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7. Multiply 471 by TT 8. Multiply 871 by 37. 9. Multiply 867 by T

OPERATION.

155. To multiply a whole and mixed number together.

Ans. 1143.

Ex. 1. Multiply 17 by 62.

OPERATION.

17

74

102

of 1712 }

1142

Ex. 2. Multiply 73 by 4.

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63

2 of 4 - 22 28

We first multiply 17 by 6, the whole number of the multiplier, and then by the fractional part,, which is simply taking of it; and add the two products.

Ans. 8. Ans. 23.

Ans. 66.

Ans. 30%.

We first multiply in the multiplicand by 4, the multiplier; thus, 4 times are 12, equal to 2, which is in effect taking of the multiplier, 4. We then multiply the whole number, 7, by 4; and add the two products. Hence, in performing like examples,

EXAMPLES FOR PRACTICE.

303

Multiply the fractional part and the whole number separately, and add the products.

.3. Multiply 98 by 5. 4. Multiply 12g by 7. 5. Multiply 9 by 811.

6. Multiply 10 by 74.

7. Multiply 11 by 8.

8. What cost 75lb. of beef at 5 cents per pound?

9. What cost 237bbl. of flour at $6 per 10. What cost 8 yd. of cloth at $5 per yard?

11. What cost 9 barrels of vinegar at $63 per

Ans. $0.37

Ans. 467.

Ans. 881.

Ans. 801.

Ans. 714.

Ans. 949.

barrel?

Ans. $141.

Ans. $ 417.

barrel?
Ans. $57.

155. The rule for multiplying a whole and mixed number together? Does

it make any difference which is taken for the multiplier ?

12. What cost 12 cords of wood at $ 6.374 per cord?

13. What cost 11cwt. of sugar at $93 per cwt.?

Ans. $103.

14. What cost 43 bushels of rye at $ 1.75 per bushel?
Ans. $7.65§.

15. What cost 7 tons of hay at $117 per ton?

Ans. $834.

Ans. $95.

Ans. $158.

16. What cost 9 doz. of adzes at $10 per doz.?
17. What cost 5 tons of timber at $3 per ton?
18. What cost 15cwt. of rice at $7.62 per cwt.?
Ans. $114.37.
19. What cost 40 tons of coal at $ 8.37 per ton?

Ans. $335.

OPERATION.

1 × 1 = 31

156. To multiply a fraction by a fraction. Ex. 1. Multiply by

=

Ans. $76.50.

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Ans. 12.

OPERATION BY CANCELLATION,

7 3 7
9 X 4 12

3

To multiply by is to take of the multiplicand, † (Art. 154). Now, to obtain of 7, we simply multiply the numerators together for a new numerator, and the denominators together for a new denominator (Art. 138). Therefore,

Multiplying one fraction by another is the same as reducing compound fractions to simple ones.

RULE.- Multiply the numerators together for a new numerator, and the denominators together for a new denominator.

EXAMPLES FOR PRACTICE.

2. Multiply by fr 3. Multiply

by 18.

NOTE. When there are factors common to the numerators and denominators, the work may be shortened by canceling those factors.

Ans.
Ans.

156. What is the first rule for multiplying one fraction by another? How does it appear that this operation multiplies the fraction of the multiplicand? What is the inference drawn from it? What is the note ?

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4. Multiply
5. Multiply +

6. Multiply

7. Multiply by 17.

8. Multiply 9. What cost

by 12.

by 18.

by 17.

10. If a man he travel in

by 3.

of a bushel of corn at of a dollar per bushel? Ans. of a dollar. travels of a mile in an hour, how far would of an hour? Ans. of a mile.

11. If a bushel of corn will buy of a bushel of salt, how much salt might be bought for of a bushel of corn?

Ans. of a bushel.

OPERATION.

43 = 22; 63 — 20.

=

4

12. If of of a dollar buy one bushel of corn, what will of of a bushel cost? Ans. of a dollar. 13. If of of of an acre of land cost one dollar, how much may be bought with of $18? Ans. 197 acres.

23×20

$

157. To multiply a mixed number by a mixed number. Reduce them to improper fractions, and then proceed as in Art. 156.

Ex. 1. Multiply 43 by 64.

