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6. Multiply by 83.
7. Multiply by 189.

8. Multiply 11 by 365.
9. Multiply by 48.

10. If a man receive of a dollar for one
will he receive for 21 days' labor?
11. What cost 56lb. of chalk at of a cent

5) 15

of a cent per lb. ? Ans. $3.24.

12. What cost 396lb. of copperas at 13. What cost 79 bushels of salt at of a dollar per bushel? Ans. $69. ART. 154. To multiply a whole number by a fraction. Ex. 1. Multiply 15 by g. Ans. 9.

FIRST OPERATION.

In the first operation we divide the whole number by the denominator of the fraction, and obtain of it. We then multiply this numerator of the fraction, and thus obtain

3 x3=9 quotient by 3, the of it, which is 9.

SECOND OPERATION.

15

3

In the second operation we multiply the whole number by the numerator of the fraction, and divide the product by the denomina455 9 tor, and obtain 9 for the answer, as before. Therefore, Multiplying by a fraction is taking the part of the multiplicand denoted by the multiplier.

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Ans. 76 Ans. 1661. Ans. 3526. Ans. 43.

day's labor, what
Ans. $73.

per lb. ?
Ans. $0.42.

RULE 1. Divide the whole number by the denominator of the frac tion, when it can be done without a remainder, and multiply the quotient by the numerator. Or,

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

EXAMPLES FOR PRACTICE.

2. Multiply 36 by J.
3. Multiply 144 by 1.
4. Multiply 375 by 19.
5. Multiply 2277 by 78.
6. Multiply 376 by

70

Ans. 28. Ans. 88. Ans. 325. Ans. 1610. Ans. 243.

QUESTIONS. Art. 154. How do you multiply a whole number by a fraction according to the first operation? How by the second? What inference is drawn from the operations? What is the first rule for multiplying a whole number by a fraction? The second?

7. Multiply 471 by 17. 8. Multiply 871 by 37. 9. Multiply 867 by 136.

OPERATION.

17

ART. 155. To multiply a whole and mixed number together.

Ex. 1. Multiply 17 by 62.

62

102

of 17 = 12₫

114

OPERATION.

72305
4

Ex. 2. Multiply 73 by 4.

Ans. 82. Ans. 2339. Ans. 636

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 together.

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Ans. 114.

Ans. 30%.

We first multiply in the multiplicand by 4, the multiplier; thus, 4 times are 12, equal to 23, which is in effect taking of the multiplier, 4. We then multiply the whole number by 4, and add the two products together.

3 of 4 = 23

28

303

RULE. Write the less number under the greater, and if the FRACTION is in the multiplier, take a part of the multiplicand denoted ly the fraction; but if it is in the multiplicand, take a part of the multiplier denoted by the fraction, and in each case add the product thus obtained to the product of the whole numbers.

EXAMPLES FOR PRACTICE.

3. Multiply 93 by 5.
4. Multiply 123 by 7.
5. Multiply 9 by 811.
6. Multiply 10 by 74.
7. Multiply 119 by 8.

8. What cost 7 lb. of beef at 5 cents per pound?

9. What cost 237 bbl. 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. 463. Ans. 88. Ans. 80. Ans. 714. Ans. 949.

Ans. $0.37.

barrel?

Ans. $141.

Ans. $ 417.

barrel ? Ans. $573.

QUESTIONS.- Art. 155. What is 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 $6.37 per cord? Ans. $76.50.

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

14. What cost 43 bushels of rye at $ 1.75

15. What cost 7 tons of hay at $113 per ton? 16. What cost 9 doz. of adzes at $10§ per

17. What cost 5 tons of timber at $3 per ton?

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Ans. $155.

18. What cost 15cwt. of rice at $7.621 per cwt. ? Ans. $114.371. 19. What cost 40 tons of coal at $8.373 per ton?

Ans. $335

ART. 156. To multiply a fraction by a fraction.
Ex. 1. Multiply 7 by .

