole day? 14. 40 is 4-sevenths of what number? 15. If 7-ninths of a hogshead of sugar be worth 49 dollars, what is the whole hogshead worth? 16. 49 is 7-ninths of what number? 17. If a rail-road car run 24 miles in 8-tenths of an hour, what distance will it run in an hour? 18. 24 is S-tenths of what number? 19. Henry is 10 years old; and his age is equal to 5-sixths of Andrew's age. How old is Andrew? Suggestion. You may perceive that 1 -fifth of Henry's Hge must be equal to 1-sixth of Andrew's age. 20. 10 is 5-sixths of what number ? 21. If 21 workmen will perform 3-fifths of a certain piece of work in a week, what number of workmen would it take to perform the whole work in a week ? 22. 21 is 3-fifihs of what number? 23. A coach-man purchased a horse, and after paying 5-eighths of the price, he still owed 30 dollars. What was the price of the horse ? Solution. If he paid 5-eighths of the price, the 30 dollars, which he still owed, was 3-eighths of the price. 30 dollars being 3-eighths of the price, 1-third of 30 dol- lars, or 10 dollars, is 1-eighth. 10 dollars being 1-eightb of the price, S times 10 dollars is the price. 24. 30 is 3-eighths of what number? 12. » RELATIONS OF NUMBERS. 57 25. If 2-fifthj be taken from the whole of a/iy thing, how many fifths are there left ? 26. While George was fishing, a pickerel broke off 2-fifths of his line; he then had 12 feet of the line left. How long was his fine at first ? 27. Suppose a laborer can earn 60 cents a day, by working 5-sixths of the time; — how much could he earn by working constantly ? 28. After 3-sevenths of a cask of wine had leaked out, the owner drew off the remainder, and found there were 48 gallons. How many gallons had he lost ? 29. A farmer improved 3-ninths of his farm in tillage, appropriated 4-ninths to pasturage, and had 18 acres of wood-land. How many acres had he in all ? 30. 2-eighths of Edward's books are bound in leather, 3-eighths of them in marble paper, and 15 of them in blue paper. How many has he of each description ? Section 12. Note to Teachers. Tliis section embraces all the operations taught in the preceding sections of this chapter. Prefixed to each example, is the number of the section in which the operation involved in the example is taught. If the pupil fail in any part of this section, he should be put back to the section whose number is prefixed ta the example in which he fails. REVIEW. 1 . ( § 1 .) James found 4-eigbths of a dollar, and earn- ed 5-eighths. How much money had he then .'* 2. (§2.) If we divide anything into 6 equal parts, and take away 4 parts, how much of the thing is left ? 3. (§ 3.) If you have 7-ninths of 1 dollar, what part of a dollar must you get to make up 1 dollar ? 4. (§ 4.) If a man can walk 1 mile in 1-fourth of axx hour, how many miles can he walk in 1 hour ? 5. (§4.) What part of 6 is 5 .^ What part of 7 is 3 .? What part of 10 is 4 ? What part of 18 is 7 ? 6. (§ 5.) If 1-fourth of an acre of land will produce 9 bushels of corn, how much will 1 acre produce ? 7 (§5.) If 1-^eighth of a barrel of oeef be allowed to 3 soldiers for a week's provision, what number of soldiers will 1 barrel supply for a week ? 58 ORAL ARITHIVIETIC. VI. 8. (§ 6.) Suppose a yard of gold w.re to be worth 28 dollars; — what is 1 -fourth of a yard worth? 9. (§ 7.) If 50 day's work is to be done by 5 men, how many day's work must each man perform ? 10. (§8.) If one pound of Hyson tea be worth 96 cents, what is T-eigbths of a pound worth ? 11. (§9.) A company of 14 men gave 84 dollars for a boat. What part of 84 dollars did 5 men pay ? 12. (§ 9.) If 8 dollars will buy 72 pounds of brown sugar, how many pounds will 6 dollars buy.'^ 13. (§ 10.) Suppose 18 yards of cloth will make 6 coats; — how many yards are required for 10 coats ^ 14. (§11.) If 5-ninths of a yard of cotton cambric be worth 50 cents, what is the value of one yard ^ 15. (§ 11.) Suppose a man by working constantly can dig 40 bushels of potatoes in a day, — how many bushels will he dig, if he be idle 2-fifths of the day ? FRACTIONS AND RELATIONS. Section 13. Note to Teachers. The remainders^ that will arise in the several examples of division in this section, must be expressed in the language of fractions. See answers under examj)le3 4th. and 8th. If the learner should not readily un- derstand the process of converting the remainders into fractions, he may be referred to section 4th. in tiiis chapter, and, after reviewing the examples therein, may return immediately to this section. 1. How much cloth, at 2 dollars a yard, can I buy for 1 dollar } How much for 3 dollars } 2. What part of 2 is 1 } How many times 2 in 3 .'' 3. How many yards of ribbon, at 2 cents a yard, can you buy for 7 cents h — I mean, — how many whole yards, and what part of another yard can you buy } 4. How many times 2 in 7 .^— I mean, — how many twos are there, and what part of 1 niore two^ in 7 } Solution. 2 is contained in 7, 3 times, and 1 over: the 1 over is 1-half of another time 2. Jlns, 3 and 1-half. 5. How many tunes 2 in 9 .^ in 12 ? in 13 ? 6. How much wine, at 3 dollars a gallon, can I buf for 1 dollar ? How much for 4 dollars ? 7. What part of 3 is I ? How many times 3 in 4 ^ 13. FRACTIONS AND RELATIONS. S9 8. If a girl by setting types can earn 1 dollar in 3 days, how much can she earn in 20 days ? Solution, She can earn as many dollars as there are threes in 20. 3 in 20, 6 times and 2 over; the 2 over is 2-lhirds of another three. Ans, 6 dollars and 2-thirds. 9. How many times 3 in 4 ? in 16 ? in 20 ? 10. If a pound of lead cost 4 cents, how much can I buy for 1 cent ? How many pounds for 9 cents ? 11. If it take a man 1 hour to walk 4 miles, how many hours will it take him to walk 15 miles ? 12. How many times 4 in 9 ? in 15 .'^ in 34 ? 13. How much coal, at 5 dollars a ton, can be bought for 1 dollar ? How much for 7 dollars ^ 14. How many hogsheads of salt, at 5 dollars a hogs- head, can be bought for 44 dollars ^ 15. How many times 5 in 7 .'' in 44 } in 58 } 16. How much ribbon, at 6 cents a yard, can you buy for 10 cents ? How much for 37 cents "^ 17. If it cost 6 cents a mile to ride in the stage, what number of miles can you ride for 50 cents ? 18. How many times 6 in 10 ? in 50 ? in 45 ^ 19. How much sugar, at 7 cents a pound, can I buy for 9 cents ? How much for 52 cents ? 20. How many pounds of shot, at the rate of 7 cents a pound, must be sold for 34 cents ? 21. How many times 7 in 9 ? in 52 ? in 34 } 22. How much hay, at 8 dollars a ton must be sold for 9 dollars ? How much for 29 dollars ? 23. How many pounds of honey, at the rate of 8 cents a pound, must be sold for 77 cents ? 24. How many times 8 in 9 ? in 29 ? in 77 ? 25. At 9 cents a pound, how much cheese must be sold for 13 cents ? How much for 31 cents ? 26. If a man work for 9 cents an hour, how many hours will it take him to earn 64 cents ? 27. How many times 9 in 13 ? in 31 ? . in 64 } , 28. How much sugar, at 10 cents a pound, can be bought for 12 cents ? How much for 64 cents ? 29. If a workman can build 10 rods of fence in 1 day, how many days will It take him to build 48 rods } 60 ORAL ARITHMETIC. VI, 30. How many times 10 in 12 ? in 64 ? in 48 ? 31. How much rice at 3 cents a pound, must be giveu for 4 quarts of milk at 5 cents a quart ? Direction. First find the vahie of 4 quarts of milk. 32. How many times 3 in 4 times 5 ? Direction. First find how much 4 times 5 is. 33. How many pounds of flour at 4 cents a pound, must be given for 5 pounds of honey at 7 cents a pound ? 34. How many pmes 4 in 5 times 7 ? 35. What quantity of butter at 10 cents a pound, wil. pay for 6 combs at 8 cents apiece ? 36. If an active man earn 7 shilhngs a day, and a lazy man 4 shillings, how many days must the lazy man work, to pay the active man for working 6 days ? 37. How many times 4 in 6 times 7 ? 3S. How many yards of cloth at 5 dollars a yard, will pay for 3 boxes of raisins at 9 dollars a box ? 39. Hov\^ many times 5 in 3 times 9 ? 40. What quantity of corn at 6 dimes a bushel, will pay for 1 1 bushels of oats at 3 dimes a bushel ? 41. How many times 6 in 11 times 3 ? Section 14. jVoie to Teachers. The lerirners must now be led to observe, that the ex pressions, l-/m//, 1-third, 2-iliirdSy Sec. are to be understood, — l-haW of one 1-third of one, 2-tliirds of one. Sec. in all cases where the number, cf whicii the fraction indicates a part, is not stated. 1 . If I should cut each of 3 sheets of paper into halves, how many halves would they make ? Solution. In 1 sheet there are 2-halves, — in 3 sheets there are 3 times 2-halves, or 6-halves. 2. How many halves are there in 1? in 2 ? in 3? 3. If I had 4 sheets and 1-half of a sheet of paper, how many boys could I supply with half a sheet apiece ? 4. How many halves are there in 4 and 1-half? 5. If 2 slate pencils should each of them be broken into thirds, how many thirds would they make ? 6. How many thirds are there in 1 .^ in 2 ? in 4 ? 7*. Suppose I had 3 pencils and 2-thirds, how many boys could I supply with 1-third of a pencil each ^ 8. How many thirds are there in 3 and 2'thirds ? 14. FRACTIONS AND RELATIONS. 61 9. If 1 -fourth of a yard of cloth will make a satchel, how many satchels will 2 yards make ? 10. How many for.rths are there in 1? in 2 ? in 3 } 11. How many quires of paper, at 1 -fourth of a dollar a quire, can you buy for 3 dollars and 2-fouiths ? Solution. I can buy as many quires as there are fourths of a dollar, in 3 dollars and 2-fourths. In 1 dollar there are 4-fourths, — in 3 dollars, 3 times 4-fourths, or 12- fourths: 12-fourths plus 2-fourths is 14-fourths. 12. If a carpenter use 1 -fifth of a board to make 1 book shelf, how many book shelves can he make of a whole board } of 2 whole boards ? of 2 boards and 3-fiftl^s ? of 4 boards and 1 -fifth ? 13. How many fifths are there in 1 ? in 2 ? in 2 and 3-fifths ? In 4 and 1-fifth ? 14. If a bunch of quills cost 1 -sixth of a dollar, how many bunches can you buy for 1 dollar ? for 1 dollar and 5-sixths ? for 3 dollars and 2-sixths ? 15. How many sixths are there in 1? in 1 and 5- sixths ? in 3 and 2-sixths ? 16. If a stagp run 1 mile in 1-seventh of an hour, what number of miles will it run in 1 hour ? in 2 hours and 1-seventh ? in 4 hours and 4-sevenths ? 17. How many sevenths are there in 1 ? in 2 and 1-seventh ? in 4 and 4-sevenths ? 18. At 1 -eighth of a dollar a yard, how many yards of ribbon can I buy for 1 dollar and 3-eighths r for 2 dollars ? for 5 dollars and 7-eighths ? 19. How many eighths are there in 1 and 3-eighths .'* in 2? in 5 and 7-eighths ? 20. How many ninths are there in 1.^ in 1 and 4- nmths ? in 2 ? in 2 and 7-mnths ? in 6 ? 21. How many tenths are there in 1.^ in 4.^ in 3 and 5-tenths ? in 5 ? in 8 and 3-tenths ? 22. A laborer earned 9 dollars and a half, working at half a dollar a day. How many days did he work ? 23. If 1-eighth of a yard of cloth cost a dollar, what will 3 yards and G-eii^hths cost .^ 24. If a man earn a dollar in 1 -sixth of a week, how much can he earn m 8 we3ks and 4-sixths ? 02 ORAL ARITHMETIC. VI 25. If it take 1-fifth of a pound of fur to make a hat, how many hats can be made of 4 pounds and 2-fifths ? 26. If 1-fourth of a yard of late cost a dollar, how much will 5 yards and 3-fourths of a yard cost ? Section 15. 1. How many dollars in 2-halves of a dollar? in 3- halves of a dollar? Jlns. 1 dollar. 1 dollar and 1-half. 2. How many dollars are there in 4 half dollars ? in 5 half dollars ? in 9 half dollars ? 3. How many whole ones in 2-halves ? in 3-halves ? in 4-halves ? in 5-halves ? in 9-halves ? 4. What will 13 pencils cost, at half a cent apiece ? 5. If 3-thirds of an orange be put together they make up 1 orange. Now, if you had 6-thirds of an orange, how many oranges could you make up ? if you had 10-thirds ? if you had 17-thirds ? 6. How many whole ones in 6-thirds ? in 10-thirds ? in 17-thirds ? 7. What cost 26 quills, at 1-third of a cent apiece ? Solution. If 1 quill cost 1-third of a cent, 26 quills will cost 26-thirds of a cent. 26-thirds of a cent are as many cents as 3 is contained times in 26. 3 in 26, S times and 2 over. *Rns. S cents and 2-thirds. 8. How many whole apples could you make up, if you had 5-fourths of an apple ? 14-fourths of an apple ? 9. What cost 31 cups, at 1-fourth of a dollar apiece ? 10. How many whole ones in 5-fourths ? in 14- fourths ? in 31 -fourths ? 11. If 1 cotton ball be given for 1-fifth of a yard of galloon, how much galloon must be given for 8 cotton balls ? for 17 cotton balls ? for 44 cotton balls ? 12. How manv whole ones are there in 8-fifths ? in l7-fifths ? in 44.fifths ? 13. If a quire of paper cost 1-sixth of a dollar, what IS the cost of 12 quires ? 17 quires ? 19 quires ? 14. How many whole ones in 12-sixths ? in 17- sixths ? in 19-sixths ? 15. How many whole ones in 18-sevenths? m 24- tenths ? in 31-eighths ? in 47-ninths ? in 25-fourtljs ^ 16. 17. FRACTIONS AND RELATIONS. 63 Section 16. 1. Ellen paid, for the Young Ladies^ Class Book^ 3- fourths of a dollar; for the Boston School Atlas^ 2-fourth3 of a dollar; and for the J^ational Spelling- Book^ 1 -fourth of a dollar. What did the whole cost ? 2. How much is 3-fourths and 2-fourths and 1 -fourth ? 3. A trader sold a piece of cloth for 19 dollars and 5-eighths, and a hat for 4 dollars and 7-eighths. How many dollars did he receive for both } Solution, 19 dols. plus 4 dols. are 23 dols. 5- eighths of a dol. plus 7-eighths of a dol. are 12-eighths of a dol., equal to 1 dol. and 4-eighths. Then, 23 dols. plus 1 dol. and 4-eighths are 24 dols. and 4-eighths. 4. A traveller rode 31 miles and 3-fifths in the fore- noon, and 25 miles and 4-fifths in the afternoon. How many miles did he ride in the whole day ? 5. What is 31 and 3-fifths plus 25 and 4-fifths ? 6. A trader bought some goods for 64 dollars and 5- sevenths, and paid 5 dollars and 3-sevenths for the post- age of them. What was the whole expense ? 7. What is 64 and 5-sevenths plus 5 and 3-sevenths ? 8. A gentleman paid 33 dollars and 7-tenths for some cloth, and 11 dollars and 6-tenths for having it made into a suit of clothes. What did the suit cost ? 9. What is 33 and 7-tenths plus 11 and 6-tenths ? 10. What is 16 and 6-ninths plus 8 and 5-ninths.^ 11. What is 40 and 5-sixths plus 41 and 3-sixths ? Section 17. 1. Suppose a rail-road car to run 2-thirds of a mile in 1 minute, what distance will it run in 10 minutes ? Solution. In 10 minutes it will run 10 times 2-thirds of a mile, or 20-thirds of a mile. 20-thirds of a mile are equal to 6 miles and 2-thirds. 2. If 3-fourths of a gallon of wine leak out of a cask in 1 hour, how much will leak out in 7 hours ? 3. How many whole ones in 7 times 3-fourths } 4. If a yard of cambric muslin cost 4-fifths of a dol- lar, how much will 9 yards cost ? 5. How many whole ones in 9 times 4-fifths ^ ft4 ORAL ARITHMETIC. VI 6. Suppose a man to eat 5-sixths of a pound of beef m one day, how many pounds will he eat in 5 days ? 7. How many whole ones in 5 times 5-sixths ? 8. If 3-sevenths of a pound of gunpowder tea cost 1 dollar, how many pounds can I buy for 8 dollars ? 9. How many whole ones in 8 times 3-sevenths ? 10. Suppose 5-eighths of a yard of cloth will make a vest, how many yards will it take to make 6 vests ? 11. . How many whole ones in 6 times 5-eighths ? 12. If 1 quire of letter paper be worth 4-ninths of a dollar, how many dollars are 7 quires worth ? 13. How many whole ones in 7 times 4-ninths ? 14. Suppose a man to walk 1 mile in 2-tenths of an hour, what time will it take him to walk 9 miles ? 15. How many whole ones in 9 times 2-tenths ^ Section 18. 1. What will 6 yards of broad-cloth cost, at 7 dollars and 3-eighths of a dollar per yard ? Solution. 6 yards will cost 6 times 7 dollars and 3- eighths. 6 times 7 dollars are 42 dollars. 6 times 3- eighths are 18-eighths, equal to 2 and 2-eighths. Then, 42 dollars plus 2 dollars and 2-eighths are 44 dollars and 2-eighths. 2. What will 4 hundred- weight of sugar cost, at 9 dollars and 2-fifths per hundred-weight ? 3. What is 4 times 9 and 2-fifi;hs ? 4. Suppose a ship to sail 10 miles and 1-half in one hour, what distance will it sail in 7 hours ^ 5. What is 7 times 10 and 1-half? 6. If a horse eat 1 bushel and 9-tenths of a bushel ot oats in a week, how much will he eat in 4 weeks ? 7. What is 4 times 1 and 9-tenths ? 8. If 1 dime will buy 3 yards and 2-thirds of a yard of ribbon, how many yards will 6 dimes buy ? 9. What is 6 times 3 and 2-thirds ? 10. Suppose the price of coal at the mine, is 3 dol- lars a ton, and the freight of it to the city is 3-fourths of a dollar a ton, w^hat will 10 tons cost at the city } 11. What is 10 times 3 and 3-fourths ? 18. 19. FRACTIONS AND RELATIONS. 65 12. Suppose a boat goes 10 miles and 6-sixths of a mile in 1 hour^ what distance will it go in 8 hours ? 13. What is 8 times 10 and 5-sixths ? 14. If 3 yards and 7-eighths of cloth will make a cloak, how many yards will it take to make 5 cloaks } 15. What is 5 limes 3 and 7-eighths ? Section 19. 1. If 1-fifth of a chest of tea be worth 6 dollars and 7-eighths, what is a whole chest worth ? Solution, A whole chest is worth 5 times as much as 1-fifth of a chest. 5 times 6 dollars are 30 dollars; 5 times 7-eighths of a dollar are 35-eighths of a dollar, or 4 dollars and 3-eighths. 30 dollars plus 4 dollars and 3-eighths are 34 dollars and 3-eighths. 2. 6 and 7-eighths is 1-fifth of what number ? 3. Suppose 1 -ninth of a kite line to be 5 yards and 3-fourths of a yard long, — how long is the whole line ? 4. 5 and 3-fourths is 1 -ninth of what number ? 5. A young man being asked his age, answered indi- rectly, that 1 -third of his age was 7 years and 2-sixths of a year. What was his age ? 6. 7 and 2-sixths is 1 -third of what number ? 7. Suppose a man can build 3 rods and 2-fifths of a rod of wall in 1-sixth of a week, — how many rods can he build in a whole week ? 8. 3 and 2-fifths is 1-sixth of what number? 9. If 1 -tenth of a bushel of corn be worth 6 cents and I -fourth of a cent, what is a bushel worth ? 10. 6 and 1 -fourth is 1 -tenth of what number? 11. 1-fourth of Edmund's kite line measures 8 yard« and 3-sevenths of a yard. How long is the line ? 12. 8 and 3-sevenths is 1-fourth of what number ? 13. If 1-half of a yard of lace cost 3 dollars and 4- fifths of a dollar, what will a yard cost ? 14. 3 and 4-fifths is 1-half of what number? 15. Suppose that 1-seventh of an acre of land will produce 6 bushels and 7-ninths of a bushel of barley how many bushels will an acre produce ? 16. 6 and 7-ninths is 1-seventh of what number ? ORAL ARITHMETIC Section 20. VI. 1. If 1 apple were divided equally among 3 boys, what part of 1 apple would 1 boy receive ? If 2 apples were thus divided, how many thirds would one boy receive ? 2. Here we see 1 -third of 2 boards, placed over 2- thirds of 1 board. Is it not plain, that 1 -third of the 2 boards together, is equal to 2-thirds of 1 board ? 3. 1-third of 2 is equal to what part of 1 ? 4. There were 3 boys, who had 1 dollar apiece; and each boy gave a decrepit soldier 1-fourth of his money. What part of 1 dollar did the poor soldier receive ? 5. Here, we see lifourth of 3 boards placed over 3- fourths of 1 board. Sup- pose the fourths seen in the 3 boards should be placed together end to end — Is it not plain, they would make 3-fourths of 1 board ? 6. 1-fourth of 3 is equal to what part of 1 ? , 7. I have 4 oranges to divide among 5 boys. — I first cut 1 orange into fifths, and give each boy 1-fifth; and thus I proceed, dividing 1 orange at a time, until they are all divided. Now, what part of a whole orange can each boy make up, by joining his fifths together ? 8. 1-fifth of 4 is equal to what part of 1 ? 9. If 1 melon were divided equally among 6 boys, what part of 1 melon would 1 boy receive ? If 2 melons were divided, how many sixths would 1 boy receive } 10. 1 -sixth of 2 is equal to what part of 1 ? Jl. If 3 barrels of fiour were divided equally among 7 men, how much would 1 man receive ? 12. 1-seventh of 3 is equal to what part of 1 ? 13. If 3 pounds of beef were divided equally among 8 soldiers, what part of a pound would 1 soldier get ? 14. 1 -eighth of 5 is equal to what part of 1 ? 20. 21. FRACTIONS AND RELATIONS. 67 f 15. An ostler has 2 bushels of oats to divide among 9 horses; — how much must he give to each horse ^ 16. 1-nmth of 2 is equal to what part of 1? 17. If 7 dollars were divided equally among 10 men, what part of 1 dollar would each man have ? 18. 1-tenth of 7 is equal to what part of 1? 19. 1-fourth of 2 is equal to what part of 1 ? 20. 1-sixth of 5 is equal to what part of 1.'^ 21 . 1 -eighth of 3 is equal to what part of 1 ? 22. There were 36 oranges in a basket and Albert was directed to take 1-fourth of them. Accordingly he cirt 1-fourth out of every orange, and took it to himself. How many fourths of an orange did he get ? He then joined his fourths together, to make them into whole oranges; — how many whole ones had he? 23. 1-fourth of 36 is equal to how many fourths of 1 ? — equal to how many whole ones ? 24. In another basket there were also 36 oranges, and Benjamin was directed to take 1-fourth of them. But, instead of cutting 1-fourth out of every orange, as Albert did, he took 1 orange from every 4 in the basket. How many oranges did Benjamin get? 25. Now tell me which is the most; — 1-fourth of 36, or 36-fourths of 1 ? 26. 1-half of 10 dollars is equal to how many halves of 1 dollar? — equal to how many dollars? 27. 1 -third of 18 oranges is equal to how many thirds of 1 orange? — equal to how many whole oranges? 28. 1 -fifth of 17 oranges is equal to how many fifths of 1 orange?— equal to how many whole oranges? 29. 1-sixth of 42 is equal to how many sixths of 1? — equal to how many whole ones? 30. 1 -seventh of 56 is equal to how many sevenths of 1? — equal to how many whole ones? Section 21. 1. If a chest of green tea be worth 27 dollars, what is 1-fourth of it worth? Solution. 1-fourth of the tea is worth 1-fourth of 27 dollars. 1-fourth of 27 dollars is 6 dollars, there being 68 ORAL ARlTHxMETIC. VI 3 dollars over. 1 -fourth of 3 dollars is equal to 3-fourths of 1 dollar. 6 dollars plus 3-fourth3 of a dollar are 6 dollars and 3-fourths Or, we may say, — One fourth of 27 dollars is 27-fourths of 1 dollar; equal to 6 dollars and 3-fourths. 2. What is 1 -fourth of 27? 3. 3 men bought a barrel of sugar for 23 dollars, and divided it equally among them, each man taking 1-third of the sugar, and paying 1-third of the price. How many dollars did each man pay? 4. What is 1-third of 23? 5. Suppose a family to eat 26 loaves of bread m a week; — what number of loaves would the family con- sume in 1 -seventh of a week, or 1 day? 6. what is 1 -seventh of 26 ? 7. Suppose 48 bushels of wheat are to be divided among 5 men; how much will 1 man receive ? 8. What is 1 -fifth of 48? 9. 6 men purchased a boat for 27 dollars: each man paid 1 -sixth of the money, and owned 1 -sixth of the boat. How many dollars did 1 man pay? 10. What is 1-sixth of 27? 11. Suppose a bag of coffee to weigh 65 pounds; — what is the weight of 1 -ninth of it ? 12. What is 1-ninth of 65? 13. If it will take a man 60 days to clear a piece of wood-land, in what time will he clear 1-eighth of it? 14. What is 1-eighth of 60? 15. A sailor was cast upon a desolate island, and subsisted 10 days upon 34 biscuit, eating an equal quan- tity each day. How many did he eat each day? 16. What is 1-tenth of 34? 17. If a bar of silver, that is worth 37 dollars, should be cut into 3 equal parts, how many dollars would 1 of the parts be worth? 18. What is 1-third of 37? 19. Suppose a party of 9 goldvhunters find a quantity of ore, which is worth 88 dollars; what is the value of each man's share? 20. What is 1-ninth of 88? 21. FRACTIONS AND RELATIONS. 69 21. If It take a man 4 months to earn 38 dollars, how much does he earn in 1 month ? 22. What Is 1 -fourth of 38? 23. If 6 barrels of superfine flour cost 35 dollars, what is the price of 1 barrel of it ? 24. What is 1-sixth of 35? 25. Suppose 39 bushels of corn to grow upon 1 acre; — how much corn will 1 -fifth of an acre produce ? 2b'. What is 1-fifthof 39? 27. If 2 dollars will pay for 13 pounds of butter, how many pounds can be bought for 1 dollar? 28. What is 1-half of 13? 29. If 8 dollars will pay for 78 pounds of cheese, how many pounds will 1 dollar pay for? 30. What is 1 -eighth of 78? 31. Suppose 10 men drink 55 gallons of beer in a month; — how much will 1 man drink in a month? 32. What is 1-tenthof 55? 33. Suppose 7 acres of land to produce 60 dollars' worth of hay; — what is the value of the hay which 1 acre of the land produces? 34. .What is 1-seventh of 60? 35. If 1 man can clear a piece of wood-land in 29 days, in what number of days would 5 men clear it? Instruction. Consider that 5 men can do 5 times as much work in a day, as 1 man can do: consequently, it will take 5 men only 1 -fifth of the time that it will take 1 man to clear the land. 36. How many days will it take 7 men to do a piece of work, that 1 man can do in 46 days? 37. If 1 man will drink a firkin of beer in 50 days, how many days will it last 6 men? 38. Suppose 24 men can hoe a piece of corn in 1 day; what number of men must be employed to hoe it in 8 days ? Suggestion. Each man, that shall be employed, can do 8 times as much work in 8 days, as he can in 1 day. 39. If 40 men can build a wall in 1 day, what number of men must be employed to build it in 4 days? 40. If a cistern can be discharged by 1 lancet m 19 hours, in what time can it be discharged by 3 faucets? 70 ORAL ARITHMETIC. VL Section 22. 1 . If a smith can make 5 cups from 12 ounces of silver, how much silver is required to make 3 cups? Direction. First find how much silver would make 1 cup; then, 3 times that quantity would make 3 cups. 2. What is 3 times 1-fifth of 12? Solution. 1 -fifth of 12 is 2 and 2-fifths. 3 times 2 is 6; 3 times 2-fifths is 6-fifths, or 1 and 1-fifth. Then 6 plus 1 and 1-fifth is 7 and 1-fifth. 3. If 22 bushels of wheat will make 4 barrels of flour, how much wheat will make 6 barrels of flour? 4. What is 6 times 1-fourth of 22? 5 . Suppose the equipments for 8 soldiers to cost 75 dol- lars; what would be the expense of equipping 5 soldiers? 6. What is 5 times 1 -eighth of 75? 7. If 29 tons of hay will keep 9 horses through the winter, how many tons would 6 horses require? 8. What is 6 times 1-ninth of 29? 9. Suppose 7 acres of pasturage to be worth 65 dol- lars; what is 3 acres of the same pasturage worth? 10. What is 3 times 1-seventh of 65? 11. If 8 acres of pasturage will keep 35 sheep, how many sheep would be sufficient for 6 acres ? 12. What is 6 times 1-eighth of 35? 13. Suppose a man to eat 50 pounds of beef in 8 weeks; what number of pounds would he eat in 9 weeks? 14. What is 9 times 1-eighth of 50? 15. If it take 36 yards of broad-cloth to make 10 suits of clothes, how many yards would make 4 suits? 16. What is 4 times 1-tenth of 36? 17. A trader gave 59 dollars for 9 barrels of flour, and sold 3 barrels of it, at the same price per barrel that he gave. For how much did he sell the 3 barrels? 18. What is 3 times 1-ninth of 59? 19. If 6 pounds of brown sugar be sold for 52 cents, what would be the price of 5 pounds of it? 20. What is 5 times 1-sixth of 52? 21. If 8 scholars use 18 quires of paper in a month, how many quires would 10 scholars use in a month? 22. What is 10 times 1-eighth of 18? ^. 23 FRACTIONS AND RELATIONS. 71 23. Suppose a stage to run 58 miles in 7 hours; what distance does it run in 6 hours ? 24. What is 6 times j -seventh of 58 ? 25. If a mill grind 17 bushels of corn in 2 hours, how many bushels will it grind in 7 hours ? 26. What is 7 times 1-half of 17 ? 27. Albert paid 61 cents for 9 writing-books, and William bought 7 writing-books, paying at the same rate. How much did Wilham's books cost him ? 28. What is 7 times 1 -ninth of 61 ? 29. Suppose a hunter gets 8 pounds of gunpowder ui exchange for 44 pounds of venison; how many pounds of venison must he give for 10 pounds of powder ? 30. What is 10 times 1-eighth of 44 ? Section 23. 1. When writing paper is sold at 20 cents a quire, what is the price of 1 -third of a quire ? 2. If 1 -third of a quire of paper is worth 6 cents and 2-thirds, what is 2-thirds of a quire worth ? 3. What is 1 -third of 20 ? 2-thirds of 20 ? 4. Suppose a yard of ribbon to be worth 23 cents; what is 1 -fourth of a yard worth ? 5. If 1 -fourth of a yard of ribbon is worth 5 cents and 3-fourths, what is 2-fourths of a yard worth ? Solution. 2-fourths of a yard is worth 2 times 5 cents and 3-fourths. 2 times 5 cents are 10 cents; 2 times 3-fourths of a cent are 6-fourths of a cent, or I cent and 2-fourths. 10 cents plus 1-cent and 2-fourths, are 11 cents and 2-fourths. 6. What is 1-fourth of 23 ? 2-fourths of 23 ? 7. Suppose a pound of white sugar to be worth 23 cents; what is 1 -fifth of a pound worth ? 8. If 1 -fifth of a pound of sugar is worth 4 cents and 3-fifths, what is 3-fifths of a pound worth ? 9. What is 1 -fifth of 23 ? 3-fifths of 23 ? Solution. 1-fifth of 23 is 4 and 3-fifths 3-fifths of 23 is 3 times 4 and 3-fifths. 3 times 4 is 12: 3 times 3-fifths are 9-fifths, or 1 and 4-fifths. 12 plus 1 and 4- fiftbs is 13 and 4-fifths. 72 ORAL ARITHMETIC. VL 10. Suppose a man can walk 34 miles in a day, what distance can he walk in 1 -sixth of a day ? 11. If a man walk 5 miles and 4-sixths5 in 1-sixth of a day, how far will he walk in 5-sixths of a day ? 12. What is 1-sixth of 34 ? 5-sixths of 34 ? 13. Suppose a bushel of corn to be worth 65 cents; what is 1 -seventh of a bushel worth ? 14. If 1-seventh of a bushel of corn cost 9 cents and 2-sevenths, what will 3-sevenths of a bushel cost? 15. What is 1-seventh of '65 ? 3-sevenths of 65 ? 16. Suppose 1 dollar will pay for 35 pounds of rice; how much rice will 1 -eighth of a dollar buy ? 17. If 1-eighth of a dollar will buy 4 pounds and 3- eighths, how much will 5-eighths of a dollar buy ? 18. What is 1-eighth of 35 ? 5-eighths of 35 ? 19. Suppose a man earns 70 cents a day; how much does he earn in 1 -ninth of a day t 20. If a man can earn 7 cents and 7-ninths, in 1 -ninth of a day, how much can he earn in 8-ninths of a day ? 21 . What is 1 -ninth of 70 ? 8-ninths of 70 ? 22. Suppose an acre of land will produce 43 bushels of oats; what will 1-tenth of an acre produce ? 23. If 1-tenth of an acre produce 4 bushels and 3- tenths, what will 7-tenths of an acre produce? ■ 24. What is 1-tenth of 43 ? 7-tenths of 43 ? 25. If a yard of cloth will pay for 30 pounds of cheese, how many pounds will 3-fourths of a yard buy? Direction, First find hov/ many pounds 1-fourth of a yard will pay for. 26. A farmer sold 4-fifths of a ton of hay, for oats, allowing 32 bushels of oats to be worth the same as a ton of hay. How many bushels of oats did he receive ? 27. What is 4-fifths of 32 ? 28. Suppose 1 dollar will pay for 38 pounds of rice; for how manv pounds will 8-tenths of a dollar pay ? 29. What'is 8-tenths of 38 ? 30. A man bought a piece of land containing 1 acre and 4-sixths, and paid at the rate of 40 dollars per acre. How much did he pay for the land ? 31. What is 40 plus 4-sixths of 40 ? 34 FRACTIONS AND RELATIONS. 78 Section 24. 1. Suppose 3-fourths of a yard of flannel to cost S2 c«nts; what does 1 -fourth of a yard cost ? What would a yard cost ? Solution, If 3-fourths of a yard cost 32 cents, 1- fourth of a yard costs 1 -third of 32 cents. 1 -third of 32 cents is 10 cents and 2-thirds of a cent 4-fourths, or a whole yard would cost 4 times 10 cents and 2-thirds. 4 times 10 cents are 40 cents; 4 times 2-thirds are 8- thirds, equal to 2 and 2-thirds. Then, 40 cents plus 2 cents and 2-thirds are 42 cents and 2-thirds. 2. 32 is 3-fourths of what number ? Solution. Since 32 is 3-fourths of the number, 1- third of 32 is 1-fourth of it. 1 -third of 32 is 10 and 2-thirds. 4 times 10 and 2-thirds are 42 and 2-thirds. 3. If 2-fifths of an acre of land will produce 9 bushels of rye; how many bushels will 1-fifth of an acre produce ^ How many bushels will an acre produce ? 4. 9 is 2-fifths of what number ? Instruction. Observe^ that 1-half of 9 must be 1-fifth of the required number. 5. If a man drink 6 gallons of beer in 5-sixths of a month, how many gallons does he drink in 1-sixth of a month ? How many galbns will he drink in a month ^ 6. 6 is 5-sixths of what number ? 7. A man, who spends 43 cents a day, finds his ex- penses to be 5-sevenths of his wages. What is 1-sev- enth of his wages ? What is the whole of his wages ? 8. 42 is 5-sevenths of what number ? 9. If 5-eighths of a dollar will pay for 24 pounds of flour, how many pounds will 1 -eighth of a dollar pay for ? How many pounds will a dollar pay for ? 10. 24 is 5-eighths of what number? 11. Suppose 6 gallons of wine to leak from a cask in 8-ninths of an hour; how much will leak out in 1-ninlh of an hour ? How many gallons in 1 hour ? 12. 6 is 8-ninths of what number ? 13. If 4-tenths of a yard of cloth be worth 33 cents, how much is 1 yard worth ? Direction, First find what I •tenth of a yard is worth- 74 ORAL ARITHMETIC. VL 14 If 3-eighths of a bale of cotton be worth 17 dol- lars, what is the whole bale worth ? 15. A laborer spent 30 cents a day, and still saved 3-sevenths of his wages. How much was his wages ? 16. Suppose that I have read 5-ninths of the pages in a certain book, and there are 35 pages more to be read; — how many pages are there in the book ? NOTATION OF FRACTIONS. Learners will now attend to the meaning of the words, FRACTION, DENOMINATOR, and NUMERATOR. A Fraction is any part of one. For example, one-half of an orange is r fraction of 1 orange; three-fourths of an orange is Ruoiher fraction of 1 orange. In this book, fractions have been expressed by a number joined with a word; thus, 4-ninths. Fractions are commonly expressed by two numbers, standing one above the other, with a line between them; thus, ^ h^f\ 1 one- 2 t^wo- I one- 4 four- 3 three- "3" third, "3 tliirds, "6 sixth, 5" fifths, 9 ninths. 17. What fraction is expressed, when there is a 4 with a 1 over it. '^ 7 with 2 over it? 8 with 5 over it.-^ 10 with 6 over it ? 18. Which is the greater fraction; ^ or ^? ^ or ^.^ Y 01 -^ .'' g^ 01 10 r 4 or Q r 19. Which is the greater fraction; ^ or f ? f or f ? 3 or 1 ? 4 ^« 7 ? 8 ^^ _3_ ? y or 5 r -Q or ^ r -^q or ^q . The Denominator of a fraction, is the number of equal parts into which a whole one is divided. For example, if a whole orange be divided into 4 equal parts, the de- nominator is 4; the parts being denominated /okW/i.9. The JsTximerator of a fraction is the number which shows how many of the equal parts the fi'action ex- presses. For example, the fraction f expresses 3 of the four equal parts; therefore 3 is the numerator. 