121. If a steamer goes 77} miles in 5 hours, what is her rate per hour ? NOTE. First divide 77 by 5, then change the remainder to thirds, and divide. 122. If 444 yards of cloth be required to make 8 suits, how many yards are required for 1 suit ? 123. If land that extends along the street 103] rods is made into 18 house lots of equal width, what is the width of each lot? 124. What is the length of one side of a square that can be enclosed by a string 898 feet long? 125. Divide 129 by 32. (130.) 473 - 10= ? 126. Divide 74 by 16. (131.) 1211 + 16 = ? 127. Divide 131 by 72. (132.) 2721 - 20 = ? 128. Divide 484 by 65. (133.) 5 -11= ? 129. Divide 93). by 9. (134.) 12404 - 90 = ? 246. To divide an Integer or a Fraction by a Fraction. In 1 there are how many fourths ? sixths ? eighths ? ninths ? ILLUSTRATIVE EXAMPLE IV. How many baskets of peaches at of a dollar a basket can be bought for $5? Explanation. - As many baskets can be 5= bought as there are times sin 5. 16 = 15 = 2 = 71 We first change 5 to thirds, making the There are as many times f in L as there are 2's in 15, or 77. Ans. 74 baskets. WRITTEN WORK. NOTE. For different analysis of this example, see Appendix, page 304. 247. Oral Exercises. a. Eight are how many times ? ? 13 ? $? ¢ ? 1} or j ? b. Twenty are how many times 1} ? 13 ? 21? 21? 10 ? 13 ? c. Divide 4 by }; 7 by 3; 9 by ; 6 by $; 8 by 17. How do you change an integer to divide it by a fraction? How do you then divide? How do you divide by a mixed number? d. If a person walks a mile in f of an hour, how many miles can he walk in 8 hours ? e. At $a pound for coffee, how many pounds can be bought for $4? f. How many chairs at $14 each can be bought for $10 ? NOTE. Change 14 to fourths. g. How many tons of coal at $6} can be bought for $25 ? h. The old shilling of New England was worth 164 cents. How many shillings made a dollar ? i. If a boy can write a page in t of an hour, how many pages can he write in 1% of an hour ? in 1 hour ? j. If a hat can be made from of a yard of velvet, how many hats can be made from 37 yards ? k. Divide to by 3; }} by ; 31 by ; 18 by Po When fractions have a common denominator, how do you divide? WRITTEN WORK. 248. ILLUSTRATIVE EXAMPLE V. Divide by s. Explanation. - $ and f changed to frac#= 1%; } = 1% tions having a common denoininator are if and 1$. 13 divided by 1f equals 12 divided ** = 1$ = 12 = 10 = 1+ by 10, or 14. Ans. 14. 1. Divide 8 by *; * by $; by #; f by $; Is by . When fractions have different denominators, how do you prepare them to divide ? In the written work of Illustrative Example V., after obtaining a common denominator we have 12 = 10, or the new numerator of the dividend divided by the new numerator of the divisor. If, in the place of these numbers, we put the factors which formed them, we shall have (4 x 3) = (5 x 2) or or * , in which the expression for the divisor, ş, is inverted, becoming j, and the answer, found by multiplying # by j, is f, or 13, as before. 249. To divide one fraction by another, we may then invert the divisor and proceed as in the multiplication of * For other explanations of division of fractions, see Appendix, p. 304. ?x3 fractions. The written work of Illustrative Example V. will then be merely ***= f = 1}. Perform the following examples 'by either of the methods illustrated above : m. How many are 4 = }? 3-4 ? 8-71? 6 = 3? n. How many are 5 = ? =$? f = 75? =$? 250. From the previous illustrations may be derived the following Rules. 1. To divide a fraction by an integer, Divide the numerator or multiply the denominator by the integer. 2. To divide an integer or a fraction by a fraction, Change the dividend and divisor to fractions having a common denominator, and then divide the numerator of the dividend by the numerator of the divisor. Or, 2. Invert the divisor, and proceed as in the multiplication of fractions. 251. Examples for the Slate. 135. Divide +39 by 6. 138. Divide 181 by gr. 136. Divide jy by 7. 139. Divide 96 by 11. 137. Divide 1: by 18. 140. Divide 108 by do 141. At $ per pound, how many pounds of rice can be bought for $11? 142. At $ } per foot for rubber hose, how many feet can be bought for $ 41 ? (143.) 18 = %= ? (147.) 4 - 14=? (144.) 21-1= ? (148.) i} = 18 = ? (145.) 98 - 10 = ? (149.) 38 = = ? (146.) 54 - * = ? (150.) 13-14? 151. How many bushels of peas at $} a bushel can be bought for $18? for $12} ? 152. At $ % per thousand ems for setting type, how many thousand ems can be set for $75 ? 153. If 1 yard of cloth can be made from 18 of a pound of wool, how many yards can be made from 5 tons of 2000 pounds each ? 154. One rod equals 161 feet. How many rods in 100 feet? 155. How many breadths of paper, each 38 of a yard wide, will reach around a room, the distance being 27+ yards ? 156. At $2$ per yard, how many yards of cloth can be bought for $ 454 ? 157. How many lengths of 71 feet are there in a fence 17064 feet long? 158. How many square rods, each containing 304 square yards, are there in 758 square yards? 159. A man had $ 1.50, which he exchanged for francs at 184 cents each. How many francs did he receive ? (160.) 38 - 435 = ? (162.) 261-37 = ? (161.) 53-6} = ? (163.) 1 - 5414 = ? 252. ILLUSTRATIVE EXAMPLE VI. Change the expres 96 sion to its simplest form. 27 WRITTEN WORK. 94 +2}= 4 + 12 = myynt = 24 = 34. 24 = 34. Ans. 37. Expressions like that above are sometimes called complex fractions. But they merely indicate division. Change the form of the following expressions, and perform the division indicated : For other examples in division of fractions, see page 123. TO FIND THE WHOLE WHEN A PART IS GIVEN. Oral Exercises. 253. ILLUSTRATIVE EXAMPLE I. If f of a ton of hay costs $16, what will of a ton cost ? what will 1 ton cost ? a. If of a certain number is 28, what is the entire number? b. 81 is i'r of what number? c. A man bought a harness for $75, which was of what he paid for his carriage. What did he pay for his carriage ? d. I paid $6 a week for board in Albany, which was of what I paid in Buffalo; this was of what I paid in Chicago; and this was of what I paid in San Francisco. What did I pay in San Francisco ? e. An exploring party having lost ț of their bread, are obliged to subsist on 14 ounces a day. What were they allowed at first? NOTE. If is lost, { remain. f. If 4 of a piece of work be performed in 24 days, how many days will it take to do the remainder ? g. A vessel, having lost of her cable, has 200 feet remaining. How many feet had she at first ? h. Mary is 24 years old, and her age is equal to once and 7 the age of her brother. How old is her brother ? NOTE. Mary's age is 8 of that of her brother. i. A mother and her son have $ 45 in a purse; the son's part is f as great as the mother's. What is each one's part ? Solution. — The mother's part must be f of itself, and her son's part added to her part must be of her part. But the two together have $ 45 ; then $ 45 is of the mother's part. j. If I sell an article for $80, and thereby gain a sum equal to } of the cost, what is the cost ? k. If I sell an article for $80, and thereby lose a sum equal to } of the cost, what is the cost ? |