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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 447 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 139 by 32. (130.) 473 +10= ? 126. Divide 44 by 16. (131.) 1211 - 16 = ? 127. Divide 131 by 72. (132.) 2727 - 20 = ? 128. Divide 284 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 ? In 2 there are how many times ? ? 10 ? 11?
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 = 1
bought as there are times f in 5. 146 = = 15 = 2 = 7}
We first change 5 to thirds, making 16.
There are as many times f in as there are 2's in 15, or 77. Ans. 74 baskets.
NOTE. For different analysis of this example, see Appendix, page 304.
247. Oral Exercises.
a. Eight are how many times š? ? 11 ? ? $? 1} or i? b. Twenty are how many times 1} ? 13? 21? 21? 17 ? 14 ? c. Divide 4 by $; 7 by *; 9 by X ; 6 by $; 8 by 14. 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 of an hour, how many miles can he walk in 8 hours ?
t. 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 $25 ?
h. The old shilling of New England was worth 163 cents. How many shillings made a dollar ?
i. If a boy can write a page in t of an hour, how many pages can he write in 1% of an hour ? in 1 hour ?
j. If a hat can be made from of a yard of velvet, how niany hats can be made from 37 yards ?
k. Divide % by io je to by zti by si z8 by jo When fractions have a common denominator, how do you
divide? 248. ILLUSTRATIVE EXAMPLE V. Divide by .
Explanation. -- and changed to frac$= 1; 3= 1% tions having a common denominator are 13
and 1. 1 divided by 1% equals 12 divided
+* + 1$ = 12 = 10 = 14 by 10, or it. Ans. 13.
1. Divide 8 by * ; t by ; by ; by $; } 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 (4x 3) = (5x 2) or 4 x 3 or 4*, in which the expression for the divisor, , is inverted, becoming }, and the answer, found by multiplying # by \, is , or 1}, 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, ste Appendix, p. 304.
fractions. The written work of Illustrative Example V. will then be merely * x3 = f = 1.
Perform the following examples by either of the methods illustrated above : m. How many are $ - *? = ? =11? = }?
How many are $ = ? f = $? f = 75? i f? 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 rumerator of the divisor. Or,
2. Invert the divisor, and proceed as in the multipticution of fractions.
251. Examples for the Slate. 135. Divide 139 by 6.
138. Divide 181 by gli: 136. Divide by 7.
139. Divide 96 by I. 137. Divide 14 by 18.
140. Divide 108 by yo 141. At $ iz 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 = 38=? (144.) 21 - Po=?
(148.) 37-18=? (145.) 98 - 10 = ?
(149.) 38 - = ? -(146.) 54= *= ?
(150.) 18 = = ? 151. How many bushels of peas at $f a bushel can be bought for $18? for $124?
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 1 of a pound of wool, how many yards can be made from 5 tons of 2000 pounds each ?
154. One rod equals 164 feet. How many rods in 100 feet ?
155. How many breadths of paper, each ft 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 $453 ?
157. How many lengths of 715 feet are there in a fence 17064 feet long?
158. How many square rods, each containing 30 square yards, are there in 75% square yards?
159. A man had $ 1.50, which he exchanged for francs at 189 cents each. How many francs did he receive ? (160.) 38 - 435 = ?
(162.) 26} = 371 = ? (161.) 51 - 64 = ?
(163.) 1 - 5417 = ? 252. ILLUSTRATIVE EXAMPLE VI. Change the expression to its simplest form.
98 = 23 = . = 23 = 60x8 244 = 37. 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 :
(164.) (165.) (166.)
2} (167.) (170.) (173.)
1914 (171.) (174.) 1%
4726 For other examples in division of fractions, see page 123.
60 10 g
TO FIND THE WHOLE WHEN A PART IS GIVEN.
253. ILLUSTRATIVE EXAMPLE I. If ß of a ton of hay costs $16, what will f 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 it 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 e 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 } 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 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 ? the age of her brother. How old is her brother ? NOTE. Mary's age is 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 pari 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 ?