Ans. $103. per bushel? Ans. $7.65§.

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Ans. $83.

doz.?

EXAMPLES FOR PRACTICE.

2. Multiply by r

3. Multiply by 1.

Ans. $95ğ.

3

To multiply by is to take of the multiplicand 7 (Art. 154). Now, to obtain of, 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.

Ans.

OPERATION BY CANCELLATIO

7 3 7
X =
4 12

RULE I.- Multiply the numerators together for a new numerator, and the denominators together for a new denominator; then reduce the fraction to its lowest terms. Or,

RULE II. Cancel all the common factors in the numerators and denominators, and then multiply the remaining factors together as before.

Ans.
Ans.

QUESTIONS.

Art. 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

second rule?

8

by 12.

19

4. Multiply
5. Multiply 18 by 8.
6. Multiply
7. Multiply by 17.
8. Multiply

by 17.

Ans. T

Ans.

by 38.

9. What cost of a bushel of corn at of a dollar per bushel ?

Ans.

of a dollar. 10. If a man travels of a mile in an hour, how far would he travel in of an hour? Ans. of a mile. of a bushel of salt, of a bushel of corn? Ans. of a bushel.

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

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.

ART. 157. To multiply a mixed number by a mixed number, it is only necessary to reduce them to improper fractions, and then proceed as in the foregoing rule.

Ex. 1. Multiply 43 by 63.

Ans. 303.

2. Multiply 7
3. Multiply 4
4. Multiply 11
5. Multiply 12
6. What cost 7

OPERATION.

43=23; 6=20.

4

23 20 92
5 3 3

=303.

EXAMPLES FOR PRACTICE.

by 83.
by 94.
by 84.
by 113.

number?

=

Ans. Ans..

Ans.

cords of wood at $5 per cord?

Ans. 60%. Ans. 45. Ans. 99 Ans. 147.

Ans. $413.

7. What cost 78yd. of cloth at $3 per yard? Ans. $ 2518. 8. What cost 6 gallons of molasses at 233 cents per gallon ? Ans. $15218.

9. If a man travel 33 miles in one hour, how far will he travel in 97 hours? Ans. 3477.

QUESTIONS.-Art. 157. How do you multiply a mixed number by a mixed

10. What cost 36111 acres of land at $253 per acre? Ans. $9167118. 11. How many square rods of land in a garden, which is 97 rods long, and 493 rods wide? Ans. 4810 rods.

DIVISION OF VULGAR FRACTIONS.

ART. 158. DIVISION of Vulgar Fractions is the process of dividing fractions by fractions, or whole numbers and fractions by each other.

ART. 159. To divide a fraction by a whole number.

Ex. 1. Divide § by 4.

Ans..

FIRST OPERATION. In this operation we divide the numerator of the 8÷4 2 fraction by 4, and write the quotient 2 over the de=g nominator.

9

It is evident this process divides the fraction by 4, since the number and size of the parts into which the whole number is divided remain the same, while only of the number of parts is expressed by the fraction. Therefore, Dividing the numerator of a fraction by any number divides the fraction by that number.

Ex. 2. Divide

SECOND OPERATION.

5

5

7X9

63

=

Ans.

by 9.

We multiply the denominator of the fraction by the divisor, 9, and write the product under the nu

merator.

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 have but of its former value. Now, 109 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, Multiplying the denominator of a fraction by any number divides the fraction by that number.

RULE I. - 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,

RULE II. Multiply the denominator of the fraction by the whole number, and write the product under the numerator.

QUESTIONS. -Art. 158. What is division of vulgar fractions? - Art. 159. How is the fraction divided by the first operation? How does it appear that this process divides the fraction? What inference may be drawn from this operation? How is a fraction divided by the second operation? Will you explain how this process divides the fraction? What inference is drawn from this operation? What is the first rule for dividing a fraction by a whole number? The second?

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