The word "of" between fractions is equivalent to the Such expressions are generally called Compound

sign x. Fractions.

12. What is the cost of of a yard of silk velvet at $5 a yard?

13. At $3 a bushel, what is of a bushel of apples worth?

14. What is the cost of of a pound of sugar at 8 cents a pound?

15. What is of a gallon of molasses worth at # of a dollar a gallon?

Solution. In of a day, he earns of a dollar. of = and 8, equal to 3 of a dollar. The result is obtained by multiplying the numerator of the multiplicand by the numerator of the multiplier and the denominator of the multiplicand by the denominator of the multiplier. Thus, & × 3 = 18 = 8.

22. If James walked of a mile and his brother as far, how far did his brother walk?

23. Mary found of a dollar and gave away of it: how much did she give away?

24. What is the cost of of a yard of cloth at of a dollar a yard ?

TO TEACHERS.-It is clear from the foregoing solutions that the usual "three cases" in multiplication of fractions, which from time immemorial have confused the beginner, should be treated as one case. This can be done by writing 1 as the denominator, when the multiplicand or the multiplier is an integer.

ART. 156. Rule for Multiplication of Fractions.-Change mixed numbers and integers to improper fractions, and then multiply the numerators together for a new numerator, and the denominators together for a new denominator.

WRITTEN EXERCISES.

NOTE. In the following problems the pupils must use their ingenuity to discover the shortest methods of solution. Cancellation should be employed wherever possible. It is often however convenient to multiply whole and mixed numbers without reducing them to simple fractions.

bushel how much did he receive?

22. At $194 a ton, how much are 171⁄2 tons of hay worth?

23. Samuel traveled

eled of that distance:

of 143 miles, and Thomas travhow far did Thomas travel?

24. Mr. A bought 35 bushels of wheat at 874 cents a bushel, and sold at 883 cents, and the remainder at 893 cents a bushel: how much did he clear by the transaction?

25. Illustrate by an original problem, multiplication of fractions.

Division of Fractions.

ORAL EXERCISES.

GENERAL NOTE ON DIVISION OF FRACTIONS.-Divide by . The divisor is of 1, or of 5. Dividing by the whole of 5, we get But we were not to divide by 5, but by of 5. Having divided by a divisor 6 times the true divisor, the quotient is only as large as it ought to be. Hence we multiply by 6 to obtain the true quotient. Inspection shows that the terms of our fractional divisor are inverted. Thus: ÷ & = 2 × ? = ! 8 = 10.

ART. 157.-1. Mary divided

8 girls: what part of a dollar

of a dollar equally among did each girl receive?

Solution. Each girl received of of a dollar. of of a dollar is of a dollar, and of of a dollar is

of a dollar, equal The same result

too of a dollar, the part received by each girl. may be obtained by writing the divisor in the form of a fraction, inverting it, and proceeding as in multiplication.

2. Andrew shared of a bushel of nuts with his 5 companions: how much did each receive?

3. If 6 books are worth

book worth?

of a dollar, what is one

4. A boy sold 8 fishes for of a dollar: how much did he receive apiece?

5. At $3 a yard, what part of a yard of cloth can be bought for of a dollar?

6. At $2 a load, what part of a load of kindling wood can be bought for of a dollar?

7. A tract of land containing § of an acre was divided into 6 equal lots: what part of an acre did cach lot contain?

8. A teacher paid of a dollar for 5 spellers: what did each speller cost?

of a dollar equally among 5 boys:

9. Alfred divided how much did each receive?

10. If 6 yards of muslin cost of a dollar, what is the cost per yard?

ART. 158.—11. At of a dollar a pound, how many pounds of tea can be bought for $3?

Solution. As many pounds can be bought for $3, as of a dollar is contained times in $3. is contained in 3, 15 times, and is contained of 15 times, or 15 times = 33. Hence, 33 pounds can be bought for $3. The same result may be obtained by inverting the divisor and proceeding as in multiplication. Thus, 3 ÷ =3x=33.

12. If 4 yards of cloth make a coat, how many coats can be made from 18 yards of cloth?

13. At $2 a day, how long will it take a man to earn $15?

14. How long will $52 pay my board at the rate of $31 a day?

15. At 3 miles an hour, how long will it take a man to walk 40 miles?

ART. 159.-16. At of a dollar a gallon, how many gallons of molasses can be bought for of a dollar?

Solution. As many gallons can be bought as of a dollar is contained times in % of a dollar. is contained in, 5 times, and is contained in, 5 times. Hence, 45 or 1% or 1 gallons can be bought for of a dollar. The same result may be obtained by inverting the divisor and proceeding as in multiplication. Thus, 10=10 × 4 = 45 = 13.

17. How many oranges at of a cent each, can be bought for 3 cents?

18. If a coat is worth $7, how many coats can be bought for $221 ?

19. At of a dollar a pound, how many pounds of tea can be bought for $93?

20. At $2 a yard, how many yards of cloth can be bought for $271⁄2 ?