137.a. The preceding cases may be summed up in the following general

RULE FOR MULTIPLICATION OF FRACTIONS.

Reduce whole and mixed numbers to improper fractions, then canceling the common factors, multiply the remaining numerators together and the remaining denominators, and the result will be the answer required.

EXAMPLES FOR PRACTICE.

1. What will 12 bushels apples cost, at of a dollar a bushel? 2. If a bushel of wheat weighs } of a hundred weight, how much will 10 bushels weigh?

3. If a man earns of a dollar per day, how much can he earn in 12 days?

4. If a family consume of a barrel of flour in a week, how much will they consume in 15 weeks?

5. If I burn of a cord of wood in a month, how much sha!! I burn in 123 months?

6. If a man can reap 11 of an acre of grain in a day how many acres can he reap in 29 days?

7. If a pound of powder is worth 6 shillings, how much are 73 pounds worth?

8. If a gallon of oil is worth 7 shillings, how much are 83 gallons worth?

9. When beef is 10 dollars a barrel, how much will 9 barrels cost?

10. What will of a firkin of butter cost, at 15 dollars a firkin?

11. At § of a dollar a cord, how much will the sawing of 20 cords of wood amount to?

12. What cost 16 pounds of cheese, at 8 cents a pound? 13. What cost 9 dozen of eggs, at 12 cents per dozen? 14. What cost 15 yards of cambric, at 15 pence per yard? 15. What cost 11 cords of wood, at 11 dollar per cord? 16. At 12 cents a pound, what cost 23 pounds of pepper? 17. At 5 shillings a pound, what cost 123 pounds of tea? 18. What cost 16 pounds of starch, at 22 cents per pound?

QUEST.-137.a. What is the general rule for multiplication of fractiona?

19. What cost 18 ounces of nutmegs, at 16 cents an ounce? 20. At 123 cents a yard, what will 17 yards of cotton cost? 21. At 31 dollars a yard, what cost 15 yards of broadcloth? 22. What cost 153 yards of ribbon, at 10 cents per yard? 23. What cost 22 penknives, at 1 of a dollar apiece? 24. At of a dollar a yard, what cost 83 yards of lace? 25. At dollar a yard, what will 97 yards of muslin cost? 26. At a dollar a bushel, what cost bushel of wheat? 27. What will pound of tea cost, at 5 of a dollar a pound? 28. What cost 66 bushels of apples, at 29. At 62 cents a yard, what cost 12

10

30. What cost 18

31. What cost 13

12

9 10

183 cents a bushel? yards of balzorine? cents per yard?

yards of lace, at 16 bushels of oats, at 183

cents a bushel?

32. What cost 31 yards of sheeting, at dollar per yard? 33. At dollar a quart, what cost 181 quarts of cherries? 34. At 33 shillings a yard, what cost 71 yards of gingham ? 35. What cost 143 bushels of potatoes, at 183 cents a bushel? 36. At 73 shillings a yard, what cost 83 yards of silk? 37. At dollar a bushel, what cost 47 bushels of pears? 38. What cost 633 pounds of sugar, at 93 cents per pound? 39. What cost 223 yards of velvet, at 33 dollars a yard? 40. What cost 393 yards of calico, at 13 shillings a yard ? 41. What cost 25 pounds of figs, at 15 cents a pound? shillings per cord?

42. What cost 353 cords of wood, at 18

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

44. What cost 383 tons of hay, at 157 dollars a ton?

45. At 42 miles a day, how far can you travel in 171 days? 46. Multiply 8539 by 3 of 19. 47. Multiply 126 by § of 33. 48. Multiply by 141. 49. Multiply 693 by 1. 50. Multiply 462 by 317. 51. Multiply 14 by 3 of 34. 52. Mult. of by 3 of 3. 53. Mult. of 9 by 3 of 7. 54. Multiply of 31 by 172. 55. Mult. of 184 by of 241. 56. Multiply 16 by of 6. 57. Multiply of 82 by 53. 58. Multiply of 91 by 82. 59. Multiply 1463 by of. 60. Mult. 256 by of 25. 61. Mult. 217 by 3 of 2 of 8. 62. Mult. 31613 by of 38. 63. Mult. 85 by 5 of 118. 64. Mult. 18 by 3% of 345. 65. Mult. 468,5 by 2 of 24. 66. Multiply of of of 18 of 11 by of§ of 45. 67. Multiply of of of of 29 by 1 of of 68. Multiply of of of 161 by 1 of 3

393.