20. What numerator, and what denominator, would express the fraction, four-fifths ? two-eighths } six- ninths .^ one-fifteenth.^ five-eighteenths.^ •35 FRACTIONS. 75 When the numerator is equal to the denominator, thus, I, then the fraction is equal to 1; as 4 -fourths of an orange, when joined together make 1 orange. When the numerator is greater than the denominator, thus, I, then the fraction is equal to as many times 1 as the denominator is contained times in the numerator. 21. How many times 1, [how many whole ones], in | ? m V' ? 20 ? 30 ? To ' 24 ? 12 • 22. How many times 1, and what fraction over, in | ^ I? inV? in¥? in II? 23. Where have you observed the numerator of a frac- tion to stand; — above^ or below the denominator ? mf? Section 25, 1. James has ^ of a dollar, and Henry has f of a dollar: — which of them has the most money ? Compare the fraction ^ with other fractions. 2. ^ is equal to how ma- ny fourths } how many six 3. ^ is equal to how ma- ny sixths } 4. i is equal to how ma- ny eighths ? 5. ^ is equal to how ma- ny tenths ? 6. ^ is equal to how many twelfths ? teenths ? how rnany twentieths ? 7. Edward broke a slate pencil into 3 equal pieces, and Albert broke one into 6 equal pieces. How many of Albert's pieces were equal to 1 of Edward's pieces ? Compare the fraction ^ with other fractions. 8. ^ is equal to how ma- ny sixths ? 9. ^ is equal to how ma- ny ninths ? 1 ^m 1 , 1 • i HH : ^^m ■■ i m^m 76 ORAL ARITHMETIC. Vl. 1§. -3- is equal to how many twelfths ? how many eighteenths ? how many thirtieths ? Suggestion, y of 1 is equal to ^ of 12-twelfths. 11. ^ is equal to how many eighths ? how many twelfths ? how many sixteenths ? 12. I is equal to how many tenths ? how many twen- tieths ? how many twenty -fifths ? 13. i is equal to how many twelfths ? how many eighteenths ? how many thirtieths ? 14. ^ is equal to how many fourteenths.^ how many twenty-eighths ? how many thirty-fifths ? 15. I is equal to how many twentieths ? Solution, ^ is equal to ^05 I ^^ equal to 3 times 5- twentieths, which is ^f . 16. f is equal to how many twelfths ? 17. A boy, who had f of an orange, cut each fifth mto 2 parts, (making tenths); his brother gave him ^ more. What fraction of an orange had he at last ? 18.. Into how many parts must you cut a sixth of an orange, to make the parts eighteenths? Why .'^ 19. f is equal to how many eighteenths ? 20. Change f to fourteenths, and then add ^ to it. 21. 1^ is equal to how many twenty-fourths ? 22. Change f to eighteenths, and then take -^ from it. Section 26. 1. What is meant by a Fraction ?* How is a frac- tion commonly expressed ?. What is the Denomina" tor of a fraction ?. What is the JSTumerator ? 2. If the denominator of a fraction be 9, and the numerator 7, how should these numbers be written ? — • and what would the fraction be called in reading it ? 3. Suppose two fractions have numerators alike, and denominators different — which is the greater fraction — that with the greater, or the smaller denominator ? 4. Suppose two fractions have denominators alike, and numerators different — which is the greater fraction — that with the greater, or the smaller numerator ? 26. 27 FRACTIONS. 77 Observation. If an orange be cut into eighths, and then 4 of the eighths be joined together, these A-eighths become l-Zia/fof an orange. And thus the fraction, |, when reduced to its lowest terms, is ^. 5. Reduce f to its lowest terms — that is, find the lowest numerator and denominator, that will express a quantity equal to f . 6. Reduce | to its lowest terms. 7. Reduce f t« its lowest terms. Reduce f. 8. Reduce f to its lowest terms. Reduce |-. 9. Reduce | to its lowest terms. Reduce f. Observation, A fraction is reduced, by dividing the numerator and denominator by any number, which will divide them both without a remainder For example, we reduce -^q thus; 2 in 6, 3 times, 3 is a new numera- tor; 2 in 10, 5 times, 5 is a new denominator. 10. Reduce each of the following fractions to its low- ^cf forme 5 4 8 6 „4_ 8 _2_ 1_0 est terms, -jq. -jq. -yq- 12\ 12' T5^- 12- is- 11. Stephen's knife cost -^^ ^^ ^ dollar, and John's cost -^Q of a dollar. Whose knife cost the most } 12. Reduce the fractions, /^ ^"^ io ^^ their lowest terms, and then add them together. 13. Reduce, and then add together, -^§ and |f . 14. Reduce, and then add together, If and 2%- Section 27. 1. -4- of a water melon was divided equally among- 3 boys. What part of the whole melon did 1 boy receive } 2. -J of :^ is equal to I what part of 1 } Solution. If \ be divided into 3 equal parts, it will take 12 such parts to make a whole one. Therefore, s- of? is ^^2- [^times 4 isy^-l 3. What part of a whole one is ^ of | ? Illustration. If -J- of an orange were cut into 6 equa. parts, it would take 5 times 6 such parts to make a whole orange. Operation, -g^ times ^ is y^ . 78 ORAL ARITHMETIC. VI. 4. What part of 1 is -^ of ^ ? i of -»- ? \ of -J ? i of i? iofi? iof^? 5. If 18^ dollars [18 and | dollars] be divided equal ly among 3 men, what will each man receive ? 6. What is I of 18i? Sol, ^ of 18 is 6; -j of ^ is y^^; 6 plus -^^ is 6-^^^. 7. Whatis jof 30-^? iof24i? jofl8-}? 8. A boy, having }j of a dollar, paid -| of his money for a knife. What part of a dollar did he pay ? 9. I of 'I is equal to what part of 1 ? Solution, One-fourth of ^ is equal to ^; ^/irce-fourths of ^ is equal 3 times I, which is f . 10. Whatpartof lisfofi.? |of|.? |ofl.? fof 1 ? 5 p.r 1 p 6 ^f 1 ? ^ r y 01 -g- r g 01 1 (^ r 11. A girl having |- of a dollar, paid -^ of her money for a book. What part of a dollar did she pay ? 12. I of I is equal to what part of 1 ? Solution, l of one-fourth [^ tin^^s -^ is -q] is equal to ^; ^ of I is 3 times i, which is |- . 1 3. Which is the greater fraction of a dollar, — -| of I of a dollar, — or, -^ of | of a dollar ? 14. What part of 1 is i of f ? i of | ? -^of|? -\ of 3? iof|? lof-/^? 15. If 4 cloaks are to be made from 12 f yards of cloth, how many yards must be put into each cloak ? 16. Whatis iof 12|? Solution, ^ of 12 is 3: ^l^ of ^ is -^^ , -} of f is ^2 • llien 3 plus ^2 ^^ ^A* 17 . What is i of 20f ? -} of 28 1 ? ^ of 45 1 c p 18. A boy having f of a dollar, paid -| of his money for a book. What part of a dollar did he pay ? 19. I of f is equal to what part of 1 ? Solution, 'I of I- is equal to -^^ 5 4 ^^ i ^s -^2 5 I ^f f is 2 times ^^ , or ^^ . -f^ is equal to ^ . 28. FRACTIONS 79 20. f of an acre of land was divided into 5 equal lots, and a gardener bought 3 of the lots. What part of an acre did he buy ? Direction. First find what part of an acre there was in one lot; — then, what part in 3 lots. 21. What part of 1 is |off ? Direction. First find what part of a w^hole one \ of I is; — then find j of f , — and then f of f . 22. What part of 1 is f off ? |of-|.? |of|.? f of f? foff.^ fof|.? 23. A merchant, who owned | of a ship, sold f of his share. What part of the ship did he sell ? 24. Whatpartof lis|of|.? fof|.J^ F^ff? | of T • 8 "* 8 • 5 ^^ To • 25. Suppose a piece of broad-cloth to contain 32| yards; — how many yards are there in | of the piece ^ Direction. First find | of 32; then find | of |. 26. Whatisf of20f .? fof36|.? i%of40f .?» fof Sof? fof54|-.? -/oOfSOf.? Section 28. 1. Suppose you have -3- of an orange and \ of an or- ange, — into how many parts must you cut the third ^ and into how many parts the fourth^ so that the parts of the third, and of the fourth shall be of equal size ? We here see, that when l is divided into 4 parts, and \- into 3 parts, the parts are all twelfth In this example 12 is found to be a Common Denomi- nator; and the two fractions -5- and \ , become -^^ and -^^ . 2. Change \ and -^ to a common denominator: that is, find how many parts a half^ and how many a third must be divided into, so that the parts shall be equal: also find how many of these parts would make a whole one. Observation. If two denominators be multiplied to- gether they will produce a common denominator. 80 ORAL ARITHMETIC. VI 3. Change \ and | to a common denominator. Solution. 3 times 5 is 15, the common denominator ^ of -i| is iV ; i of -^1 is -^j . Answer, -^j and i\ . 4. Change \ and ^ to a common denominator, 5. Change ^ and ^ to a common denominator. 6. Change ^ and 3^ to a common denominator. 7. Change ^ and :5^ to a common denominator. 8. Change y and |^ to a common denominator. ^ 9. Change |^ and \ and | to twelfths. 10. Change -I and ^ and |- to twenty-fourths. 11. Change \ and -5^0 and xs to thirtieths. 12. Change ^^ and f to a common denominator. Solution, 4 times 5 is 20, the common denominator 4 "^ 20 ^^ 20 ' y ^^ 20 ^^ 20^ 5" ^^ '^ umes 20? so- ls. Change,^ and | to a common denominator. 14. Change f and ^ to a common denominator. .• ' 15. Change | and -5- to a common denominator. 16. Change ^ and y to a common denominator. 17. Change f and f to a common denominator. 18. Change f and y to a common denominator. 19. Change -^ and -f to a common denominator. 20. Change |- and |- to a common denominator. 21. How much is f and f added together. Solution, ^ is equal to ^V? ^"^ f is ^f : y is equal to jV and f is if. ^1 plus ^f is ff , equal to ^|. 22. How much is \ and 4 added too;ether ? 23. How much is f and f added together 24. How much is f and f added together ? 25. How much is f and | added together ? 26 How much is f and | added together ? 27. ^ and \ and -j^^ are how many twelfths ? 28. f and ^ and -j^ are how many twelfths ? ' 29. :^ and j and -^j are how many sixteenths ? * 30. If 4 be taken from f , how mudi will remain .•* Solution, [y times y is ^5] . y is equal to ^ , | is |^ . ^ is equal to ^, | is f}. Then H minus ff is ^. 29 FRACTIONS. M 53. Take f from |, — how much remains ? 34. Take f from 1^, — how much remains ? 35. Take f from |, — how much remains ? 36. Take J from f, — how much remains ? 37. Take f from ^, — how much remains ? SECTION 29. 1. A farmer gathered 21 f bushels of apples from one tree, and lOf from another. How many bushels did he gather from both trees ? r Direction. First add together the whole bushels, then change the fractions of a bushel to a common de- nominator and add the new numerators. 2. If a bonnet cost 5| dollars and a shawl 5-{q dollars, how much do they both cost ? 3. On a certain day, I travelled 30| miles in a stage, 154 miles in a gig, and 10 miles on horseback. How many miles did I travel that day ? 4. A farmer sold a cow for 23|- dollars, and a calf for 4 1 dollars. How much did he get for both ? 5. Three soldiers shared a loaf of bread as follows :-^- the first man took f of it, the second took ^ of it, and the third took the remainder. What part of the loaf did the third soldier get ? 6. Three men. A, B, and C, are to reap a field of wheat — A is to reap | of it, B ^% of it, and C the re- mainder. What part of the field is C to reap ^ 7. A trader, having 2 barrels of flour, sold | of a barrel to one man, and f of a barrel to another man. What part of a barrel had he remaining ? 8. A man, having 10 dollars, paid away 4\ dollars for a hat, and 3^ dollars for a pair of boots. How many dollars had he left ? 9. A miller, having 20 bushels of corn, sold 6f bush- els to one man, and 9|- to another. How many bushels had he remaining ? 10. A man paid 25 f dollars for a watch, and 2^^^ dol- lars for having it repaired, and then sold it so as to gain 8 dollars. For how much did he sell it.^ est ORAL ARITHMETIC. VI. Section 30. 1. Suppose I had 4 oranges, — to how many boys could I give f of an orange apiece ? Direction, Find how many thirds of 1 orange in 4 oranges, then find how many times 2-thirds there are. 2. How many times is f contained in 4 ? Solution. 1 is equal to |, and 4 is equal to 4 times I or ^y : then f is contained in ^-^ , 6 times. 3. How many pairs of gloves can I buy for 6 dollars, the price being f of a dollar a pair ? 4. How many times is | contained in 6 ? 5. Suppose a man to walk 1 mile in | of an hour, — what distance will he walk in 1 hour ? 6. How many times is f contained in 1 ? Solution. 1 is equal to |-. f in -I, 4^ times. 7. How many yards of cloth, that is sold for f of a dollar a yard, can be bought for 4 dollars ? 8. How many times is f contained in 4 ? 9. How many pounds of tea, that is sold for f of a dollar a pound, can be bought for 4-^ dollars ? 10. How many times is f contained in 4^? 11. If I of a barrel of biscuit will last a ship's crew 1 week, how many weeks will 3 1 barrels last them } 12. How many times is y contained in 3f ? 13. How many yards of cloth, at ^ of a dollar per yard, can be bought for | of a dollar ? Solution, -J of a dollar is equal to -^j of a dollar; | of a dollar is equal to -jf of a dollar. As many yards can be bought as -^^ is contained times in ^. 14. How many times is ^ contained in | ? 15. If a man can hoe -j of a field of corn in 1 day, in how many days can he hoe -J of the field ? 16. How many times is ^ contained in | ? 17. How many times is f contained in f ? Direction. Change both fractions to a common de« nominator; then divide one numerator by the other 18. How many times is f contained in ^} 19. How many times is f contained in ^ ^ 20. How many times is | contained in -^^ ? 30. 31. • FRACTIONS. 83 21 V Suppose that 6 cloaks are to be made from 22| .yards of broad-cloth; — what number of yards must be put into each cloak ? Solution, Each cloak must contain -g^ of 22 1 yards. \ of 22| is 3, there being 4f over. 4| is equal to y . ^of \ is 2V 9 ^^^ i ^^ V is 1^ times as much, or ^|. Then 3 yards plus H of a yard, are 3J| yards. 22. If 30f pounds of bread will supply a family for 1 week, how many pounds will supply the family for 1 day : 23. What is ^of 30f ? 24. If 8 yards of cloth cost 51 f dollars, what will 1 yard cost } What will 3 yards cost } 25. What is I of 51 1 ? What is | of 51 f .? Section 31* Note to Teachers, Thissectionfurnishesatestof the learner's knowledge of the several operations taught since the Review in Section 12. Should tlie learner fail in any of these examples, he must be put back to the section, whose number is prefixed to the example in whicli tlie failure appears. REV IE W. 1. (§ 13.) How many hom's will it take you to read a book of 75 pages, if you read 9 pages an horn* } • , 2. ( § 14.) If a bushel of oats be given in exchange for I of a bushel of grass seed, how many bushels of oats must be given for (>| bushels of grass seed ? 3. (§ 15.) If a man drink \ of a gallon of beer in a day, how many gallons will he drink in 33 days } 4. (§ 16.) Suppose a watch to cost 17 1- dollars, and a chain 1 1- dollars, — what is the cost of both ^ 5. ( § 17.) If 1 quire of letter paper cost y^ of a dol- lar, what will 7 quires cost, at the same rate ? 6. (§ IS.) Suppose a fire engine to throw from its pipe, 4 1 barrels of water in 1 minute, — what number of barrels will it throw in 10 minutes ? 7. (§19.) A farmer sold -} of a ton of hay for 3|- dollars. What is the price ol a ton at the same rate ? 8. ( § 20.) There were 9 men, who performed a piece of work, for which they received 6 bushels of wheat. What part of a bushel was the share of each man } §4 ORAL ARITHMETIC. VI 9. (§21.) A ship's crew'used 14 casks of water, during a passage of 5 months, from Calcutta to New York How much did that quantity allow them per month ? 10. ( § 22.) If 7 barrels of flour cost 30 dollars, what will 9 barrels cost at the same rate ? 11. (§23.) A man purchased a farm, containing 95 acres; but not being able to pay for the whole, he sold ofli Yo of the land. How many acres did he sell .'^ 12. (§ 24.) If a mill grind 9 bushels of corn in f of an hour, how many bushels will it grind in 1 hour ? 13. (§25.) If pen-knives are worth -} of a dollar apiece, and pencils gV ^^ ^ dollar apiece, how many pencils must be given in exchange for 3 knives ? 14. (§26.) Reduce -Jf to its lowest terms. How do you reduce a fraction to its lowest terms ? 15. ( § 27.) A man, owning -| of an acre of land, sold f of what he owned. What part of an acre did he sell ? 16. (§ 2S») Change | and f to a common denomina* tor. How do you change fractions to a com. denom. ^ '17. (§ 2S.) John gave y of a dollar for a book, and \ of a dollar for a slate, and then sold them both for | of a dollar. Did he gain or lose ? — How much ^ IS. (§29.) A farmer cut 18| tons of hay, and sold 2f tons of it. How many tons had he left ? 19. (§ 30 ) When coffee is f of a dollar per pound, how many pounds can be bought for f of a dollar ? 20. (§ 30.) A tenant raised 58 | bushels of corn, and gave his landlord -|- of it for the use of the land. How many bushels had the tenant for himself.'! Section 32. MISCELLANEOUS EXAMPLES. 1. The Gulf Stream is a current in the ocean, running 3 miles an hour. If a steam boat, whose engine propels her 12^ miles an hour, should run in the stream, with the current, what distance would it move in 8 hours ? 2. If the above steam boat were running against the current, what distancie would it move in 8 hours ? 82. MISCELLANEOUS EXAMPLES. 85 3. A trader bought 25 barrels of flour, paying 7 dol* lars a barrel for 1 1 barrels of it, and 9 dollars a barrel for the remainder. What did the whole cost ? 4. What sum of money must be divided among 10 men, so that each man shall receive 19} dollars ? 5. Suppose a man can perform a journey iji 8 days, travelling 10 hours a day, — in how many days can he perform it, travelling 12 hours a day.'* 6. Henry reads 12 pages in the same time that Wil- liam is reading 7 pages; — how many pages will Henry read while William is reading 60 ? 7. If 72 dollars be divided equally among 9 sailors, how many weeks' board, at 3 dollars a week, will each sailor's share pay for } S. A man failed in trade, and could pay only 4 dollars on every 9 dollars that he owed. How much did he pay on a debt of 100 dollars ? 9. There is a pole standing in a pond, so that | of it is under the water, and 3| feet of it is above the water. How long is the pole ? 10. A pole is standing so that I- of it is in the mud, f of it is in the water, and 2 ^ feet of it is above the wa- ter. What is the length of this pole ? 11. If A borrow of B, 8 bushels of wheat, when the price is 9 shillings a bushel; how much wheat must A return, when the price is 7 shillings a bushel ? 12. A trader, having 100 dollars, laid out -/q of his money for narrow-cloth at 5 dollars a yard, and the remainder for broad-cloth at 7 dollars a yard. How many yards did he buy of each sort ? 13. If 2 2" bushels of apples will fill a barrel, how many bushels will it take to fill 8 barrels ? 14. John can pick a quart of berries in an hour; Ann can pick twice as fast; — how many can both pick in an hour ? In what time would they pick 10 quarts ? 15. How many bushels of corn must a miller grind, to get 1 bushel for himself — allowing, that he takes 2 quarts from every bushel before grinding it. and, that 32 quarts make a bushel ? Suggestion^ He gets2(|ts. for grinding less than a bush. 86 ORAL ARITHMETIC. VI 16. If 1 monitor can mend 2 pens in a minute, how long will it take 3 monitors to mend 28 pens ? 17. It is worth as much to pasture 1 cow, as 5 sheep. If I pay 1 dollar a month for pasturing a cow, what must I pay for pasturing 35 sheep, 7 months ? 18. If 3 horses eat 1 ton of hay in 1 month, how long will 5 tons last 4 horses ? 19. A drover sold a cow for 20 dollars, and, in so doing, he gained a sum equal to -3- of what he had paid for the cow. How much had he paid for her ? 20. Suppose a man can dig a trench in 4 days, and a boy in 6 days; — what part of it can each dig in 1 day .'* What part of it can both together dig in 1 day } In what time can they both finish it ? 21. Suppose a cistern has one tap that will discharge it in 5 hours, and another in 7 hours, — in what time will they both discharge it ? 22. If 1 man can perform a piece of work in 35 days, m what time can 6 men perform it ? 23. If 4 men drink a barrel of cider in 20 days, in what time will 9 men drink the same quantity ? 24. If 9 men can do a piece of work in 5 days, in how many days will 7 men do the same w^ork ? ' 25. A farmer kept his sheep in four pastures — In the first pasture he had -^^ of his flock; in the second, 1%; in the third, ^q; and in the fourth he had 32 sheep* How many sheep had he ? 26. There is a school, in which ^ of the scholars read in the Classical Reader^ i read in the JSTational Reader^ \ read in Plerponfs Introduction^ and 36 httle boys read in the Young Reader. How many scholars are there in the school ? 27. If a post 4 feet high cast a shadov/ 3 feet, ai noonday, what is the height of a steeple, that casts a shadow 90 feet, at the same time. ; 28. A and B are laborers — A earns 10 dollars a month, and B 9; but A gives \ of his earnings to B. What will each lay up in 3 months .'' 29. If 12 men can perform a piece of work in 6 days, m w*hat time would 10 men perform the work ? 82. MISCELLANEOUS EXAMPLES. 87 30. How many men must be employed, to dig a trench in 3 days, that 6 men can dig in 4 days ? 31. Suppose 2 men start from the same place, and travel in opposite directions, one at the rate of 5 miles an hour and the other f as fast; — how far apart will they be in 11 hours ? 32. A fox has 35 rods the start of a greyhound, but the hound runs 10 rods while the fox runs 7. How • many rods must the hound run to catch the fox } ^ 33. A started on a journey, and travelled 5 miles an hour — B started on the same journey, 2 hours after, and travelled 7^ miles an hour. In how many hours did B overtake A ? 34. A jockey paid 9 times as much for his horse as he did for his saddle; he paid 3 times as much for his saddle as he did for his bridle; and for his bridle he paid 5 dollars. What did the whole cost? 35. Suppose a man can rea}» ^ of a field of wheat in a day, and his son can reap ^ cS it in a day; — what part of it can they both reap in a day ? In what time can they both reap the whole ? 36. A boy being asked how much money he had, replied — ' If I had as much more, and ^ as much more, and I as much more as I really have, I should then have 70 cents.' How much must he have had ? 37. A gentleman paid 85 dollars for 5 weeks' board of himself, his son, and one servant, at a hotel — His own board cost twice as much as his son's, and his son's cost three times as much as the servant's. What was the expense of each, per week } 88 NOTE TO TEACHERS. The teacher should i>ow be provider! with " A KEY TO THE NORTH AMERICAN ARITHMETIC," otliervvise he must lose much time in examin- ing operations. Tlie Key is a small book designed exclusively for teachers, and contains answers to all the examples in the Written Arithmetic. If the Key cannot be obtained at every place where the Arithmetic is for sale, it may still be obtained from the publisliers of the Arithmetic, and from the principal book-stores in the larger cities and towns. A variety of expedient methods may be pursued, in examining written opera- tions in arithmetic; and perhaps no one system can be adopted, from which it will not be found advantageous, occasionally, to depart. My own practice for several years, with occasional variation, has been as follows. A certain number of examples having been assigned for a lesson the day pre- vious, each scholar is supposed to be })repared with the solutions upon his slate, and the class are paraded for recitation. Every scholar passes his slate into the hands of the scholar next above him, except the head scholar, who hands his to tlie foot scholar. The first scholar then reads from the slate he holds, the answer to the first example; and the teacher, holding the key, declares the answer to be right, or wrong. When the answer has been pronounced right, it is the duty of every scholar who finds a different answer upon the slate he holds, to signify it, and the error is noted against the owner of the slate. The first ex- ample being disposed of, the answer to the second example is read by the second scholar, and disposed of in like manner. Thus the readino^ of answers goes through the class, and each scholar detects the errors of his neighbour. Individ- ual scliolars are occasionally called upon to explain their work in a particular example, and to give their reasons for the operation adopted. By this mode of examination, the work of a large class is particularly inspected, in nearly the same time that would be required to inspect the work of one scholar. Be- sides the advantage of despatch in this mode of examination, the exercise it- self is beneficial to the pupils. — Each scholar acts the part of an inspector — he is interested to be critical — he acquires a facility in deciphering the work of others — and thus his perceptive powers are cultivated, and a habit of alertness IS attained. Before the learners attempt to perform operations by figures, they should be able to write figures with facility, and to arrange them regularly. To attain this object, the arrangement of figures below, may be repeatedly copied upon the slate, until a good degree of despatch and accuracy is acquired. ^2 3Jj-5 6 ^23A5 6 ^ 2 3 Jj- 5 6 7890^2 7890^2 7890^2 3j^5 6 7 8 3^5 6 7 8 3^5 6 7 8 9 0^23^ 90^23J^ 9 0^23Ja 567890 567890 567890 ^23Ji-56 ^23A56 ^ 2 3 J^ 5 6 7890/2 7890/2 7890/2 3^5 67 8 3A5 6 78 3^5 67 8 80 WRITTEN ARITHMETIC. CHAPTER I. ]VUMERATIO]V. Section 1. The unit, which is the first thing to be considered in numeration, signifies One. The figure 1 stands for one unit; 2, for two units; 3, for three units; 4, for four units; 5, for five units; 6, for six units; 7, for seven units; 8, for eight units; 9, for nine units. The TEN is a number which is made up of ten units. One ten is expressed thus, 10; two tens, thus, 20; three tens, thus, 30; four tens, thus, 40; &c. The HUNDRED is a number which is made up of ten tens. One hundred is expressed thus, 100; two hun- dreds, thus, 200; three hundreds, thus, 300; &c. Suppose the balls below, which are arranged in three places, to represent 8 units, 3 tens, and 1 hundred. HUNDRED TENS UNITS # # # m m # 138 Learn from the figures above, that the first or right hand figure expresses units, the second figure expresses tens, and the third figure expresses hundreds. H=^ 90 WRITTEN ARITHMETIC. I. The THOUSAND is a number, which is made up of ten hundreds. One thousand is expressed thus, 1000; two thousand, thus, 2000; three thousand, thus, 3000; &c. Observe, that a figure expresses thousands, when it stands in the fourth place from the right; therefore ten thousand is expressed thus, 10000; and a hundred thousand, thus, 100 000. Examine the following Mimeration Table. Begin at the right hand, and observe, that every three figures may- be viewed by themselves; — the first three express so many units ^ tens and hundreds; the second three, so many Thousands; the third three, so many Millions; the fourth three. Billions; the fifth three, Trillions.* .2^ £.§ ^.2,.^ la -^£2 ^'-sg ^sg ^■B< ^ Cr;^ CrJi-J CrjM 5c^ »^Cn^ HJ^E- ajE-M SE^S KHH KE^;^ 472 156 795 841 526 To read the line of figures in this table, begin with the left hand figure, and proceed as follows. -§ Ij S-^^ §1 11 .^ c ^ ^ , cS s -^^ *> ►L^ -^ fi tJ^aa^ C-vJ^M wC«C cui:2oy3-M3 472 156 795 841 526 This character, 0, called nought^ or cipher^ expresses nothing of itself — it stands only to occupy a place, where there is none of the denomination belonging to that place to be expressed. For example, in the num- ber 240, there are no units; therefore a cipher stands in the units' place. In the number 407, there are no tens; therefore a cipher stands in the tens' place. ♦ The old method of embracing six figures in a period, is of late abandoned ' 8. NUMERATION. 91 Note to Teachers. Require the learners to copy upon their slates tlie fol- lowing: ligures expressing niunbers. Then require tliem to read from their siutes tlie severe, numbers expressed. (Ex. 1.) 508 (19) 1 000 001 (2) 3 861 (20) 90 040 (3) 1050 (21) 107090 (4) 27 400 (22) 6 000 304 (5) 13 008 (23) 77 010 000 (6) 29111 (24) 100100 011 (7) 112 600 (25) 220 002 (8) 30 030 (26) 11333111 (») 206 209 (27) 216 090 900 (10) 500 OSS (28) 10 000004 (11) 7 432 040 (29) 8 000 000 500 (12) 200 005 (30) 50 000 000 036 (13) 9 070 638 (31) 1 000 700 007 (14) 3 018 103 (32) 8 400 052 000 600 (15) 16 974 036 (33) 8 631008 000 (16) 340 007 140 (34) 22 000 004 (17) 31 031 032 (35) 919 000 000 060 (18) . 9 908 000 (36) 86 000 001100 018 Section 2. Note to Teachers. The following numbers written in words, are to be written upon the slate in figures. If the learner meet with difficulty in denoting the larger numbers, he may be instructed to repeat the Numeration Table, from units up to the highest denomination in the number to be denoted; and, while repeating the table, he may make a dot for each denomination, arranging the whole in a line. Then, the figure to express the highest denomination may be written under the left hand dot, and there will be no difficulty in arranging the figures of other denominations under their respective dots. 1. Seventy. 2. Forty-eight. 3. One hundred and twenty-four. 4. Six hundred and nine. 5. Three thousand, and six hundred. 6. Two thousand, four hundred and fifty. 7. Nineteen thousand, and sixty-eight. 8. Five thousand, seven hundred and tbirty-one. 99 WRITTEN ARITHMETIC. I 9. Thirty-six thousand, seven hundred and forty. 10. Two hundred and sixty-eight thousand. 11. Nine hundred five thousand, and one hundred. 12. Eighteen thousand, seven hundred and thirty-five 13. Seven hundred thousand and nine. 14. Thirteen million, sixteen thousand, and nineteen. 15. One hundred five million, two thousand, and one. 16. Six billion, forty million, and six thousand. 17. Twenty-one bilHon, and one hundred million. 18. Five trillion, fourteen billion, seventy million, one thousand, two hundred and thirty-six. 19. One hundred twenty-two trillion, eight hundred and forty-seven thousand. 20. Ten bilhon, nine hundred eighty-seven thousand, seven hundred and thirty. 21. Seven hundred trillion, and thirty-six thousand. 22. Twelve billion, eight hundred forty-two thousand, seven hundred and eighty. 23. Twenty-nine trillion, eight hundred nine billion, one thousand, and eighteen. 24. Eight hundred twenty-three billion, ten million, eight thousand, and fifteen. Questions to be answered Orally, (1) What is a unit? (2) What is the greatest number, that can be expressed by one figure alone ? (3) In what situation must the figure 9 stand, to express 9 tens ? (4) What is the greatest number that can be expressed by two figures ? (5) Recite the several denominations of numbers, from units to trillions J as they stand in the Numeration Table. (6) What denominations are expressed in the 1st. three places of figures ? (7) What denominations are expressed in the 2nd. three places ? (8) Where must the figure 7 stand to express 7 tens of thousands — that is, seventy thousand ? (9) What denomina- tions are expressed in the 3rd. three places ? (10) Where must the figure 2 stand, to express two hundred thousand ? 1. 2. ADDITION M CHAP. II. ADDITIOIV. Section 1. 1. What is the whole sum of 5312 dollars, 8032 doK lars, 601 dollars, and 7123 dollars ? w ^ We first write the nunibers under one g *g another, so that all the units may stand in a I'l «^ column on the right hand. We then add 6 3 12 ^^^ six, and 2 are eight; and we write 8 8 3 2 under the column of units. We next add 5 1 the column of tens, and, finding their sum 7 12 3 to be 6, we write 6 under the column. In ■ - ck i^ Q ^^® same manner we add the hundreds, and ^^ -^^^ the thousands. Find the sum of the numbers in each of the following examples, by addition upon the slate. (2). 51 (3). 733 (4). 6243 (5). 24031 4 120 4123 1320 60 12 9401 40214 43 634 130 34314 Section 2. 1. Add the following numbers into one sum. 4638 and 216 and 8329 and 1212. ^ ^ Finding the sum of the units to be 25, or §1 2 tens and 5 units, we write only the 5 units, I"? gS and presently add the 2 tens in with the hE^5 column of tens. In adding the hundreds, 4 6 38 we find their sum to be 13. Now if we 216 should write down 13, the 3 would stand 8 32 9 under the column of hundreds, and the 1, 12 12 under the column of thousands; therefore 14 3 9 5 we write the 3 only, and presently add the • 1 in with the thousands. 94 WRITTEN ARITHMETIC. H In the following examples, observe, that when the sum of any column amounts to more than 9, you must set down only the right hand figure of it, and, must add the left hand figure to the next column. (2). 6214 (3). 5221 (4). 7420 (5). 3150 2403 7540 612 216 590 1368 2541 8481 8732 520 9103 275 1217 5648 430 8610 2464 ^7300 1000 2541 RULE FOR ADDITION. W^'ite the numbers^ units iindet unitSj tens under tens^ ^c, Md each column separately^ beginning with the column of units. When the sum of any column is not more than 9, lorite it under the column: when the sum is more than 9, write only the units^ figure under the column^ and carry the number of tens to the next column. Finally^ write down the whole sum of the left hand column. 6. Add together the numbers, 143 and 8 and 56 and 7. 7. Add together the numbers, 3 and 96 and 5 and 984. 8. What is the whole sum of 26, 9, 18, 153 and 728 ? 9. What is the whole sum of 8, 6, 42, 728 and 4105? 10. What is the whole sum of 44, 536, 827 and 3480 ? 11. What is the whole sum of 1118, 6004, and 84932 ? 12. What is the whole sum of 61297, 58 and 389166 > 13. Find the sum of 423, 315, 531, 414, 612, 234, 621, 414, 711, 144, 621 and 918. 14. Find the sum of 314, 90, 246518, 7, 1101, 47, 3430, 8601520, 2004 and 5674. 15. Find the sum of 1728, 26510, 34, 100, 3261, 9, 245, 1640831, 6733 and 40000000. 16. A clerk received from one man 94 dollars, from another 361 dollars, and from another 113 dollars. What was the v?4iole sum of moQey received ^ 17. A merchant sent to the bank at one time 301 dol- lars; at another 214; at another 1109; at another 109 How much did he send in all ? U ADDITION. vim IS. A certain lot of land has been divided into three farms; one of the farms contains 112 acres, another 123 acres, and the other 147 acres. How many acres were there in the original lot ? 19. If you start on a journey, and travel on Monday 42 miles, on Tuesday 57, on Wednesday 49, on Thurs- day 54, on Friday 63, and on Saturday 75, how far will you have travelled at the end of the week ? 20. Suppose 477 dollars are in one bag, 8509 in another, 1965 in another, and 956 in another; what sum of money is there in the four bags ? 21. A merchant bought a quantity of sugar for 2075 dollars, and then sold it so as to gain 415 dollars. For how much did he sell the sugar ? 22. There are four numbers, the first of which is 532, the second 895, the third 240, and the fourth as much as the other three. What is the sum of them all ? 