18. Find the value of (41⁄2 × 7) + 1§ × (33—1%). ́19. Find the value of 28 + (73 — 23) × & × ($ + 1).

NOTE 3. The word of between fractions is equivalent to the sign of multiplication; and such an expression is sometimes called a compound fraction. Find the values of the following indicated products:

24. 1⁄2 of 3 of 2 of 1⁄2 of § of § of } of § of %.

In the following examples, cancellation may be employed by the

26. What is the value of 250

× 2881 1887?

3403 6499

7379 9409
X

27. What is the value of 3 × 8403 × 8539?

9991

9797

Ans. 53 101 Ans. 6.

Ans. 13031

28. What will 7 cords of wood cost at $35 per cord?

29. What is the value of (3) × 19 × (1)?

18079

Ans. $25.

Ans. 21873.

Ans. 25.

30. If 1 horse eat of a bushel of oats in a day, how many bushels will 10 horses eat in 6 days? 31. What is the cube of 127?

32. At $9 per ton, what will be the cost of of of a ton of hay? Ans. $4.

33. At $ a bushel, what will be the cost of 1 bushels of

corn?

1 6

34. When peaches are worth $7 per basket, what is of a basket worth?

35. A man owning of 1563 acres of land, sold of of his share; how many acres did he sell?

Ans. 47.

36. What is the product of (3)3 × (1)2 × (1%)2 × (31)*?

Ans. $15.

37. If a family consume 17 barrels of flour a month, how many barrels will 6 families consume in 89 months? -38. What is the product of 150-( of 121+ of 483)-75, multiplied by 3 × ( of 13 × 4) — 24 ? Ans. 342,63.

39. A man at his death left his wife $12,500, which was of of his estate; she at her death left of her share to her daughter; what part of the father's estate did the daughter receive? Ans.

25

40. A owned & of a cotton factory, and sold of his share to B, who sold of what he bought to C, who sold of what he bought to D; what part of the whole factory did each then own? Ans. A, ; B, 8; C, 45; D, 24·

5

5

16

8

5

41. What is the value of 21× 1+§ of 44 × (?)2+(33)3—(33)2?

In the second operation we first multiply 15 by 7, and then divide

observe that the result,, is obtained by multiplying the denominator of the given dividend by the numerator of the divisor, and the numerator of the dividend by the denominator of the divisor. Hence, in the second operation, we invert the terms of the divisor, 3, and then multiply the upper terms together for a numerator, and the lower terms together for a denominator, and obtain the same result as in the first operation. In the third operation, we shorten the process by cancellation.

We have learned (107) that the reciprocal of a number is 1 divided by the number. If we divide 1 by g, we shall have 1 ÷ 3 = 1 × 3 = §. Hence

195. The Reciprocal of a Fraction is the fraction inverted. From these principles and illustrations we derive the following general

RULE. I. Reduce integers and mixed numbers to improper fractions.

II. Multiply the dividend by the reciprocal of the divisor.

NOTES.-1. If the vertical line be used, the numerators of the dividend and the denominators of the divisor must be written on the right of the vertical.

2. Since a compound fraction is an indicated product of several fractions, its reciprocal may be obtained by inverting each factor of the compound fraction.

16

27

7. Divide by 8, 13 by, and 3 by 5.

8. Divide 17 by 11.

Ans. 13.

9. Divide 114 by 35.

Ans. 15.

NOTE 3.

The second method of solution given above has two advantages. 1st, It gives the answer by a single operation; 2d, It affords greater facility for cancellation.

11. Divide of 5 by 8 of 5.

12. Divide of by of

1/2

3

21

13. Divide 2 × 71 by 31 × 3,3%.

14. Divide 11 by x 51 x 7.

15. Divide 31 x 19 by x 73 x 15.

16. Divide 18 by 1 × × 5 × 3 × 91.

7

18. Divide by 31 × 38 × 84.

Ans. 1. Ans. 1

19. Divide × × × by × × × × 1%.

ANALYSIS. The fractional form indicates division, the numerator being the dividend and the denominator the divisor, (168, II); hence, we reduce the mixed numbers to improper fractions, and then treat the denominator, 22, as a divisor, and obtain the result, 14, by the general rule for division of fractions.

51 V 4%

NOTE 4.-Expressions like and are sometimes called complex fractions.

22

5. In the reduction of complex fractions to simple fractions, if either the numerator or denominator consists of one or more parts connected by + or —, the operations indicated by these signs must first be performed, and afterward the division.

21. What is the value of ?

Ans. .

27. If 7 pounds of coffee cost $3, how much will 1 pound cost? 28. If a boy earn $3 a day, how many days will it take him to earn $61? Ans. 17.

29. If of an acre of land sell for $30, what will an acre sell for at the same rate?

Ans. $67.

30. At of of a dollar a pint, how much wine can be bought

for $10
$?

Ans. 2 pints.

31. If of a barrel of flour be worth $21, how much is 1

barrel worth?

32. Bought

Ans. $73.

of 4 cords of wood, for of of $30; what

Ans. $4,8.

was 1 cord worth at the same rate? 33. If 235 acres of land cost $1725g, how much will 1251 acres cost?

part?

Ans. $918116.

Ans. 311.

7

143

34. Of what number is 261 the 35. The product of two numbers is 27, and one of them is 25; what is the other? 36. By what number must you multiply 1611 to produce 148 ? 37. What number is that which, if multiplied by 3 of will produce?

of 2,

Ans. 111.

5

38. Divide 720-(§ 28-71) by 401 +(%) x (1)*. 39. What is the value of (31 × (3)2 + 1 of 1)3 ÷ (173 — §

3