2. I have paid for a knife $ §, for a Common School Arithmetic $, for a slate $, and for stationery $ §; what did I pay for the whole?

3. R. Howland travelled one day 20 miles, another day 19 miles, and a third day 22 miles; what was the whole distance travelled? Ans. 62 miles.

4. I have bought 6 Cumberland coal, and 3 quantity purchased?

tons of anthracite coal, 19 tons of tons of cannel coal; what is the whole Ans. 30 tons.

5. There is a pole standing the remainder above the water; water?

in the mud, & in the water, and

what portion of it is above the

6. F. Adams, having a lot of sheep, sold at one time of them, and at another time of the remainder; what portion of the original number had he then left? Ans.

7. From a piece of calico containing 314 yards there have been sold 11 yards, 9 yards, and 3 yards; how much remains?

of 7

8. From a cask of molasses containing 843 gallons, there were drawn at one time 4 gallons, at another time 11 gallons; at a third time 26 gallons were drawn, and gallons returned to the cask; and at a fourth time 13 were drawn, and 3 gallons of it returned to the cask. How much then remained in the cask?

gallons

Ans. 35gal.

9. A merchant had 3 pieces of cloth, containing, respectively, 19 yards, 36 yards, and 33§ yards. After selling several yards from each piece, he found he had left in the aggregate 718 yards. How many yards had he sold? Ans. 18.

MULTIPLICATION OF COMMON FRACTIONS.

234. MULTIPLICATION of Fractions is the process of multiplying when the multiplier, or multiplicand, or both, are fractional numbers.

NOTE. If the multiplier is less than 1, only such a part of the multiplicand is taken as the multiplier is of 1. Therefore, the product resulting from multiplying a number by a proper fraction is not larger, but less, than the multiplicand.

235. To multiply when one or both of the factors are fractions.

Ans.

Ex. 1. Multiply by 9.

=

FIRST OPERATION.

=

It is evident that the fraction

3 = 34 Ans.is multiplied by 9 by multiplying its numerator by 9, since

the parts taken, 63, are 9 times as many as before, while the parts into which the unit of the fraction is divided remain the same.

SECOND OPERATION.

=

It is evident, also, that the fraction 78 × 97 31 Ans. is multiplied by 9 by dividing its denominator by 9, since the parts into

which the unit of the fraction is divided are only as many, and consequently 9 times as large, as before, while the parts taken remain the same. Therefore,

Multiplying the numerator or dividing the denominator of a fraction by any number multiplies the fraction by that number (Art. 217).

SECOND OPERATION.

14 3

42÷7 6 Ans.

& of 14

=

=

6, as before.

By multiplying the whole number, 14, by 3, the numerator of the fraction, we obtain 42, a product 7 times as large as it should be, as the multiplier was not 3, a whole number, but, or 37; hence, we divide the 42 by 7; and thus obtain Therefore,

Multiplying by a fraction is taking the part of the multiplicand denoted by the multiplier.

3. Multiply by 3.

OPERATION.

=

Ans..

To multiply by is to take of the Ans. multiplicand, . Now, to obtain of, we Ans. multiply the numerators together for a new numerator, and the denominators together for a new denominator (Art. 226). Therefore,

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

When either of the factors is not a fraction, as in examples first and second, it may be reduced to a fractional form, and then the operation may be like that in the last example. Hence the general

-

RULE. · Reduce whole or mixed numbers, if any, to improper fractions. Multiply the numerators together for a new numerator, and the denominators together for a new denominator.

NOTE. When there are common factors in the numerators and denomina tors, the operation may be shortened by cancelling those factors.

29. At of a dollar per foot, what cost 7 cords of wood?

30. Multiply 1611 by 1911.

Ans. 313633.

37. Multiply 12 by .

38. Multiply 15 by fr

39. A merchant owning of a ship sells to A. What part is that of the whole ship?

40. Multiply 37 by 104.

Ans. 84. Ans. 8.

of his share

Ans. 39. Ans. 49133

236. When one of the factors is a whole number, and the other a mixed number, we may

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

8. What cost 76lb. of beef at 5 cents per pound?
9. What cost 237bbl. of flour at $6 per barrel?

Ans. $141.

Ans. $ 417.

10. What cost 8gyd. of cloth at $5 per yard?
11. What cost 9 barrels of vinegar at $ 63 per barrel?

Ans. $578.

12. What cost 12 cords of wood at $ 6.37 per cord?

13. What cost 11cwt. of sugar at $93 per 14. What cost 43 bushels of rye at $ 1.75

Ans. $76.50.

cwt.?
per bushel?

Ans. $7.65§.

15. What cost 7 tons of hay at $113 per ton?

Ans. $83.

16. What cost 9 dozen of adzes at $10 per dozen? 17. What cost 5 tons of timber at $3

per ton?

Ans. $158.

18. What cost 15cwt. of rice at $7.624 per cwt.?

Ans. $114.371.

19. What cost 40 tons of coal at $8.37 per ton?

Ans. $335.

DIVISION OF COMMON FRACTIONS.

237. DIVISION of Fractions is the process of dividing when the divisor or dividend, or both, are fractional numbers.

NOTE. If the divisor is less than 1, the quotient arising from the division will be as many times the dividend as the divisor is contained times in 1. Therefore, the quotient arising from dividing a whole or mixed number by a proper fraction will always be larger than the dividend.

238. The reciprocal of a fraction is the number resulting from taking its numerator as denominator, and its denominator as numerator, since any two numbers, whose product is 1, are Thus, the reciprocal of 1o is that

the reciprocals of each other.
fraction inverted, or, since 1×=1.

239. fractions.

To divide when the divisor or dividend, or both, are

Ex. 1. Divide 14 by 7.

FIRST OPERATION.

÷7 Ans.

Ans.

It is evident that the fraction 14 is divided by 7 by dividing its numerator by 7, since the size of the parts, as denoted by the denominator, remains the same, while the number of parts taken is only as large as before.

SECOND OPERATION.

14÷7

Ans.

taken, as denoted by the size of the parts is only

It is evident the fraction is also divided by 7 by multiplying its denominator by 7, since the number of parts numerator, remains the same, while the as large as before. Therefore,

Dividing the numerator or multiplying the denominator of a fraction by any number divides the fraction by that number (Art. 217).

2. Divide by 1

Ans. 2.

Since the fractional units of the two

2 Ans. fractions are of the same kind, it is evident that 12 thirteenths contain 6