23. A broker, by selhng a note for 836 dollars, lost 140 dollars. What must he have paid for the note ? 24. A capitalist gave to one of his sons, 13427 dol- lars; to another, 13025 dollars; to another, 12947 dol- lars. What did he give to all of them ? 25. Sacred history shows, that the time, from the creation of the world to the Deluge, was 1656 years; thence to the building of Solomon's temple, 1344 years; thence to the birth of Christ, 1004 years. How old is the world the present year ? 26. George Washington was born in the year 1732, and lived to be 67 years old. In what year did he die } ,27. Three men united in trade; — the first man had 5136 dollars, the second had 1562 dollars, and the third had 756 dollars. How much had they all ? 28. A ti-ader bought four pieces of cloth: the first piece contained 86 yards; the second, 55 yards; the third, 87 yards and the fourth 91 yards. What was the cost of the whole, at 1 dollar per yard ? 20. A gentleman pur(;hasecl a farm fo^^ 8257 dollars; paid 953 dollars for having it fenced, and 300 dollars for having a barn built upon it. For how much must he sell it, in order to 2;ain 100 dollars ? 96 WRITTEN AKlTHxMETlC. ]J 30. A dfover paid 300 dollars for 100 sheep, 525 dollars for 150 sheep, and 1000 dollars for 250 sheep. How many did he buy ? What did the whole cost ? 31. What is the sum of two million, five hundred thirty-one thousand, one hundred and twenty, — -fourteen thousand, — thirty thousand and twenty-four, — five hundred und sixty, — and seven hundred and two ? 32. The inhabitants of the British Islands are stated thus: England, 11260 555; Wales, 717 103; Scotland, 2092014; Ireland, 6 846 949 ; Army and Navy, 310 000; Isle of Man, 40 981 ; Guernsey, 20 827 ; Jersey, 28 600 ; Scilly Isles, 2 614. What is the whole number? 33. The inhabitants of the United States, by the census of 1840, were stated thus: Maine, 501 793; New Hamp- shire, 284 574; Massachusetts, 737 699; Rhode Island, 108 830 ; Connecticut, 309 978 ; Vermont, 291 948 ; New York, 2 428 921; New Jersey, 373 306; Pennsylvania, 1 724 033 ; Delaware, 78 085 ; Maryland, 470 019; Vir- ginia, 1 239 797; North Carolina, 753 419; South Carolina, 594 398 ; Georo;ia, 691 392 ; Alabama, 590756 ; Mississippi, 375 051; Louisiana, 352 411; Tennessee, 829 210; Ken- tucky, 779 828; Ohio, 1 519 467 ; Indiana, 685 866; Illi- nois," 476 183 ; Missouri, 383 702 ; Arkansas, 97 574 ; Mi- chigan, 212 267; Florida, 54 477; Wiskonsan, 30 945; Iowa, 43 112 ; District of Columbia, 43 712. What was the whole number ? Questions to he ansi/mred Oralhj, (1) When you have several numbers to add to- gether, in what order do you write them ? (2) Which column do you add first ? (3) Do you add all other columns in tbe same manner that you add the first ? (4) When the sum of any column is less than 10, where is it to be written? (5) When the sum of any column is more than 9, what is to be done? (6) Why do we carry as many ones to the next left hand column, as there are tens in any column that we have added ? (7) Recite the rule for addition. I. 2. SUBTRACTION. 97 CHAP. III. SUBTRACTIOIV, Section 1. 1. Subtract 632 from 1847; that is, take 632 from 1347, and find what number remains. 1847 ^^ ^^^^^ write the smaller number under ^32 " the greater. Then, take 2 units from 7 units, ^ ^ - 3 tens from 4 tens, 6 hundreds from 8 hun- .__J — L dred, and nothing from 1 thousand. Subtract the smaller number from the greater in each of the folio w^ing examples. (2). 25 (3). 639 (4). 4258 (5). 705684 12 213 3215 4261 6. A farmer having 359 sheep, sold 136 of them, and kept the remainder. How many did he keep ? 7. A trader having 2748 dollars, laid out 2616 dollars for goods. How many dollars had he remaining ? Section 2. 1. Subtract the number 1528 from the number 8473. We unite 1 of the 7 tens with the 3 units, 847 3 rnaking 13 units, and say, 8 from 13, leaves .?_?_?. 5. Then, having used 1 of the 7 tens, w^e 694 5 take 2 tens from 6 tens. In the same way ■ we take 5 hundreds from 14 hundreds. Do not pass from the abov^ example without under- standing it. Whenever an upper figure is smaller than the figure under it, we use 1 from the next upper figure, and this 1 becomes 10 w^hen considered with the right hand figure. Arithmeticians commonly call this process, borrowing 10; and, instead of reckoning the figure from which they have borrowed to be 1 less than it stands, they pay 1 to the figure under it — reckoning the lower figure to be 1 morp* than it stands. I 98 W.RITTEN ARITHMETIC. IL. Perform subtraction in the following examples. (2). 1853 (3). 5264 (4). 2657 (5). 6807 1370 762 349 4096 6. Subtract 1268 from 1503. In subtracting the 8 units, we use a ten^ 15 03 ^i^r^^ y^Q obtain by supposing 1 of the 5 hun- ^^^^ dreds, (which is 10 tens,) to be where the 235 is. Then, having used 1 of the 10 tenSj we presently subtract 6 tens from 9 tens 7 Subtract 1146 from 2601. 8. Subtract 5428 from 8019. 9. Subtract 258 from 34307. RULE FOR SUBTRACTION. Write the smaller number under the greater^ placing units under units^ ^c. Begin with the units^ and subtract each figure in the lower number from the figure over it. When a figure in the upper number is smaller than the figure under it^ consider the upper figure to be 10 more than it is^ and the next upper figure on the left hand^ to be 1 less than it is. PROOF, '^dd together the remainder and the smaller number : their stun will be equal to the greater number ^ if the work be right. 10. Find the difference between 39 and 64, by sub- tracting the smaller number from the greater. 11. What is the difference between 464 and 502 ? 12. What is the difference between 99 and 200 ? 13. What is the difference between 35720 and 9100 ? 14. Subtract 44 from 10000. 15. I deposited 1450 dollars in the bank, and I have since drawn out 835 dollars. How many dollars have I remaining in the bank ? 16. Suppose a man owes 1634 dollars, and possesses property to the amount of 8150 dollars; how much will he have left, after paying his debts ? 17. Subtract sixty-two thousand five hundred and seven, from one million eighty thousand and forty-four. O^ SUBTRACTION. 99 IS. The number of inhabitants in the city of London is 1 250 000; the number in the city of Paris is 750 000 How many more are there in London, than in Paris .^ 19. The population of Great Britain and Ireland is 21 500 000; the population of France is 32 000 000. How many more inhabitants are there in France, than in Great Britain and Ireland? 20. The Rocky Mountains, in North America, rise 12 500 feet above the level of the ocean; the Andes, in South America, rise 21440 feet. How many feet higher are the latter, than the former? 21. A merchant paid 13 745 dollars for a ship, and sold it for 15 150 dollars. What did he gain? 22. A farmer sold a piece of wood-land for 396 dol- lars, which was 78 dollars more than he gave for it. How much did he give for the land? 23. Columbus discovered America in the year 1492. How many years is it since the discovery? 24. The United States declared Independence in the year 1776. How many years since the declaration? 25. A man bought 20 casks of wine, containing 2459 gallons, and sold 14 casks containing 1682 gallons. How many casks, and how many gallons were left? 26. There are two numbers, whose difference is 758; the greater number is 1524. What is the smaller number? Q^uestions to be answered Orally, (1) How can you find what the difference is be- tween two numbers? (2) When one number is to be subtracted from anothegjttfcvhat order must the numbers be written? (3)^[ff what place do you begin to perform the subtraction? (4) When a figure in the upper number is smaller than the figure under it, what is to be done? (5) Where does the remainder appear, after the subtraction is performed? (6) Recite the rule for subtraction. (7) How can you prove that an operation in subtraction is per- formed correctly? 100 WRITTEN ARITHMETIC. I. Section 3. MISCELLANEOUS EXAMPLES. 1. A man owing 379 dollars, paid at one time 47 dollars, at another 23, at another 84, and at another, 143. How much did he still owe? 2. There are 1000 dollars in 4 bags; the first bag contains 230 dollars, the second 245, the third 270 What is contained in the fourth bag? 3. Suppose the world to have been created 4004 years before the Christian era, how old is it at this date? 4. A man having in his desk 2000 dollars, took out 120 dollars to pay a debt, and afterwards put in 75 dols. How much w^as there remaining in the desk? 5. A merchant bought a ship for 11 240 dollars, paid 305 dols. for repairing it, and sold it so that he lost 95 dols. For how much did he sell it? 6. What is the sum of 58, 45, and 70? Then, if you subtract 43 from this sum, what will be the remainder? 7. A merchant, who had 650 barrels of flour, sold 95 barrels to one man, 38 to another, and 225 to another. How many barrels had he left? 8. A jockey bought a horse for 115 dollars; he ex- changed him for a better horse, paying 23 dollars, and then sold the better one for 137 dollars. Did he gain or lose? — and how much? 9. If 654 be subtracted from 10000, and then 29670 be added to the remainder, what will be the sum? 10. A gentleman gave 972 dollars for a carriage and two horses; the carriage was valued at 525 dollars. What was the value ofj^j^orses? 11. Dr. Franklin diWW the year 1790, and he was 84 years old when he died. In what year was he born? 12. A clerk went out with 240 dollars, to settle some accounts: he paid 126 dollars to one man, received 37 dollars from another, and paid 94 dollars to another. How many dollars had he then? 13. Add together two hundred, sixteen thousand, thirteen million, and seven billion; and then subtract ten thousand from the sum 1.2. MULTIPLICATION 101 CHAP. IV. MULTIPLICATIOJV. Section 1. 1. If a gunner shoot 72 pigeons every time he goes a gunning, how many will he shoot m going 3 times ? We might here obtain the answer by adding together, 72 and 72 and 72; but we shall obtain it more readily by multiplying 72 by 3; that is, by finding 3 times 72. Multiplicand 72 We write 72, and write 3 under Multiplier 3 it. Then we multiply the 2 units Product 216 ^."^ '^l ^ '^f separately, thus, 3 times 2 are 6; 3 times 7 are 21. Observe, that the number which we multiply is called the multiplicand; the number by which we multiply is called the multiplier; and the number which we obtain by multiplication is called the product. Find the product in each of the following examples. (2). 61 (3). 524 (4). 9132 (5). 420121 4 2 3 4 6. If a farm produce 230 bushels of wheat a year, how many bushels will it produce in 3 years ? 7. Multiply 512 by 4; — that is, find 4 times 512. Section 2. 1. Multiply 743 by 6; — that is, find 6 times 743. 743 6 times 3 are 18, or 1 ten and 8 units; we (3 write only the 8 units, (as in addition), and ' , - Q proceed; — 6 times 4 are 24 and 1 we carry are 25; we write the 5 and proceed. Find the product in each of the following examples. (2). 5236 (3). 1908 (4). 6175 (5). 3640 4 2 .5 8 102 WRi^TTEN ARITHMETIC. IV. 6. What will 3 books cost, at 31 cents apiece ? 7. What will 4 slates cost, at 24 cents apiece ? 8. What will 5 baskets cost, at 17 cents apiece? 9. What will 6 ^ows cost, at 25 dollars apiece ? 10. What will 7 horses cost, at 115 dollars apiece? 1 1 How niany are 8 times 9 ? 12. How many are 9 times 16 ? 13. How many are 5 times 342? 14. How many are 7 times 6453 ? 15. How many are 3 times 42 90S ? 16. How many are 6 times 704 370? ^ 17. Multiply 251 by 8, — that is, find 8 times 251. 18. Multiply 475 by 4. 19. Multiply 3086 by 6. 20. Multiply 15 350 by 8. 21. Multiply 430 039 by 7. 22. Multiply 7 000 005 by 9. 23. Multiply 42 862 000 by 5. 24. Multiply 928 064 309 by 4. 25. Suppose 8 to be a multiplicand, and 6 the multi plier; how much will be the product? 26. Suppose 35 to be a multiplicand, and 7 the multi plier; how much will be the product? 27. Suppose 491 to be a multiplicand, and 5 the mul- tipher; how much will be the product? Section 3. 1. Multiply 657 by 24. We first multiply by the 4 units. Then ^ we multiply by the 2 tens, and since this ZZ product must be ten times greater than it 2628 would be if the 2 were 2 units, we set the 1314 product one place to the left. At last, we 15768 ^^^ ^^^ ^^^'^ products together, and the sura • is the whole product of 657 by 24. 2. Multiply 75 by 16. 3. Multiply 6?4 by 45.. 4. Multiply 3291 bv 63. 6. Multiply 71 538 by ''7. 6. Multiply 428 601 by 81. 3. MULTIPLICATION. 103 RULE FOR MULTIPLICATION. Write the multiplier un- der the multiplicjnd^ placing units under units^ <^c. When there is but one figure in the multiplier^ begin with the units J multiply each figure in the multiplicand separately^ and place each product under the figure in the multiplicand from which it arose; observing to carry the tens to the left as in addition. When there is more than one figure in the multiplier j multiply by each figure separately^ and tcrite its product in a separate line^ placing the right hand figure of each line under the figure by which you multiply; and final" ly^ add together the several products. The sum will be the whole product. 7. Suppose 5 476 208 to be a multiplicand, and 3942 the multiplier; how much will be the product.'^ 5476208 3942 10952416 21904832 49285872 16428624 21587211936 8. Suppose 73 054 to be a multiplicand, and 548 th.'' multiplier ij how much will be the product.^ 9. Suppose 295 to be a multipHcand, and 486 the multiplier; what will be the product } 10. What is the product of 9351 by 765 ? 11. What is the product of 3008 by 254 ? 12. What is the product of 5603 by 6448 ? 13. How many are 74 times 6580 ? 14. How many are 236 times 3759 ? 15. There is an orchard containing 9 rows of trees, and there are 57 trees in each row. How many trees are there in the orchard ? 16. A merchant bought 75 pipes of wine, at 145 dol- lars a pige. What did the whole cost } 17. A merchant bought 37 mules, for shipping, at 52 dollars per head. What did the whole cost } 104 WRITTEN ARITHMETIC. IV 18. A man travelled 26 days, travelling 47 miles a flay. How far did he travel in the whole time? 19. A merchant sold 342 tons of iron, at 142 dollars per ton. What was the price of the whole? 20. If a coach wheel turn round 346 times in 1 mile, how many times will it turn round in the distance from New York to Philadelphia, it being 95 miles? 21. A prize was divided among 47 men, and each man received 25 dollars. How much was the prize? 22. What sum of money must be divided among 4*^ men, so that each man shall receive 59 dollars? 23. A merchant bought 7 bales of cloth, each balu containing 11 pieces, and each piece, 29 yards. How many pieces, and how many yards were there? 24. A trader bought 9 pieces of cloth, each piece containing 42 yards, at 6 dollars a yard. How many yards were there, and what did the whole cost? 25. If hats are worth .7 dollars apiece, w^hat are 15 boxes of hats worth, each box containing 24 hats? 26. The distance from Washington to Boston is 436 miles; and in each mile there are 320 rods. How many rods is it from Washington to Boston? 27. The distance from Washington to New-Orleans is 1255 miles. How many rods is it? 28. What is the value of the hay, that is produced on 16 farms; allowing each farm to produce 62 tons, and allowing the hay to be worth 12 dollars a toU? 29. There are 24 hours in a day, and 365 days in a year. If a ship sail 7 miles in an hour, how many miles will she sail in a year? 30. How many days' work can 9 men do in 24 days? 31. How many days will it take 1 man to perform a piece of work, that 9 men will perform in 24 days? 32. How many days will it take 1 man to build a piece of road, that 13 men can build in 47 days? 33. How many men must be employed to do a piece of work m 1 day, that 11 men can perform in 18 days? 34. Suppose that a ship's crew of 13 men will drink 82 gallons of water in 14 days, how long \fould the same quantity of water last 1 man? 4. MULTIPLICATION IQB Section 4. ABBREVIATIONS. When there are ciphers standing between figures^ in ^he multiplier^ they may be disregarded, 1. What is the product of 12318 12318 multiplied by 7004 ? 7004 49272 86226 86275272 2. What is the product of 9651 multiplied by 304? 3. How many are 1001 times 57 906 ? 4. How many are 905 times 820 437 ? Ciphers on the right hand of the multiplier or multi" plicand may be disregarded till the multiplication is per- formedy and then placed on the right hand of the product* 5. What is the product of 5763 5763 multiplied by 3600 ? 3600 34578 17289 20746800 6. What is the product of 158 multiplied by 350 ? 7. How many are 800 times 369 ? 8. How many are 40 times 4728 ? Ciphers on the right hand of the multiplier and muU tiplicand both^ may all be disregarded in multiplying^ and finally placed on the right hand of the product* 9. What is the product of 46000 46000 multiplied by 340. 340 184 138 15640000 10. What is the product of 8370 multiplied by 240? 11. How many are 90 times 761000? 12. How many are 5700 times 6800? 106 WRITTEN ARITHMETIC. lY, When the multiplier is 10, 100, 1000, ^c merely place the ciphers of the multiplier on the right hand of the multiplicand^ and it becomes the product, 13. What is the product of 5 multiplied by 10 ? 14. What is the product of 17 multiplied by 100.^ 15. What is the product of 49 multiplied by 1000.^ 16. In 1 dollar there are 100 cents. How many cents are there in 6 dollars ^ 17. How many cents are there in 25 dollars ? 18. If 1 box of lemons cost 7 dollars, how many cents will it take to pay for 10 boxes ? When the multiplier is a number^ that can be produc- ed by multiplying two smaller numbers together^ muU tiply the multiplicand first by one of the smaller numbers j and the product thence arising by the other. 19. Find the price of 32 horses, at 96 dollars apiece. 96 price of 1 horse. 8 768 price of 8 horses. 4 3072 price of 4 times 8 horses, or 32 horses. Observe in the above example, that 32 can be pro- duced by multiplying 4 and 8 together. The 4 and the 8 are called the factors of 32. 20. A merchant bought 24 hogsheads of molasses at 19 dollars a hogshead. What did the whole cost ? In this example we consider 24 to be the multiplier. For 24, we can find several different sets of factors; viz. 3, 8; alsQ, 4, 6; also, 2, 3, 4; also, 2, 2, 6. Either set of these factors may be used. 21. If a ship sail at the rate of 129 miles a day, how many miles will she sail in 72 days ? 22. If 1 man can dig 41 bushels of potatoes in a day, now many bushels can 28 men dig ? 23. Multiply 425 by 36, using the factors of 36. 24. How many are 63 times 540 ? 25. How many are 45 times 2807 7 MULTIPLICATION. \m Questions to be answered Orally, (1) What is meant by multiplicand? — what by multiplier? — and what by product? (2) When we say, 5 times 8 are 40, which of these numbers is the muhipHcand ? — which the multipHer ? — and which the product ? (3) Con you obtain the product of any two numbers, by means of addition ? (4) Re- cite the rule for multipHcation. (5) When there are ciphers between figures in the multiplier, what may be done ? (6) When there are ciphers on the right of the multipher, or multiplicand, or on the right of both, what may be done with them ? (7) In what manner can you multiply by 10, by 100, by 1000, &c.? (8) What is meant by the factors of a number? (9) Name two factors of 24. (10) Name three factors of 24. (11) Name two factors of 36. (12) Name three factors of 36. Perform the following examples by either of the fore- going methods, which may be found convenient. 26. What is the value of a farm consisting of 200 acres of land, at 40 dollars an acre ? 27. Suppose a book to contain 235 pages, 45 lines in each page, and 50 letters in each line; — how many letters are there in the book ? 28. Suppose an orchard to consist of 109 rows, 126 trees in a row, and 1007 apples on a tree; — how many trees, and how many apples are there ? 29. Suppose a crew of fifty men have provision for 30 days, allowing each man 20 ounces a day; — how many days would it last, if each man ate 1 ounce a day ? 30. Suppose a crew of fifty men have provision for 30 days, allowing each man 20 ounces a day; — how many men would it serve for the same time, if each man ate one ounce a day .'' 31. How many fishes would be caught by 14 boats, employed for 30 days, each boat drawing a net 15 times a day, and taking 13 fishes each draught ? 32. What is the product of 90042 multiplied by 9009 ? 108 WRITTEN ARITHMETIC. y. CHAP. V. DIVlSIOJf. Section 1. I. How many yards of cloth, at 3 dollars a yard, can be bought ^or 396 dollars ? Here we must find how many times 3 dollars there are in 396 dollars: that is, we must divide 396 by 3. 3)39 6 We first divide the 3 hundreds, then the Y32 ^ tens, and then the 6 units; thus, 3 in 3, ~ once; 3 in 9, 3 times; 3 in 6, 2 times. Observe in the above example, that the 3 which we first divide, means 3 hundred; and the 1 which we place under it means 1 hundred, showing that 3 is contained in 300, 100 times. The 9 means 9 tens^ and the 3 which we place under it means 3 tens, showing, that 3 is contained in 90, 30 times. A Dividend is a number which is to be divided; such is the number 396 in the above example. A Divisor is a number by which we divide; such is the number 3 in the above example. The Quotient is the number of times which the divisor is contained in the dividend; such is the number 132 in the above example. Find the quotient m each of the following examples. (2). 4)8 (3). 2)46 (4). 3)936 (5). 4)4884 ' 6. A man laid out 69 dollars for sheep, paying 3 dol- lars a head for them. How m.any did he buy ? 7. If 4 bushels of wheat will pay for 1 barrel of flour, how many barrels will 848 bushels pay for ? Section 2. 1. How many times is 4 contained in 3684 ? 4)3684 tn this example we find that 4 is not " go 1 contained in 3, therefore we join the 3 ' with the 6, and say, 4 in 36, 9 times. . 2. DIVISION. 109 2. How many times is 7 contained in 56? 3. How many times is 9 contained in 639? 4. How many times is 5 contained in 405? 5. How many times is 4 contained in 3248? 6. How many times is 3 contained in 1569? * 7. If 4 horses are required to draw 1 wagon, how many wagons might be drawn by 168 horses? 8. How many yards of broad cloth, that is sold at 6 dollars a yard, can be bought for 492 dollars? 9. If a man can travel 5 miles an hour, how many hours will it take him to travel 205 miles? 10. Suppose 69 to be a dividend, and 3 a divisor; what is the quotient? 11. Suppose 128 to be a dividend, and 4 a divisor; what is the quotient? 12. Suppose 486 to be a dividend, and 6 a divisor; what is the quotient? 13. How many times is 4 contained in 872? 4)8 7 2 4 in 8, 2 times; 4 in 7, 1 time, and rrr there is 3 over; (we join this 3 with the 2, zll making 32,) then 4 in 32, 8 times. 14. How many times is 6 contained in 726? 15. How many times is 8 contained in 896? 16. How many times is 5 contained in 1605? 17. How many times is 7 contained in 924? 18. How many times is 4 contained in 6732? 19. Suppose 1585 to be a dividend and 5 the divi- sor; what is the quotient? 20. Suppose 4518 to be a dividend and 6 the divi sor; what is the quotient ? 21. How many times is 7 contained in 742? 7)742 The divisor not being contained once in rr~ the ten's place of the dividend we write a in the ten's place of the quotient. 22. How many times is 3 contained in 609? 23. How many times is S contained in 1624? 24. How many times is 5 contained in 4015? 25. How many times is 9 contained in 2880? 26. How many times is 7 contained in 10500'* K 110 WRITTEN ARITHMETIC. \\ 27. If I bad 78 dollars to lay out for flour, and the flour was 6 dollars a barrel, how many barrels could 1 buy for all the money? 23. A drover received 268 dollars for sheep, that he sold at 4 dollars a head. How many were there .'^ 29. If 1 ton of hay be worth 9 bushels of corn, how many tons of hay are 576 bushels of corn worth? 30. If 3 bushels of wheat will pay for a yard of cloth, how many yards will 105 bushels pay for? 31. How many soldiers may be clothed from 5708 yards of cloth, allowing 4 yards to make a suit? 32. How^ many muskets can be purchased for 2952 dollars; the price being 6 dollars apiece? 33. If 76 dollars should be divided equally among 4 men, how many dollars would each man receive? If there were only 4 dollars to be divided, each man w^ould receive just 1 dollar: therefore each man must receive as many dollars as there are fours in 76. 34. Suppose 5 men have to pay a bill of 95 dollars, how many dollars must each man pay? 35. If 171 biscuit be divided equally among a crew of 9 sailors, how many does each sailor receive? 36. A farmer planted 354 trees, in 6 equal rows. How many were there in 1 row? 37. A fisherman hired a boat, agreeing to give the owner 1 fish of every 7 that he might catch: he caught 434. How many should he give the owner? 33. 8 sailors received 1576 dollars for retaking their ship. How much did each sailor receive? 39. A man intending to go a journey of 336 miles, wishes to perform it in 6 days. How many miles must he travel each day? 40. 9 men have agreed to make up a purse of 2178 dollars. How many dollars must each one put in? 41. Suppose A to spend 3 dollars as often as B spends 1 dollar; how many dollars will B spend while A is spending 89004 dollars? 42. Suppose 3656 dollars have been equally divided among a number of men, and each man has received 8 dollars; how many men were there? 2 DIVISION. iil 43. A number of men contributed 9 dollars apiece, and thereby made up a purse of 54 dollars. How many men were there ? 44. Suppose 9 has been multiplied by some number, and the product is 54; what was the multiplier.'* 45. 5 men paid equal shares of a debt of 80 dollars. How much did each man pay .? 46. Suppose some number has been multiplied by 5, and the product is 80; what number was multiplied ? 47. Two numbers have been multiplied together, and their product is 126: one of the two numbers multiplied is 7; — what is the other ? 48. Divide 348 by 4; then prove the work to be right, by multiplying the quotient and divisor together } 4)348 We find by the quotient, there are 87 '~~Qj times 4 in 348: therefore we know that 87 ^ times 4, or 4 times 87, must make 348. ■ Had our quotient been wrong, our product ^^^ and dividend would not be equal. 49. Divide 72 by 8, and prove the work to be right 60. Divide 5890 by 5, and prove the work to be right 51. Divide 39781 by 7, and prove the work to be right. 52. Divide 90048 by 8, and prove the work to be right. 53. Divide 17604 by 9, and prove the work to be right. 54. A hatter has 130 hats finished; and, in order to send them to market, he must pack them in boxes, that will hold 8 hats apiece. How many full boxes can he send; and how many hats will remain on hand ? 8)1 30 We have 2 units over. This 2 is a re- 1^2 'inainder; it shows that there are 2 hats, • which cannot be divided into eights. 55. How many sheep, at 4 dollars a head, can a butcher, who has 747 dollars buy; and how many dol- lars will he have remaining ? 56. If 5 yards of cloth will make a suit of clothes, how many suits can be made from 96 yards; and how many yards will there be over ? 57. How many times is 6 contained in 4637; and how many are there over ? 112 WRITTEN ARITHMETIC. V. 58. How many times is 8 contained in 9150; and how many are there over ? 59. Suppose 568 to be a dividend, and 7 the divisor; what is the quotient, and the remainder ^ 60. Suppose 1953 to be a dividend, and 7 the divi- sor; what is the quotient, and the remainder ? 61. Divide 564 by 7, and prove the work to be right. The remainder, in division, is an undivided part of the dividend: therefore, the remainder must be added to the product of the divisor and quotient, to make the product equal to the dividend. 62. Divide 109 by 6, and prove the work to be right. 63. Divide 817 by 5, and prove the work to be right. Section 3. The method of dividing taught m the two preceding sections, is called Short division: the method taught in this section; is called Long division. In long division, we place the quotient on the right hand of the dividend, and perform some operations under the dividend, here- tofore performed in the mind. 1. How many times is 4 contained in 95307 ? tJ b ^ Perceiving that 4 is contained I] I.* § in 9, twice, we place 2 in the § §^ §"• quotient, multiply the divisor by 1, S. 2, and subtract the product (8) 4)95307(23826 ^^^"^ ^- This is the same as 8 saying in short division, ' 4 in 9, "TT 2 times, and 1 over.' Now, y 2 since the 1 over must be joined . with the 5, we bring the 5 down 33 to the right of the 1: and then, 3^ perceiving that 4 is contained in 10 15, 3 times, we place 3 in the 8 quotient, multiply the divisor *^ by 3, and subtract the product c^. as before. Thus we proceed to bring down every figure of the Remainder 3 dividend, and unite it with the previous remainder. DIVISION 113 Perform the following examples by long division. 2. How many times 5 are there in 7163 ? 3. How many times 7 are there in 88 704 ? 4. How many times 6 are there in 97 547 ? 5. How many times 3 are there in 8 057 251 ? . 6. How many times 4 are there in 8 708 983 ? 7. How many times 5 are there in 6 457 080? 8. How many times 8 are there in 25 648 ? {c^c\c\r The divisor noi being contain- 8)25 648(3200 ^^ ^^^^ j^ ^^^^ l^^-^ j^^^j ^^^^^ ^^ of the dividend, we join this fig- 1 6 ure with the next. After bring- 1 6 ing down the 4, we find the divi- ' ^ sor is not contained in it; there- ^ g fore, we place a in the quotient, ■ — and bring down the next figure. 9. How many times 5 are there in 43 906 ? 10. How many times 9 are there in 70 223 ? 11. How many times 6 are there in 901 500 ? 12. How many times 7 are there in 161 635 ? 13. How many times 24 are there in 3762 ? 24')3762fl56 "^^^^^ operation is performed ' . in the same manner that it w^ould have been, if the divisor had 136 consisted of only one figure. 1^0 The two following examples 152 will show the method of deter- 144 mining when a figure placed in T^ the quotient is too great, and — when it is too small. 1*4. How many times is 18 contained in 12 532? In this example, we have 1 8)12532(697 chosen 7 for the last figure of the 108 quotient; but it appears, that 7 j^«-Q times 18 are more than 112; ^ />2 therefore 18 is not contained 7 ~ times in 112. Tht 7 and the ^ ^ ^ ^ product arising from it must be ^?.^ rubbed- out, and a smaller figure must be placed in the quotient. 114 15. WRITTEN ARITHMETIC. v: How many times is 35 contained in 45 817 ? Here we have chosen 8 for the last figure of the quotient; 35)45817(1308 35 108 105 317 280 3? 16. How many times 17. How many times 18. How many times 19. How many times How many times How many times How^ many times How many times How many times 20. 21. 22. 23. 24. but, after subtracting 8 times 35 from 317, there remains, 37. This remainder will contain 35, once more; therefore, we must rub out the 8 and the work resulting from it, and must put 9 in the place of %S. s 47 contained in 804 ? 53 contained in 1625 ? s 68 contained in 94 605 } s 71 contained in 661 419 ? s 108 contained in 216 ? s 325 contained in 7134.^ s 476 contained in 92 107 ? s 504 contained in 1008 ? s 651 contained in 43 126 ^ RULE FOR DIVISION. When the divisor does not ex* ceed 9, draio a line tinder the dividend^ find how many times the divisor is contained in the left hand figure^ or two left hand figures of the dividend^ and write the figure expressing the number of times underneath: if there be a remainder over^ conceive it to be prefixed to the next fig-^ ure of the dividend^ and divide the next figure as before* Thus proceed through the dividend. When the divisor is more than 9, find how many times it is contained in the fewest figures that will contain it^ on the left of the dividend^ write the figure expressing the number of times to the right of the dividend^ for the first quotient figure; multiply the divisor by this figure^ and subtract the product from the figures of the dividend considered. Place the next figure of the dividend on the right of the remainder^ and divide this number as before. Thus proceed through the dividend. PROOF. JVIultiply the divisor and quotient together, and to the product add the remainder: the sum will bb mutl to the dividsnd^ if the work be right* DIVISION U5 25. Divide 46242 by 252, and prove the operation. 252)46242(183 252 252 183 2104 756 2016 2016 882 ^^L^ 756 ^^^ 126 46242 26 Divide 74 201 by 625, and prove the operation. 27. Divide 408 732 by 9, and prove the operation. 28. Divide 15 362 by 88, and prove the operation. 29. Divide 57 026 by 492, and prove the operation. 30. Divide 982 700 by 53, and prove the operation. 31. Divide 162 941 by 256, and prove the operation* 32. Divide 648 035 by 14, and prove the operation* 33. Divide 106 401 by 333, and prove the operation* 34. Divide 62 509 by 4423, and prove the operation* 35. Divide 1 071 400 by 29, and prove the operation. 36. How many acres of land, &t 22 dollars an acre^ can be bought for 8514 dollars ? 37. Suppose a man to earn 35 dollars a month; how many months will it take him to earn 490 dollars ? 38. If a man travel 48 miles a day, in how many days will he perform a journey of 3264 miles ? 39. If 774 dolhirs be divided equally among IS sail* ors, how many dollars will each sailor receive ? 40. If a man's income be 2555 dollars a year, how much is it a day, there being 365 days in a year ? 41. The income of the Chancellor of England, is 99 280 dollars a year. How much is it per day .'' 42. G3 gallons of water will fill a hogshead. How 4Tiany hogsheads will 5166 gallons fill? 43. How many hogsheads can be filled from 19 721 gallons ? — and how many gallons will there be left.'' 44. Suppose a regiment of 512 men have 8192 pounds of beef; how^ many pounds are there for each man } 45. If a dividend be 46 319, and the divisor 807^ what is the quotient ? — and what the remaiader .'^ 116 WRITTEN ARITHMETIC. V Section 4. ABBREVIATIONS. When there are ciphers on the right hand of a divisor,^ cut them off J and omit them in the operation; also cut off and omit the same number of figures from the right hand of the dividend. Finally^ place the figures cut off from the dividend^ on the right of the remainder. 1. How many times 900 are there in 741 725 ? . 9100)74 17125 ^^? ^''^'^^ T^V ^^ ^' ^^'T • ^ remams 1 , to which we annex the 824 125 25, making the true rem. 125. 2. How many times 70 are there in 8 563 512 ? 3. How many times 300 are there in 6374 ? 4. How many times 5000 are there in 46 578 ? 5. How many times 40 are there in 80 603 ? 6. How many times 600 are there 675 700 ? 