Ans. 303.

=

92

3 3

Ans. 4.

Ans.

Ans.

Ans. T

Ans.

303

EXAMPLES FOR PRACTICE. 2. Multiply 7 by 83. 3. Multiply 4 by 91. 4. Multiply 114 by 8.

5. Multiply 12 by 118. 6. What cost 7

cords of wood at $5 per cord?

Ans. 60
Ans. 45.

Ans. 99.
Ans. 147.

Ans. $413. 7. What cost 7 yd. of cloth at $3 per yard? Ans. $ 251%. 8. What cost 64 gallons of molasses at 234 cents per gallon? Ans. $1.5218.

9. If a man travel 3 miles in one hour, how far will he travel in 97 hours? Ans. 34.

157. How do you multiply a mixed number by a mixed number :

10. What cost 36111 acres of land at $ 253 per acre?

Ans. $9167113.

11. How many square rods of land in a garden, which is 97% rods long, and 49 rods wide? Ans. 4810 rods.

DIVISION.

158. Division of Fractions is the process of dividing when the divisor or dividend, or both, are fractions.

159. To divide a fraction by a whole number.

Ex. 1. Divide § by 4.

Ans..

We divide the numerator of the fraction by 4, and write the quotient, 2, over the denominator.

It is evident this process divides the fraction by 4, since the size of the parts into which the whole number is divided, as denoted by the denominator, remains the same, while the number of parts taken is only many as before. Therefore,

as

FIRST OPERATION.

8

9

4

Dividing the numerator of a fraction by any number divides the fraction by that number.

Ex. 2. Divide

2

9

Ans..

by 9.

We multiply the denominator of the fraction by the divisor, 9, and write the product under the numerator, 5.

It is evident this process divides the fraction, since multiplying the denominator by 9 makes the number of parts into which the whole number is divided 9 times as many as before, and consequently each part can but have of its former value. Now, if each part has but of its former value, while only the same number of parts is expressed by the fraction, it is plain the fraction has been divided by 9. Therefore,

SECOND OPERATION.

5

7

9 =

5

63

Multiplying the denominator of a fraction by any number divides the fraction by that number.

RULE. Divide the numerator of the fraction by the whole number, when it can be done without a remainder, and write the quotient over the denominator. Or,

Multiply the denominator of the fraction by the whole number, and write the product under the numerator:

158. What is division of common fractions?-159. How is the fraction divided by the first operation? What inference may be drawn from this operation? How is a fraction divided by the second operation? What inference is drawn from this operation? The rule?

EXAMPLES FOR PRACTICE.

3. Divide

by 3.

4. Divide 18 by 6.

5. Divide

by 12.

6. Divide

by 8.

7. Divide

8. Divide

-9. Divide

10. Divide by 12.

by 9.

by 15.

by 75.

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13÷

g.

11. John Jones owns of a share in a railroad, valued at $117; this he bequeaths to his five children. What part of a share will each receive? Ans. 4.

160. To divide a whole number by a fraction.

Ex. 1. How many times will 13 contain ?

3

Ans.

12. Divide by 15. 13. Divide by 28.

Ans. 5.

14. James. Page's estate is valued at $10,000, and he has given of it to the Seamen's Society; of the remainder he gave to his good minister; and the remainder he divided equally among his 4 sons and 3 daughters. What sum will each of his children receive? Ans. $6807.

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OPERATION.

13 X 7 91

3

3

2. Divide 18 by 7.

3. Divide 27 by 11.

4. Divide 23 by 1.

Ans.

Ans.

Ans. 137

Ans.

=

Ans.

Ans.

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Ans..

Ans. g.

Ans. 30.

30, Ans.

13 will contain as many times as there are sevenths in 13, equal 91 sevenths. Now, if 13 contain 1 seventh 91 times, it will contain as many times as 91 will contain 3, or 30.

RULE.- Multiply the whole number by the denominator of the fraction, and divide the product by the numerator.

EXAMPLES FOR PRACTICE.

Ans. 204. Ans. 29.

Ans. 92.

160. The rule for dividing a whole number by a fraction? The reason for the rule?

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