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16. Subtract 84 from 124.

SOLUTION. We cannot subtract & from, so we take 1 from 12, which from leaves, and 8 from 11 leaves 3;

added to equals 1 or

hence the difference is 34.

;

17. Subtract 8 from 121.

18. Subtract 57 from 10%.

19. Subtract 10% from 207. 20. Subtract 12,7 from 248. 21. Subtract 2011 from 30. 22. Subtract 407 from 60.

WRITTEN PROBLEMS

Ans. 31.

Ans. 41.

Ans. 98.

Ans. 1171.

Ans. 98. Ans. 1918.

IN ADDITION AND SUBTRACTION OF FRACTIONS.

1. A has $51, B has $41, and C has $64; how much money have they all?

Ans. $16.

2. A miller ground 7 bushels of corn for A, 94 for B, 10 for C; how much did he grind in all?

Ans. 2717 bu. 8. A lady bought material for a wrapper costing $11, and buttons costing $; what change should she receive from a $5 bill? Ans. $3.

4. A lady went shopping with $100 and paid $12 for a bonnet, $32 for a dress, and $52g for a cloak; how much money did she bring home? Ans. $2.12.

5. A boy gave 12 cents for a slate, 18 cents for a knife, 87 cents for a grammar, and 62 cents for what did they all cost?

an arithmetic; Ans. $1.31

6. A merchant bought two pieces of muslin, each containing 41 yd.; after selling 57 yd. from them, how many yards remained? Ans. 25 yd.

7. Mr. Weeks finds that his family burned last winter 1 tons of coal in December, 2 tons of coal in January, 2 tons of coal in February, and in March 13 tons; how much was burned during the four months? Ans. 71 tons.

8. Four loads of hay weighed upon the scales 49, hundredweight, 43 hundredweight, 39, hundredweight, and 458 hundredweight; what was the weight of the hay, weight of the wagon being 15, hundredweight?

Ans. 11538 hundredweight.

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MULTIPLICATION OF FRACTIONS.

173. Multiplication of Fractions is the process of finding a product when one or both factors are fractions. 174. There are Three Cases: 1st. A fraction by an integer; 2d. An integer by a fraction; 3d. A fraction by a fraction.

CASE I.

175. To multiply a fraction by an integer.

MENTAL EXERCISES.

are 18, which reduced to its lowest terms equals 3.

1. How many are 5 times?

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SOLUTION 2d.-5 times equal, or; if 5 times equal }, 5 times or; hence 5 times equals .

equals 2 times

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10. Since 5 times equals, how may this result be obtained by omitting the analysis?

11. How then may a fraction be multiplied by omitting the analysis?

WRITTEN EXERCISES.

1. Multiply by 4

SOLUTION.-Multiplying the numerator (Prin. 1), we have 4 times equals ; or, dividing the denominator (Prin. 4), we have 4 times 11, or 34.

OPERATION.

11x4=11

Rule. To multiply a fraction by an integer, multiply the numerator, or divide the denominator.

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CASE II.

176. To multiply an integer by a fraction.

MENTAL EXERCISES,

1. Multiply 12 by i.

SOLUTION.-12 multiplied by one equals 12, hence 12 multiplied by equals of 12, which is 3, and 12 multiplied by equals 3 times 3, which are 9.

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10. In multiplying 12 by, by what do we divide 12, and by what do we multiply the quotient?

11. How then may we multiply a whole number by a fraction?

WRITTEN EXERCISES.

1. Multiply 7 by .

SOLUTION.-7 multiplied by one equals 7, hence 7 multiplied by equals of 7, which is 3, and 7 multiplied by equals 4 times 3, which equals 28. Therefore, etc.

2. Multiply 28 by 7.

SOLUTION.-Multiplying 28 by 3 and dividing by 5, we have 28× equals 163; and then multiplying 28 by 7, we have 196; adding 196 to 16, we have 2123. Therefore, etc.

OPERATION. 7x=3&

OPERATION.

28

7용

5)84

16告 196

212 Ans.

NOTE.-This method of multiplying by a mixed number is more convenient than the one usually presented.

Rule.-Multiply the integer by the numerator of the mul tiplier, and divide the product by the denominator; or divide

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CASE III.

177. To multiply a fraction by a fraction.

MENTAL EXERCISES.

1. What is the product of by ?

SOLUTION. multiplied by one equals, hence & multiplied by equals of, which is, and multiplied by equals 3 times, which is . Therefore, etc.

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10. In multiplying by, what do we multiply together for the numerator of the product and what for the denominator?

11. How then do we multiply a fraction by a fraction?

WRITTEN EXERCISES,

1. What is the product of by ?

SOLUTION.- multiplied by one equals, hence multiplied by equals of, which is, and multiplied by equals 7 times, which is . Therefore, etc.

OPERATION.

{x}=}{

SOLUTION 2D.-times equals of 5 times 7; 5 times equals (Prin. 1); and of equals(Prin. 3).

Rule.-Multiply the numerators together and the denomi

nators together, cancelling common factors.

NOTE.-Practically this case is the same as finding a fractional part of a fraction, but theoretically the two cases are entirely distinct.

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MENTAL EXERCISES.

1. How much will 4 yards of muslin cost, at 6 cents a yard? 2. If you give to 8 boys each 24 apples, how many apples will it take?

3. How many dollars does that man give away who gives to each of 10 beggars of a dollar?

4. How much will 15 inkstands cost at the rate of 2 inkstands for of a dollar?

5. What will 12 Letter-writers cost at the rate of 3 for of a dol lar?

6. If a man can dig a rod of ditch in 7 of a day, how many rods can he dig in 4 days?

7. Multiplying by is the same as taking what part of a number? 8. Show that a fractional part of an integer equals the product of the integer by the fraction.

9. What will be the cost of 2 yards of cloth at $34 a yard? 10. What will 7 bushels of shelled corn cost at 25 cents a bushel? 11. A lady bought 4 dozen spoons at $6 a dozen; what did they cost her?

WRITTEN EXERCISES.

Required the cost of

1. 96 lb. of paper, at 25 cts. per pound?
2. 475 lb. of beef, at 64 cts. per pound?
3. 130 yd. of muslin, at 183 cts. a yard?
4. 387 sheep, if 1 sheep cost 8 dollars?
5. 180 tons of coal, at $10.62 per ton?
6. 7293 lb. of sugar, at 6 cts. a pound?
7. 83 barrels of flour, at $63 a barrel ?
8. 6 yards of cloth, at $5 a yard ?
9. 3 bushels of apples, at $§ a bushel ?
10. 24 handkerchiefs, at 87 cents apiece?
11. 6 tons of hay, at $63 a ton?
12. 83 cords of wood, at $4 a cord?
13. 15 barrels of vinegar, at $4 a barrel ?
14. 16 quarts of beans, at $0.18 a quart?
15. What is the value of 2×××?!!?
16. What is the value of X??×411×???
17. Value of ??×?×?×?

992 1089

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NOTE.-In examples like 15, 16, 17, and 18, resolve the numerators and denominators into their factors and then cancel.

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