7. How many times 8000 are there in 16 000 ? 8. Divide 65 237 by 50, and prove the operation. 9. Divide 567 289 by 400, and prove the operation 10. How many times 570 are there in 35 871 ? 11. How many times 280 are there in 6423 ? 12. How manv times 4200 are there in 91 621? 13. How many times 9060 are there in 287 000 .? When the divisor is 10, 100, 1000, «^c., cut off as many figures from the right hand of the dividend^ as there are ciphers in the divisor; the other figures of the dividend will be the quotient^ and the figures cut off will be the remainder. 14. How many times 10 are there in 240 ? 15. How many times 10 in 435; and how many over } 16. How many times 100 are there in 4000? 17. How many times 100 in 748 ; and how many over ? 18. 100 cents are equal to 1 dollar. How many dollars are there in 5400 cents ? 19. In 642 cents, how many dollars are there; and how many cents over ? 20. In 1937 cents, how many dollars are there; ami liow many cents over '^ 4. DIVISION. 117 When factors of the divisor can be founds (that i>, when two numbers can be founds which^ being multiplied together^ produce the divisor^) you may divide the divi- dend by one of the factors^ and the quotient thence arising by the other: the last quotient will be the true one, 21. In a certain school there are 36 scholars, among whom 540 quills are to be equally divided. How many will 1 scholar receive ? Let u& suppose the school to be divided into 4 classes, allowing 9 scholars to be in each class. Then we will find how many quills 1 class will receive, and from this number, find how many 1 scholar will receive. 4)540 number of quills for the school. 9)135 number of quills for 1 class. 15 number of quills for 1 scholar. Observe in the above example, that the divisors 4 and 9, are the factors of 36: and, if we had divided first by the 9, and then by the 4, our last quotient would have been the same it now is. 22. Divide 11 376 by 72; using the factors of 72. 23. If 1024 dollars be divided equally among 64 men, how many dollars will 1 man receive ? 24. How many times is 42 contained in 1176 i 25. If 27 yards of cloth cost 216 dollars, how many dollars does 1 yard cost ? 26. Suppose 1952 to be a dividend, and 32 the divi- sor; what is the quotient.'* To obtain the true remainder^ where factors have been used as divisors^ multiply the last remainder by the first divisor^ and to the product add the first remainder. 27. Suppose 622 to be a dividend, and 35 the divi- sor; what is the quotient; and what the remainder.'* 28. Suppose 99 to be a dividend, and 25 the divisor; what is the quotient; and what the remainder ? 29. Suppose 4862 to be a dividend, and 81 the divi- sor; what is the quotient; and what the remainder ? 30. Divide 1739 by 56. U8 WRITTEN ARITHMETIC. Questions to be answered Orally, (!) When we say, ^3 is contained in 20, 6 times, and 2 over,' which of these numbers have we for the dividend ? — Which for the divisor ? — Which for the quotient ? — Which for the remainder ? (2) What is meant by the dividend ? (3) W^hat is meant by the divisor ? (4) What is meant by the quotient ? (5) What is meant by the remainder ? (6) Can the remainder ever be equal to, or greater than the divi- sor ? — Why ? (7) Suppose you have a number of dollars to divide among a number of men; which num- ber do you make the dividend ; — and which the divisor ? — If there be a remainder, will it be so many dollars, or so many men ? (8) Recite the rule for division. (9) How do you proceed when there are ciphers on the right hand of the divisor? (10) How do you divide by 10, 100, 1000, &c.? (11) How can you divide by means of factors ? (12) When you have divided by the factors of the divisor, how do you find the true remainder.'^ (13) How do you prove an operation in division ? Perform the following examples by either of the fore- going methods, which may be found convenient. 31. Suppose it takes 7 bushels of apples to make a barrel of cider, how many barrels of cider can be made from 945 bushels of apples ? 32. Suppose an acre of land properly cultivated, to produce 38 bushels of corn; how many acres must be cultivated to produce 4902 bushels ? 33. If 50 dollars will pay for an acre of land, how many acres can be bought for 6900 dollars ^ 34. How many days will a ship be in sailing from New York to Liverpool; allowing the distance to be 3000 miles, and the ship to sail 100 miles a day ^ 35. A vintner wishes to put 6615 gallons of wine mto hogsheads that will hold 63 gallons apiece; — how many hogsheads must he have ? 4. RETROSPECTIVE OBSERVATIONS. 119 36. If you had 118 dollars, how many hats could you pay for, at 5 dollars apiece; and what number of dollars would you have left? 37. Suppose a drover has 2130 dollars; how many oxen can he pay for, at 47 dollars apiece; and how many dollars will he have left.^ 38. In 668 360 yards of cloth, how many pieces, and how many bales; there being 35 yards in each piece, and 56 pieces in each bale? 39. If 4810 dollars be shared equally among 130 men, how much will each man receive? 40. A farmer planted 2072 trees in 14 equal rows. How many did he plant in a row? 41. A gentleman wishes to spend 136 days in per- forming a journey of 3264 miles. How many miles must he travel each day? 42. If a man vvhose property is valued at 21 148 dol- lars be worth 17 times as much as his neighbor, how much is his neighbor worth? RETROSPECTIVE OBSERVATION'S. In the course of the last four chapters, you have practised four kinds of operations on numbers: viz. Addition, Subtraction, Multiplication, and Division. These operations should be perfectly understood — the effect of each should be distinctly perceived; for, it is on their proper application, that the solution of all ques- tions in arithmetic depends. Addition is the operation by which two or more numbers are united in one sum. Subtraction is the operation by which the difference between two numbers is found. Multiplication is the operation by which a number is produced, equal to as many times one given number, £s there are units in another given number. Division is the operation by which we find how many times one number contains another, — and, by which we divide one given number into as many equal parts, as there are units in another given number. 120 WRITTEN ARITHMETIC. Questions to be answered Orally* (1) How many kinds of operations are practised on numbers? (2) What are they called? (3) What is Addition? (4) What is Subtraction? (5) What is Multiplication? (6) What is Division? (7) Pro- pose a question that you would solve by addition. (8) Propose a question that you would solve by sub- traction. (9) Propose a question that you would solve by multiplication. (10) Propose a questioti that you would solve by division. (11) How can a question in multiplication be solved by addition? (12) How can a question in division be solved by subtraction? Section 5. MISCELLANEOUS EXAMPLES. 1. The population of the world has been estimated to be as follows. North America, twenty-six millions; South America, twelve millions; Europe, two hundred and twenty millions; Asia, five hundred millions; Africa, thirty-eight millions; Australia, four millions. What is the whole number? 2. In 1830, the national debt of the United States was 48 565406 dollars; in 1831 it was 39123191 dol- lars. How much was paid in one year? 3. The national debt of England cannot be less than 1900 000 000 dollars. How many years would it take to pay this debt, allowing ten millions of dollars to be paid annually? 4. What would be the expense of laying a rail-way from Louisiana to Maine; the distance being 1800 miles, and the rail-way costing 14 000 dollars a mile? 5. In how many days could a passage be effected from Maine to Louisiana, on the proposed rail-way; allowing a car to run 25 miles an hour, day and night? 6. How many days would it take a man to ride on horseback from Maine to Louisiana, riding 5 miles an hour, and 10 hours a day? 5 MISCELLANEOUS EXAMPLES. 121 7. Light passes from the sun to the earth — a distance of 95 milhons of miles — in about 8 minutes. What distance does light move in a minute? 8. The diameter of the earth is 7912 miles; and the diameter of the sun is 1 12 times as great. What is the diameter of the sun? 9. The income of the Bishop of Durham, in England, IS 106 560 dollars per annum. How many clergymen would this support, on a salary of 800 dollars per annum? 10. Five men and three boys found a sum of money, and divided it so that each man had 43 dollars and each boy 26 dollars. What sum did they find? 11. If a trader buy 558 barrels of flour at 5 dollars a barrel, and pay 14 dollars for storage, for how much must he sell the flour, to gain 160 dollars? 12. Suppose 5 bushels of wheat to make a barrel of flour, how many barrels of flour can be made from 12 bins of wheat, each bin containing 95 bushels? 13. In 12 times 95, how many times 5? 14. If a farmer sell 45 acres of land at 38 dollars an acre, and divide the money equally among 4 sons and 1 daughter, what is each one's share? 15. A man, who owned 520 acres of land, purchased 376 acres more, and then divided the whole into 8 equal farms. How many acres did each farm contain? 16. In 520 plus 376, how many limes 8? 17. If a man's income be 1349 dollars a year, and his expenses 3 dollars a day, how much will he lay up in a year; there being 365 days in a year? 18. A merchant gave 39 240 dollars for a cargo of sugar, and after selling it, found he had gained 1671 dollars. For how much did he sell it? 19. A merchant gave 18 dollars a hogshead for 245 hogsheads of molasses, and then sold the whole for 4000 dollars: did he gain or lose; — and how much? 20. A lot of land was divided into 8 farms, and each farm contained 150 acres. How many acres were there in the whole lot? 21. If a man's expenses are 2 dpi. a day, and his income 17 dol. a week; what will he save in 7 weeks? L 122 WRITTEN ARITHMETIC. V. 22. Three men bought a ship : the first man paid 2274 dollars; the second paid 3 limes as much as the first, and the third paid as much as the first and second both. What was the price of the ship? 23. A hogshead holds 63 gallons. How many gallons of wine are there in 20 hogsheads; allowing that each hogshead wants 5 gallons of being full? 24. If a man earn 36 dollars a month, how many months will it take him to earn 576 dollars? 25. If a man earn 40 dollars a month, and spend 13 dollars a month, how many months will it take him to lay up 297 dollars? 2G. A farmer having 20 barrels of pork, sold 9 barrels at 22 dollars a barrel, and the remainder at 19 dollars a barrel. What did he get for the whole? 27. If a trader, who has 152 barrels of flour, should lay out 1370 dollars in buying more flour, at 5 dollars a barrel, how many barrels would he have? 28. A trader hired 650 dollars, and in 6 months paid all but 92 dollars. How much did he pay? 29. What is the value of 139 yards of broad-cloth, at 7 dollars per yard? By the method of reasoning heretofore practised, we should say in this solution, 139 yards are worth 139 times 7 dollars; and thus w^e should make 7 the multi- plicand, and 139 the multiplier. But since it is more convenient to make the smaller number the multiplier, we reason thus, — If the value of 1 yard were 1 dollar, the value of 139 yards would be 139 dollars; since the value of 1 yard is 7 dollars, the value of 139 yards is 7 times 139 dollars: and accordingly we make 139 the multipli- cand, and 7 the multiplier. 30. A trader bought 240 sheep, at 4 dollars a head, and paid for them in cows, at 20 dollars a head. How many cows did he give? 31. If I pay 6 dollars an acre for the ploughing of 18 acres of land, and 100 dollars for having the whole planted and hoed, what does the cultivation cost? 32. How many cows at 19 dollars a head, will pay for 38 sheep at 4 dollars a head? 5. MISCELLANEOUS EXAMPLES. 123 33. A farmer bought a fields valued at 150 dols., for which he gave 9 cows, valued at 14 dols. apiece, and the rest in money. How much money did he pay? 34. What number must be added to 9 times 14, m order that the sum shall be 150? 35. If a stage travel 13 miles in the same time that a wagon travels 5 miles, how many miles will a stage travel while the wagon is travelling 65 miles? 36. Suppose that 9 bushels of wheat will fill a hogs- head; how many hogsheads can be filled from a heap containing 149 bushels; and how many bushels will be left in the heap ? 37. Charles and Joseph are studying arithmetic. Charles is 322 examples in advance of Joseph, but Joseph performs 55 examples in a day, and Charles, 41. In how many days will J. overtake C? 3S. Two men started together and travelled on the same road, at the rate of 7 miles an hour: but one of them rested 1 hour in every 3 hours, and the other rested 1 hour in every 4 hours. How far apart were they, at the end of 12 hours? 39. A drover, having 599 dollars, wishes to buy all the oxen he can pay for, at 34 dollars a head, and then lay out the remainder of his money for sheep, at 3 dollars a headw How many of each must he buy? 40. A, B, and C made up a purse of 500 dollars. A put in 16 dollars, and B put in 3 times as much. How much did C put in? 41. A merchant bought 64 tons of hemp at 215 dol- lars a ton. How many ten-dollar bank notes did it take to pay for the hemp? 42. A merchant paid 9600 dollars for 43 tons of hemp At how much must he sell the hemp per ton, in order to gain 247 dollars ? 43. What number must be subtracted from 7342, in order that the remainder shall be 456? 44. What number must be multiplied by 30, in order that the product shall be 2130? 45. What number must be divided by 15, in order that the quotient shall be 640? 124 WRITTEN ARITHMETIC. V. Section 6. FEDERAL MONEY. Federal money is the national currency of the United States. Its several denominations are, — the MILL, the CENT, the DOLLAR, and the EAGLE. 10 mills are equal in value to 1 cent. 10 cents are equal to 1 dime. 10 dimes, or 100 cents, are equal to 1 dollar. 10 dollars are equal to 1 eagle. In commerce, we express eagles in dollars, and dimes in cents. For example, instead of saying, 2 eagles and 5 dollars, we say, 25 dollars: and instead of saying, 3 dinpes and 4 cents, we say, 34 cents. 1. How many cents are there in 86 dollars? (See method of multiplying by 100, in page 106.) 2. How many cents in 7 dollars and 58 cents .-^ 3. How many dollars are there in 3700 cents .^ (See method of dividing by 100, in page 116.) 4. Howmany do4s. and how many cts. over, in 534 cts. ? This character, ^, placed before a number, shows the number to express dollars. For example, $12, is 12 dollars. When dollars and cents are expressed in one sum, they are separated by a point, thus, $4.16; to be read, 4 dollars and 16 cents. Observe, there must be two places of figures for cents: therefore, if the cents be less than 10, a cipher must be placed on the left hand of the figure which expresses them. For example, 56 dollars and 9 cents is written thus, $56.09. 5. What is the whole sum of $34.25, $ 18.04, $ 142, $176.81, and 58 cents? 34.25 ^^ writing these numbers for addition, 1 8.04 ^^^ place dollars under dollars, and cents 142 under cents. We then add up each col- 1 7 6.8 1 umn, just as we add the columns of sim- ^5 3 pie numbers. Finally, we point off two ' _ figures on the right of the sum for cents, ^ 71. oa ^^^ ^Yie other figures are dollars. 6 FEDERAL MONEY. 196 6. What IS the sum of $57.20, $6.02, and $81.16.? 7. Add together $538, $1.52, $5.07, and 63 cents. 8. Add together 18 cents, $70.19, $56, and 7 cents. 9. Add together 36 dollars, 7 dollars and 45 cents, 86 cents, 130 dollars and 6 cents, and 340 dollars. JO. Add together 9 dollars, 1 dollar and 70 cents, 13 dollars and 7 cents, 50 cents, and 10 cents. 11. Add together 47 cents, 62 dollars, 9 dollars and 12 cents, 5 dollars and 5 cents, and 3 dollars. 12. Add together 37 dollars, 4 dollars and 17 cents, 96 dollars and 1 cent, 99 cents, and 2 dollars. 13. What is the expense of one quarter's schooling, allowing $ 19 for board, $9 for tuition, $3.75 for books, and 92 cents for stationary ? 14. A sailor paid $16.35 for a hogshead of molasses, in New Orleans, and also paid $3.40 for the freight of the molasses to Boston. For how much must he sell it in Boston, in order to gain $4 .'^ 15. Subtract $4.35 from $6.48; taking cents from cents, and dollars from dollars. 16. Subtract $7.18 from $48.50. 17. Subtract $251.12 from $546.18. 18. Subtract $47.56 from $319. 319.00 ^^ writing these sums of money for 47.56 subtraction, we supply the places of ^' 1 A A cents in the greater sum, by ciphers, ^ ! and then proceed to subtract. When either of the sums of Federal money presented for subtraction has no cents expressed^ the places of cents may be supplied by two ciphers. 19. Subtract $654 from $783.48. 20. Subtract $31.12 from $5390. 21. Subtract 42 cents from $51. 22. Subtract 7 cents from $1. 23. Subtract 5 cents from $754. 24. Subtract 4 cents from $ 1. 25. What is the difference between $3.06, And $9 f 26. What is the difference between $6, and 7 cents? 126 WRITTEN ARITHMETIC. V. 27. A lady having $3, paid $1.15 for a yard of cam- bric. How much money had she left ? 28. A farmer sold a barrel of pork for $21.50, taking in payment a hogshead of salt at $5, and the rest in money. How much money did he receive ? 29. A trader began business with $648, and at the end of 2 years, had $911.06. What did he gain? 30. A traveller having no money, sold his horse for $92.75, and his gig for $78, and then paid $ 17 for pas- sage home. How^ much money did he bring home ? 31. A jockey gave $120 for a horse, and then ex changed for another horse, receiving $15.30 for differ ence of value, and then exchanged again, paying $28.50 How much did the last horse cost him ? 32. How much is 18 times $4.72? $4.72 is the same as 472 cents: there • 4.72 fQYe we multiply it as 472 cents, and the ^^ product is 8496 cents. Now to change 37 7 6 these cents to dollars, we must divide them 47 2 by 100: this we do, by pointing off two ^^S4~96 figui'Gs for a remainder. The quotient is dollars, and the remainder is cents. 33. How much is 4 times $ 1.08 ? 34. How much is 7 times $52.31? 35. How many dollars are 8 times 75 cents ? 36. How many dollars are 32 times 25 cents ? 37. How much is 19 times 43 cents ? 38. .How much is 241 times $654.12? 39. What is the value of 6 pounds of Hyson tea, at $ 1.20 cents per pound ? 40. What is the value of 10 yards of flannel, at 64 cents per yard ? 41. What is the value of 6 hats, at $6.47 apiece? 42. What will a laborer receive for 25 days' work, at $1.15 per day? 43. How much must be paid for 30 pounds of coffee, when the price is 16 cents a pound? 44. How much must be paid for 12 drums of figs, when the price is $ 1.55 a drum ? 6 FEDERAL MONEY. 1S7 45. What is the value of 147 bushels of apples, at 8 cents per bushel ? It is more convenient in this example, to make the number of bushels the multiplicand and the number of cents the multiplier. For method of reasoning, see re marks in section 5, under example 29. ' 46. If a man spend 2S cents a day, how much will he spend in 365 days, or 1 year ? 47. What is the cost of 430 pounds of chocolate, at 20 cents per pound ? 48. At 6 cents a pound, what is the value of a quarter of beef, weighing 214 pounds ? 49. At $2.30, [230 cents] an acre, what is the value of 4748 acres of wild land ? When the price of a single article is given in Federal money^ and the value of any number of that article is required^ either the price may be multiplied by the nttm- ber of articles J or the number of articles by the price; the product loill be the answer, 50. At $ 1.72 per pound, what is the value of 5 chests of tea, each chest containing 64 pounds.'^ 51. A trader gave $5.16 a barrel for 2170 barrels of flour, and sold it so as to gain $100.50 on the whole. For how much did he sell it ? 52. A mar> bought 30 yards of cloth at $1.32 per yard, and 30 yards at 86 cents per yard. How much more did the first piece cost, than the last ? 53. If I pay 22 cents a gallon for 72 hogsheads of rnolasses, each hogshead containing 63 gallons, and then sell the whole for $936, how much do I lose } 54. A man having $ 350, took a journey of 700 miles, paying 6 cents a mile for stage passage, and $ 14 for board. How much money did he bring home ? 55. If a man earn $ 1.02 a day, and spend 36 cents a day, how much will he lay up in 75 days .'' 56. If a man get §8.35 for every 6 days' work, how much will he get by working 510 days ? 57. Suppose 42 casks to contain 46 gallons of wine .each; what is the value of the whole, at $ 1. 11 per gal. ? 128 WRITTEN ARITHMETIC. V. 58. How many times 7 cents are there in $430.78? 7)4 30.7 8 ^^ divide 5^430.78 as if the figures stood to express the whole in cents. 1 o 4 ryY^^ quotient is the number of times. 59. How many times 6 cents are there in j^20.22? 60. How many times 15 cents are there in J^ 11.10 ? 61. How many times 90 cents are there in ^27.00 ? 62. How many times $4.06, [406 cents,] are there in $190 146.04, [19014604 cents] ? 63. How many lead pencils can you buy for $3.44, when they are sold at 8 cents apiece ? 64. How many pounds of butter, at 21 cents per pound, can be bought for $3.57 ? 65. A laborer earned $53.75, by working at $1.25 a day. How many days did he work ? 66. If 84 cents should be divided equally among 6 boys, what would each boy receive ? 67. If $28.71 [2871 cents] be divided equally among 9 men, what will each man receive ? 68. If $205 58 be divided equally among 38 men, what will each man receive ? 69. If $637 be divided equally among 24 men, what will each man receive ^ After dividing the number of dollars by the number of men, it appears from the quo- tient and remainder, that each man can have $26, and still $13 will remain undivided. We change $13 to cents, by annexing two ciphers, and then divide the cents by the number of men. From this quotient and remainder it ap- pears, that each man will have 54 cents, and 4 cents will re- main undivided. 70. If $7640 be divided equally among 61 men, what will each man receive ? 24)637(26 48 157 144 24)1300(54 120 100 96 4 Answer y $26.54. Remainder^ 4 cents. 6. FEDERAL MONEY 129 71. 8 men received $230 for performing a piece of work. What was each one's share of the money ? 72. An insurance office, whose stock was owned in 1000 shares, divided among the stock-holders, §1586. How much was paid on one share ? 73. The expense of a village school, for 6 months, was 5^466.80; and it was paid in equal shares by 40 gentlemen. What was each one's share ? 74. Add together $9.87, 50 cents, $705.30 and $390: subtract from this sum, 606 dollars and 7 cents: multiply the remainder by 45: divide the product by 37. What is the quotient, and the remainder ? 75. 'A shoe-maker paid $1.58 apiece for 10 calf-skins, and 22 cents a pound for 3 sides of sole leather, each side weighing 35 pounds. From this stock he made 48 pairs of shoes, which he sold at $1.75 a pair. What did he get for his work ? 76. Suppose a man, whose income is $400 a year, should spend $3.90 a week, how much would he save in 2 years; there boing 52 weeks in 1 year ? 77. Suppose wheat to be worth $1.05 per bushel, and rye 70 cents per bushel: how many bushels of rye must be given for 550 bushels of wheat ? Q^uestions to be answered Orally. (1) What is Federal money? (2) State the denominations of Federal money. (3) State the number of mills in a cent, the number of cents in a dime, &c. (4) How m_any cents make a dollar.'^ (5) By what short muthod do you find the number of cents in any number of dollars ? (6) How do you distinguish the number of dollars, that there are in any number of cents? (7) in writing dollais and cents together, how many figures express the cents ? (8) When the cents to be written with dollars are less than 10, what is to be done ? (9) Suppose you are dividing dollars, and a remainder occurs, what is to be done, in order to divide the remainder ? 130 WRITTEN ARITHMETIC. V Section 7. TABLES OF COMPOUND NUMBERS. ENGLISH MONEY is the national currency of England 4 farthings (qr.) .... make 1 penny. d 12 pence make 1 shilHng. s. 20 shillings make 1 pound. £. TROY WEIGHT is used in weighing gold and silver. 24 grains (gr.) make 1 pennyweight. dwt 20 pennyweights .... make 1 ounce. oz 12 ounces make 1 pound. lb. AVOIRDUPOIS WEIGHT is the common weight, used in weighing groceries, and all coarse commodities. 16 drams (dr.) make 1 ounce. oz* 16 ounces make 1 pound. lb. 28 pounds make 1 quarter. qr. 4 quarters make 1 hundred-weight. cwt« 20 hundred-weight . . . make 1 ton. T. APOTHECARIES' WEIGHT is used for the purpose of compounding medicines, but not in selHng them. 20 grains (gr.) make 1 scruple. 9 3 scruples make 1 dram. 5 8 drams make 1 ounce. § 12 ounces make 1 pound. ib CLOTH MEASURE is used in measuring cloth, ]ace, &c. 4 nails (na.) make 1 quarter. qr. 4 quarters make 1 yard. yd- 5 quarters make 1 Enghsh ell. E. e 5 quarters make 1 French ell. Fr. e. 3 quarters make 1 Flemish ell. Fl. e. DRY MEASURE is used in measuring grain, salt, &c. 2 pints (pt.) make 1 quart. qt- 8 quarts make 1 peck. pk 4 pecks make 1 bushel. bu 7 COMPOUND NUMBERS. 131 WINE MEASURE is used by grocers and others, for measuring wine, oil, molasses, and most other liquids. 4 gills (gi.) make 1 pint. pt 2 pints make 1 quart. qt. 4 quarts make 1 gallon. gal. 31^- gallons make 1 barrel. bi. 42 gallons make 1 tierce. tier. 63 gallons make 1 hogshead. hhd, 84 gallons make 1 puncheon. pun. 126 gallons make 1 pipe or but. p. 2 pipes, or 4 hlids. . . make 1 ton. T. BEER MEASURE is used in measuring malt liquors 2 pints (pt.) make 1 quart. qt. 4 quarts make 1 gallon. gal. 9 gallons make 1 firkin. fir. 2 firkins make 1 kilderkin. kil 2 kilderkins make 1 barrel. bl. LONG MEASURE is applied to length, distance, &c. 3 barley-corns make 1 inch. in. 12 inches make 1 foot. ft. 3 feet make 1 yard. yd. 5^ yards or 16^ feet . . make 1 rod or pole. r. 40 rods make 1 furlong. fur. 8 furlongs make 1 mile. m. 3 miles make 1 league. 1 9f furlongs make 1 geographical mile. 60 geographical miles . make 1 degree. deg. 360 degrees the earth's circumference. SQUARE MEASURE is used in measuring land, floor- ing, boards, tihng, and all other surfaces whatever. 144 inches make 1 foot. ft. 9 feet make 1 yard. yd. 30^- yards, or 272^- ft. . . make 1 rod or pole. r. 40 rods make 1 rood. R. 4 roods make 1 acre. A. 640 acres ^ake 1 mile ml. 132 WRITTEN ARITHMETIC. CUBIC MEASURE is used in measuring solid bodies, and in finding the capacity of rooms, boxes, &c. 1728 inches make 1 foot. ft. 40 feet of round timber . make J ton. T. 50 feet of hewn timber . make 1 ton. T. 16 cubic feet make 1 foot of wood. ft.w. 8 feet of wood make 1 cord of wood. C TIME is naturally divided into days^ by the revolution of the earth upon its axis; and into years j by the revo- lution of the earth round the sun. 60 seconds make 1 minute. m. 60 minutes make 1 hour. h. 24 hours make 1 day. d 365 days make 1 year. Y. The earth revolves round the sun once in 365 days, 5 hours', 48 minutes, and 48 seconds: this period is therefore a Solar year. In order to keep pace with the solar year in our reckoning, we make every fourth year to contain 366 days, and call it Leap year. The year is divided into 12 months. The number of days in each month is commonly learned thus, — '30 days hath September, April, June, and November ; February hath 28 alone, And all the rest have 31. Leap year comes 1 year in 4 ; Then February hathl day more.' 'The 4th. 11th. 9th. and 6th., Have 30 days to each affixed ; And every other, 31, Except the 2nd. month alone, Which has but 28, in fine, Till leap year gives it 29.' Questions to he answered Orally. (1) What is English money .^ (2) Recite the ta- ble, (3) How many shillings are there in 2 pounds ^ (4) How many pence in 2 shillings ? (5) How ma- ny pence in 8 farthings } (6) How many pence in 36 farthings } (7) How many farthings in 6d. 3qr. ^ (8) Wha^ is the use of Troy Weight.^ (9) Re- cite the table. (10) How many ounces in 2 pounds .^ j (11) How many penny-weights in 4 ounces ^ f COMPOUND NUMBERS. 133 (12) What is the use of Avoirdupois Weight? (13) Recite the table. (14) How many ounces in 4 pounds ? (15) How many hundred-weight in 3 tons } (16) How many hundred-weight in 24 quarters.^ (17) How is Apothecaries' Weight used ? (18) Recite the table. ^19) How many scruples m 4 drams ? (20) How many drams in 27 scruples ? (21) What is the use of Dry Measure ? (22) Re- cite the table. (23) How many pecks in 12 bushels and 2 pecks ? (24) How many bushels, and how many pecks over in 35 pecks ? {2o) What is the use of Cloth Measure .? (26) Recite the table. (27) In 5 yards how many quar- ters ? (2S) In 32 nails how many yards ? (29) How many English ells in 14 quarters ? (30) To what is Wine Measure ap'plled ? (31) Recite the table. (32) In 2 gallons of vinegar how many quarts ? — how many pints ? — how many gills ? (33) In 32 gills how many gallons ? (34) What is measured by Beer Measure? (35) Recite the table. (36) How many gallons are there in 6 firkins ? (37) How many kilderkins in 19 fir- kins ? (33) How many frrkins in 1 barrel ? (39) What is Long Measure applied to ? (40) Recite the table. (41) What number of inches are there in 1 yard ? (42) How many feet in 38 inches ? (43) In 6 furlongs how many miles ? (44) What is the use of Square Measure ? (45) Recite the table. (46) How many square feet in 4 square yards ? (47) How many rods in 2 roods ? (48) How many acres in 20 roods ? (49) What is the use of Cubic Measure ? (50) Recite the table. (51) How many cubic feet are there in 3 feet of wood ? (52) In 48 feet of wood, how many cords of wood ? (53) How is time naturally divided? (54) Re- cite the table. (55) Wfiat is a Solar year ? (56) How many 'months in a year ? (57) Recite the lines that tell the number of days in each month. 134 WRITTEN ARITHMETIC Section 8. V. REDUCTION OF COMPOUND NUMBERS. Reduction is the operation of changing any quantity from its number in one denomination, to its number in another denomination. For ^"nstance, if we change an admeasurement from 2 feet to 24 inches, that is, if we find how many inches there are in 2 feet, the opera- tion is called reduction. Again, if we change 24 inches to 2 feet, this operation is also called reduction. ENGLISH MONEY. 2. How many pounds in 128S2 farthings ? many farthings 1. How in Jei3 8s. 4d. 2qr. £ s. d 13 8 4 20 qr. 2 268 shillings. 12 3220 4 pence. Ans. 12882 farthings. Jins. In this example, we con- sider, that there are 20 times as many shillings as pounds in any sum; there- fore we multiply the 13 pounds by 20, and add the S shillings to the product. Then, since there are 12 times as many pence as there are shillings, we mul- tiply the shillings by 12 and add the 4 pence to the pro- duct. Lastly, since there ,the are 4 times as many farth- mgs as pence, we multiply the pence by 4, and add 2 farthings to the product. qr. 4) 12882 1 2 )3220 2qr. 2|0) 26i8 4d. 8s. £13 £ 13 d. qr. 4 2 This example is the re- verse of the first example. We here consider, that ev- ery 4 farthingS: make 1 pen- ny; therefore, we find by division how many times 4 there are in the number of farthings: the quotient is pence, and the remainder is farthings. Then, since ev- ery 12 pence is 1 shilling, we divide the pence by 12; quotient is shillings, and the remainder pence. Lastly, since every 20 shil- lings make 1 pound, we di- vide the shillings by 20 8 COMPOUND NUMBERS. 135 RULE FOR REDUCTION. When a greater denomina- tion is to be reduced to a smaller^ multiply the greater denomination^ by that number which is required of the smaller^ to make ONE of the greater; adding to the product so many of the smaller denomination as are ex- pressed in the given sum. Perform a like operation on this product^ and on each succeeding product. When a smaller denomination is to be reduced to a greater^ divide the smaller denomination by that number which is required of the smaller^ to make ONE of the next greater: the quotient will be of the greater denomination^ and the remainder will be of the same denomination with the dividend. Perform a like operation on this quotient, and on each succeeding quotient, 3. How many farthings are there in 18s. 7d. 3qr. ? 4. How many pounds are there in 9207 farthings ? 5. How many pence are there in £5 Os. lid. ? 6. How many shilhngs are there in 647 farthings .'' 7. How many times 8 pence are there in £3 5s. } TROY WEIGHT. 8. 1 51b How many grains in lloz. 18dwt? lb. 15 12 191 20 3838 24 15352 7676 oz. 11 dwt. 18 •5ns. 92112 grains 9. How many pounds in 92112 grains? 2|0) 24)92112(38318 72 201 192 91 72 192 192 191 18 12)191(15 12 71 60 11 Ans, 151b. lloz. 18dwt 10. How many penny-weights in 91b. 13oz. 16dwt. / 1 1 . How many pounds of silver in 829 penny-weights ^ 12. How many grains in lOoz. 19dwt. 12gr. ? 13. How many pounds in 23641 grains ? 136 WRITTEN ARITHMETIC. V. AVOIRDUPOIS WEIGHT. 14. How many pounds are there in 1 ton ? 15. How many drams are there in 7 tons, 3 quarters, 27 pounds, 5 ounces, and 18 drams ? 16. How many tons are there in 31122 pounds ? 17. What will 3 hundred-weight, 3 quarters, and 17 pounds of indigo cost, at ^2.67 per pound ? 18. A wealthy farmer wishes to put down 3T. IScwt. 2qr. 81b. of butter, in firkins, containing 50 pounds apiece. How many firkins will it require ? apothecaries' weight. 19. How many scruples are there in 1 pound .'' 20. How many pounds. are there in 1395 drams ? 21. In 3ib 9§ 05 19 lOgr. of epecacuanha, how many doses are there; each dose containing 30gr. .'* 22. If it take 1 ounce of salts for a dose, what will 75 pounds amount to, at 4 cents a dose ? 23. If it take 10 grains of calomel and 1 scruple of jalap for a dose, how many doses are there in iBb 15 45 of such a mixture ? cloth measure. 24. How many nails are there in 1 English ell ? 25. How many yards are there in 16240 nails ? 26. In 320 yards, and 3 quarters, how many quarters ? How many Flemish ells ? 27. How many more nails are there in 75 English ells, than there are in 93 yards ? 28. A shop-keeper sold cloth enough in one day to gain £6 Is. 8d., at a profit of 2 farthings on every yard. How much did he sell ? , DRY MEASURE. 29. How many pints are there in 1 bushel ? 30. How many pints are there in 58 bushels, 3 pecks, 7 quarts, and 1 pint? 31 . How many bushels are there in 8240 quarts ? 32. If 3 bushels and 2 pecks of corn will fill a barrel, what quantity of corn will 20 barrels hold ? 33. Suppose it takes 3 pecks of salt to preserve a barrel of pork, how much salt would be necessary to preserve 351 barrels of pork? 8. COMPOUND NUMBERS. 