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222. Oral Exercises.
are how many ninths? Ans. f.
4 of a day and 4 of a day.
f. Find the difference between g. What must be added to to make When the minuend and subtrahend are like fractions, how do you subtract?
Z-3=272 = 7 Ans.
e. 1†-Á¬Á = what?
223. ILLUSTRATIVE EXAMPLE. If of a yard of velvet is cut from a piece containing of a yard, what part of a yard will be left?
Explanation. That the subtraction may be performed, these fractions must be changed to equivalent fractions having a common denominator. The least common denominator is 12. =121⁄2 and = 12.
h. -? -? -? -? - ? - ? i.
1. 号一号? 号一号? 朵一番? 是一子?
j. 2-f? 8-7? 11-3? 9-31? 7-2? 8-33?
k. How many yards will be left if from a piece containing
6 yards there be taken 14 yards?
1. What is the difference in the height of two boys, one
being 33 feet, the other 23 feet high?
m. A pole is standing so that the mud, and the rest in the air.
of it is in the water, in What part is in the air? n. How much will be left of a piece of cloth containing 7 yards, after cutting from it 2 vests and a coat, allowing of a yard for a vest and 4 yards for a coat?
o. From a bin containing 23 bushels of wheat there were taken out 33 bushels at one time and 4 bushels at another. How much remained?
224. From the previous illustrations we may derive the following
To subtract one fraction from another:
1. If they have a common denominator, find the difference of their numerators.
2. If they have not a common denominator, change them to equivalent fractions which have a common denominator, and then find the difference of their numerators.
225. Examples for the Slate.
(44.) 11⁄2-2= ?
(50.) 12-2 = ?
(48.) 19-21= ?
(49.) 75-15= ?
For other examples in subtraction of fractions, see page 123.
66. From 8 trees I gathered apples as follows: 2 barrels, 33 barrels, 5 barrels, 44 barrels, 32 barrels, 13 barrels, 34 barrels, and 2 barrels. If I sold 151⁄2 barrels of the apples to one man and 2 to another, how many had I left?
67. A lady who had $50 received $81 more, spent $174, lost $4%, and collected $15 of a debt. How much money in
dollars and cents had she?
68. A man having a sum of money spent of it for a house, of it for furniture, 3 of it for horses and carriages, and & of it to build a church. What part of his money had he left? 69. How much more is the sum of 12 and 67% than their difference?
For other examples in addition and subtraction of fractions, see page 123.
To multiply a Fraction by an Integer.
227. ILLUSTRATIVE EXAMPLE I. If it takes
of a yard of cloth to make 1 apron, how much will it take to make 2
Solution. -If it takes of a yard to make 1 apron, to make 2 aprons it will take 2 times, or of a yard, equal to 14 yards.
In multiplying the fraction by 2, which term of the fraction was multiplied? How will you multiply any fraction by an integer?
228. Oral Exercises.
a. Multiply by 2; § by 3; by 7; % by 4; 5 by 5; by 8; # by 6.
b. How many are 3 times ? 4 times ; 5 times f? 8 times? 9 times?
C. How many are 3? 7×2? × 4? 15×2? × 6 ? 1×5? 3×9? 3 × 20?
d. If 2 pounds of cane are required to seat 1 chair, how many pounds will be required to seat 12 chairs ?
NOTE. Multiply the integer and the fraction separately.
At $183 a dozen, what is the cost of 5 dozen lamps? f. If 7 men can build a dam in 43 days, in what time can 1 man build it?
g. At $10 each in currency, what is the value of 5 gold eagles ?
h. In a piece of land 1 foot long and 1 foot wide there is 1 square foot. How many square feet are there in a piece 82 feet long and 1 foot wide? in a piece 183 feet long and 5 feet wide?
i. If a man receives $3 for shoeing a horse and $ for shoeing an ox, how much will he receive for shoeing 4 horses and a yoke of oxen?
229. Examples for the Slate.
ILLUSTRATIVE EXAMPLE II. Multiply by 56.
3 × 56
70. If a man can mow of an acre of meadow in 1 hour, now much can he mow in 38 hours?
71. How many yards of cloth are required for 6 suits, each suit requiring 7
72. What is the width of 18 house lots, each 53 rods wide? 73. What distance can a vessel sail in 33 hours, going at the rate of 5 miles an hour?
74. There are 16 in 40 rods? in 320 75. How much ivory worth $1 a pound can be bought for the same sum that will pay for 15ğ pounds worth $12 a pound?
76. One quart dry measure contains 67 cubic inches. How many cubic inches are there in a bushel, or 32 quarts?
77. If by working 11 hours a day a piece of work can be done in 45 days, in what time can it be done by working 1 hour a day?
feet in a rod. How many feet are there rods, or 1 mile?
78. If 17 men can shear a lot of sheep in 9 days, in what time can 1 man shear the lot? 79. Multiply 144 by 9. 80. Multiply 16 by 7. 81. Multiply 237 by 11.
To multiply an Integer by a Fraction.
230. ILLUSTRATIVE EXAMPLE III. What is of 2 inches?
of 2 =}.
a. What is b. What is
82. Multiply 365 by 39.
84. Multiply 37633 by 21.
If of each of the 2 inches is taken,
we shall have of an inch. Ans. † of an inch. (See illustration; also Art. 196.)
231. Oral Exercises.
232. ILLUSTRATIVE EXAMPLE IV. What is
Solution. of 2 inches is of an inch, and 3 times, or of an inch, equal to 1 inches.
of 7? of 6?
d. Multiply 8 by 3. e. Multiply 6 by 7.
of 5? 1 of 5?
of 2 inches?
of 2 inches must be Ans. 1 inches.
c. What is of 7? of 6? % of 4?
of 5? of 9?
In finding the fractional part of a number, as in the example above, what was the first operation? Ans. Dividing the number by the denominator of the fraction. By what was the result multiplied? How then will you find the fractional part of a number?
233. The process by which the fractional part of a number is found is called multiplying by a fraction.
ILLUSTRATIVE EXAMPLE V.
Multiply 11 by
Solution. To multiply 11 by
is to take of 11. of 11 is, and of 11 must be 4 times, or 44 = 84. Ans. 84.
f. Multiply 10 by 4.
g. Multiply 12 by .