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SUBTRACTION.

222. Oral Exercises.

a. f less fare how many ninths ? Ans. $. b. 37-1= what?

d. 35-13= what? c. % - 5o = what?

e. fi--11 = what? f. Find the difference between 24 of a day and 34 of a day. g. What must be added to go to make }? 18?

When the minuend and subtrahend are like fractions, how do you subtract?

WRITTEN WORK.

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 ģ- = = 1, Ans.

equivalent fractions having a common denominator. The least common denominator is 12. f='s and free 19-=Ans.

h. }-{? £-7? |-}? 1-1? {-}? - to?
i. - *? -? $-$? 4-7? -4? 1- ?
j. 2- ? 8-1? 11 - ? 9-3}? 7-21 ? 8-37 ?

k. How many yards will be left if from a piece containing 63 yards there be taken 14 yards ?

1. What is the difference in the height of two boys, one being 34 feet, the other 24 feet high ?

m. A pole is standing so that of it is in the water, 1 in the mud, and the rest in the air. 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 i of a yard for a vest and 4} yards for a coat?

0. From a bin containing 23} bushels of wheat there were taken out 3 bushels at one time and 41 bushels at another. How much remained ?

224. From the previous illustrations we may derive the following

Rule.
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.) - = ?

(50.) 12}- = ? (45.) 13 - 3 = ?

(51.) 174 -12% = ? (46.) 36-37 = ?

(52.) 263-18 = ? (47.) 36 – 3 = ?

(53.) 103 -51% = ? (48.) 19-2} = ?

(54.) 171-28 = ? (49.) 75 – 15} = ?

(55.) 181 - 1513 = ? For other examples in subtraction of fractions, see page 123.

226. Examples in Addition and Subtraction. (56.) - + 3 = ?

(60.) 20-5} + 8 = ? (57.) + t - 33 = ?

(61.) 8.4-23 +7% = ? (58.) 4- +++ - 138 = ? (62.) 7- (79- 2) = ? (59.) 8-}-4-719 = ? (63.) 5-(3 + 3) = ?

64. Two men start at the same place and travel in opposite directions, one at the rate of 3,9 miles per hour, the other at the rate of 43} miles per hour. How far apart were they at the end of an hour ?

65. Two boats are 5280 feet apart and rowing towards each other, one at the rate of 320.6 feet per minute, the other at the rate 30937 feet per minute. How far apart are they at the end of one minute ?

66. From 8 trees I gathered apples as follows: 21 barrels, 3} barrels, 5% barrels, 47 barrels, 37 barrels, 17 barrels, 34 barrels, and 2} barrels. If I sold 15} barrels of the apples to one man and 2} to another, how many had I left ?

67. A lady who had $ 50 received $84 more, spent $174, lost $ 43, and collected $ 154 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, To of it for furniture, sto 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 125 and 670 than their difference ?

For other examples in addition and subtraction of fractions, see page 123.

MULTIPLICATION.

To multiply a Fraction by an Integer. 227. ILLUSTRATIVE EXAMPLE I. If it takes į of a yard of cloth to make 1 apron, how much will it take to make 2

aprons ?

Solution. — If it takes of a yard to make 1 apron, to make 2 aprons it will take 2 times ž, or f- of a yard, equal to 15 yards.

In multiplying the fraction of 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; g by 7; lo by 4; š by 5; by 8; 4 by 6.

b. How many are 3 times ? 4 times ; 5 times a ? 8 times %? 9 times ?

c. How many are 13 x 3 ? 16 2? f x4? 15 2? jo 6 ? Yo x 5 ? x 9? 20 ?

d. If 27 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.

e. At $18 a dozen, what is the cost of 5 dozen lamps ?

f. If 7 men can build a dam in 4 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 89 feet long and 1 foot wide ? in a piece 18 feet long and 5 feet wide ?

i. If a man receives $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 io by 56.

WRITTEN WORK.

7 3 x 56

21

10} Ans. 16 2

18 70. If a man can mow us 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 75 yards ?

72. What is the width of 18 house lots, each 52 rods wide ?

73. What distance can a vessel sail in 33 hours, going at the rate of 58 miles an hour ?

74. There are 16} feet in a rod. How many feet are there in 40 rods ? in 320 rods, or 1 mile ?

75. How much ivory worth $1 a pound can be bought for the same sum that will pay for 15% 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 453 days, in what time can it be done by working 1 hour a day?

78. If 17 men can shear a lot of sheep in 92% days, in what time can 1 man shear the lot ?

79. Multiply 148 by 9. 82. Multiply 3654 by 39. 80. Multiply 1636 by 7. 83. Multiply 2567 by 18. 81. Multiply 23% by 11. 84. Multiply 37633 by 21.

To multiply an Integer by a Fraction. 230. ILLUSTRATIVE EXAMPLE III. What is 1 of 2 inches ?

1

ILLUSTRATION.

+

If } of each of the 2 inches is taken, } we shall have fe of an inch. Ans. g of an }) inch. (See illustration; also Art. 196.)

of 2 = $.

231. Oral Exercises.

a. What is } of 2? 4 of 8? } of 4? } of 7 ? of 6 ? b. What is to of 3? of 2? I of 9? of 5? i of 5? 232. ILLUSTRATIVE EXAMPLE IV. What is 2 of 2 inches ?

Solution. — of 2 inches is of an inch, and of 2 inches must be 3 times ], or of an inch, equal to 17 inches. Ans. 11 inches.

c. What is of 7? of 6? of 4? 4 of 5? f 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 1, and 6 of 11 must be 4 times 11, or 44 = 84. Ans. 84.

d. Multiply 8 by f. f. Multiply 10 by 4. e. Multiply 6 by 3. g. Multiply 12 by %.

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