137 WINE MEASURE. 34. How many gills are there in 1 hogshead ? 35. How many hogsheads are there in 90S4 pints ^ 36. If 3 tierces of molasses be sold at 12 cents a quart, what will the whole amount to ? 37. What would 2 pipes of Madeira wine amount to, at 67 cents per quart ? 3S. A certain toper drank 1 gill of rum every forenoon, and 1 in the afternoon, for 6 years; in consequence of which, he died. How many hogsheads did he drink ? BEER MEASURE. 39. In 2 barrels and 1 firkin, how many pints } 40. In 6538 quarts, how many kilderkins ? 41. How many bottles, holding 6 gills apiece, will be required, to bottle 6 barrels of porter ? 42. A man retailed 4 barrels of ale, and received for it $69.12. At what price did he sell it a pint .'^ 43. Suppose a retailer to sell 3 quarts of porter every day for 1 year, excepting 52 Sabbaths, how many bar- rels would he sell in the year } LONG MEASURE. 44. In 35 yards, 2 feet, 10 inches, how many inches .'* 45. In 29578 barley-corns, how many yards ? 46. In 16 leagues and 2 miles, how many rods ? 47. Ho\V many geographical miles would a ship sail, in going round the globe ? 48. In 2541 inches of wire, how many yards? 49. Suppose 7 inches of wire to make 1 link of a chain, and 4 links to measure 1 foot; how many yards of wire would make a chain 8 feet long } SQUARE MEASURE. To find the number of square inches^ feet^ or rods^ in any surface wJiich has four sides^ and four equal angles^ [corners^'] multiply the length and breadth together, 50. How many square inches are there in a slate, that is 13 inches long, and 8 inches wide ? 51. How many square rods are there in a field 28 rods long, and 16 rods wide ? How many acres ^ 52. How many square yards of carpeting will cover a floor 36 feet long, and 18 feet wide } 138 WRITTEN ARITHMETIC. V CUBIC MEASURE. A cube may be illustrated by a solid block, having 6 equal sides. Let us suppose we have before us a num- ber of small blocks, representing cubic inches. If we lay 144 of these blocks together upon the table, they will cover a square foot. Then, if we cover this layei of blocks with another layer, and thus continue till wf« have piled up 12 layers, the pile will contain 12 times 144 cubic inches, or 1 cubic foot. Therefore, to find the cubical contents of any things multiply its lengthy and breadth J and depth together, 53. How many cubic inches are there in a brlck^ that is 8 inches long, 4 inches wide, and 2 inches thick ? 54. How many cubic feet in a box, that is 25 inches long, 20 inches broad, and 11 inches deep .^ 55. How many cubic inches in 1 ton of hewn timber ? 66, How many cubic feet in a pile of wood 15 feet long, 4 feet wide, and 5 feet high .'' How many feet of wood .'* How many cords ? 57. How many cubic feet in a cord of wood ? TIME. 58. How many seconds are there in a common year .^ How many in a leap year ? How many in a solar year ? 59. How many minutes are there in 57 days ? 60. If your pulse beat 73 times in a minute, how many times will they beat in the month of January ? 61. How many years and days, from the 1st day of January, 1830, to the 1st day of October, 1834 ? Questions to be answered Orally. (1) What is meant by reduction ? (2) How do you reduce shilHngs to pence ? (3) How do you reduce pence to shillings r (4) How do you reduce Avoirdupois ounces to pounds ? — Why? (5) How do you reduce pounds to ounces ? — Why ? (6) How do you reduce yards to nails ? (7) How do you reduce nails to yards ? (8) Recite the general rule for reduction. 1 w hat is the wl I2s. Od. Iqr., £60 £7 10s. , lOd., Is. 8( £. s. d. qr 13 7 10 2 4 12 1 60 11 3 19 2 116 7 10 10 1 8 3 76 278 12 5 3 COMPOUND NUMBERS. 139 Section 9. COMPOUND ADDITION. ENGLISH MONEY. lole sum of £13 7s. lOd. 2qr., £4 Os. lid. 3qr., 19s. Od. 2qr., £116, i. 3qr., and £76? . The sum cf the column of farthings is 11 ; equal to 2d. 3qr. We write the 3qr. and add the 2d. to the column of pence. The sum of the pence is 41; equal to 3s. 5d. We write the 5d . and add the 3s. to the column of shillings. The sum of the shillings is 52; equal to £2 12s. We write the 12s. and add the £2 to the column of pounds. RULE FOR COMPOUND ADDITION. Write the num- bers so that each denomination shall stand in a separate column. Md the numbers of the lowest denomination together^ and divide their sum by that number which is required of this denoviination to make 1 of the next high- er: write the remainder under the column added^ and .carry th^ quotient to the next column. Thus proceed with every denomination. 2. What is the sum of £4 18s. 9d., £100 7s. Od. Iqr., 16s. 4d., 3s. 6d. 2qr., £20, and £9 7s. 4d. .? 3. What is the sum of Us. Od. 3qr., £33 2s. 6d., 8s. 7d. Iqr., £450, £9 17s. 8d. 3qr., and £37 9s..?' 4. A man in London paid for a hat, £1 18s. 6d.; for a coat, £9 8s. 4d., for a vest, £1 10s.; for pantaloons, £3; for boots, £1 2s. What did the suit cost.^ TROY WEIGHT. 5. Add together these quantities of silver. 4lb. 9oz. 16dwt., lOoz. Idwt. 22gr., and 31b. 4oz. Odwt. 6gr. 6. Add together lloz. 15dwt. 18gr., 21b. lOoz. ISdwt. 23gr., 91b. Ooz. 17dwt. 3gr and 5oz- 12dwt 140 WRITTEN ARITHMETIC. V. AVOIRDUPOIS WEIGHT. 7. Add together 14T. lOcwt. 2qr, 231b. 4oz., 27T. 4cwt. 2qr. 241b. 14oz., and 3qr. 01b. 15oz. lldr. C. Add together 16cwt. Iqr. 111b. 6oz. IGcvvt. 2qr. 201b., 5T. Ocvvt. 3qr. 5lb.' 13oz. 2dr., and 2T. apothecaries' weight. 9- What is the weight of a mixture containing 5Bb log 53 19 8gr., 63 29, 53 19 18gr., and 2ft> 4§ .^ 10. What is the weight of a mixture containing 1 ii 3§ 13 29, 75 53 19 15gr., and 4 Jb 0§ 63 ? cloth measure. 11. Add together 19yd. 2qr. 3na., 14yd. 2qr. Ina., 32yd. Oqr. Ina., 2qr. 2na., and 57yd. 3qr. 2na.' 12. Add together 15E.e. 4qr. 2na., 6E.e. 3qr. Ina., 45E.e. 3qr. 3na., 230E.e., and 4E.e. 4qr. d'ry measure. 13. Add together 25bu. 2pk. 5qt., 240bu. Opk. 6qt , 316bu. 3pk. 7qt. Ipt., and 650bu. 2pk. 5qt. 14. Add together 635bu. Opk. 3qt., 247bu. 3pk. Oqt. Ipt., 2bu. 3pk. 6qt., 56bu., and 31bu. Opk. 2qt. wine measure. 15. How many hogsheads are 12hhd. 42gal. 3qt. Ipt., 548hhd. 62gal. 3qt., and Shhd. 9gal. Iqt. ? 16. How many tons are IT. Ip. 116gal. 3qt., Ip 48gal., 5T. Ip. 86gal. 3qt., 102gal., and 4T. ? BEER MEASURE. 17. Add together 5bl. Ikil. Ifir. 8gal. 3qt., Ifir. 5gal. 2qt. Ipt., 16bl. Okil. Ofir. 4gal., and 25bl. Ikil. 18. Add together Ifir. 7gal. 3qt., 24bl. Okil. Ifir. 6gal. 2qt Ipt., and 20bl. Ikil. Ifir. 4gal. LONG MEASURE. 19. How many yards are 45yd. 2ft. llin., 13yd. Oft. 9in., 1ft. lOin., and 20yd. ift. Sin. 2b.c. ? 20. How many miles are 10m. Ofur. 36rd., 58m. 7fur. 13rd., 38rd., 16m. 4fur. 21rd., and 6fur. ? SQUARE MEASURE. 21. How many yards are 36yd. 7ft. 126in., 3yd. 6ft., 130in., 71yd. 5ft. 140in., and 10yd. 4ft. 21in. ? 22. How many acres are 34A. 3R. 32rd., 86A. OR QUd,, 381A. 2R., and 46A. IR. 25rd. ^ j| 10^ COMPOUND NUMBERS. 141 CUBIC MEASURE. 23, How much hewn timber is 7T. 45ft. 1712 in., 8T. 39ft. 1698in., and lOT. 29ft. 800in..? 24. How many cords of wood are 9C. 7ft. w. 15c. ft, 4C. 6ft. w. 12c.ft., and 14C. 7ft. w. llc.ft. ? fe TIME. * 25. Add together 2Y. 250d. ISh. 51m. 15s., lY. iSd. 7h. Om. 55s., and 240d. Oh. 37m. 29s. 26. Add together 4Y. 141d. lOh. Om. 5s., 12Y. I94d. 20h. 49m., and 2Y. 280d. Oh. 55m. 38s. Section 10. COMPOUND SUBTRACTION. ENGLISH MONEY. 1. An Enghsh merchant gave £9176 16s. 8d. Iqr. for a ship's cargo, and then sold the same cargo for JE9607 4s. 5d. 3qr. How much did he gain ? To subtract the 8d., we unite 1 of the 4s. with the 5d., mak- ing 17d., and take 8 from 17 Then, having used 1 of the 4s., we unite £1 with 3s., making 23s., and take 16 from 23. RULE FOR COMPOUND SUBTRACTION. Write the several denominations of the smaller quantity under the same denominations of the greater quantity: then^ begin with the lowest denomination^ and perform subtraction on each denomination separately. Whenever a number expressing a denomination in the upper line is smaller than the number under it^ increase the upper number by as many as make 1 of the next higher denomination^ and consider the number of the next higher denomination in the upper line^ to be 1 less than it stands. 2. Subtract £4 lis. 6d. from £61 14s. 5d. 3. If an Enghsh servant receive £ I per month, and spend 13s. 4d. 3qr. pcjr month, what does he lay up.'^ 4. Subtract c£75 18s. 7d. Iqr. from £856 14s. 9d ] £ 9607 9176 s. 4 16 d. 5 8 qr 3 1 430 7 9 2 142 WRITTEN ARITHMETIC. V TROY WEIGHT. 5. Subtract Ub.O oz. 19dwt. from 21b. lloz. 9dwt 6. A silver-smith having 41b. 3oz. of silver, worked up lloz. 14dwt. of it. How much had he left -^ AVOIRDUPOIS WEIGHT. 7. From 8T. 12cwt. Iqr. 17 lb. take 7cwt. 3qr. 21b. . 8. A farmer laid in 68T. of hay, and used 55T. I4cwt. in wintering his stock. How much had he left ? apothecaries' weight. 9. From Ift 8§ 53 take 7§ 73 29 16gr. 10. A mixture weighing 3g 29, contains 89 of jalap, and the rest is rhubarb. How much rhubarb ? CLOTH measure. 11. Subtract 3qr. 2na. from 46yd. Iqr. Ina. 12. If 7yd. 2qr. 2na. be cut from a piece of cloth containing 46yd. Iqr. 3na., how much will be left.'* DRY MEASURE. 13. Subtract 4bu. Ipk. 7qt. Ipt. from 87bu. 14. A farmer raised lOObu. of corn, and sold 46bu. 3pk. of it. How much had he remaining .'' wine MEASURE. 15. From 2hhd. 15gal. take Ihhd. 20gal. 3qt. 16. If from a tierce of molasses 7gal. 2qt. Ipt. leak out, how much will remain in the tierce ? BEER MEASURE. 17. From 4bl. Ikil. Ifir. take Ifir. 7gal. 3qt. 18. A brewer having 26bl. Ikil. of beer, sold 12bl. Okil. Ifir. How much had he remaining ? LONG MEASURE. 19. Subtract 4yd. 2ft. 9in. from 5yd. 1ft. lOin. 20. John rode 16m. 5fur., and Henry rode 20m. Ifur. 8rd. How much further did H. ride, than J ^ SQUARE MEASURE. 21. A farmer owning 94A. of land, sold off a piece, 48 rods long, and 20 rods wide. How many acres had he remaining ? (See Square Measure, page 137.) CUBIC MEASURE. 22. If a piece of timber 9 feet long, 2 feet wide, and 1 foot thick, be taken from 2T. 14ft. of hewn timber, how much will be left.'* (See page 138.) n. COMPOUND NUMBERS. 143 TIME. 23. Subtract 3Y. 45d. 61i. 50m. from 5Y. 14d. 12h 24. A ship went to India and returned, in 321 d. 7h How much less than a year was she in the voyage ^ Section II. COMPOUND MULTiriJCATION. ENGLISH MONEY. 1. What is the value of 8 yards of English broad cloth, at £2 Os. 5d. 3qr. per yard ? £ s. d. qr. ^ times 3qr. are 24qr., equal to 2 5 3 ^^' ^ times 5d. are 40d., and 6 3 we carry are 46d., equal to 3s. lOd. 8 times Os. is Os., but we carry 3s. 8 times £2 are £16. 16 3 10 RULE FOR COMPOUND MULTIPIJCATION. Begin uith the lowest denomination^ and multiply each denomi- nation separately ; divide each product by the number which is required of its own denomination to make 1 of the next higher; write the remainder under the denomi- nation miiltiplied^ and carry the quotient to the product of the next higher denomination. 2. Multiply £529 13s. lOd. 3qr. by 5. 3. What is the value of 7 tons of hemp, at £50 18s lOd. per ton. 4. Multiply £7529 ISs. Od. Iqr. by 6. 5. Multiply £250 16s. lid. by 24. In examples like this, it is most convenient to multi« ply hy factors of the multiplier. 6. Multiply £57 8s. lOd. 2qr. by 45. 7. What cost 34 cows, at £3 9s. 6d. apiece? Here find the price of 32 cows by the factors of 32, and to the product add the price of 2 cows. 8. Multiply £1746 14s. lOd. 2qr. by 46. 9. What is the value of 29 yards of Irish hnen, at 7s. 9d. 2qr. per yard. 10. Multiply 18s. 4d. by S3. 144 WRITTEN ARITHMETIC. V TROY WEIGHT. 11. Multiply 141b. Ooz. 8dvvt. llgr. by 7. 12. What is the weight of 11 Federal dollars; the weight of 1 dollar being ITdwt. 8gr. ? AVOIRDUPOIS WEIGHT. 13. Multiply 7T. 12cwt. Iqr. 141b. by 8. 14. What is the weight of 25 hogsheads of fish; each hogshead containing 5cwt. 3qr. 15 lb. ? CLOTH MEASURE. 15. Multiply 29yd. 2qt. 3na. by 9. 16. How many yards of broad-cloth are there in 35 pieces; each piece containing 47yd. Iqr. 2na? DRY MEASURE. 17. Multiply 33bu. 3pk. 6qt. Ipt. by 5. 18. How nnany bushels of corn are there in 18 bags, each bag containing 2bu. 2pk. 5qt. Ipt. ^ WINE MEASURE. 19. Multiply Ip. Ihhd. 52gal. 2qt. Ipt. by 4. 20. How many hogsheads of wine are there in 13 casks; each cask containing 49gal. 3qt. ^ BEER P/IEASURE. 21. Multiply 6bl. Ikil. Ofir. 6gal. 2qt. Ipt. by 7. 22. If 1 man drink 2gal. 3qt. Ipt. of beer in a week, how much will 38 men drink in a week ? LONG MEASURE. 23. Multiply 5lea. 2m. 6fur. 36rd. by 8. 24. If a man travel 55m. 5fur. 17rd. a day, for 18 days, how many miles will he have travelled. SQUARE MEASURE. 25. Multiply 36A. 3R. 27rd. by 6. 26. How many square yards are there in 14 rolls of carpeting; each roll containing 52sq. yd. 3 sq. ft, CUBIC MEASURE. 27. Multiply IT. 34ft. i200in. of round timber by 3. 28. There are 4 piles of wood; each containing 3C. 6ft. w. 12c. ft. How much wood is there m all. TIME. 29. Multiply 4Y. 255d. 16h. by 9. 30. If a ship alter her latitude 1 degree m sailmg 16h. 40m., in what time will she alter it 15 degrees ? 12 COMPOUND NUMBERS. |I5 Section 12. COMPOUND DIVISION. 1. If £2047 13s. 9d. be divided equally among 6 men how much will each man receive ? We divide the pounds, and £ g^ ^ there remains <£ 1 . ThisXl 6)2047 13 9 ^® reduce to shillings, and — — — — — unite it with the 13s. making 34 1 5 7 2 qr. 333 We divide the 33s., re- duce the remainder to pence, and proceed as before. 2. If 19s. lid. 3qr. be divided equally among 3 men, how much will each man receive ? 3. Divide £16 14s. lOd. 3qr. equally among 5 men. 4. Divide £3 Os. 8d. equally among 7 men. 5. Divide £59 18s. 4d. equally among 25 men. ^ J x- 1 This operation is in long £ s. d. £ s. Q. J. . . WT £ .. J- • 1 25)59 18 4(2 7 11 ^^'^'^'««- ^e first divide ^ rr. the pounds: the quotient is £2, and the remamder, £9. 9 We then reduce the £ 9 to ^Q shillings, adding in the 18s., 25)198(7s. and divide this sum [198s.] 175 as before: the quotient is 23 7s. and the remainder, 23s. 12 We then reduce the 23s. to ^^ pence, adding in the 4d., and 25)280(lld. divide this sum as before. ^^ 5d. remain undivided. Ob- 30 serve, that, in every instance, 25 the quotient and remainder are of the same denomination with the dividend. 5d. remaining. RULE FOR COMPOUND DIVISION. Divide each denomination separately^ beginning with the highest. Wlienever a remainder occurs^ reduce it to the next Zow- er denomination^ add it to the number expressed in the lower denomination^ and divide it thereunth. J* 146 WRITTEN ARITHMETIC. V. 6. Divide £460 5s. lOd. equally among 37 men. 7. If 15cwt. 3qr. 181b. of flour be packed equally in 9 barrels, how much will each barrel contain ? 8. If it take 15 yards of cloth to make 6 coats, how much does it take to make 1 coat ? 9. If an army consume 492bu. Opk. 6qt. of wheat in 42 days, how much does it consume in 1 day .^ 10. Divide 3qt. Ipt. of wine equally among 7 men. 11. If 30hhd. 13gal. 2qt. of cider will fill 18 casks of equal size, how much does each cask hold ? 12. Divide 58m. 2fur. 32rd. into 8 equal distances. 13. Suppose a man is to travel 339m. 4fur. 20rd. in 6 ilays; what distance must he travel each day? 14. If a field containing 22A. 2R. 12rd. be divided into 4 equal lots, what will each lot contain .^^ 15. Suppose a township, containing 17715 acR ? of land, should be divided into 80 equal farms, how many acres would each farm contain ? 16. Suppose a rail-way car to perform 4 trips in 5d 16h. 9m., in what time does it perform 1 trip ? Q^uestions to be answered Orally. (1) Which of these numbers is a compound num- ber,— £356, or £2 18s. ? (2) Why is it called a compound number ? (3) Recite the rule for addi- tion of compound numbers. (4) Suppose the sum of a column of numbers expressing furlongs to be 37; what must be written under the column, and what must be carried to the next column ? — Why ? (5) Recite the rule for subtraction of compound num- bers. (6) Recite the rule for multiplication of compound numbers. (7) Recite the rule for divi- sion of compound numbers. I. FRACTIONS. 147 CHAP. VI. FRACTIONS. Fractions have been exemplified in page 44, and the mode of expressing them has been defined in page 74. A proper fraction expresses a quantity, less than a unit. Therefore, the numerator of a proper fraction, must be less than the denominator: for example, f . An improper fraction expresses a quantity, equal to a unit, or greater than a unit: therefore its numerator must equal, or exceed its denominator: thus, f . |. A compound fraction is a fraction of a fraction — a part of a part of a unit: for example, f of ^ . NOTE. The written operations requii-ed in the several sec- tions of this chapter, con*espond with the mental operations involved in sections of the same number, in chapter VI, Oral Arithmetic. Learners will be enabled to perceive the written process to be adopted, by recun-ing to the oral examples. Section I. I. What is the sum of -^^ ^'^^ T2 ^^^ T2 - _3_ These fractions have a common denom- 5_ inator; that is, they all have the same de- 2 nominator. We add the numerators only, and under the sum of the numerators, place 12 T2 the common denominator. 2. What is the sum of | and ^ and | and ^ ? 3. now nmcii is -f^ mm ij anu -fj auu -fj anu -^j i 4. A man paid j-^ of a dollar for breakfast, -f^ of a dollar for dinner, and -^-^ of a dollar for supper. What part of a dollar did he spend ? 5. How much is 23- and -^j and ^^ and ^V and 23- - 6. A merchant sold ^f of a ship to one man, and -j^ to another. What part of the ship did he sell } 7. Add together ^f 'and j\ and f| and jf and jf ? 8. How much is f and ^ and f and f ? 9. How much is i}^ and -^-^ and yta and -^^ } life WRITTEN ARITHMETIC. VI. Section 2. Write upon the slate, the several fractions required in the following examples. 1. If you divide a bushel of corn into 8 equal parts, and then put 6 of the parts into a sack, what fraction of a bushel will there be in the sack ? 2. If an acre of land be divided into 20 equal lots, and 14 of the lots be enclosed by a fence, what fraction of an acre will there be in the enclosure ? 3. Suppose any thing to be divided into 45 equal parts; what fraction will express 26 of the parts .'' 4. Suppose 1 dollar to be divided into 100 equal parts ; what fraction will express one of the parts ^ 2 of the parts ^ 6 parts ? 25 parts ^ 99 parts ^ Section 3. 1. If -^ be subtracted from yj, what will remain ? 11 Both of these numerators express ff- 6 teenths; therefore w^e merely subtract one ■ numerator from the other, and under the TT remainder, place the denominator. 2. If yq be subtracted from -j^q , what will remain.^ 3. What is the difference between |- and f ? 4. If -^ be subtracted from \j what will remain ? 5. What is the difference between ^\ and |f ? 6. A farmer divided a ton of hay into 20 equal parts, and gave 14 parts to his cows, and the rest to his sheep. What fraction of a ton did the sheep get ? 7. Subtract ^ from 1, — that is, subtract f from the number of eighths that there are in a w^hole one. 8. Subtract ys fi'om a whole 1. 9. What is the difference between -j^?? and 1 ? 10. Subtract |q|^ from a whole 1. 11. A merchant owning a ship, sold y| of her to one man, ^ to another, and ^\ to another. What part of the ship did he still own ? 12. A boy having 1 dollar, paid away -^-^-q of it, and lost -^-^Q, What fraction of a dollar had he left .^ 13- Subtract f|^- from a whole I. 2, 3. 4. 5. RELATIONS OF NUMBERS. 149 RELATIONS OF NUMBERS. Section 4. We frequently have occasion to view one number as a certain part of another number; and thus we notice the relation that exists between the two numbers. In order to state what part one number is, of another, we m«ke the number which is the part a numerator, and the other number a denominator. State the fractions, which answer to the following ex- amples, upon the slate. 1 . What part of 5 cents is 1 cent ? is 3 cents ? 2. What part of 10 is 1 ? is 2 ? is 5 ? is 9 ? 3. What part of £1 or 20 shillings, is 1 shilling.^ is 6 shillings ? is 14 shilhngs ? 4. What part of 35 is 1 ? is 8 ? is 11 ? is 34 ? 5. What part of $ 1 or 100 cents, is 1 cent? is 2 cents ? is 9 cents ? is 46 cents ? is 94 cents ? 6. What part of 6 pence is 1 penny ? is 5 pence } 7. What part of 1 shilling is 1 penny ? is 7 pence } 8. W^hat part of 1 peck is 1 quart ? is 7 quarts ? 9. What part of 1 hogshead is 1 gallon } is 18 gals, r 10. If ^V ^^ ^ hogshead of wine be worth $1, what is -5^ of a hhd. worth ? What is Ihhd. worth ? 11. What part of 1 year is 1 day ? is 10 days .'^ is 40 days ? is 100 days ? is 275 days ? 12. If a man spend $ 1, in 3-^-5 of a year, how much will he spend in -^-^ of a year ? in ^|f of a year } How much will he spend in 1 year ? 13. What part of 2016 is 1 ^ is 84 } is 759 .? Section 5. 1. Suppose ^ of a ship to be worth $4703; what is the whole ship worth .^ 2. 4703 is ^ of what number } 3. If \ of an acre of land produce 71 bushels of po- tatoes, how many bushels will 1 acre produce ? 4. 71 is -3- of what number } 5. 875 is 4- of what number ? 6. 1900 is 5 of what number .'^ N* iiSO WRITTEN ARITHMETia VI 7. If 230 men will lay -^ of a mile of rail-way a week, how many men will it take to lay 1 mile in a week t 8. 230 is \ of what number ? 9. 44 is Y of what number ? 10. 6902 is \ of what number ? 11. li-ii of a pound of silver be worth $1.09, what is 1 pound of silver worth ? 12. If a ship sail 17 miles in ^V of a day, what dis- tance would she sail in the whole day ? 13. 204 is 3^ of what number ? 14. If "x^o ^^^ P^P^ of wine be worth $1.15, what is the whole pipe of wine worth ? 15. Suppose \- of the sugar in a hogshead to weigh Icwt. 2qr. 121b.; what does the whole weigh? Section 6. 1. If 1 acre of land will produce 126 bushels of po- tatoes, how many bushels will \ of an acre produce } 2. What is -J of 126 ? 3. Suppose 38406 needles can be made from a bar of steel; how many can be made from \ of the bar ? 4. What is -^ of 38406 ? 5. If 1 dollar will pay for 316 quills, what number of quills will '\ of a dollar pay for ? 6. If you eat 1095 meals in 1 year, what number of meals do you eat in ^ of a year ? 7. What number of cubic inches are there in -g^ of a cubic foot? (See Cubic Measure, page 132.) 8. If 1 week's board cost $3.64, what does ^ of a week's board cost ? 9. Suppose a packet ship to be worth $17841.50; what is -iQ of her worth ? 10. A man, having $205.12, paid -j^ ^^ his money for a piece of land. What was the price of the land ? 11. A man gave $2568 for a house, and then paid ^i part as much for having it repaired. For how much must he sell the house, in order to lose nothing ? 12. What is yV of 1800? 13. Suppose a piece of cloth to contain 60yd. 2qr. how much cloth is there in \ of the piece ? 3 7. 8. RELATIONS OF NUMBERS. I5l Section 7. 1. Suppose that 12 men are to pay a debt of $420, iti equal shares; what must 1 man pay? Solution. 1 man is j2 ^^ ^^ men; therefore 1 man must pay -^^ of $420. -^~ of 420 is — 2. If a prize of $3936 be divided equally among 8 men, what part of the money will 1 man receive ? How many dollars will 1 man receive ? 3. 27 men own 864 acres of land together. What part of 864 acres does 1 man own ? What number of acres does 1 man own ? 4. If $ 135.45 will pay for Ihhd. of wine, what part of the money would pay for 1 gallon ? What would be the price of 1 gallon ? 5. If 170 acres of land produce 6630 bushels of corn, what part of 6630 bushels does 1 acre produce ? How many bushels does 1 acre produce ? 6. If 6 yards of broad-cloth be worth J£ll lis. 9d., what part of the money is 1 yard worth ? What is the value of 1 yard, in pounds, shillings, &c. } 7. A black-smith paid $ 63 for 1 5 tons of coal. What did the coal cost him per ton ? Section 8. 1. A man purchased a farm for $5642, and paid j of the price in cash, and gave his note for the remainder. How many dollars did he pay down ? Direction, First find y of $5642, by dividing this «um by the denominator of the fraction; then find 4- sevenths, by multiplying the quotient by the numerator. 2. Whatisf of 1905? 3. If an acre of land will produce 14870 ears of corn, how many ears will f of an acre produce ? 4. What is I of 19064 ? 5. Suppose an acre of land to be worth $48.16; what is the value of | of an acre of the same land ? 6. If 1 dollar will pay for 270 quills, what number of quills will j^Q of a dollar pay for ? 7. If 72 gallons of wine leak from a pipe in 1 dav^ now many gallons leak out in ^ of a day ? 152 WRITTEN ARITHMETIC. VI. 8. Suppose a hogshead of sugar to be worth J£20 4s. 3d. ; what is the value of ^ of the sugar ? 9. What is if of 4720 ? In the several foregoing examples in this section, the earner has probably divided the given number by the denominator of the fraction, and multipHed the quotient Dy the numerator. It is, however, sometimes more convenient, to multiply the given number by the nume- rator, and divide the product by the denominator. 10. What is I of 32 .? (Here are the two methods.) 4)32 8 3 First Method, ""is i of 32. Second Method. 32 3 4)96 is 3 times 32. 24 is 3 times | of 32, which is 1 of 32. 24 is 1 of 3 times 32, which is 1 of 32. We may see why these two methods of operation produce the same result, in the following illustration. Here is | of 32 units arranged in one line, and \ oi S times 32 units arranged in three lines. The number of units [®] in the two arrangements is the same. a®®®o®®® ^®oeoe®o ®#oe9oe® oooooooo ©®®®®©®® oooooooo oooooooo oooooooo •••«<»«©« oooooooo oooooooo oooooooo •••®<^©®® oooooooo oooooooo oooooooo 11. Find -/tj of 60156, by each of the above methods. 12. Find sf y of 10849, by the second method. 13. A laborer worked | of a year, at 92 cents per day. What did his wages amount to? 14. In \^ of a pipe of wine, how many gallons? 15. What is 1-^0 of $1491? After multiplying by 6 and dividing by 100, reduce the remainder to cents, and divide the cents. 16. A borrowed of B, $ 758, promising to pay it in one year; and, in addition thereto, he agreed to pay a sum, equal to yf o ^^ ^^^ ^"^^ borrowed, for the use of the money. How much must B receive? 17. Whatisy^o of$2S? 9. 10. RELATIONS OF NUMBERS. 1W Section 9. 1. If $686.56 should be divided equally among 8 men, what part of the money, — and what number of dollars and cents, would 3 men receive ? 2. Suppose that $33 will pay for 198 yards of cloth; what part of 198 yards, — and how many yards can be bought for $ 14 ? 3. If 15 acres of land produce 283bu. Opk. 4qt. of wheat, what part of this quantity, — and how many bush- els will 9 acres produce ? 4. If 540 barrels of flour will supply an army for 30 days, how many bushels will supply it for 19 days ? Solution. 19 days are ^-| of 30 days; therefore the army will consume ^^ of 540 barrels. 5. If a man can build 256 rods of fence, in 60 days, how many rods can he build in 45 days ? 6. If 72 tons of hemp cost $ 13680, what will 20 tons cost, at the same rate ? 7. If it take a man 31 days to travel 1178 miles, how many miles can he travel in 25 days ? 8. If 24 English watches are worth ^6108 18s. , what IS the value of 7 watches of the same kind ? Section 10. 1. If 16 rilen can fell 208 trees in a day, how many trees can 35 men fell in the same time ? 2. What is 35 times ^V of ^^^ ? 3. If 10 barrels of flour cost $59.30, how much will 33 barrels cost, at the same price per barrel ? 4. What is 33 times iV of $59.30 ? 5. If 64 soldiers eat 448 pounds of beef in a week, how many pounds will 250 soldiers eat in a week ? 6. What is 250 times -J4 of 448 } 7. If 12 gallons of linseed oil be sold for $13.44, what should be the price of 52 gallons of linseed oil ? 8. What is 52 times -^^ of $ 13.44 ? 9. If a man earn $91.70 in 7 months, how much can he earn in 2 years ? 10. If 48 pounds of feathers can be bought for $16, how many pounds can be bought for $25 ? 154 WRITTEN ARITHMETIC. VI Section 11. 1. If f of an acre of land will produce 28 bushels of potatoes, how many bushels will ^ of an acre produce ? How many bushels will 1 acre produce ? 2. If I of a hogshead of sugar be worth $22.50, what s ^ of it worth ? What is the whole worth ? 3. $22.50 is I of what sum of money ? 4. Suppose a ship to sail 105 miles in -^2 of a day; what distance will the ship sail in -^^ ^^ ^ ^^Y - What distance will she sail in the whole day ? 5. 105 is y2 ^f ^^hsit number? 6. If -^j of a chest of tea be worth $23, what is yV of it worth ? What is the whole of it worth ? 7. $23 is lY of what sum of money? 8. If f of a bag of coffee be worth $38.46, what is the whole bag of coffee worth ? 9. Suppose a rail-way car to run 198 miles, in -^-^ of a day; what distance will it run in 1 day ? 10. If 192 men will perform f of a certain piece of work in a week, what number ojf men will it take, to perform the whole of the work in a week ? 11. 192 is f of what number ? 12. A man purchased a farm, and after he had paid ^ of the price, he still owed $1288. What must have been the price of the farm ? 13. A trader purchased a pipe of wine, and after ^ of it had leaked out, he sold the remainder at $1.15 per gallon. How much did it amount to ? Questions to be answered Orally. (1) What is meant by a common denominator of two or more fractions ? (2) How do you add frac- tions, that have a common denominator ? (3) How do you subtract one fraction from another; the two fractions having a common denominator ? (4) When a certain fractional part of a number is known, how do you find the whole of the number? (5) When the whole of a number is known, how do you find any certain fractional part of it ? 11. 12. 13. RELATIONS OP NUMBERS. 155 Section 12. REVIEW. 1. (§1.) If you should pay -jYo ^^ ^ dollar for a quire of paper, -^-^^ of a dollar for a slate, and ^q^q for a book, what fraction of a dollar would you spend ? 2. (§2.) If a ton of hay be rolled uy in 20 equal heaps, what fraction of a ton will 14 heaps be ? 3. (§ 3.) Suppose a young man to lay out \^ of his money for a farm; what part of his money has he left ? 4. (§ 4.) Suppose a school to be allowed 15 minutes for recess; what fraction of an hour is the recess ? 5. (§ 5.) 2V ^f ^" ounce, or Idwt. of pure gold, is sufficient ^to gild a silver wire, 65 miles in length What length of wire may be gilded with 1 ounce ? 6. (§6.) If a man's income be ^^193 a year, how much is his income for ^2 of a year, or 1 month ? 7. ( § 7.) If $ 14 will pay for 70 books, what part of 70 books,— and how many books, will ^ 1 buy ^ 8. (§8.) If a man's income be ^803 a year, how much is his income for 3W of a year, or 30 days ? 9. (§9.) Suppose $93.66 to be paid for 14 yards of broad-cloth; what part of the money does 6 yards cost? How many dollars do 6 yards cost.'^ 10. (§ 10.) If 11 barrels of flour are worth $59.07, what is the value of 25 barrels, at the same rate ? 11. (§11.) If 18 shillings be given for --^^ of a hun- dredweight of fish, what must be given for 1 cwt. ? FRACTIONS AND RELATIONS. Section 13. 1. Suppose you can read 12 pages in an hour; how many hours will it take you to read 160 pages ? 12)160(13-^- We find, by division, it will take 12 13 hours, and still 4 pages remain ""TTT to be read. Now, since it takes OQ 1^2 ^f ^" ^^^"^ ^^ ^'^^^ ^ P^g®? ^t will take 1^2 ^^ ^" ^^^^ ^^ ^'^^^ ^ 4 pages. Ans. 13^^ hours. 156 WRITTEN ARITHMETIC. VI Jlny REMJilKDER, which appears after the operation of division^ is the numerator of a fraction^ the divisor being the denominator; and this fraction forms a part of the quotient. Therefore, place the remainder and the divi' sor^ as a fraction^ to the right of the quotient, 2. At |i2 per yard, how many yards of cloth can be bought for ^49; that is, how many whole yards, and what part of another yard can be bought ? 3. How many times 2 are there in 49 ? 4. How much flour can be bought for ^639, at $5 per barrel; that is, how many barrels, and what part of another barrel can be bought ? 5. In 639, how many times 5; — that is, how many fives, and what part of another 5, in 639 ? 6. How much salt can be bought for 87 shillings, at 4 shillings per bushel ? 7. Suppose a rail-way car to run 16 miles an hour' in how many hours will it run 350 miles ? 8. How many times 12 are there in 1049 ? 9. How many times 39 are there in 76800 ? 10. In 438 shillings, how many pounds are there ? Observe that Is. is gV of "^l? therefore the remain- der in this example, may be expressed as a fraction. 11. How many yards of cloth can be bought for 549 shillings, at <£1 per yard? 12. Suppose a ton of hay to be equal in value to 34 bushels of oats; how many tons of hay must be given for 450 bushels of oats ? 13. How many times 17 are there in 23 times 31 ? 14. How much rice, at $4 per cwt., must be given for 6201b. of cheese, at 10 cents per pound ? Section 14. CHANGE OF WHOLE NUMBERS TO FRACTIONS. 1. How many thirds are there in 14.^ . . In 1 there are 3-thirds, o therefore, there are 3 times ^ as many thirds as whole ones, 42-thirds. *Ans. 3 . in any whole number. 14 15. FRACTIONS AND RELATIONS. 157 RULE. To change a whole number to an improper fraction., multiply the whole number by the denominator^ and the product will be the numerator. 2. If you cut 17 sheets of paper into half sheets, how many halves will there be } 3. How many fifths of a dollar are there $ IG .^ 4. In 31 pounds, how many sixths of a pound ? 5. In 73 yards, how many eighths of a yard } 6. Change 641 to ninths. Change 641 to tenths. 7. If a stage run 1 mile in \ of an hour, how many miles would it run in 126 hours ^ ' 8. How many fourths are there in 15-|.^ j^_3 In this example, we add the A^ 3-fourths to the fourths produc- • ed by the multiplication of 15 6 3-fourths. i^^ 4^ ^^^^ ^^^^ ^l^^^j^ 6_3 ^ J^ote. A whole number and a fraction expressed to- gether, thus, 15f , is called u. mixed number. 9. How many eighths of a mile in 57^ miles "^ 10. Change Q6\y ^o an improper fraction. 11. Change 4-^^ to an improper fraction. 12. If 3^ of a dollar will pay for 1 gallon of beer, how much beer can be bought for $6^^ 13. If |- of a dollar will pay for 3 yards of ribbon, how many yards can be bought for $6^? Section 15. CHANGE OF FRACTIONS TO WHOLE NUMBERS. 1. How many whole ones are there in "^-^ ? 3)42-thirds. S^^^^^ f make a whole 1, there — are as many whole ones m y 14 wholes. j^g ^YiQ^,Q ^^^ tinies 3 in 42. RULE. To change an improper fraction to a whole number^ divide the numerator by the denominator^ and the quotient will be the whole number. 2. How many whole sheets of paper must be cut into halves, to make "^^ of a sheet ^ i5S WRITTEN ARITHMETIC. VI. ' 3. In ^1^ of a dollar, how many dollars ? 4. How many pounds ai-e there in J-|^ of a pound ? 5. In ^-%-^ of a yard, how many yards are there ? 6. If a stage run 1 mile in ^ of an hour, how many hours would it be in running 126 miles ? 7. Change ^4^ to a whole number. 