52. Multiply 1314 by 231.

131

23

651

171 393

262

30961

EXPLANATION. -Beginning at the right, we multiply first by the fraction, and then by the whole number. Thus, 1× =} }× 131 = 65. 23 × = 2, or 17. Then multiply 131 by 23 as in ordinary multiplication. Adding all of the partial products, we have 30961, the entire product.

The method just shown is adapted to larger mixed numbers. When both mixed numbers are small, reduce them to improper fractions, and solve by the general rule.

180. 1. If a man earns $12 per day, how much can he earn in 26 days?

2. At $17 dollars each, how much will 80 Bibles cost?

3. Mr. C. sold 15 loads of hay at $8 a load. How much did he receive for the hay?

4. When hay is worth $15 per ton, how much will of a ton cost?

5. Mr. Jones owns of a farm valued at $13465. What is the value of his share of the farm?

6. What must be paid for 20 bushels of apples, at $5 a bushel? 7. How much will 3 pieces of muslin cost at 64 cents a yard, if each piece contains 28 yards?

8. A man bought 15 tons of coal at $4 a ton. How much did the coal cost him?

9. Mr. H. bought of a piece of cloth containing 19 yards, at $44 per yard. How much did it cost him?

DIVISION.

181. Division of Fractions is the process of dividing when one or both of the terms are fractional.

182. To divide a fraction by an integer.

PRINCIPLE.-Dividing the numerator or multiplying the denominator of a fraction by any number divides the value of the fraction by that number (Prin. 2, Art. 156).

1. If 3 pounds of tea cost $1%, what will 1 pound cost?

*

1

÷ 2

=

$

1

1

2 X 2 4 Or,

10

3

10

EXPLANATION -If 3 pounds of tea cost $%, 1 pound will cost as much as 3 pounds, or of $1%, or $1%.

Since dividing the numerator of a frac tion divides the fraction (Prin.), we divide the numerator of $ by 3. Hence, the result is $1.

equally between 2 boys. What part of a

EXPLANATION.--If $ was divided equally between 2 boys, each boy received † of † or t. The numerator of the fraction $ can not be divided by 2 an exact number of times; but

$1 ÷ } = $1 × 1 = $since multiplying the denominator of a fraction divides the fraction (Prin.), we multiply the denominator of $1 by 2. Hence the result is $t.

12. If 4 brooms cost $%, what is the cost of each broom?

13. A girl paid $ for 5 dozen eggs. What was the price per dozen?

$2

183. To divide an integer by a fraction.

1. How many pencils can by bought for $2 at $1 each?

2 X 4

1

Or,

8

EXPLANATION.-At $ each, $2 will buy as many pencils as $ is contained in $2. As is contained 4 times in 1, in 2 it is contained 2 times 4 or 8 times. Hence, at $1 each, for $2 you can buy 8 pencils.

Or, since in $2 there are $, as many pencils can be bought as $1, the price of one pencil, is contained times in $, which

are 8 times, or 8 pencils.

2. Divide 6 by, or find how many times is contained in 6.

3

6 X 5

30

6÷

=

: 10

5

3

3

Or,

3 6

6÷

X
1 3

= 10

EXPLANATION.-As is contained 5 times

in 1, in 6 it is contained 6 times 5 or 30 times. If is contained is 6 30 times, }, which is 3 times, is contained as often, or of 30 times, which are 10 times.

In these examples, it may be seen that the result in each case is obtained by mul

tiplying the integer by the denominator (which reduces integer and fraction to a common denominator), and dividing this product by the numerator of the fraction; that is, by multiplying the integer by the fraction inverted.

12. How many books at $5 each can be bought for $15?

13. If a boy should spend $ daily, in how many days would he spend $21?

184. To divide a fraction by a fraction.

1. Divide

by, or find how many times

is contained in .

EXPLANATION.-1 is contained in,

times; and is conSince is contained times, or times.

X=tained in 3, 4 times 3 times, or times. in, times, is contained in }, } of Or, reduce the fractions to a common denominator and divide the numerator of the dividend by the numerator of the divisor. Thus, ÷ { = & ÷ ; or following the rule given below, f1⁄2÷£21⁄2=†1⁄2 × 1,2 (cancelling the common factor 12). Inverting the divisor and multiplying is therefore a short process of reaching the same result.

RULE.-Reduce all integers and mixed numbers to improper fractions.

Invert the divisor and proceed as in multiplication of fractions.

26. Divide 21 by 4.

4) 213

512

EXPLANATION.-4 is contained in 213, 5 times with a remainder of 13, or; and divided by 4 equals. Hence, the quotient is 512.

This method is often better than to solve by the general rule.

EXPLANATION.-It is often convenient in dividing by a mixed number to reduce both the divisor and dividend to the denominator of the fraction, and then to divide by the numerators.

15 124

12 12

EXPLANATION.-Since multiplying both dividend and divisor by the same number does not change the 182) 1497 (8 quotient (Art. 156, Prin. 3), we multiply both by 12, the least common denominator of the fractions, and divide as in whole numbers.

1456

41 182

41

182

This method is especially convenient, when the numbers are large and the denominators of the frac

PRACTICAL PROBLEMS.

185. 1. If a cord of wood cost $31, how many cords can be bought for $291?

2. A man raised 56 bushels of potatoes on of an acre of land. What was the yield per acre?

3. How many quarts of vinegar can be bought for 75 cents, at 18 cents a quart?

4. If 44 yards of ribbon cost 95 cents, how much will 3 yards cost?

5. How much must be paid for 21 books at the rate of $9 per dozen?

6. If 7 yards of cloth cost $13.44, what will be the cost of of a yard?

7. Mr. Knox bought of a ton of hay for $10. At this rate, what would be the cost of 6 tons?

8. If a bushel of potatoes cost $5, how many bushels can be bought for $414?

9. A farmer sold of a bushel of seed corn for $3. How much at this rate would he get for 27 bushels?

.

10. Mr. B. bought 5 tons of

coal for $304. How much would

16 tons cost at the same rate?

11. A lady bought 9 yards of silk for $11.25. yards of the same goods cost?

What would 6%

12. If 7 barrels of apples cost $114, what will be the cost of 37 barrels?

13. Mr. Earnest earns $2 a day, and his expenses are $7 a week. How many weeks must he labor to pay for a lot worth $1384?

14. A house was bought for $5000 and sold for above cost. For how much was it sold?

15. Mr. M. had $1280. He bought goods with of it, and a horse with of the remainder. How much money had he left?

16. If $573 will buy 12 tons of iron, how many tons will $1237 buy?