25. At of a dollar a yard, what will muslin come to?

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

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

of a yard of

of a bushel

of a dollar

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 184 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 84 quarts of cherries?

34. At 34 shillings a yard, what cost 7 yards of ging ham?

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

36. At 73 shillings a yard, what cost 8 yards of silk? 37. At of a dollar a bushel, what cost 474 bushels of peaches?

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

39. What cost 2 yards of velvet, 32 dollars a yard? 40. What cost 93 yards of calico, at 1 shillings a yard? 41. What cost 25 pounds of figs, at 15 cents a pound? 42. What cost 353 cords of wood, at 18 cord?

shillings per

43. What cost 175 bushels of corn, at 3 of a dollar a bushel ?

44. What cost 83 tons of hay, at 15% dollars a ton? 45. If a man can travel 42 miles in one day, how far can 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?

Analysis.-1 is one 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 2 sons 12 of a dollar: how many twelfths did each receive?

4. A little girl bought 5 lead pencils for 1 of a shilling how much did she give apiece for them?

5. A father gave 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 1 dollar, in 4 dollars there are 4 times 2 halves; and 4 times 2 halves are 8 halves. Ans. 4 dollars will buy 8 yards.

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

8. At of a 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 quills as is contained times in ; and is contained in §, 3 times. Ans. 3 quills.

10. How many yards of cloth can I buy for of a cord of wood, if I give of a cord for a yard of cloth?

EXERCISES FOR THE SLATE.

CASE I.

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

Analysis-1 is 1 third of 3; therefore, 1 bushel will cost 1 third part as much as 3 bushels. 1 third of is .

Operation. +3=3. Ans.

Ans. of a dollar.

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.

In this case we cannot divide 5÷4=354, or 54. Ans. the numerator of the dividend by 4 the given divisor, without a remainder. We therefore multiply 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.

13. Divide by 3.

First Method. +3=3, or 4. Ans. 14. Divide by 6. 16. Divide 15 by 7.

Second Method.
§÷3=4, or 4. Ans.
15. Divide by 8.
17. Divide 17 by 12.

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

}

CASE II.

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 3. Now is contained in 3, 3 times. Ans. 3 pounds. 23. At of a dollar a bushel, how much barley can be bought for of a dollar?

First Operation.

We first reduce the given fractions to a common denominator; (Art. 125;) then divide the numerator of the dividend 15÷=17. Ans. 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 becomes ; and 2×5-15, or 17, which is the same as before.

Again, we may also illustrate the principle thus:

Second Operation.
3+2=3
3X5=15

Dividing the dividend by 2, the quotient is 3 (Art. 113.) But it is required to divide it by only 3 of 2; consequently the is 5 times too small for the true quotient. Therefore multiplied by 5 will be the quotient required. Now 3×5-15, or 17, which is the same result as before.

And 15-17. 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 occurring 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. Now 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 4 by 12.

Solution. of 4, and 1-4. Now +-X, or. 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?