8. Suppose 8 pounds of sugar can be bought for $1; how many pounds will -ff- of a dollar pay for ? 9. Change ^/ to a mixed number ? rNgy We divide the S7-fifths by 5, and ^ obtain 17 whole ones: then, there are I'^f 2-fifths over, making 17-|. 10. Change -f- to a mixed number. 11. How many dollars are there in -^f^ of a dollar.^ 12. How many gallons in ''-f^ of a gallon ? 13. If -5- of a dollar will pay for 1 pound of coffee, how many dollars wdl 312 pounds of coffee cost ? 14. If 1 pound of butter cost \ of a dollar, what would be the price of 491 pounds, at the same rate ? Section 16. The sum of these fractions will be an improper frac- tion, and it must be changed to a mixed number. 2. What is the sum of 491 -i\, 75fx and SSG^^y ? 49]_9.. In this example, we add together 7 5_B_ the elevenths, and find their sum to g3^_b be ff; which is equal to 2-i\. The -^1 we write down, and carry the 2 to 1 404-^- 11 c x-t\j-x^-^ the column 01 units. 3. What is the sum of 419f, ISf, 12f , 8573f , 9j, 251 f, 141-1, and 25-|? 4. Add together 336^^5, 14i-|, 970Uf, 28, 158^-?-, 1240^5, l\, lOO^V^ and -i|. 6. Subtract 1876| from 2258|. 2 c Q 3 • We cannot take | from ^ , therefore, 1075! we join 1 unit with the |-, making ^/, ** and take | from ^f . We then proceed to take 6 units from 7 units. ' 9 16. 17. 18. FRACTIONS AND RELATIONS. 15& 6. Subtract 46031| from 71706}. 7. Subtract 609 |i- from 5420067^. 8. If 17| yards of cloth be cut from a piece contain- ing 49 yards, how much will be left ? 9. A retailer put into a firkin, 28-]^^ pounds of butter at one time, 19^ pounds at another, and SSyg- pounds at another, and then sold out 25 ^| pounds. How many pounds still remained in the firkin ? Section 17. 1. Suppose a rail-road car to run f of a mile in 1 minute; what distance will it run in 47 minutes ? 2. How many whole ones in 47 times f ? 3. If ^ of a yard of broad-cloth will make 1 jacket, how many yards will it take to make 18 jackets ? 4. How many whole ones in 18 times |-? 5. If I of a pound of gunpowder tea cost 1 dollar, how many pounds can be bought for 50 dollars ? 6. How many whole ones in 50 times ^} Section 18. I 1. What is the product of 86f , multipHed by 9? QQ4 We multiply f by 9 thus, 9 times y g^ is ^j ; eq^al to 5-^. Then we write this Y under the |, and carry the 5 to '^'^^T the product of the 6 units. 2. What is the product of 41|, multiplied by 7 ? 3. What is the value of a field, which contains 5 acres, allowing it to be worth 54^ dollars per acre .-^ 4. How much is 4 times 35 ^§ ? 5. How much is 9 times 14731 1|? 6. How much is 28 times 54 f? 54f Here we are obhged to multiply by 28 8 units and 2 tens separately, and we 432 cannot well bring in the product of the 108 fraction by the 2 tens. Therefoce, 132 we first multiply the whole numbers, Y" — -J and then find 28 times |, m a separate ^oc^K)^ operation, which is not here written. 160 WRITTEN ARITHMETIC. VI 7. How much is 92 times 2051 -j^q ? 8. How much is 100 times M^j? 9. How many gallons of wine are there in 31 casks, each cask containing 54 -j^^ gallons ? 10. A merchant paid $75 -j^^ apiece, for 47 mules. What did the whole amount to ? 11. If a steam-boat run 236^ miles in 1 day, what distance will it run in 16 days ? 12. How much is 15 times f^? 13. What cost 75 books, at -jYo ^^ ^ dollar apiece } 14. If a horse eat -|| of a bushel of oats a day, how many bushels will he eat in 365 days ? Section 19. 1. If I of a chest of tea be worth $6.87^, what is the whole chest worth ? 2. If j\^ of a dollar will pay for travelling 27| miles on a turnpike road, how far can you go for $i? 3. 790 1 is jV ^f ^vhat number ? 4. If 4. of a kite line be 25 1 yards in length, what is the whole length of the line ? 5. 305-/3 is ^ of what number ? 6. If a man.can earn 12^ cents in ^ of a day, what sum of money can he earn in 1 day ? 7. If ^ of a yard of gold wire be worth y| of a dollar, what is the value of 1 yard of the wire ? Section 20. 1. A boy having $2, gave | of his money for a knife. What fraction of 1 dollar did the knife cost ? 2. -5- of 2 is equal to what part of 1 ? 3. 6 men divided 5 barrels of flour equally among them, each man taking ^ of the flour in each barrel What fraction of a barrel did each man get ? 4. "3^ of 5 is equal to what part of 1 ? 5. If you should take ^-^ of a bushel of corn from each of 10 bushels, what fraction of 1 bushel would you obtain ? 6. What part of 1 is ^j of 10 ? 7. What part of 1 is 3V of 2 ? is ^ of 3 } is ^\o{ 4 ? is -sV of 18? is 3V of 38.^ I 19. 20.21. FRACTIONS AND RELATIONS. 16! 8 A tenant raised 28 bushels of corn, and gav^e his landlord ^ of it. What improper fraction of a bushel, [how many thirds of a bushel,] did the landlord receive ^ How many bushels did the landlord receive ? 9. -^ of 23 is equal to what improper fraction ? Then ^ of 28 is equal to how many whole ones ? 10. I of 42 is equal to what improper fraction ? Then 5 of 42 is equal to what mixed number ? 11. -5^3 of 29 is equal to what improper fraction.'* Then -^^3 of 29 is equal to what mixed number ? 12. If $721 should be divided equally among 6 men, how many sixths of a dollar would each man have ? How many dollars would each man have ^ Section 21. 1. Suppose a hogshead of brown sugar to be worth ^115; what is the value of -} of the sugar.'* 2. What is -}of 115? 8. 5 men from Connecticut, bought 793 acres of land in Michigan, and divided it into 5 equal farms. How many acres were there in each farm ? To find J of 793 acres, we divide 793 by 5. The quotient is 158 acres, and there is a remainder of 3 acres. To divide these 3 acres, we take -\ of each acre for each farm. -]- of 3 acres is f of 1 acre. 4. What is ■} of 1315 ? -J- of 530 ? -J of 8201 ? 5. Suppose 12 men to share equally in a prize of $551.20; what is each man's share ? G. What is 1^5 of 55120 ? ^V ^f 967 .? ^V of 700 > 7. The Young Ladies'' Class Book^ which consists of 408 pages of select reading lessons, has been read through by 25 scholars; each reading an equal portion. How many pages did 1 scholar read ? 8. \ of o€l \\ of 20s.] is how many shillings, and what fraction of a shilling ? 9. Find \ of £7 in shillings — that is, reduce J£7 to Bhillings, and find \ of the number of shillings. 10. -g^ of 5 shillings is how many pence ? 11. How many grains in -^^ of G pennyweights ^ 12. -^ of 3 ounces is how many drams ^ 162 WRITTEN ARITHMETIC. VI, 13. 11 men divided 9 hogsheads of molasses equally among them. How many gallons had each man i 14. Y5 of 7 furlongs is how many rods .'' 15. Y of 6 square feet is how many square inches ? 16. How many seconds are there in -^^ ^f ^^ hours? 17. A trader sold -4- of a hogshead of wine at 37 cents for every -^ of a gallon. What did it amount to ? Section 22. 1. If 34 barrels of flour be made from 147 bushels of wheat, how much wheat will make 9 barrels of flour ? 2. What is 9 times -^-^ of 147 ? 3. If 8 yards of broad-cloth cost ^^38, what will 13 yards cost, at the same rate ? 4. Wh^t is 13 times ^ of 38 ? 7 times ^V ^f ^^4 } 5 times jV of 270 ^ 21 times iV of 40975 ? 5. If it cost $1.25 to ride 20 miles in a stage, how much will it cost to ride 32 miles ^ 6. If ^ 16 will pay for 85 pounds of butter, how many pounds will $25 pay for ? 7. If 12 cords of wood cost $75, what will be the cost of 19 cords, at the same rate .'^ 8. Suppose 21cwt. of flour to be worth the same as 65 bushels of salt; how many bushels of salt must be given in exchange for IScwt. of flour.'* 9. If 8 barrels will hold 19 bu. 3pk. 4qt. of corn, how much corn can be put into 15 barrels ? Section 23. 1. When writing paper is sold at $5.42 per ream, what is the price «f -]- of a ream ? 2. If ^ of a ream of paper is worth $ 1.35| , what is I of a ream worth ? 3. Suppose a hogshead of sugar to be worth $93; what is -^ of it worth ? 4. If I of a hogshead of sugar is worth $13,284, what is 4 of it worth ? 5. Wiiat is i of .3765 ? What is f of 3765 ? 6. How many gallons are there in ^ of a pipe of wine f How many gallgns ia ^ of a pipe ? e2. 23. PERCENTAGE. 163 In the following examples, it will be most convenient to multiply by the numerator of the fraction, before dividing by the denominator. See Second Method of operation, exemplified in page 152. 7. The highest point of the Andes, is 21440 feet: Mont Blanc, of the Alps, is |- as high. What is the height of Mont Blanc } 8. Virginia contains 06000 square miles: Rhode Island is only ^ as large. How many square miles are there. in Rhode Island ? 9. If a man's income be $ 1000 per year, what is his income for 9 months, or -{^ of a year } 10. Suppose a man by constant industry can earn $1.50 per day; what will he earn in 10 days, allowing him to rest f of the time } 11. How much must be paid for \^ of a ton of Russia hemp, when the price is $210 per ton } 12. A man having 4 miles to go, rode ^ of the way, and walked the remainder. How many rods did he walk } 13. Suppose 45000 pounds of iron to be sufficient to lay the track on 1 mile of rail-way; how many pounds of iron are required to lay the track on 1| mile } 14. How much is 4500 plus -^- of 4500 .? 15. Suppose a rail- way car to run 350 miles a dav; what distance will it run in 5 1 days } 16. How much is 5-| times 350.^ — That is, — how much is 5 times 350 and | of another time 350 ^ 17. How much is 6| times 91.^ S^ times 146^ 3| times 244 } 12| times 379 } lO-j^o times 978 } PERCENTAGE. The term, per cent.^ is an abbreviation of per centum, and signifies, by the hundred. I per cent, of any num- ber, is y-J-Q of that number; 2 per cent, is -jfo-; 3 per- cent, is Yoo? ^ P^'* cent, is -j^^; and so on. 18. What is 5 per cent, of 360 dollars.^ 3 Q Q Since 5 per cent, is ^f^ j we multiply 5 by 5, and divide by 100. To divide ■ by 100, we merely point off two figures $18.00 fj.Qj^ |.j^g right, as' taught in page 116. 164 WRITTEN ARITHMETIC. VI. 19. What is 1 per cent, of 100 ? 2 per cent, of 100 ? 20. What is 2 per cent, or i^q of 350 dollars ? 21. A merchant, who has $3875 deposited in the oank, wishes to draw out 4 per cent, of his deposit. How manj dollars must he draw ? 22. What is 6 per cent, or i|^ of 4250 dollars ? 23. What is 6 per cent, of $92.50, or 9250 cents ? 24. What is 4 per cent, of $ 132.75 ? 25. A merchant having 2513 gallons of wine on hand, lost 1 per cent, of the whole, by leekage from the casks. How many gallons did he lose ? 26. Find 6 per cent, of 128 dollars. ^28 After multiplying and dividing, our " Q quotient is $ 7^^^^ . Now, since j^q of a — — dollar is 1 cent, -^^-^ of a dollar is 63 $7.68 cents: therefore the answer is $7.68. 27. Find 7 per cent, of 2517 dollars. 28. Find 18 per cent, of 20 dollars. 29. What is 4 per cent, of $70.14, or 7014 cents ? In this example, after multiplying by 4, and dividing by 100, there is a remainder of 56, And since the quotient is cents^ this remainder is yoq of a cent. 30. What is 9 per cent, of $470.46 ? 31. What is 55 per cent, of $964.07.? 32. A and B have $500 apiece. If A should give B 6 per cent, of his cash, what would each then have ? 33. What is ^ of 1 per cent, of 62 dollars ? 34. What is 4 7] per cent, of 62 dollars .? 35. What is ■} per cent, of 246 dollars } 36. What is 5 J- per cent, of 246 dollars ? • 37. A merchant paid $491 for a quantity of salt: for how much must he sell it, to gain 9 per cent..'^ 38. A trader paid $230 for a piece of cloth, contain- mg 46 yards, and sold it so as to lose 4 per cent. At how much did he sell it per yard ? 39. If I pay $525 for 90 barrels of flour, at what price per barrel must I sell it, to gain 7 per cent..'' 40. Suppose a merchant to pay $85 per ton for 6 tons of iron; at what price must he sell it per hundred- weight, in order to gain 12 per cent. .'^ . 23 INTEREST. 165 41. A merchant failed in business, and was able to pay his creditors only 65 per cent, of their demands. What did he pay on a demand of ^ 534 r INTEREST. Interest is money paid for the use of money that has been owed. For instance, suppose that A lends B $ 100 for one year, and at the end of the year, B pays, not only the $ 100, but also pays $ 6 for the use of the $ 100: in this case, $ 6 is the interest. The money for which interest is paid, is called the Principal The sum per cent, paid for one year's in- terest, is called the Rate, The principal and interest added together, are called the •Amount. RULE FOR COMPUTING INTEREST. Mnltiply the principal by the rate per cent.^ and divide the product by 100: the quotient will be the interest for 1 year, 42. What is the interest of $ 100, for 1 year, at 5 per cent. ? What is the amount ? 43. What is the interest of $ 1 or 100 cents, for 1 year, at 5 per cent..'^ What is the amount ? 44. What is the interest of $354, at 6 per cent, for 1 year r for 2 years ? for 3 years ? for 4 years ? What is the amount for 4 years ? 45. What is the interest of |^ 40. 50, for 4 years, at 6 per cent. ? What is the amount ? 46. What is the interest of $18, for 3 years, at 7 per cent. ? What is the amount ? 47. What will $8110 amount to in 15 years; the rate of interest being 4 per cent. ? 48. What is the interest of $6470, for 3 years, at 5^ per cent.? What is the amount ? When interest is to be computed for any number of ^ months^ — First find the interest for 1 year; then take -^^ of a yearns interest for 1 month; j^ or \ for 2 months; h ^^ h fi^ ^ months; and so on, 49. What is the interest of $35, for one month, at 6 per cent, per annum ? What is the amount } 106 WRITTEN ARITHMETIC Vl 50. What is the interest of ^21, for 2 months, at 7 per cent, per annum ? 51. What is the interest of 5^4291, for 3 months, at 5 per cent, per annum ? 52' At 4 per cent, per annum, what is the interest of jjj 122.75 for 4 months ? for 5 months ? for 6 months ? for 7 months ? for 8 months ? for 9 months } for 10 months ? What is the amount for 11 months ? 53. What is the interest of $ 14.50, for 1 year and 1 month, at 6 per cent..^ 54. What is the interest of $19.25, for 3 years and 2 months, at 8 per cent.? 55. What is the amount of $458, for 2 years and 3 months, at 7 per cent..'* 56. What is the amount of $8.75 for 5 years and 4 months, at 4 per cent. ? 57. What is the amount of $91.50, for 2 years and 7 months, at 8 per cent..'^ 58. What is the interest of $81, from February 7, 1832, to August 7, 1835, at 6 per cent..^ 59. Suppose a promissory note of $ 145, to be dated, January 15, 1 831 ; what will be the amount of that note, October 15, 1834; the rate being 6 per cent.."^ 60. A owed B $96, on interest at 6 per cent. At the end of 2 years, A paid the interest then due, and $25 of the principal: at the end of 3 years and 11 months, he paid the whole debt. What was each payment ^ When interest is to be computed for any number of days^ — First find the interest for 1 month; then take -j^ of a month'' s interest for 1 day; -f^ or yj for 2 days; io or i^for 3 days; 3% or -f^ for 4 days; -^% or ^ for 5 days; -^^ or \ for 6 days; and so on. In the following operations, in this section, all fractions of a cent may be disregarded: this being the common practice in business. 61. What is the interest of $231, for 7 days, at 6 per cent, per annum ? Direction. First find the interest for 1 year; then tor i;^2 of a year or 1 month; and then for -^q of a month. I ^3 INTTEREST. 167 (12. What is the interest of $75, for 10 days, at 6 per cent, per annum ? 63. What is the interest of $254 for 21 days, at 6 per cent, per annum ? 64. What is the interest of $ 110, for 5 months, and .8 days, at 6 per cent, per annum ? 65. What is the interest of $34 for 1 year, 3 months, and 25 days, at 6 per cent, per annum ? 66. What is the interest of $91.18, for 3 years, 2 months, and 13 days, at 6 per cent, per annum } Several other methods are practised by merchants, in computing interest; among which, are the following. When the rate is 5 per cent. — Divide the principal by 20, and the quotient is the interest for 1 year. 61. What is the interest of $4207, for 2 years, at 5 per cent, per annum } 63. What is the interest of $951.17, for 4 years, at 5 per cent, per annum ? When the rate is 6 per cent. — Multiply the principal by half the number of months in the time., divide the •product by 100, and the quotient is the interest. 69. What is the interest of $119, for 16 months, at 6 per cent., per annum ? 70. What is the interest of $96.48, for 10 months, at 6 per cent, per annum ? 71. What is the amount of $27.56, on interest 6 months, at 6 per cent, per annum ? 72. What is the interest of $133.24, for 11 months, at 6 per cent, per annum ? To find the interest for days, the rate being 6 per cent.— Multiply the principal in dollars by the number of days., divide the product by 6, and cut off one figure from the right of the quotient. The rest of the quotient figures express nearly the interest^ in cents. 73. What is the interest of $249, for 75 days, at 6 per cent, per annum ? 74. What is the interest of $5824, for 21 days, at 6 per cent, per annum ? 168 WRITTEN ARITHMETIC. VI. 75. What difference will it make to the man who pays interest on $ 100 for 1 year, whether it be computed by daysy or according to true rule in page 165? DISCOUNT. Discount is an ^ibatement of a certain part of a debt, when the debt is paid before it becomes due. For in- stance ; suppose that A is bound to pay B $ 106, in one year from the present time ;. but B', wanting the money now, agrees to receive $ 100 for the debt, on condition of present payment: in this case, $100 is the present worth of the debt, and $6, is the discount. The present worth of any debt due at a future period, is that Slim of money, which, if put at interest, would amount to the debt by the time it becomes due. There- fore^ when the interest is 5 per cent., that is, j-^-^ of the pnncipal, then the discount is ^^^ of the principal. RULE FOR COMPUTING DISCOUNT. Multiply the principal by the number of cents found to be the interest of one dollar for the time, and divide the product by the number which results from adding 100 to the multiplier. The quotient will be the discount. For example, at the rate of 12425 6 per cent., the discount on ' y\ $124.25, due in 1 year and — — ■ 10 months, is found thus— HI) 1366.75(13.21i\\ 76. What is the discount on $48.51, due in 3 years; the rate of interest being 5 per cent. ? 77. What is the discount on $247, due in 1 year, the rate of interest being 6 per cent. ? 78. What is the present worth of $150, due in 1 year, the rate of interest being 6 per cent. ? Find the discount, and subtract it from the debt. 79. What is the present worth of $1640, due in 2 years, the rate of interest being 5 per cent. ? 80. Find the difference between the discount and the interest of $ 100 for 1 year, the rate being 6 per cent. 81. Find the present worth of $75, due in 2 years and 9 months, [2f years], interest being 6 per cent. 24. 25. 26. FRACTIONS 169 Section 24. 1. Suppose I of a piece of broad-cloth to be worth $118.87; what is \ of the piece worth .^ What is the whole piece worth ? 2. 11887 is I of what number? 3. If the interest of $ 100 be $3.50 for -f^ of a year, what is the interest of $100 for -^^ of a year.'* Then what would be the interest for 1 year ? 4. If -J-| of an acre of land produce 133 bushels of potatoes, how many bushels does t^-^ of an acre produce ^ Plow many bushels would 1 acre produce ? 5. 9071 is ^Q of what number.^ 6. If a man earn $190 a yenr by working y^^ of the time, how much could he earn by working constantly ^ 7. $ 14 is 8 per cent, or yf^- of what sum of money .'* Section 25. CHANGE OF THE TERMS OF FRACTIONS. The numerator and denominator of a fraction, are called the two terms of a fraction. These terms maybe changed, and the fraction may still express the same quantity. For instance, the terms 2 and 3, in the frac- tion f , may be changed to 4 and 6, and the fraction will become -J, which is still equal to f . 1. I is equal to how many twenty-fourths ? Direction. 8-eighths are equal to 24-twenty-fourths; therefore, find |- of 24, and this number will be the re- quired numerator of 2-4- 2. Y is equal to how many fourteenths ? 3. Change f to eighteenths and add y% to it. 4. J is equal to how many forty-fifths ? 5. Change ^^ to fortieths, and then take -}^ from it. Section 26. REDUCTION OF FRACTIONS TO LOWER TERMS. When a number can be found, that will divide both terms of a fraction, without a remainder, the two quo- tients arising from the division, will express the fraction redxiced to lower terms. For example, both terms of p 170 WRITTEN ARITHMETIC. VI the fraction -^2 ^^" ^® divided by 3, and the reduced fraction will be f . Again, both terms off can be divid- ed by 2, and the reduced fraction will be ^ . Thus any fraction may be reduced to its lowest terms^ by repeat- edly dividing the terms, until no number will divide them both without a remainder. 1. Reduce each of the following fractions to its low- oct tPrmc 3 6 4 9 10 6 18 12 est teims. -g-. 9. y^- 12- T^; is- 2Y- 24\ 2. Reduce each of the following fractions to its low- *»Gt t^rmc 1'^ 20 100 6.0 1_0_ 4.5. _30_ CbL LC1111&. 5-Q-. -g-Q . Xoo . 9 00* Too* 90* 120 • Only once dividing the terms of a fraction, will reduce it to its lowest terms, if we use the greatest common di- visor^ that is, the greatest number that will divide both terms without a remainder. TO FIJVD THE GREATEST COMMON DIVISOR of twO numbers^ — Divide the greater number by the smaller^ then divide the divisor by the remainder; and thus con- tinue dividing the last divisor by the last remainder^ till nothing remains. The divisor used last of all^ will be the greatest common divisor, 3. Find the greatest common divisor of 91 and 117. 91)117(1 This operation is perform- 91 ed according to the direction '26)9\(S above, and 13 is found to be •J Q the greatest common divisor; — ^ ^ or the greatest number by 13)26^2 which 91 and 117 can be di- z^ vided without a remainder. 4. Find the greatest common divisor of 15 and 235. 5. Reduce -^ to its lowest terms, by using the great- est common divisor of the two terms. 6. Reduce to their lowest terms, |ff , ^|-|, and ||f Section 27. COMPOUND FRACTIONS. A compound fraction arises from dividing a unit mto a certain number of equal parts, and then dividing one of these parts into other equal parts. 27. FRACTIONS. ^71 TO REDUCE S COMPOUJVD FRACTION TO A SIMPLE FRACTION^ — Multiply all the numerators together for a new numerator^ and all the denominators for a new denom- inator: then reduce the new fraction to its lowest terms. 1. Reduce f of ^ to a simple fraction. 2. ^ of a water melon was divided equally among 6 boys. What fraction of the melon did 1 boy receive .'* 3. Reduce f of f to a simple fraction. 4. J of an acre of land was divided into 4 equal lots. What fraction of an acre did 2 lots contain ? 5. Reduce |- of ^ to a simple fraction. 6. I of 1^4 is equal to what part of 1 ? 7. Reduce -^-j of i\ to a simple fraction. 8. 1 penny is what part of Is.? what part of £1? 9. 7 pence is what simple fraction of J£l .'^ Suggestion. 7 pence is -12 of 1 shilling, and 1 shilling IS ^ of £1. Therefore, 7 pence is j^ ^^ 2V of ^1» 10. Reduce 10 grains to the fraction of an ounce; that is, reduce ^^ of -^q to a simple fraction. 11. Reduce 3 nails to the fraction of ^ yard. 12. Reduce 4 inches to the fraction of a yard. 13. Reduce 25 seconds to the fraction of an hour. 14. Reduce f of :j of |^ to a simple fraction. 15. 4 of f of -j^Q is equal to what part of 1 .'* 16. Reduce -^^ of ^f ^^ ts ^^ ^ simple fraction. When the lower denominations of a compound number are to be reduced to the fraction of a higher denomina- tion^ — First ^ reduce the given quantity to the lowest de- nomination mentioned^ and this number will be the numerator: then reduce a unit of the higher denomina- tion^ to the same denomination with the numerator^ and this number will be the denominator. 17. Reduce 14s. lOd. 2qr. to the fraction of ,£1. 14s. lOd. 2qr. £\ is 20s. The denomin- 12 12 ^^or and numer- T^A 77777^ ator here. found, ^'f ^''f make IB: this . fraction,whenre- Awm.714qr. Denom. 960 qr. duced, is -^. 172 WRITTEN ARITHMETIC. VI 18. Reduce 2s. 7d. Iqr. to the fraction of c£l. 19. Reduce lid. 3qr. to the fraction of a pound. 20. Reduce 15s. Od. 3qr. to the fraction of a pound 21. Reduce lOd. Iqr. to the fraction of a shiUing. 22. Reduce 2s. 9d. 3|^qr. to the fraction of a pound. Direction, Find the number of fifths of a farthing in 8s. 9d. 3f qr., for a numerator; then find the number of fifths of a farthing in £1, for a denominator. 23. Reduce Sj pence to the fraction of a pound. 24. Reduce 5qt. Ipt. to the fraction of a bushel. 25. Reduce 9gal. 3qt. Ipt. to the fraction of Ihhd* 26. Reduce 6 rods 3yd. 2ft. to the fraction of a mile. 27. Reduce 35 -jq seconds to the fraction of a day. When the fraction of a higher denomination is to be reduced to its value in whole numbers of lower denomi- nation^ — Multiply the numerator by that number of the next lower denomination which is required to make a unit of the higher^ and divide the product by the denominator; the quotient will be a whole number of the lower denomi- nation^ and the remainder will be the numerator of a frac- tion. Proceed with this fraction as before^ and so on, 28. Reduce f of £1 to its value in shillings &c. 2 Since f of £ 1 is the same as f of 20 20 shillings, we find f of 20 shillings, »7\^Q in shillings and the fraction of a shil- — ling; — itis 5|shillings. Then, since ^ ^ • I of 1 shilling is the same as ^ of 12 ^^ pence, we find f of 12 pence; — it is 7)60 8f pence. Then, since f of 1 pen- "~^ 4 ny is the same as | of 4 farthings, 4 we find ^ of 4 farthings; — it is 2f — farthings. Thus by finding one de- '^/l^^ nomination at a time, we finally ob- 2f tain, 5s. 8d. 2fqr. 29. Reduce f of ^ 1 to its value in shillings &c. 30. I of £1 is how many shillings, pence, &c..'* 31. In I of a shilling, how many pence, &c..^ 32. Change £15 f to pounds, shiUings, pence, Slc S3. Reduce ^ of Icwt. to quarters, pounds, &c. 28. FRACTIONS. 173 34. Change 9^^ pounds, to pounds, ounces, and drams. 35. Reduce | of a mile to furlongs, rods, feet, &c. 36. In lOf acres, how many acres, roods, rods, &c. 37. How many dimes, cents, and mills, in-/2 of $1 .^ 38. In -j\ of a dollar, how many cents and mills ? 39. Suppose sugar to be $ 12 per hundredweight; what quantity can be purchased for $ 113 .^ Section 28. \ COMMON DENOMINATORS. When two or more fractions have the same number for a denominator, this number is called their Common Denominator, Fractions having different denominators, must be reduced to a common denominator, before ad- dition or subtraction can be performed on them. RULE FOR REDUCING FRACTIONS TO A COMMON DENOMINATOR. Miiliiply each numerator into all the denominators except its own^ for a new numerator. Then multiply all the denominators together for a common de- nominator^ and place it under each new numerator. 1. Reduce |, f 5 and f, to a common denominator. 5 4 6 8 . _9 ^ __8_ 9 45 32 48 72 _7 _I _^ Z. 315 224 432 ^04- 2. Reduce f , 1% , and \ to a common denominator. 3. Reduce \\ and -^-^ to a common denominator. 4. Reduce ^, :! , and -3-^ to a common denominator. 5. Reduce -^, -j, f, and \ to a common denominator. 6. How much is f and y added together ? 7. How much is ^-f and i| added together ? 8. How much is f and | and -{^ added together } . 9. If ^5 be taken from |, how much will remain ? 10. If f be taken from |f , how much will remain } 1 1 . Which is greater, -/g or i\ ? — how much greater ? 174 WRITTEN ARITHMETIC. VI. Section 29. 1. A farmer raised 142 1 bushels of corn m one field, and 237y^2 bushels in another. How many bushels did he raise in both fields ? The learner may reduce the fraction in his answer, to Its lowest terms, in this, and all future examples. 2. A farm is divided into three lots; — the first lot con- taining 46 1- acres, the second 50 f acres, and the third 62yq acres. How many acres are there in the farm.'' 3. Add together 441| and 65^ and 2556^. 4. If 6f be taken from 8 f how much will remain } 5. Subtract 437| from 1659 f. 6. If 6 I gallons of wine should leak from a cask con- taining 53 "I gallons, how many gallons would remain ? 7. Add together 623 1 and 113j^2 5 ^^^^ ^^en subtract from the sum 450 f. 8. Three soldiers shared a loaf of bread as follows: — the first took f of it, the second took yj of it, and the third took the remainder. What fractional part of the loaf did the third soldier receive ? 9. A trader having 25 barrels of flour, sold 8|- barrels to one man, and 9|| barrels to another. What quantity of flour had he then remaining ? Section 30. 1. Suppose I have 16 dollars; to how many men can I give f of a dollar apiece ? 2. How many times is f contained in 16 ? 3. How many pairs of gloves can I buy for 18 dollars, the' price being | of a dollar a pair? 4. Divide 18 by | ; that is, reduce 18 to fourths^ and find how many times 3-fourths is contained therein. 5. Divide 46 by f ; that is, find how many times the fraction f is contained in 46. 6. If a man walk 1 mile in ^% of an hour, what dis- tance will he walk in 4 hours ? 7. How many times is jq contained in 4? 8. How many times is f contained in ^ ? Direction. Reduce f and -^ to a common denomina- tor, and then divide one n imerator by the other. 29. 30. 31. FRACTIONS. 175 9. How many times is -^j contained in -^ ? 10. How many barrels of flour can be bought for 38 dollars, at 4| dollars per barrel ? 1 1 . How many times is 25 f contained in 91 f .'^ 12. How many times is 6| contained in 423-? Direction. Reduce the two fractions to a common denominator; then reduce the mixed numbers to impro- per fractions, and divide one numerator by the other. 13. If a barrel of cider will last a man 3| monlhsj how manv barrels will he drink in lOf months ? Section 31. REVIEW. 1. ( § 13.) Suppose a man to earn 95 cents per day; how many days would it take him to earn $43.16 ? 2. ( § 14.) How many pounds of sugar can be bought for $ 14|, when the price is |^ of a dollar a pound ? 3. (§15.) If 1 yard of silk cord cost -^^ of a dollar, What is the price of 75 yards, at the same rate ? 4. (§16.) Ifl4|- yards of cloth be cut from a piece containing 52 1- yards, how much will be left ? ^* (§ ^'^O ^f ^ pound of shot cost -^q of a dollar, what will be the cost of 17 pounds, at the same rate ? 6. (§18.) How much corn will grow on 140 acres of land; allowing each acre to produce 34 1 bushels ? 7. (§ 19.) If a man's expenses be $46.24yq for -^ of a year, what will be his expenses for 1 year ? 8. (§20.) ^ of 71 is equal to what improper frac- tion ? Then ^ of 71 is equal to what mixed number ? 9. (§21.) Suppose 42 men to share equally in a prize of $ 1000; what is the share of one man .'' 10. ( § 22.) If a man can cut 46 cords of wood in 14 days, how many cords can he cut in 60 days ? 11. (§23.) Suppose a man can build a mile of wall in 310 days; in what time can he build -^ of a mile ? 12. (§24.) A man bought a quantity of flour, for domestic use, and in 36 days he found that ^ of it was consumed. How long would the whole last ? 176 WRITTEN ARITHMETIC. VI 13. (§25.) f is equal to how many ninths.? how many thirty-sixths ? how many seventy-fifths ? 14. (§ 26.) Reduce the two fractions, ^ and ff, to their lowest terms, and then add them together. 15. (§27.) Suppose a farm to contain 98^^^ acres of land; how many acres are there in | of the farm ? 16. (§ 28.) Add together, -^^ and -j^ and \i', then subtract j^ froni the sum; — what is the remainder ? 17. (§29.) If f and -f^ of a number be subtracted from itself, what part of that number is the remainder ? 18. (§ 30.) If the mail-stage run 9^^ "^iles in 1 hour, how many hours will it be in running 175 f miles ? RETROSPECTIVE OBSERVATIOjYS. A fraction is rendered greater by increasing the nu- merator, and smaller by increasing the denominator. To multiply a fraction by a whole number^ — Either multiply the numerator^ or divide the denominator. To divide a fraction by a whole number^ — Either di' vide the numerator^ or multiply the denominator. When a number is multiplied by 1, the product is equal to the multiplicand. Therefore, when a number is multipHed by a fraction, which is less than 1, the pro* duct must be less than the multiplicand. To multiply a whole number by a fraction^ — Multiply by the numerator^ and divide by the denominator. Dividing a number by 1, gives a quotient equal to the dividend. Therefore, dividing a number by a proper fraction, must give a quotient greater than the dividend, because, the fraction being less than 1, is contained a greater number of times in the dividend. To divide a whole number by a fraction^ — Multiply by the denominator,, and divide by the numerator. To multiply a fraction by a fraction^ — Multiply nu- merator by numerator,, and denominator by denominator. To divide a fraction by a fraction,, — Multiply the nu- merator of the dividend by the denominator of the divisor^ for a numerator; and multiply the denominator of the div' idcnd by the numerator of the divisor^ for a denominator 32 MISCELLANEOUS EXAMPLES. 177 19. Multiply the fraction, yV^? t>y 3S- 20. Divide the fraction, y^, by 145. 21. Multiply S706 by the fraction, |f|. 22. Divide 611 by the fraction, -^^Iq . 23. Multiply the fraction, -^ff , by the fraction, -^. 24. Divide the fraction, 2^, by the fraction, ^f . 25. What is the product of 608 f multiphed by 8-^.^ 26 What is the quotient of 45 f divided by 3|? Section 32. MISCELLANEOUS EXAMPLES. 1. Suppose a man can perform a journey in 14 days and 3 hours, travelling 9 hours a day; in what time can he perform the journey, travelling 11 hours a day? 2. A trader gave 5^75 for 56 gallons of wine, and lost 11 gallons by leakage. At how much per gallon must he sell the remainder, to get the whole cost? 3. Suppose a retailer to pay $165 for a ton of sugar, at what price must he sell it per pound, in order to gain 10 per cent, on the cost? 4. What quantity of salt, worth 62 cents per bushel, must be given in exchange for 258 pounds of pork, worth 9 cents per pound? 5. What is the profit on 400 hogsheads of molasses, purchased in New Orleans at 12-^ cents per gallon, [63 gal. in each hhd.], freighted to New York at 5^3.50 per hhd., and sold at 24 cents per gallon; 3gal. 2qt. having leaked from each hhd. on the passage? 6. If a pint of rum a day will kill a man in a year and a half, how many men would^ a cargo of 600 hogsheads kill in the same time? 7. If 1 1 young men can become fools by drinking 6 bottles of wine, at $ 3 a bottle, what would it cost a dinner party of 25, to become fools in like manner? 8. If a man's expenses be $1.40 a day, and his in- come $700 a year, what will he lay up in 7 years? 9 A and B are laborers — A earns $19.50 a month, and B earns $ 16.25; but A gives B -y\ of his earnings What will each lay up in 14 months? 