943

10

of of 49.

DIVISION OF FRACTIONS.

CASE I.-Dividing a fraction by a whole number.

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

Operation. ÷3 Ans.

Suggestion.-1 is 1 third of 3; therefore, 1 bushel will cost 1 third part as much as 3 bushels. Now, dividing the numerator of the fraction by 3, and setting the quotient over the denominator, the result, is the answer required.

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

Suggestion. In this case we cannot divide the numerator by 4, with- 5 out a remainder. We therefore mul- 6 tiply the denominator by the 4, which divides the fraction. (Art. 113.) Hence,

-4=

Operation.

5

5

or

Ans.

6x4' 24

138. To divide a fraction by a whole number.

Divide the numerator by the whole number, when it can be done without a remainder. Or, if this cannot be done, Multiply the denominator by the whole number.

CASE II.-Dividing a fraction by a fraction.

12. At of a dollar a pound, how many pounds of honey

can be bought for of a dollar?

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

Suggestion.-Since

of a dollar will buy 1

pound, of a dollar will buy as many pounds

as is contained times in 3. Now 1 fourth Ans. 3 pounds. is contained in 3 fourths 3 times; that is, di

viding one numerator by the other, the quotient is 3. Therefore, of a dollar will buy 3 pounds.

13. At of a dollar a bushel, how much barley can le bought for of a dollar?

Suggestion. Since these fractions have different denominators, the parts denoted by their numerators are of different value; consequently one numerator cannot be divided by the other. We therefore reduce them to a common denominator; then divide the numerator of the dividend by the numerator of the divisor as above.

First operation. 2=18

8

Second operation. ÷= 3 × 3 =

Ans. 15, or 17 bu.

OBS. 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, we simply multiply the numerator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor, as in reducing fractions to a common denominator. Or, what is the same in effect, we invert the divisor as in the sec. operation, and proceed as in multiplication of fractions. (Art. 135.) 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.

OBS. 1. Compound fractions must be reduced to simple ones, and mixed numbers to improper fractions.

2. When two fractions have a common denominator, it is plain one numerator can be divided by the other, as well as one whole number by another; for the parts denoted by their numerators, are of the same kind or value.

3. When the fractions do not have a common denominator, the reason that inverting the divisor, and proceeding as in multiplication, will produce the true

QUEST. 139. How is one fraction divided by another when they have a common denominator? How, when they have not a common denominator? Obs. What must be done with compound fractions and mixed numbers? When frac tions have a common denominator, how does it appear that dividing one merator by the other will give the true answer?

answer, is because this process, in effect, reduces the two fractions to a common denominator, then the numerator of the dividend is divided by the numerator of the divisor.

We do not multiply the two denominators together for a common denominator; for no use is made of a common denominator when found.

The object of inverting the divisor is simply for convenience in multiplying.

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.

139.a. The reason of the preceding rule may also be explained in the following manner:

Operation.

Dividing the dividend by 2, the quotient is. (Art. 113.) But it is required to divide it by only of 2; consequently is 5 times too small for the true quotient. To correct this, we multiply by 5, and the result will be the answer required. Now 3×5=5, or 17, Ans.

14. Divide of # by 11.

And 17 Ans.

Solution. of, and 1. Now. Ans.

Operation.

$1

85

12

Suggestion. We may arrange the numerators, (which answer to dividends,) on the right of a perpendicular line, and the denominators, (which answer to the divisors,) on the left; then canceling the factors 3 and 2, which are common to both sides, (Art. 91.) we multiply the remaining factors in the numerators together, and those remaining in the denominators.

2 | 3
8|5=1⁄2 Ans.

QUEST.-When the fractions have not a common denominator, how does it appear that inverting the divisor and proceeding as in multiplication will give the true answer? What is the object of inverting the divisor? Note. What is meant by inverting a fraction?