25. At

of a dollar a yard, what will of a yard of muslin come to?

26. At of a dollar a bushel, what cost 1 of wheat?

of a bushel

27. What will of a pound of tea cost, at § of a dollar a pound?

28. What cost 66 bushels of apples, at 184 cents a bushel?

29. At 62 cents a yard, what cost 12 yards of balzorine ?

30. What cost 18 yards of tape, at 64 cents per yard? 31. What cost 13 bushels of oats, at 183 cents per bushel ?

32. What cost 31 yards of sheeting, at of a dollar per yard?

33. At of a dollar a quart, what cost 8 quarts of cherries?

34. At 3 shillings a yard, what cost 7 yards of gingham?

35. What cost 143 bushels of potatoes, at 183 cents a bushel?

36. At 7 shillings a yard, what cost 8 yards of silk? 37. At of a dollar a bushel, what cost 47 bushels of peaches?

38. What cost 634 pounds of sugar, at 93 cents per pound?

39. What cost 2

40. What cost 93

41. What cost 25

42. What cost 35

cord?

yards of velvet, at 32 dollars a yard? yards of calico, at 13 shillings a yard ? pounds of figs, at 15 cents a pound? cords of wood, at 18

shillings per

43. What cost 1751⁄2 bushels of corn, at of a dollar a bushel?

44. What cost 83 tons of hay, at 15 dollars a ton?

45. If a man travel 421⁄2 miles in one day, how far will he travel in 17 days?

DIVISION OF FRACTIONS.

MENTAL EXERCISES.

Ex. 1. A man divided of a pound of honey equally among his 3 children: what part of a pound did each receive?

Suggestion. 1 is 1 third of 3; therefore 1 child must have received 1 third of 6 sevenths. 1 third of 6 sevenths is 2 sevenths. Ans. Each child received of a pound. 2. If 4 pounds of loaf sugar cost of a dollar, how much will 1 pound cost?

3. A father gave his two sons 19 of a dollar : how many twelfths did each receive?

4. A little girl bought 5 lead pencils for 19 of a shilling: how much did she give apiece for them?

5. A father gave 32 parts of a vessel to his 6 sons: what part of the vessel did each receive?

6. At dollar a yard, how many yards of French muslin can you buy for 4 dollars?

Suggestion. 4 dollars will buy as many yards as 1 half is contained times in 4, or as there are halves in 4 dollars. Now since there are 2 halves in a whole dollar, in 4 dollars there will be 4 times 2; and 4 times 2 halves are 8 halves. Ans. 4 dollars will buy 8 yards.

7. Atcent apiece, how many apples can I buy for 6 cents?

8. At dollar a pound, how many pounds of almonds can you buy for 12 dollars?

9. How many quills, at of a penny apiece, can you buy for of a penny?

Suggestion. of a penny will buy as many as is contained times in g, and in §, 3 times. Ans. 3 quills.

10. How many yards of cloth can a farmer buy for of a cord of wood, if he gives of a cord for a yard of cloth?

EXERCISES FOR THE SLATE.

CASE I.

11. If 3 bushels of oats cost of a dollar, what will 1 bushel cost?

Suggestion. 1 is 1 third of 3; therefore, 1 bushel will cost third part as much as 3 bushels. 1 third of is 3. Ans. of a dollar.

Operation. 3. Ans.

We divide the numerator of the fraction, which is the whole cost, by 3 the whole number of bushels, and place the quotient 2 over the given denominator.

12. If 4 yards of calico cost of a dollar, what will 1 yard cost?

Operation.

5

4 or Ans.

In this case we cannot divide the numerator of the dividend by 4 the given divisor, without a remainder. We therefore mul

tiply the denominator by the 4, which is in effect dividing the fraction. (Art. 113.)

Hence,

138. To divide a fraction by a whole number.

Divide the numerator by the whole number, when it can be done without a remainder; but when this cannot be done, multiply the denominator by the whole number.

QUEST.-138. How is a fraction divided by a whole number?

22. At of a dollar a pound, how many pounds of honey can be bought for of a dollar?

Suggestion. Since of a dollar will buy 1 pound, of a dollar will buy as many pounds as is contained times in . Now is contained in 2, 3 times. Ans. 3 pounds.

23. At of a dollar a bushel, how much barley can be bought for of a dollar?

First Operation.

8

20

17. Ans.

We first reduce the fractions to a common denominator; (Art. 125;) then divide the numerator of the dividend by the numerator of the divisor, as above.

OBS. 1. After the fractions are reduced to a common denominator, it will be perceived that no use is made of the common denominator itself. In practice, therefore, it is simply necessary to multiply the numerator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor, in the same manner as two fractions are reduced to a common denominator; or, what is the same in effect, invert the divisor, and proceed as in multiplication of fractions. (Art. 135.)

Note. To invert a fraction is, to put the numerator in the place of the denominator, and the denominator in the place of the numerator. Thus, in the example above, inverting the divisor, it becoines §; and 2×5=15, or 17, which is the same as before.

Again, we may also illustrate the principle thus.

Second Operation.
1÷2=1
×5=15

Dividing the dividend by 2, the quotient is . (Art. 113.) But it is required to divide it by only of 2; consequently the is 5 times too small for the true quotient. Therefore multiplied by 5 will be the quotient required. Now ×5=15, or 13, which is the same result as before.

And 1517. Ans.

OBS. 2. By examination the learner will perceive that this process is precisely the same in effect as the preceding; for in both cases the denominator of the dividend is multiplied by the numerator of the divisor, and the numerator of the dividend, by the denominator of the divisor. Hence,

139. To divide a fraction by a fraction.

I. If the given fractions have a common denominator; Divide the numerator of the dividend by the numerator of the divisor.

II. When the fractions have not a common denominator; Invert the divisor, and proceed as in multiplication of fractions. (Art. 135.)

OBS. 1. Compound fractions occuring in the divisor or dividend, must be reduced to simple ones, and mixed numbers to improper fractions

2. The method of dividing a fraction by a fraction depends upon the obvious principle, that if two fractions have a common denominator, the numerator of the dividend, divided by the numerator of the divisor, will give the true quotient. Multiplying the numerator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor, is, in effect, reducing the two fractions to a common denominator. The object of inverting the divisor is simply for convenience in multiplying.

24. Divide of by 1.

8

Solution. of =2, and 1=3. Now ÷=

16

X3, or 1. Ans.

QUEST.-139. How is one fraction divided by another when they have a common denominator? How when they have not common denominators? Obs. How proceed when the divisor or dividend are compound fractions or mixed numbers? Upon what principle does the method of dividing a fraction by a fraction, depend? Why multiply the numerator of the dividend by the denominator of the divisor, &c.? Why invert the divisor?