10. Find the difference between | of 91, and |- of 91 32, MISCELLANEOUS EXAMPLES. 179 On the opposite page, 30 cities and towns are exhib- ited in their respective situations, relative to each other; and the number of miles, by mail-road from town to town, is noted in figures. 11. Find the distance from Washington, through the mtermediate towns, to Augusta, Me from Washing- ton to Detroit from Washington to St. Louis from Washington to Natchez from Washington to New Orleans from New Orleans to Augusta, Me. 12. Suppose a citizen in each of the places on the opposite page, to start for Washington, and travel 7 miles an hour, 10 hours in each day; how long will each one be in performing his journey ^ 13. How long would it take you to walk from your school-room to Washington; allowing that you could walk 3^ miles an hour, 7 hours in each day .'^ 14. Two men started at the same time — one of them from New Orleans, and the other from Augusta, Me. — and travelled towards each other, with equal speed. Between what two towns, and what distance from each of these towns did they meet ? 15. Mr. A. went from Portland to Baltimore, travel- ling 5 miles an hour, and 10 hours a day, Mr. B. per- formed the same journey; but started 1 day later, and tiuvelled 7 miles an hour. Where did B. pass A.? 16. Divide $1000 among A, B, and C, giving B twice as much as A, and C twice as much as B. 17. Gunpowder is composed of 5 parts sulphur, 7 parts charcoal, and 38 parts nitre. How many pounds of each ingredient, in 100 pounds of powder ? IS. A and B purchased a cow for $16. A paid $9 of the price, and B paid $7. They sold the cow for $21. What was each one's share of the gain ? Solution, Since A paid -ij of the price, and B -^. A must have -/^- of the gain, and B -f^, 19. C and D traded in partnership; C owned $450 of the stock in trade, and D $290. They gained $ 146 What was each one's share of the gain ? 20. Suppose $1000 stock in trade to gain $230: what is the gain on $351 of that stock .'^ 180 WRITTEN ARITHMETIC. VI 21. E and F purchased 245 acres of land, for $2600. E paid $1200 of the money, and F paid the remainder. How much land should each one have? 22. The national debt of England is not less than $ 1 900000 000. Allowing 5 percent, interest to be paid on this sum, how^ many families would it support, each family spending 5J5400 per annum? 23. If a man can dig a trench in 15 days, and a boy can dig the same trench in 18 days, in what time can they both dig it? (See example 20, Oral sec.) 24. How many days will it take 17 men to perform a {>iece of work, that 1 man can perform in 95 days? 25. How many days will it take 30 men to perform a piece of work, that 4 men can perform in 50 days? 26. How many days will it take 25 men to perform a piece of work, that 6 men can perform in 40 days? 27. If 15' yards of carpeting, which is one yard wide, will cover the floor of a room, how many yards 3f car- peting, 3-quarters wide will cover the same floor? Direction. Find the number of square quarters con- tained in 15 yards of the wider carpeting; then divide this number, by the number of square quarters contained in one yard of the narrower carpeting. 28. Suppose 3^ yards of broad-cloth 5-quarter9 wide, to be made into a cloak; how many yards of silk 3-quarters wide, will it take to line the cloak? 29. How many yards of carpeting that is 5-quarters wide, will cover the floor of a room which is 19 i feet in length, and 15 feet in width? 30. How many bricks will it take to build a vrall, 1 foot thick, 5 feet high, and 24 feet long; each brick be* mg 8 inches long, 4 inches wide, and 2 inches thick? 31. If a man can hoe fg ^f an acre of corn in a day, and a boy ^ of an acre, how much can they both hoe in a day? In what time can they both hoe 9 acres? 32. There is a cistern, having 3 pipes; the first pipe will discharge the cistern in 4 holirs, the second in 5 hours, the third in 6 hours. What part of the contents of the cistern would all the pipes together let ofl^in 1 hour. In what time would they all discharge the cistern^ 32. MISCELLANEOUS EXAMPLE?. IQl 33. What is the height of a steeple, whose shadow is 148 feet 4 inches, when a shadow 5 feet 4 inches long is projected from a post 6 feet 4 inches high ? 34. A trader failed in business, owing $11000, and having only $ 5000 to divide among his creditors. How much did he pay on a debt of $ 95.20 ? 35. A fox has 50 rods the start of a greyhound, but the hound runs 15 rods while the fox runs 9-^. How many rods must the hound run, to catch the fox? 36. A cubic foot of air weighs 1^ ounce. How many pounds of air does a iXDom contain, which is 16 feet long, 14 feet wide, and 10 feet high? 37. What number must that be, which, being increased by its half, and its third, becomes 88 ? 38. A and B hired a pasture for § 30. A turned in 3 cows, and B turned in 12 sheep. Allowing 5 sheep to be equal to 1 cow, what must each pay ? 39. Suppose London has 1 500 000 inhabitants. New York 350 000, Philadelphia 220 000, New Orleans 115 000, Baltimore 110 000, and Boston 105 000; how many times greater is London, than each of the others? When B. scholar has reached this point, it will be well to con- sider how much more time he is likely to devote to study. If he have but a few months more to spend in school, the Supplement will furnish for him tlie suitable exercises, with which to finish his course of study in arithmetic. If, however, he is likely to continue at school for several years, he may omit the Supplement, and enter immediately upon the exercises of Part Third. In the preceding chapters, departments of business are not ar- ranged under distinct heads. Tlie arrangement is strictly anth- Tnetical, and business examples are made incidental to tlie course. £n the Supplement, departments of business are separately pre- sented, in distinct articles. These articles, although brief, arc rendered sufficient, by the learner's previous familiarity with the operations they require. 182 SUPPLEMENT. Article I. f INDICATIVE CHARACTERS OR SIGNS. -}- (Plus,) standing between numbers, indicates that they are to be added together; thus, 3-\-2 is 5. — {Minus,) indicates that the number after it is to be subtracted from the number before it ; thus, 5 — 2 is 3. X (Into,) indicates that one number is to be muUiplied into another ; thus, 4X3 is 12. ~ {By,) indicates that the number on the left is to be divided by the number on the right; thus, 12-7-3 is 4. = (Equal to,) indicates that the number before it is equal to the number after it ; for example, 4-[-2=i6. 6~2z=i4. 5X3=:15. 15~3=i5. CANCELLATION OF FACTORS. The cancellation of factors is the excluding of such factors from an operation as balance each other. Any two equal factors, one being a Hictor of a dividend, and the other a factor of the divisor, or, one a factor of a numerator, and the other of the denominator, may be can- cclled, that is, crossed and omitted. For example, ^ of | of i is reduced to a simple fraction, as follows — Here we cancel the two threes, _L_z?_2_ i and multiply 1 by 1, and 4 by 2. X 4 2 """ ®' When one of two opposite factors will divide the other without a remainder, both may be cancelled, and the quo- tient retained in the place of the factor divided. For in- stance, let us find what is | of |- of {J- ^^^i ^^ ^^^ 2 1 90 XX ^ 1 20 ^X 11 X 1 ■" ^"^ "" ""^^ 3 1. Reduce |- of -| of \^ of § to a simple fraction. 2. What is I of f of I of 1-1 of -X. of \ of 100 ? 3. Reduce f of | of | of -{^ to a simple fraction. When all of a term is cancelled off, the new term must be 1 11,; DECIMALS. 183 4. A merchant owning ^ of |- of f of a ship^ sold f of his share. What part of the ship did he s^ll ? 5. 3 men owned equally a saw-mill ; one sold | of J of I of his share. What part of the mill did he sell ? II. DECIMAL FRACTIONS. A DECIMAL FRACTION IS a fraction whose denominator is 10, or 100, or 1000, &c. The denominator of a decimal fraction is never written : the numerator is written with a point prefixed to it, and the denominator is understood to be a 1, with as many ciphers annexed as there are figures in the numerator. Thus, .3 is -j^^ ; .31 is -^-^^'y .316 is 1. Write upon the slate, the decimals expressing the fnl 1 nwi n o- fi'flPt inn <2 3 46 708 1642 _9_6J)_4JL loiiowmg nactions. -j^. ^^q. -jooo* T(5ooo- Tooooo- When a whole number and a decimal are written to- gether, the decimal point is placed between them. Thus, 24.6 is 24j%; 5.71 is Sf^V; 48.364 is 48^3_6_4_. 2. Write the following mixed numbers, expressing the fractions decimally. SS-j^^. 516^%%. S^V^c. 24^Vo¥o- In whole numbers, any figure, wherever it may stand, expresses a quantity ^^ as great as it would express, if it were written one place further to the left. For instance, in the number 1111, the 1 hundred is -^^ of a thousand ; the 1 ten is -j^^ of a hundred, or j^q of a thousand ; the 1 unit is ^Q of a ten, or j-^^^-^ of a thousand. In decimals, this system is continued below the place of units. For example, in the number 1.111, the ^-S 1 next to the right of the unit is l-tenth, l| that is, j^^ of a unit ; the 1 next to the right .ii f c § of the 1-tenth is -^^ of a tenth, or l-hun- oS^S dredth of a unit; the one next to the right § § g § of the 1-hundredth, is -J^ of a hundredth, 1111 or I'thousandih of a unit. Ciphers placed on the right hand of decimal figures, do not alter the value of the decimal ; because, the figures 184 SUPPLEMENT. II. remain unchanged in their distance from the unites place. For instance, .5, .50, and .500 are of equal value ; being each equal to ^. But every cipher placed on the left of a decimal, renders it ten times smaller, by removing the fig- ures one place further from the unit's place. Thus, if we prefix one cipher to .5, it becomes .05 [j^q] ; if we pre- fix two ciphers, it becomes .005 [to^oI ' ^"^ ^^ ^"' 3. Write upon the slate, decimals expressing the fol- lowing fractions. ^^. ^^^^^, ^f ^^. ^^^%^^. io otooo- To READ DECIMAL FRACTIONS, — Enumer&te and read the figures, as if they were whole numbers, and conclude by pronouncing the name of the lowest denomination, 4. Copy upon the slate, and read the following decimals. .06 .065 .0007 24.02 .008 .409 .06264 5.763084 .013 .207862 .10809 160.052 .0514 .5004 .6500171 712.3005 5. Write in decimals the following mixed numbers. Q 14 7 3 oi_ 'Toooo ^OO^oW ^Toooooo Ofi 15 '^^10000000 ADDITION OF DECIMALS. 6. Add the following numbers into one sum. 63.75 and 524.0764 and .23 and 261.803. 63.75 524.0764 .23 26L803 849.8594 In arranging decimals for addition, we place tenths under tenths, hun- dredths under hundredths, &c. We then begin with the lowest denomina- tion, and proceed to add the columns as in whole numbers. 7. What is the sum of 2.164, 870.31, 756, 9.18, 157.0008, 26.104, and .3728? 8. What is the sum of 2706, 58.2, .2065, 6.441, 75, 14.2, and 990.752? In Federal Money, the dollar is the unit ; that is, dol- lars are whole numbers ; dimes are tenths, cents are hun dredths, and mills are thousandths. See page 124. II. DECIMALS. Ig5 9. Add together $ 24.6, $ 9.07, $ 5.009, and 5 cents. 10. Write the following sums of money in the form of decimals, and add them together. $ 46 and 9 cents, 14 cents, $7 and 8 mills, 6 dimes, 8 dimes and 7 mills. SUBTRACTION OF DECIMALS. 11. Subtract 52.6087 from 406.91. 4Q591 After placing tenths under tenths, 52.6087 ^^'J ^^^ subtract as in whole numbers. ' J. . r. ^ , o The blank places over the 7 and 8, - *^ *' — ^ are viewed as ciphers. 12. Subtract 943.076 from 8270.54. 13. Subtract 1084.72 from 5603.0626. 14. Subtract 146.1706 from 16094. 15. Find the difference between .8 and .08, by sub- tracting the smaller decimal from the greater. 16. Find the difference between .45 and .31067. 17. What is the difference between 1 and .046? 18. Write 4 dollars and 8 mills in decimal form, and subtract therefrom, 6 dimes and 5 mills. 19. Subtract 7 cents and 3 mills from 10 dollars. MULTIPLICATION OF DECI?JALS. Multiplying by any fraction, is taking a certain part of the multiplicand for the product; consequently, mul- tiplying one fraction by another, must produce a fraction smaller than either of the factors. For example, -^^ mul- tiplied by -^Q is -f^Q , or, decimally, .9 multiplied by .8 is .72. Hence you may observe, that the number of decimal figures in any product, must be equal to the number of decimal figures in both the factors. 20. Multiply 531 by .52. 65.7 by .43. 7.06 by .24. .439 by .38. .149 by .26. 531 65.7 7.06 .439 .149 .52 .4 3 .24 .38 .26 1062 1971 2824 3512 894 2655 2628 1412 1317 298 276.12 28.251 1.6944 .16682 .03874 186 SUPPLEMENT. IL RULE FOR MULTIPLICATION OF DECIMALS. Mul- tiply as in whole numbers ; and in the product, point off as many figures for decimals, as there are decimal places in both factors. If the number of figures in the product be less than the number of decimal places in both factors, prefix ciphers to supply the deficiency, 21. Multiply 1608 by .4,— that is, find .4 of 1608. 22. Multiply .45 of a dollar by 8. 23. How much is 36 times .495 of a dollar ? 24. What cost 18 yards of cloth, at $4,072 per yd.? 25. What cost 28.7 yards of cloth, at $ 9 per yd. ? 26. What cost 9.3 acres of land, at $8.41 per acre? 27. If 1 yard of silk cord cost 7 mills, [.007], what is the price of .9 of a yard ? . 28. What is 6 per cent, or .06 of 340.4? 29. Multiply 42.863 by 70.28. 30. Multiply 2046 by .932. 31. Multiply .7253 by .0423. 32. Multiply 6.5431 by .402. 33. What is the product of .04 multiplied by .07? 34. What is the product of .005 by .009 ? 35. Multiply 7 and t>-hundredths by 6-thousandths. DIVISION OF DECIMALS. RULE FOR DIVISION OF DECIMALS. Divide as in whole numbers ; and in the quotient, point off as many figures for decimals, as the decimal places in the dividend exceed those in the divisor; that is, make the decimal places in the divisor and quotient counted together, equal to the decimal places in the dividend. If there be not figures enough in the quotient to point off, prefix ciphers *o supply the deficiency. When there are more decimal places in the divisor, than in the dividend, render the places equal, by annex^ ing ciphers to the dividend, before dividing. After dividing all the figures in the dividend, if there be a remainder, ciphers may be annexed to it, and the division continued. The ciphers thus annexed, must be counted with the decimal places of the dividend. II. DECIMALS. 187 36. Divide 64.395 by 40.5. Divide 5.8674 by 127. 40.5)64.395(1.59 127) 5.8674 (.0462 405 508 2389 787 2025 762 3645 254 3645 254 37. Divide 2033.1 by .324. Divide 1383.2 by 60.8. .324)2033.100(6275 60.8)1383.2(22.75 1944 1216 891 1672 648 1216 2430 4560 2268 4256 1620 3040 1620 3040 38. How many times is .27 contained in 1.224? The sign of addition, or .27) 1.224(4.533+ more, here shows, that the true 108 quotient is more than the pre- 244 ceding figures express. We jgf^ might continually annex ci- — -— phers to this remainder, and ^^ carry on the division, but we should never arrive at a com- 90 plete quotient. For the pur- 81 poses of business, it is seldom 9 necessary to extend the quo- tient below thousandths. 39. How many times is 1.23 contained in 3021.741 ? 40. How many times is 1243.4 contained in 5.37148? 41. How many times is .204 contained in 77112? 42. How many times is 4.2 contained in 194.334 ? 43. How many times is 30.02 contained in 94.657 ? 44. How many times is .44 contained in .1606? 45. What is the quotient of 42.65 divided by 36 ? 46. What is the quotient of .8 divided by 8 ? 188 SUPPLEMENT. H, CHANGE OF COMMON FRACTIONS TO DECIMALS. RULE. Annex ciphers to the numerator, and divide it by the denominator : the quotient will be the decimal. 47. Change -^-^ to a decimal. By annexing four ciphers, we ob- 12) 7.0000 ^^\j^ fQuj. decimal figures. We might, .5833+ however, annex more ciphers, and carry the decimal still lower. 48. Change ^ to a decimal. 49. Change the following fractions to their respective Hppimak 2 1 3 13 5 _5_ 14 26 aecimais. .3-. 4. 4. ^o • TF' 12' TT' ^Yl' 50. Change -j^^ of a dollar to a decimal ; that is, find how many cents and mills there are in ^ of a dollar. 51. Change $48| to a decimal expression. 52. Change <^316-^| to a decimal expression. CHANGE OF COMPOUND NUMBERS TO DECIMALS. To reduce the lower denominations of a compound number to the decimal of a higher .denomination. RULE. Reduce the given quantity to a common fraction, then change this fraction to a decimal. See page 171. 53. Reduce 7s. 6d. to the decimal of a £. 54. Reduce 15 shillings to the decimal of a £. 55. Reduce 6d. 3qr. to the decimal of a shilling. 56. Reduce 2s. lid. 3qr, to the decimal of a £. 57. Reduce 1 farthing to the decimal of a shiUing. 58. Reduce £ 18 2s. 7d. to a decimal expression. 59. Reduce I4dwt. 18gr. to the decimal of an oz.Troy 60. Reduce 4qt. Ipt. to the decimal of a bushel. 61. Reduce 3qt. Ipt. 2gl. to the decimal of a gallon. 62. Reduce lOr. 3yd. 2ft. to the decimal of a mile. 63. Express 29yd. 2qr. 3na. of cloth decimally, and find its cost, at $7,625 per yard. CHANGE OF DECIMALS TO COMPOUND NUMBERS. To reduce the decimal, of a higher denomination, to its value in whole numbers of lower denomination. RULE. Multiply the decimal by that number of the next Hi DECIMALS. 109 lower denomination, which makes a unit of the higher^ and the product will be of the lower denomination. Proceed thus with the decimal in each succeeding product. 64. Reduce .6526 of a .£ to its value in shillings, &c. .6526 We multiply the decimal of a ^ 20 by 20, to find the shillings, because, 13.0520 there are 20 times more shillings J 2 than pounds in any sum, whether the ^ sum be a whole number or a decimal. .0^40 rp^g g^jjjg reasoning also applies, in Z finding the pence, and the farthings. 2.4960 Answer, 13s. Od. 2qr.+ 65. Reduce .4039 of a <£ to its value in shillings, &c. QQ. Reduce .857 of a shilling to pence and farthings. 67. Reduce .76 of a ton to cwt. qr. lb. &;c. 68. In .2094 of a day, how many hours, minutes, &c. ? 69. In .57 of an acre, how many roods, rods, &c. ? 70. Reduce .£15.2908 to its proper expression in pounds, shillings, pence, and farthings. EXCHANGE OF CURRENCIES. In New England, Virginia, Kentucky, and Tennessee, ■| of a dollar is called a shilling. In New York and North Carolina, | of a dollar is called a shilling. In Pennsylvania, New Jersey, Delaware, and Maryland, j^y of a dollar is called a shilling. In South Carolina and Georgia, -j\ of a dollar is called a shilling. In Canada, | of a dollar is called a shilling. In Great Britain, the shilling, of the Sterling currency, IS equal to f of a dollar. 71. How many cents and mills, that is, what decimal of a dollar, in a New England shilling ? in 2 shillings ? in 3 shillings? in 4 shillings? in 5 shillings? 6) 1.000 6) 2.000 , .i^f ^^^ """^^^^ expressing ifi^ qoqi shilhngs, expresses an equal •ioo^ .666-^ value in decimals of a dollar. 190 SUPPLEMEJNT. It. 72. How many cents and mills in a New York shil- ling ? in 2s. ? in 3s. ? in 4s. ? in 5s. ? in 6s. ? in 7s. ? 73. How many cents and mills in a Pennsylvania shil- ling ? in 2s. ? in 3s. ? in 4s. ? in 5s. ? in 6s. ? lX^=.133i '2xf5 = .266| 74. How many cents and mills in a Georgia shilling? in 2s. ? in 3s. ? in 4s. ? 75. How many cents and mills in a Canada shilling? in 2s, ? in 3s. ? in 4s. ? in 5s. ? 76. How many cents and mills in a shilling, Sterling, of Great Britain ? in 2s. ? in 3s. ? in 4s. ? To change the currencies of pounds, shillings and pence, of every variety of value, to Federal money. RULE. Reduce the pounds, if there be any, to shillings. Denote the shillings as units, reduce the pence and far- things to the decimal of a shilling, and multiply the sum by that fraction of a dollar which is equal to one shilling. 77. Change 13s. 6d., of the old currency of New Eng- land, to Federal money. 13s. 6d. = 13.5s. 13.5 X i = 2.25. 78. Change ,£42 19s. 4^d., of the old currency of New England, to Federal money. £42 19s. 4 id. = 859.3758. 79. Change 13s. 6d., of the old currency of New York, to Federal money. 80. Change £ 25 17s. 8 1 d., of the old currency of New York, to Federal money. 81. Change 18s. lid., of the old currency of Pennsyl- vania, to Federal money. 82. Change ,£ 14 7s. 6 ^ d., of the old currency of Penn- sylvania, to ^cjderal money. 83. Change 16s. lOd., of the old currency of Georgia, to Federal money. 84. Change .£54 12s. l^d., of the old currency of Georgia, to Federal money. 85. Change £21 9s. 3|d., of the currency of Canada, to Federal money. 86. Change ,£ 5 12s. 4d. Sterling, of Great Britain, to Federal money. in PERCENTAGE. 191 III. PERCENTAGE. Percentage has already been explained iii page 163. Since per cent, indicates hundredths, it is properly ex- pressed in the first and second decimal places, taken to- gether. Thus, 6 percent, is .06 ; 12 percent, is .12. A fraction of 1 per cent, is expressed in decimals lower than hundredths. Thus, ^ per cent, is .005 ; \ per cent, is .0025 ; 6^ per cent, is .065; 12 1 per cent, is .1275. Multiplying by a decimal, produces such a part of the multiplicand, as the decimal indicates. Therefore, — To FIND THE Percentage on any sum, — Multiply the sum by the decimal which denotes the rate per cent, 1. A merchant having $1426 in the bank, drew out 5 per cent, of it. What sum did he draw ? 14 26 Since 5 per cent, of any quantity is -j^q of .0 5 that quantity, the question in this example is, t;l , on ^^^^^t is -j§^ of 1426 dollars ? Or, decimally, ^^ ^"^^ What is .05 of 1426 dollars ? 2. V/hat is 1 per cent, of $ 100? of $834 ? 3. What is 3 per cent, of $ 100 ? of $42? 4. What is 7 per cent, of $ 100 ? of $ 1085 ? 5. What is 9 per cent, of 354 dollars ? 6. What is 24 per cent, of 1852 dolIai*s? When the rate is a ii-action of 1 per cent. — First, re- duce the rate to a decimal, by multiplying .01 by the frac* tion. Then multiply by the decimal rate as before. 7. What is | per cent, of 234 dollars ? .01 X I— .0075. Then 234x .0075 nn 1.755. 8. What is \ per cent, of 524 dollars ? 9. What is | per cent, of 190 dollars? 10. What is 2i per cent. [.025] of 50 dollars? 11. What is 6i per cent, of 75 dollars? 12. What is 10 1 per cent, of 2C0 dollars? 192 SUPPLEMENT. Ill 13. Find 7^ per cent, of 344 dolls. When there is a fraction in the 3444-100=3.44 rate per cent, which cannot be ex- 7^ actly expressed by a decimal^— as ^408 in this example — we first find 1 per 114-^ cent, of the given sum, by divid- ¥qTo"o2 ing it by 100, and then multiply this ^x2 0.^^3 quotient by the mixed number that expresses the rate. 14. What is 4 1 per cent, of 624 dollars ? 15. What is 6f per cent, of 38 dollars ? 16. What is 3i per cent, of 2310 dollars? 17. What is 9| per cent, of 17 dollars? 18. What is 7 per cent, of 24 dolls. 32 cts. ? Here we have cents [decimals] in the num- 24 3 2 ber on which the percentage is to be taken. *q^ We however multiply as usual in decimals, r — and the first two decimal figures in the prod- ^ ^ '* ^z^"* uct express cents, the third mills, the fourth tenths of ^. mill* 19. What is 14 percent, of S641.94? 20. What is 4} percent, of $37.26? 21. What is 11 -J- per cent, of 1 150.75 ? To find what per cent.*a smaller number is of a larger,—* Consider the smaller number as a numerator, and the larger as a denominator of a fraction ; then reduce this fraction to a decimal. See page 188. 22. If a man, having $ 94 deposited in bank, draw out $ 25, what per cent, of his deposit does he draw ? 25 is fl of 94. Thenf|z=.26f|-. Ans. 26 |f per cent. 23 What per cent, of 240 dollars is 32 dollars? 24. What per cent, of 12 dollars is 7 dollars ? 25. What per cent, of $95.21 (9521 cts.), is §4.22? To find a percentage of a compound number, — Multi- ply by the rate per cent., as a ivhole, or mixed number, and divide the product by 100, or the factors of 100. 26. What is 6 per cent, of ^22 10s. 9d. ? 27. What is 4 per cent, of .£41 15s. 6d. ? 28. What is 3^ per cent, of £8 16s. 8d. ? Ill PERCENTAGE: 193 COMMISSION. Commission is the compensation made to factors and brokers for their services in buying or selling. It is rec- koned at so much per cent, on the money employed in the transaction. 29. What is the commission on 8500, at 2-^ per cent. ? 30. If I allow my factor a commission of 3 per cent, for disbursing 725 dollars 50 cents, on my account, what does his commission amount to ? 3 1 . How much does a broker receive for his services on a sale of stocks amounting to 52648 dollars, allowing his commission to be ^ of 1 per cent. ? STOCKS. Stock is a property, consisting in shares of some estab- lishment, designed to yield an income. It includes gov- ernment securities, shares in incorporated banks, insurance offices, factories, canals, rail-roads, &c. The par value of a share, is what it originally cost ; and the real value, at any time, is what it can be sold for* When it will sell for more than it originally cost, it is said to be above par, and the excess is stated at so much per cent, advance. When its real value is less than the original cost, it is below par, and is sold at a discount. 32. Sold 10 shares in the Manufacturers' Insurance Company, at 5 per cent, advance, the par value of a share being 100 dollars. How much did I receive? 33. Bought 15 shares in the Boston Bank, at | of 1 per cent, advance, the par value being 50 dollars a share. How much did I give for them ? 34. Sold 64 shares in the State Rail-road, at \\ per cent, discount, the par value being 100 dollars a share. How much did I receive for them ? INSURANCE. Insurance is security given, to restore the value o' ships, houses, goods, ^c, which may be lost at sea, c R 194 SUPPLEMENT. lY by fire. The security is given in consideration of a j^rc mium paid by the owner of the property insured. This pretnium is a percentage on the value of the property. The written instrument, which is the evidence of the contract of indemnity, is called a policy. 35. What is the amount of premium for insuring 19416 dollars, at 2 -^- per cent., on a ship from Liverpool ? 36. I effected an insurance of 3460 dollars on my dwelling-house for one year, at | of 1 per cent. What did the premium amount to? 37. If you obtain an insurance on goods valued at $7325, at i of I per ceiit., what will the premium amount to ? iv. INTEREST. Interest has already been defined, and rules f6r com-^ puting it without decimals have been given, in Chap. VI., Sect. 23. The rules are repeated in this article, with such modifications as provide for the use of decimals. To compute interest for one or more years. RULE. Multiply the principal by the decimal that ex-^ presses the rate, and the product will he the interest for 1 year. Multiply the interest for one year by the number of years o 1. Find the interest of $ 87.41 j for 3 years, at 6 per cent* 87.41 X .06 X 3=15.7338. Ans. $ 15.73+ In the answers, fractions of a cent, may be omitted. 2. Find the interest of f 644,^ for 4 years, at 6 per cent, 3. Find the interest of 92 cents, for 7 years, at 6 percent. 4. Find the interest of f 7.50, for 2 years, at 4 per cent* 5. Find the interest of i 2.91, for 3 yeai^, at 4 ^ percent. 6. Find the interest of $ 9.53, for 4 years, at 5 1 per cent. 7. What is the interest of $752.25, for 3 years, at 5 per cent. ? What is the amount ? 8. What is the interest of £ 16 8s. 6d., for 1 year, at 6 per cent. ? What is the amount ? iV. INTEREST. 19(5 To compute interest when there are months in the time. RULE. First find the interest for the years , if there be any. Then take ^^^ of a yearns interest for 1 month ; T2 ^^ 6 fi^ ^ months ; -^^ or \ for 3 months ; and so on, 9. What is the interest of 224 dollars for 7 months, at 6 per cent, per annum ? 10. What is the interest of 75 dollars and 50 cents, for 5 months, at 6 per cent. ? 11. What is the interest of 145 dollars, for 1 year and 3 months, at 6 per cent. ? 12. What is the interest of 95 dollars and 25 cents, for 2 years and 8 months, at 5 per cent. ? 13. What is the interest of $351.09, for 3 years and 9 months, at 7 per cent. ? What is the amount ? To compute interest, when there are days in the time. RULE. First find the interest for the years and months , if there he any. Then take -^^ of a month's interest for 1 day ; f^ or ^-^ for 2 days ; -^^ or -^^ for 3 days ; and so on. 14. What is the interest of $ 1000 for 1 year, 1 month and 1 day, at 6 per cent. ? 15. What is the interest of $356.75 for 8 months and 10 days, at 6 per cent. ? 16. What is the interest of $76.81 for 5 years, 2* months and 18 days, at 4 per cent. ? 17. What is the interest of $ 250 for 1 year and 29 days, at 6 per cent. ? What is the amount ? 18. What is the amount of $92.86 for 3 years, 7 months and 14 days, at 7 per cent. ? 19. What is the interest of $ 175.63, from May 19, 1842, to January 4, 1844, at 6 per cent. ? We find the time between the two 1844 1 4 dates by subtracting the first from the 1842 5 19 last, as in compound subtraction ; the 7 YTE months being denoted numerically. 20. What is the interest of $208.90, from June 2, 1843, to August 4, 1845, at 5i per cent. ? 21. What is the interest, at 6 per cent., on a note of $110, dated Sept. 7, 1843, and paid July 9, 1846? What is the amount ? 196 SUPPLEMENT. IV PARTIAL PAYMENTS. In the settlement of notes, which have been partly paid, at dates previous to the settlement, the common method of computing the interest operates justly in cases where inter- est has not been running for more thtm one year. But for longer periods of interest, this method is not strictly just to the creditor, and ought not to be adopted. THE COMMON METHOD. Compute the interest on the original debt, from the date when interest commenced to the date of the settlement ; also, compute the interest on each payment, from the date of the payment, to the date of the settlement. Then, subtract the amount of all the payments from the amount of the original debt, and the remainder will be the balance due. The United States' Court, and the Courts of the several States, in which decisions have been reported — with the exception of Connecticut, Vermont, and New Jersey — have established a uniform legal rule for the computation of interest, when partial payments have been made. THE LEGAL RULE. Compute the interest on the principal of the note to the earliest date when a payment was made, which, either alone, or together with preceding payments, exceeds the interest then due. Add this inter^ est to the principal, and from the sum. subtract the pay- ment or payments thus far made. The remainder be- comes a new principal, with which proceed as with the principal of the note. (22.) Boston, January 14th, 1843. For value received^ I promise to pay Wm. Rich, or or- der, one hundred and forty-one dollars and eight cents, in three months, with interest after. John Lang. On the back of this note were the following endorsements. May 1st, 1843, received seventy-five dollars. September 14th, 1843, re- ceived forty-five dollars. What is the balance Jan. 14th, 1844, the interest being 6 per cent., computed by the common method? First payment, $ 75. Int, 8 m. 14 d., 3.17 Sndpayt. $45. Int., 4 m. .90 Amount, $78.17 Amount, $45.90 78.17 Amount of payments, $124.07 Principal, $ 141.08 Int., 9 m. 6.34 Amount, 147.42 124.07 Balance, $2a35 IV. INTEREST. 197 (23.) New York, May 25th, 1843. For value received, I promise Joseph Day to pay him or order, the sum of three hundred and one dollars and forty-seven cents, on demand, with interest. Attest, John Smith. Samuel French. On the back of this note, the following endorsements were made. July 1st, 1843, received sixty-seven dollars and fifty cents. Janu- ary 4th, 1844, received forty-eight dollars. April 11th, 1844, re- ceived thirty-nine dollars. What is the balance, June 21st, 1844 ; interest being 6 per cent, computed by the common method ? (24.) Philadelphia, March 4th, 1842. For value received, I promise to pay to the order of Harper h Jones, one thousand two hundred dollars, on demand, with interest. Charles Train. The following endorsements are on this note. June 10th, 1842, received one hundred sixty-nine dollars and twenty cents. Oct 22d, 1842, received twenty dollars. March 30th, 1843, received twenty-eight dollars., Nov. 5tli, 1843, received six hundred eigh- teen dollars and five cents. If 6 per cent interest be computed by the legal rule, what is tlie balance due, March 5th, 1844 ? Principal, - $1200. Interest from Mar. 4, to June 10, (3 m. 6 d.), - - 19.20 First Amount, - - 1219.20 First payment, 169.20 Balance, forming a new principal, - _ . . 1050.00 Interest from June 10, to Oct 22, (4 m. 12 d.), $23.10 Second payment, 20. Leaving interest unpaid, - . - _ 3.10 Interest from Oct 22, to Mar. 30, (5m. 8 d.), 27.65 30.75 Third payment, 28.00 Leaving interest unpaid, _ - - - 2.75 Interest from Mar. 30, to Nov. 5, (7 m. 6 d.), 37.80 40.55 Second Amount, - - 1090.55 Fourth payment, - - - - - - - 618.05 Balance, forming a new principal, - - - - 472.50 Interest from Nov. 5, to Mar. 5, (4 m.), - - - 9.45 Balance due on taking up the note, - - - $ 481.95 (25.) New Orleans, January 1, 1841. For value received, I promise to pay William Lock or order, one thousand dollars,' on demand, with interest, 6 per cent. Edward Smith. 19S SUPPLEMEISIT. IV Five partial payments are endorsed on Smith's note : viz. Fe^. Ist, 1842, received seventy- five dollars. June 1st, 1842, received twenty dollars. August 1st, 1843, received twenty dollars. October 1st, 1843, received seven hundred and fifty dollars. Feb. 1st 1844, received one hundred dollars. The balance of this note was paid June 1st, 1844. How much was it, by the legal rule ? COMPOUND INTEREST. Compound Interest is that which is paid not only for the use of the principal, but also for the use of the interest after it becomes due. The period of interest, that is, the term of time at the end of which interest is due, may be a year, a quarter, or any other term agreed upon. What- ever be the period, the following rule is applicable. RULE. Find the amount for the first period^ and consider it the principal for the second period; find the amount for the second period, and consider it the prin- cipal for the third period ; and. thus proceed through the whole number of periods. Subtract the first principal from the last amount, and the remainder will be the compound interest, 26. What is the compound interest of $ 100 for 3 years, at 6 per cent. ; the interest being due annually ? 1st year. 2nd year. 3rd year. Answer. 100 106 112.36 119.10 + .06 .06 ^06 10 6.00 6.36 6.7416 $19.10 + 100 106 112.36 106.00 112.36 119.1016 27. Wliat is the compound interest of 355 dollars, for 6 years, at 6 per cent, per annum ? 28. What is the compound interest of 250 dollars, for 4 years, at 7 per cent, per annum ? 29. To what sum will 450 dollars amount, in 5 years, at 5 per cent, per annum, compound interest ? 30. At compound interest, what will 600 dollars amount to in 1 i year, at the rate of 6 per cent, a year, interest payable quarterly ? V. VI BANKING. 199 DISCOUNT. Discount is sufficiently defined in page 168; and we have now only to apply decimals to the operations. RULE. Divide the debt by the amount of 1 dollar for the time, and, the quotient is the present worth. Subtract the present worth from the debt, and the remainder ivill be the discount, 1. What is the present worth of 125 dollars, due in 18 months, when interest is 6 per cent, per annum ? $1 amounts to $1.09. 125^1.09=114.66+ 2. What is the present worth of $ 456, due in 15 months, when money is worth 5 per cent, per annum ? 3. What is the discount on 3465 dollars for 6 months, when interest is 7 per cent, a year ? 4. What is the present value of a note for 2448 dollars and 50 cents, payable in 8 months, when interest is 6 per cent, per annum ? VI. BANKING. The interest on money hired from a bank is paid when the money is taken out. That is, the bank computes the interest on the principal of the note it receives, to the time the note is to be paid, deducts this interest from the princi- pal, and advances the remainder to the hirer. Hence, bank mterest is called discount; and the note received by the bank is said to be discounted. Bank discount is always computed for three days — Q^Wedi days of grace — more than the time stated in the note for payment; and the hirer is not required to pay. until the last of these three days. 1. Find the bank discount on $585 for 60 days and grace. (2^^^ months,) at the rate of 6 per cent, a year 200 SUPPLEMENT. VH 2. What is the bank discount on 900 dollars for 90 days, and grace, at the rate of 6 per cent, a year ? 3. How much is received on a note for 2540 dollars 80 cents, payable in 4 months, and grace, discounted at a bank, when interest is 4^ per cent, a year ? 4. A note for 452 dollars, payable in 7 months, and grace, is discounted at a bank, when interest is 6 per cent, per annum. What sum is received on it ? VIT. PROFIT AND LOSS. The ascertaining what is gained or lost in buying and selling, and the adjusting of the price of goods so as to gain or lose a certain sum, or a certain per cent., come un- der the head of Profit and Loss, 1. Bought a piece of broadcloth containing 28 yards for 112 dollars, and sold it at 5 dollars 25 cents a yard. How much, and what per cent., was my profit ? (See Percent- age, Art. III., Example 22.) 2. Bought 3 pieces of broadcloth, containing 28 yards each, at 5 dollars 25 cents a yard. At what price per yard must I sell it, to gain 20 per cent. ? 3. Bought cloth at 4 dollars 60 cents a yard, which, not proving so good as I expected, I sold at 3 dollars 91 cents a yard. What per cent, did I lose ? 4. Bought 1250 barrels of flour for 6250 dollars. At what price per barrel must I sell it, to make a profit of 12^- percent.? 5. Bought wheat at 75 cents a bushel ; at what price per bushel must I sell it, to gain 20 per cent. ? 6. A merchant received from Lisbon 180 casks of rai- sins, containing 80|-lb. each, which cost him 2 dollars 18 cents a cask. At what price per cwt. must he sell them, to gain 25 per cent. ? 7. If I sell sugar at 8 dollars per cwt., and thereby lose 12 per cent., what per cent, do I gain or lose, bv selling the same at 9 dollars per cwt. ? VIII PARTNERSHIP. 201 VIIL PARTNERSHIP. Partnership is the union of two or more individuals in trade. The company thus associated is called a firm : and the amount of property, which each partner puts into the firm, is called his stock in trade. When each partner's stock is in the firm an equal length of time, the profit or loss is shared in proportion merely to each one's stock. But when the stock of the several part^ ners is employed unequal terms of time, the profit or loss is shared in proportion both to stock and time, 1. A, B, and C entered into partnership, and the stock of each was in the firm one year. A put in 240 dollars, B 360 dollars, and C 120 dollars. They gained 350 dollars. What was each partner's share of the gain ? Solution. The whole capital $720. A's stock was $240, and he must have f|§ of the gain. B's stock was $360, and he must have ||g. C's stock was $ 120, and he must have ^|§. Observe the following statement. 24 --2 . and f of $350. is 116 f dollars, A's share. 3 60 -.3 . and I of $350. is 175 dollars, B's share. 120 =1-^; and | of $350. is_58j- dollars, C's share. $350 Proof. 2. W, S, and L formed a connexion in business. W put into the firm 2500 dollars ; S 2000 dollars ; and L 1500 dollars. The stock of the several partners was m trade the same term of time, and they gained 1500 dol- lars. What was each partner's share of the profit ? 3. A, B, C, and D traded together one year. A put in 800 dollars, B 500 dollars, C 300 dollars, and D 150 dollars ; but by misfortune they lost 350 dollars. What loss did each partner sustain ? 4. Three merchants bought a ship, for 8000 dollars. A paid 2850 dollars, B 1980 dollars, and C the rest. In her first voyage she cleared 6400 dollars. How much of the profit had each partner ? 202 SUPPLEMENT. VIII. When the time is unequal, we compute on the prin- ciple, that $1 for 2 months is equal to $2 for 1 month. For example, A, B, and C traded in company ; A put in $200 for 3 months, B $180 for 5 months, and C $70 for 10 months : they gained $ 132. Now we say, that A's $200 for 3 months was the same as $600 for 1 month; B's $ ISO for 5 months the same as $900 for 1 month; and C's $70 for 10 months the same as $700 for 1 month ; therefore the relation is the same as if A had put in $600, B $900, and C $700, all for an equal term of time. These sums added together make $2200; therefore, A had ^6_o_o^ of the gain, B ^%%, and C ^.}^. These fractions, reduced, are ^^, -^-^^ and ■^■^, J^ of $132 is $6; then A had 6 times $6, B 9 times $6, and C 7 times $6. RULE. Multiply each partner^ stock by the time it was in the firm ; make each product the numerator of a fraction^ and the sum of the products a common denomi- nator ; then multiply the whole gain or loss by each of these fractions, for each partner^ s share. 5. A, B, and C traded in company. A put in 400 dollars for 9 months, B 300 dollars for 6 months, and C 200 dollars for 5 months : they gained 320 dollars. What was the gain of each ? 6. X, Y, and Z formed a partnership. X put into the firm 500 dollars for 18 months, Y 380 dollars for 13 months, and Z 270 dollars for 9 months ; but they lost 818 dollars 50 cents. What was the loss of each ? 7. Gould and Davis entered into partnership for one year. Gould's stock, at first, was only 500 dollars, but at the end of 5 months he put in 150 dollars more. Davis's stock, at first, was 600 dollars, but at the end of 9 months he took out 200 dollars : at the end of the year, it was found they had gained 682 dollars 50 cents. What was the gain of each partner ? 8. Three farmers hired a pasture at 60 dollars 50 cents for the season. A put in 5 cows 4^ months, B 8 cows 5 months, and C 9 cows 6^ months. What rent did each pav ? IX, ASSESSMENT OF tAXES. 203 IX. ASSESSMENT OF TAXES. Taxes are imposts paid by the people for the support of government. Tiiey are assessed on the citizens in proportion to their property ; except the poll tax, which is assessed by the head without regard to property. An inventory of the taxable property of every citizen is the first thing to be obtained by the assessors. When a tax is on property and polls, we deduct the amount which the polls pay from the sum to be raised, and apportion the remainder according to each man's property. To effect the apportionment, we find what per cent, of the whole property to be taxed, the sum to be raised is; and if we multiply each man's inventory by that per cent, expressed in decimals, the product is his tax. Assessors find it more expedient, however, to make a table, which shall exhibit at once the tax on all sums, from $ 1 up to any amount required. The table is made by multiplying the per cent, which the tax amounts to, by the several numbers, 1, 2, 3, 4, and so on. The following is a table of taxes to be made when 1^ per cent, is to be raised on the valuation of property. $1 pays .015 $20 pay .30 $200 pay $3.00 2 " .03 30 " .45 300 " 4.50 3 " .045 40 " .60 400 " 6.00 4 " .06 50 « .75 500 " 7.50 5 « .075 60 " .90 600 " 9.00 6 « .09 70 " 1.05 700 « 10.50 7 « .105 80 " 1.20 800 " 12.00 8 " .12 90 " 1.35 900 « 13.50 9 " .135 loo " 1.50 1000 " 15.00 10 " .15 1. By the al Jove tabi e, what \v luld be the tax on $6425 real estate, and $2346 personal estate ? 2. By the above table, what would be the tax of a freeholder, whose real estate is valued at $9842, and personal estate, at $ 1 5066 ; poll tax $ 1.25 ? 204 SUPPLEMENT X. X. RATIO, PROPORTION, RULE OF THREE. Ratio Is the mutual relation of two numbers to one another. By finding how many times one number is con tained in another, or what part one number is of another, we obtain their ratio. Thus, the ratio of 2 to 4 is 2, be- cause 2 is contained 2 times in 4 ; and the inverse ratio is |, because 2 is | of 4. Both these expressions of the ratio of 2 to 4 amount to the same thing, which is, that one of the numbers is twice as great as the other. A ratio is denoted by two dots, similar to a colon : thus, 3 : 9 expresses the ratio of 3 to 9. The former term of a ratio is called the antecedent, and the latter the consequent. Thus, 6 : 12 expresses the ratio of 6 to 12, in v/hich 6 is the antecedent, and 12 the consequent. Since a ratio indicates how many times one number is contained in another, or what part one number is of another, it is a quotient, resulting from the division of one of the terms of the ratio by the other, and may be ex- pressed in the form of a fraction : thus, the ratio 6 ; 3 may be expressed, by the fraction |, or conversely |. The equality of two ratios is /Called a Proportion ; and the terms are called proportionals. Thus, 2:4 = 3:6 express a proportion, signifying, that the ratio of 2 to 4 is equal to the ratio of 3 to 6. In a proportion, the first and fourth terms, that is, the antecedent of the fii*st ratio and the consequent of the second, are called the extreme terms ; and the second and third terms, that is, the consequent of the first ratio and the antecedent of the second, are called the mean terms. Thus, in the proportion 3 : 9 = 4 : 12, 3 and 12 are the extreme terms, 9 and 4 the mean terms. It is to be observed that, if four numbers be in propor- tion, the product of the extreme ter^ns is equal to the product of the mean terms. X. proportiOjV. 205 Since the product of the extremes in every proportion is equal to the product of the means, one product may be taken for the other. Now, if we divide the product of the extremes by one extreme, the quotient is the other extreme ; therefore, if we divide the product of the means by one extreme, the quotient is the other extreme. To apply these principles to practice, let it be asked — If 64 yards of cloth cost 304 dollars, what will 36 yards cost ? In the first place, the ratio of the two pieces of *?loth is 64 : 36 ; and secondly, the prices are in the same ratio ; that is, 304 dollars must have the same ratio to the price of 36 yards, that .64 yards have to 36 yards. Now, if we put A. instead of the answer, we shall have the fol- lowing proportion, 64 ; 36 =3 304 : A. Here, the product of the means is 10944, which, divided by 64, one of the extremes, gives the quotient 171, the other extreme, which was the term sought, and the answer. Of the four numbers in a proportion, two are of one kind, and two of another. In the preceding example, two of the terms are yards, and two are dollars. From the principles of ratio and proportion, we deduce The Rule of Three — an ancient rule, by the opera- tion of which, having three numbers given, we find a fourth, which has the same 'ratio to the third that the second has to the first. RULE OF THREE. MaJce the number, tuJiich is of the same kind ivith the answer, the third term. And if, from, the nature of the question, the fourth term or answer must be greater than the third term, maJce the greater of the tioo remaining terms the second term, and. the smaller the first; but, if the fourth term must he less than the third, make the less of the two remaiMng terms the second, term, and. the greater the first. Multiply the second, and third terms together, and divide the product by the first teim: the quotient will be the fourth term., or answer. If there are different denominations In the first two terms, they must both be reduced to the lowest denomination in either of them ; and the third term must be reduced to the lowest denomination mentioned in it. 206 SUPPLEMENT. X Operations corresponding to the Rule of Three have already been taught, in Relations of Numbers, Chap. VI. To show the correspondence, suppose it to be asked — If 3 yards of cloth cost 4 dollars, what will 9 yards cost ? In Relations of Numbers, the question stands thus — What is 9 times -J- of 4? 3)^ 1 9 H 12 Ans, In the Rule of Three, the question stands thus — 3:9 — 4: what number ? 3 :9=:4 : A _9 3 )36" 12 Ans. 1. If I buy 871 yards of cotton cloth for 78 dollars 39 cents, what is the price of 29 yards of the same ? 371 : 29^:78.39 : A The statements of this 29 ' question may be read thus 7 0551 — The ratio of 871 to 29 15573 is equal to the ratio of QT 1 \oo-yo oi /o ^ 1 .f 78.39 to the answer. Or 871 )2273.3 1(2.6 1^47/5. . . Q-y, i • . on ^ TjACf ^ thus — As 871 yd. is to 29 LLijL, yd., so is $78.39 to the ^313 answer. The operation 5226 amounts to nothing more 871 than the multiplication of 871 78.39 by eVr . 2. If 1-| yard of cotton cloth cost 42 cents, what will 87^ yards cost, at the same price per yard ? 1.75 : 87.5 :=: .42 : A 3. If I can buy li yard of cotton cloth for 6\ pence, how many yards can I buy for £10 6s. 8d. ? 6d. Iqr. : ^10 6s. 8d.= 1yd. Iqr. : A 4. If I buy 54 barrels of flour for 297 dollars, what must I give for 73 barrels, at the same rate ? 5. If 7 workmen can do a piece of work in 12 days, how many can do the same work in 3 days ? 6. If 20 horses eat 70 bushels of oats in 3 weeks, how many bushels will 6 horses eat in the same time ? 7. If a piece of cloth containing 76 yards cost 136 dol- lars 80 cents, what is that per ell English ? X. PROPORTION. 207 8. If a staff 4 feet long cast a shadow 7 feet in length, on level ground, what is the height of a steeple, whose shadow at the same time measures 198 feet ? 9. How many yards of paper, 2^ feet wide, will hang a room, that is 20 yards in circuit, and 9 feet high ? 10. A certain work having been accomplished in 12 days, by working 4 hours a day, in what time might it have been done by working 6 hours a day ? 11. If 12 gallons of wine are worth 30 dollars, what is the value of a cask of wine, containing 31^ gallons? 12. If 8 1 yards of cloth cost 4 dollars 20 cents, what will 13^ yards cost, at the same rate? 13. How many yards of cloth ^ yard wide, are equal to 30 yards 1^ yard wide ? 14. If 7 pounds of sugar cost 75 cents, how many pounds can I buy for 6 dollars ? 15. If 2 pounds of sugar cost 25 cents, and 8 pounds of sugar are worth 5 pounds of coffee, what will 100 pounds of coffee cost ? 16. A merchant owning f of a vessel, sold f of his share, {j X ^,) for 957 dollars. What was the vessel worth, at that rate ? 17. A merchant failing in trade, owes 62936 dollars 39 cents ; but his property amounts to only 38793 dollars 96 cents, which his creditors agreed to accept, and discharge him. How much does the creditor receive, to whom he owes 2778 dollars 63 cents ? 18. Bought 3 tons of oil, for 503 dollars 25 cents ; 85 gallons of which having leaked out. I wish to know at what price per gallon I must sell the residue, that I may neither gain nor lose by the bargain. 19. If, when the pried of wheat is 6s. 3d. a bushel, the penny loaf weighs 9 oz., what ought it to weigh, when wheat is at 8s. 2|d. a bushel ? 20. If 15 yards of cloth | yard wide cost 6 dollars 25 cents, what will 40 yards, being yard wide, cost ? 21. Borrowed of a friend 250 dollars for 7 months; and then, to repay him for his kindness, I loaned him 300^ dollars. How long must he keep the 300 dollars, to oalance the previous favor? ^s SUPPLEMEIMT, XI 22. If 4^ cwt. be carried 36 miles for § 5^, how many pounds can be sent 20 miles for the same money ? 23. A person owning | of a coal mine share for 570 dollars. What is the whole mine worth ? 24. If the discount on § 106, for a year, be $6, what is the discount on $477, for the same time ? sells I of his 111 XI MEASUREMENT OF SURFACES, SOLIDS AND CAPACrnF.S. It has already been taught, that surfaces are measured m squares, and, that solid bodies are measured in cubes. A SQUARE is a figure, that has four equal sides, and four equal angles. Its angles are called right angles : angles more pointed are called acute angles ; and those .less pointed, obtuse angles. To find the area of a square, in smaller squares — Multiply one side into itself, 1. How many square feet are there in a table that meas- ures 4 feet on every side? How many square inches? A PARALLELOGRAM is a four-sided figure, having opposite sides equal, and having four right angles. To find the area of a parallelogram — Multiply the length into the breadth. 2. How many square rods in a garden measuring 4 rods in length, and 3 in breadth ? How many square feet? A TRIANGLE is a figure, that has three sides and three angles. A triangle, which has one right angle, is called a RIGHT-ANGLED TRIANGLE. To find (he area of a right-angled triangle — Multi- ply the base by half the perpendicular, 3. How many square rods are there triangular field, measuring 98 rods on rods on the perpendicular? How many acres? right-angled the base, and 75 XL MKASUREMEINJT. 209 A CIRCLE is a plane surface, bound- ed by one curve line, called the circum- a ference. The diameter being known, . to find the circumference — Multiply 'r the diameter by 3.14159. Then, to ^ find the are^ — Multiply half the cir- cumference by half the diameter. 4. How many square inches are there in the head of a barrel, the diameter of which measures 17 inches ? A CUBE is a regular solid body, hav- ing six equal, square sides. To find its contents in smaller cubes — Multiply the breadth of a side twice into itself Tlie product of the length, breadth, and thickness is the contents of any thing, whose opposite sides are equal. 5. How many cubic inches are there in a box measur- ing 34 inches in length, 26 in width, and 18 in depth ? A CYLINDER is a rouud body, with equal, circular ends. To find its cubical ^^ contents — Find the area of one end, and multiply this by the length, 6. How many cubic inches are there in a drum meas- uring 16 inches across the head, and 18 inches in lengdi ? PLASTERING and PAVING are charged by the square yard. Their surface is first found in square feet, and then reduced to square yards. 7. How many square yards of plastering in the ceiling and four sides of a room, that is 15 feet long, 12 feet wide, and 10 feet high; deducting two doors, 7 by 4 feet each, and four windows, 5 by 3^^^ ^^^^ ^^^^^ ^ 8. How many bricks are required to pave a cellar, that is 48 feet long and 30 feet wide ; allowing each brick to be 8 inches long, and 3.8 inches wide ? Here find the area of the cellar in square inches, and divide it by the square inches in the area of a brick. SHINGLES AND CLAPBOARDS are of various dimen- sions. Tliercfore, to know how many are requisite to cover 210 SUPPLEMENT. XI. a building, we find the number of square inches in the roof or side to be covered, and divide this number by the number of square inches, that one shingle or clapboard will rover. 9. If shingles 4 inches in width be laid so that 6 inches of their length is exposed to the weather, how many are required to cover a roof 45 by 32 feet ? 10. How many clapboards, each covering 46 by 4 inch- es, are sufficient for the side of a house 45 by 22 feet ? BOARDS are sold by the thousand square feet, and each board is measured thus — Multiply the length in feet by the ividth in inches, and divide the product hy 12; the quotient will be square feet. 11. How many square feet are there in 17 boards, each board being 21 feet long, and 18.5 inches wide ? 12. How many square feet of boards will floor a room 14 by 18 feet, allowing -^^ of the stuff for waste? PLANK AND JOIST are measured by finding how many square feet of boards, one inch in thickness, they are equal to. Therefore — Multiply the length in feet by the width in inches, and this product by the depth in inches; then divide the last product by 12, for the square feet, 13. How many square feet in a plank that is 9 feet in length, 14 inches in width, and 2.4 inches in depth? 14. How many square feet in a joist that is 13 feet long, 4 inches wide, and 8.2 inches deep ? TIMBER is sold by the cubic ton. To measure hewn timber — Multiply the length in feet by the width in inches, and this product hy the depth in inches ; divide by 144, for the cubic feet, and then by 50 for the tons. To measure round timber — Take the circumference in inches, by girding the log, one-third of the way from the but to the top ; then multiply the length in feet, by the square of \ of the circumference ; divide by 144 /or the cubic feet, and then by 40 for the tons, 15. How much hewn timber in a stick measuring 25 feet in length, 19 inches in width, and 20 inches in depth ? 16. How much round timber in a log, 30 feet long, anr 55 inches in circumference ? XI. MEASUREMENT. o|i CELLARS, WELLS, and other pits, are measured by the cube of six-feet side ; and this cube is called a square OF EARTH. To measure a cellar — Add together the depth of the four corners^ divide the sum by 4, multiply the quo- tient by the length, and this product by the ividth, all in feet^ for the cubic feet ; then divide by 216 for the squares. To measure a well — Proceed as ivith a cylinder to find the cubic feet, and divide by 216 for the squares, 17. How many squares in a cellar, the length being 30 ft., width 22 ft., depth at corners, 12 ft., 9 ft., 7 ft., and 4 ft.? 18. At $1.08 a square, what is to be paid for digging a well, 60 feet deep, and 8 feet in diameter ? STONE WALLS are measured by the percA, of 24| cu- bic feet. To measure a straight wall — Multiply, in feet, the length by the height, and this product by the thickness, for the cubic feet ; then divide by 24,1 5 for the perches. To measure a circular wall — Take the diameter, to the centre of the thickness of the wall, and compute the cir- cumference in feet. Then multiply the circumference, height, and thickness together, all in feet, for the cubic feet, and divide by 24.75, /or the perches, 19. How much wall, of 2 feet thickness, and 8 feet height, in a cellar measuring 36 feet on every side within the clear r 20. How much wall in a well 40 feet deep ; the wall being 2 feet thick, and the diameter being 4.5 feet? BINS, BOXES, &:c., holding commodities sold by the gallon or bushel, are measured thus — Find the contents in cubic inches, as already taught ; then divide by 231 fo% wine gallons, or, by 2150.4 /or bushels. 21. How many gallons in a vat, measuring 60 inches in length, .36 inches in breadth, and 72 inches in depth ? 22. How many bushels of grain in a bin, 84 inches in length, 32 inches in breadth, and 48 inches in depth ? CYLINDRIC VESSELS, such as tubs and cisterns for holding watei, are measured thus — Multiply, m inches, the diameter of one end into itself, and this product into the height ; then divide by 294 for the wine gallons. 212 SUPPLEME.XT. XIl. If the ends of the vessel be unequal — Multiply the greater diameter by the less, and to the product add i of the square of their difference ; rnidtiply this sum by the height, and divide by §94, for the gallons, 23. How many gallons will a tub hold, the diameter of which is 18 inches, and the height 22 inches ? 24. How many gallons of water will a cistern hold, measuring 72 inches across the bottom, 60 inches across the top, and 84 inches in height ? THE CAPACITY OF CASKS is found as follows — Take the interior dimensions as nearly as possible, Sub^ tract the diameter of the head from the diameter at the bung. Multiply the difference by .7, if the staves be much curved ; or by .6, if little curved ; or by .65, if they be o/ MEDIUM curve. Add the product to the head, diameter, and the sum will be the mean diameter. Square the mean diameter ; multiply the square by the length of the cask, and divide this product by 294, for wine gallons. 25. Find the number of gallons in a cask of medium curve, 47 inches in length, *3\ inches diameter at the bung, and 26 inches diameter at the head. 26. What is the capacity of a cask, much curved, meas- uring 32.5 inches in length, 19 inches at the bung, and 15.4 inches at the head ? xn. DUODECIMALS. Duodecimals are compound numbers, the value of whose denominations diminishes in a uniform ratio of 12. They are applied to square and cubic measure. The denominations of duodecimals are the foot, (f), the prime or inch, Q, the second, (''), the third, ('''), the fourth, (''''), the fifth, (^^^^0? ^^^ ^^ ^"* Accordingly, the expression, 3 V T* 9"' ^^"' denotes 3 feet 1 prime 7 sec- onds 9 thirds 6 fourths. The accents, used to distinguish the denominations be* low feet, are called indices. Xn. DUODECIMALS. 21? The foot being viewed as the unit, duodecimals present the following relations. 1' = -J^ of 1 foot. V' =z ^^ of -ji^ of 1 foot =1 -^ of Koot 1'^' = ^L of -5-I5 of -^2 ^f 1 f^O^- • • = TT2Q ^f 1 foot* 1-' := ^^ of J^ of T-V of j\ of 1 foot. -20T^of 1 foOt. &c. Addition and subtraction of duodecimals are performed as addition and subtraction of other compound numbers ; 12 of a lower denomination making one of a higher. Multiplication, however, when both the factors are duo- decimals, is peculiar, and will now be considered. When feet are multiplied by feet, the product is in feet. For instance, if required to ascertain the superficial feet . in a board 6 feet long and 2 feet wide, we multiply the length by the breadth, and thus find its superficial, or square feet to be 12. But when feet are multiplied by any number of inches, [primes], the effect is the same as that of multiplying by so many twelfths of a foot, and therefore the product is in twelfths of a foot, or inches. Thus a board, 6 feet long and 6 inches wide, contains 36 mches, because the length being multiplied by the breadth, that is, 6 feet by -^^ of a foot, the product is || of a foot, or 36^=3 feet. When feet are multiplied by seconds, the product is in seconds. Thus 6 feet multiplied by 6 seconds, that is, -| of a foot by -^^ of tV of a foot, the product is -^ of a foot, or 36^'= 3 inches. Feet multiplied by feet, produce feet. Feet multiplied by primes, produce primes. Feet multiplied by seconds, produce seconds. Feet multiplied by thirds, produce thirds. &c. Primes multiplied by primes, produce seconds. Primes multiplied by seconds, produce thirds. Primes multiplied by thirds, produce fourths. &c. Seconds multiplied by seconds, produce fourths. Seconds multiplied by thirds, produce fifths. Seconds multiplied by fourths, produce sixths he. 6f. 4' 8" 4 6' 5" 2' 7" 11"/ 4"" 3 2' 4" 0"' 25 6' 8" 214 SUPPLEMENT. XII. If we would find the square feet in a floor 6 f. 4' 8^^ in length, and 4 f. 6' h" in breadth, we proceed as follows. We begin on the right hand, and multiply the whole multi- plicand, first by the seconds in the multiplier, then by the inches, and lastly by the feet. We then add the results together, and thus 28f.ll' 7" 11-4"" obtain the answer. We are now led to a general rule for the multiplication of duodecimal numbers. RULE. P/flfce the several terms of the multiplier under the corresponding ones of the m;ultiplicand. Beginning on the right hand, multiply the several terms of the mul- tiplicand by the several terms of the multiplier successive- ly, placing the right hand term of each of the partial products under its multiplier. Then add the partial prod- ucts together; observing to carry one for every twelve, both in multiplying and adding. The sum of the partial products ivill be the answer. Questions in duodecimals are very commonly performed by commencing the multiplication with the highest denom- ination of the multiplier, jmd placing the partial products as in the first of the two following operations. The result is the same, whichever method is adopted. The second operation, however, is according to the rule we have given, and is more conformable to the multiplication of numbers accompanied by decimals. 3f. 2' 1" 2f. 6' 4" 6 5' 2" I 7' 3" &" 1' 0" 10'" 4"" 3f. 2' 1" 2f. 6' 4" 1' 0" W" 4"// 1 r 3" 6"' 6 5' 2" 8f. I' 6'' A''' A"'' 8f. 1' 6'^ A"' A'"' When there are not feet in both the factors, there may not be any feet in the product ; but, after what has been said, there will be no difficulty in determining the places of the product. XII. MISCELLANKOUS EXAMPLES. 215 1. Multiply 14 r. 9' by 4f. €'. 2. What are the contents of a marble slao, whose length IS 5f. V, and breadth If. 10'? 3. How many square feet are there in the floor of a hall, 48 f. 6' long, and 24 f. 3' wide? 4. Multiply 4f. T 8^' by 9f. 6^ 5. How many square feet are there in a house lot, 43 f 3' in length, and 25 f. 6' in breadth? 6. What is the product of 10 f. 4' 5'' by 7f. 8' 6"? 7. Calculate the square feet in an alley 44 f. 2' 9^' long, and 2f. 10' 3'^ 2''M'''' wide. 8. How many square feet are there in a garden, 39 f. 10' 7" long, and 18 f. 8' 4" wide ? 9. What is the product of 24 f. 10' 8" 7'" 5"" by 9f. 4' 6" ? 10. Compute the solid feet in a wall, 53 f. 6' long, 12 f. 3' high, and 2f. thick. 11. The length of a ixx)m is 20 feet, its breadth 14 feet 6', and its height 10 f. 4', How many yards of painting are there in its walls, deducting a fire place of 4f. by 4f. 4'; and two windows, each 6i\ by 3f, 2'? 12. How many yards of carpeting, yard wide, will be required for a room 21 f. 6' long, and 18 f. wide? 13. What will the plastering of a ceiling come to, at 10 cents a square yard, supposing the length 21 feet 8 inches, and the breadth 14 feet 10 inches? 14. How many yards of papering on the four walls of a hall, 58 f. 8' long, 21 f. 4' wide, and 13 f. 9' high ; de- ducting 2 dooi*s, each 7 f. 6' high and 4 f. wide ; 7 windows, each 6f. 2' high and 3f. 10' wide; and a mop-board, 9 inches wide around the hall ? END OF PART SECOND. / 216 INDEX. ORAL ARITHMETIC. Numeration , 5 Addition 9 Subtraction , iQ Correspondent Examples in Addition and Subtraction 19 Multiplication 24 Division 31 Correspondent Examples in Multiplication and Division 37 Connected Operations 38 Fractions 44 Relations of Numbers , 48 Fractions and Relations 58 Notation of Fractions 74 WRITTEN ARITHMETIC. Numeration 89 Addition 93 Subtraction 97 Multiplication 103 Abbreviations in Multiplication 105 Division 108 Abbreviations in Division ; 116 Federal Money 124 Compound Numbers 130 Fractions 147 Relations of Numbers 149 Fractions and Relations 155 Change of Whole Numbers to Fractions , 156 Change of Fractions to Whole Numbers 157 Percentage ' 163 Interest 165 Discount 168 Change of the Terms of Fractions 169 Fteduction of Fractions to Lower Terms 169 Compound Fractions 170 ' SUPPLEMENT. Cancellation of Factors 182 Decimal Fractions 183 Change of Common Fractions to Decimals 188 Ch. of Comp. Num. to Dec. . . 188 Ch. of Dec. to Comp. Num. . . 188 Exchange of Currencies .... 189 Percentage 101 Commission 193 Stocks 193 Insurance 193 Interest , . , 194 Compound Interest 198 Discount 199 Banking 19;) Profit and Loss 20C Partnership 201 Assessment of Taxes 203 Ratio and Proportion 20:) Rule of Three 205 Measurement 208 Duodecimals 21 2 -\ 1? 1 iot mi c J> 'm^ •7 d T EMERSON'S SECOND PART. Williams College, Oct. 2d, 1832. To Mr. Frederick Emerson. Sir,— 1 have received the I \rsl artft Second Farts o£ yoni Nort^ American Arithmetic; and ?m highly pleased v ith the plan of the work, and the manner of its execution ihv ^ ^ar. It uniVes simplicity with fulness, and will thus be sure to interest the beginner, whilst it furnishes, at the same time, an ample guide to the more advanced pupil. Respectfully and truly yours, ALBERT HOPKINS. [Professor of Mathematies and Natural Philosophy in Williamstown College.'] Union College, Schenectady, Feb. 8th, 1833. Mr. F. Emerson. Sir, — I have received the First and Second Parts of the North American Arithmetic, and examined them with pleasure. The relations of numbers are happily illusti^ated by the relations of sensible objects. This renders the work more entertaining and instructive, especially at that age to which arithmetic is better adapted than many of those studies which for want of a system like this are too oflen introduced. Here the student will acquire not merely rules to guide his hand, but principles to enlighten his understanding. He is not furnished with a mere mill for grinding numbers into a certain result under cover. Wishing you continued success, I am, sir. respectfully, your ob't ser't, B. F. JOSLIN. [Professor of Mathematics and Natural Philosovhy in Union College.] Burlington, 15th February, 1833. {Condtision of a l^'ter to the Author.] I should think it hardly possible that a child could be faithfully conducted through these two works, [First and Second Parts], withoiit being vastly better acquainted with the subject than children fo^.aerly were. Being judiciously compelled in some measure to invent 1 heir own rules, they can scarcely fail of being able to assign a proper reason for the process, as well as to recollect it for future use. Indeed, i do not know any one particular in which, for the use of very young pupils, they could be improved. Yours respectfully, JAMES DEAN [Professor of Mathematics 07id Natural Philosophy , in the University of Vermont.] Athens, Oh^- If'ebruary the 1st, 1833. Mr. Emerson. Dear Sir, — I havb carefully and attentively examined the •' First and Second Parts" of The North Amer- ican Arithmetic;" and can say, with pleasure, that the work meets my cordial approbation. You have, perhaps, re " fered Arithmetic as highly interesting to the youthful mind aa the nature of the subject will admit. Your method of illustrating the fundamental priticiples of Fractions is clear, forcible, and peculiarly happy in its adaptation to the minds of youth. 1 am, yours, &c. WlLLlAxM WALL. [Professor of Mathematics , in the Ohio University.] 55_ 5S