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32. When 1 barrel of apples can be bought for $2, how many barrels can be bought for $578?

33. How many days will it take a man who earns $3 per day, to earn $198?

34. How many hours will it take a man who travels at the rate of 4 miles per hour, to travel 133 miles ?

35. Joseph can learn 5 pages of history per day. How many days will it take him to learn 276 pages ?

36. Edward sells 168 marbles at the rate of 7 marbles for 1 cent. How much does he receive for them?

37. When rice was worth 6 cents per pound, a person bought enough to come to 5783 cents. How many pounds did he buy?

38. If a man advances 3 feet at a step, how many steps must he take to advance 1785 feet?

39. A man drew from a bank $9474 in three-dollar bills. How many bills did he receive ?

40. How many five-dollar bills will it take to pay a debt of $3675 ?

41. Among how many men can $5698 be divided, if each man receives $11?

42. How many bottles, each holding 3 pints, can be filled from 8753 pints of cider?

43. 5433 pints equal how many bushels, pecks, quarts, and pints ?

WRITTEN WORK. 2)5433 pt. — 1 pt.

SOLUTION. — Since 2 pints equal 1 quart, 5433 pints must equal as many quarts as there are times 2 in 5433, which are 2716 times, with 1 remainder. Hence 5433 pt. 2716 qt. 1 pt.

Since 8 quarts equal 1 peck, 2716 quarts must equal as many pecks as there are times 8 in 2716, which are, etc.

Therefore, 5433 pints = 84 bu. 3 pk. 4 qt. 1 pt.

8)2716 qt. — 4 qt. 4)339 pk. 3 pk.

84 bu.

Note:-The reduction of numbers from a lower denomination to a higher is called Reduction Ascending. The last example and the seven which follow are illustrations of it.

44. 5975 nails equal how many yards, quarters, and nails ? 45. 96452 equal how many ib, 3, etc.?

46, 3794 farthings equal how many pounds, shillings, pence, and farthings?

47. 1247 furlongs equal how many leagues, miles, and furlongs? 48. 5786 gills equal how many gallons, quarts, etc.?

49. 587 days equal how many months, weeks, and days, allowing 4 weeks to a month ?

50. 645 inches equal how many yards, feet, and inches ? 51. If 7 acres cost $5943, how much will 1 acre cost?

Reasoning Process. — If 7 acres cost $5943, 1 acre will cost ļ of $5943, which, by dividing 5943 by 7, is found to be $849.

52. If 9 barrels of flour weigh 1764 pounds, how much does 1 barrel weigh?

53. Joseph can buy 588 marbles for 8 dimes. How many can he buy for 1 dime?

54. A man who owned 7359 square feet of land, divided it into 9 equal house-lots. How many square feet were there in each lot?

55. Arthur's travelling expenses during a journey of 8 days were $67.52. What were his average expenses per day?

56. In how many days can 11 men do a piece of work which 1 man can do in 137 days ?

57. If 1 man can do a piece of work in 584 days, in how many days can 6 men do it?

58. How much will a peck of wheat cost at the rate of $1.876

per bushel ?

59. If 2 bu. 1 pk. or 9 pk. of cranberries cost $10.71, how much will 1 peck cost?

60. If it takes a man 7231 seconds to walk 2 leagues and 1 mile, how many seconds will it take him to walk 1 mile ?

61. How many bottles of ink at $.06 per bottle can be bought for $57?

Solution. — If one bottle of ink can be bought for 6 cents, as many bottles can be bought for $57, or 5700 cents, as there are times 6 in 5700, which (found by division) is 950 times. Hence, 950 bottles of ink can be bought.

Note. — In solving problems like these, both divisor and dividend should be reduced to the lowest denomination mentioned in either. In the above example, both were reduced to cents.

62. How many sheets of drawing paper, at $.08 per sheet, can be bought for $9.76?

63. How many pounds of soap, at $.09 per pound, can be bought for $65.97 ?

64. How many pounds of sugar, at $.08 per pound, can be bought for $114?

65. How many quills, at $.006 each, can be bought for $75 ?

66. How many steel pens, cošting $.008 each, can be bought for $147 ?

54. Division by Factors.

(a.) When the divisor is the product of factors, we may find the required quotient by dividing by the factors of the divisor.

1. What is the quotient of 1752 • 24?

SOLUTION. — Since 24 = 8 X 3, 8)1752 we may find the required quotient

3)219 = quotient by 8. by first dividing by 8 and then by 3.

73 = quotient by 8 X 3, or 24. Dividing by 8 gives 219 for a quotient, i. e. 219 eights. Dividing this by 3 gives 73 for a quotient, i. e. 73 units each equal to 3 eights or to 24.

(b.) In the same way perform the following:

2. 8528 • 16? 3. 3456 • 18? 4. 14352 = 48? 5. 47328 – 32 ?

6. 56592 • 27 ? 7. 30156 = 28 ? 8. 25785 ; 45 ? 9. 79344 • 72?

10. What is the quotient of 7391 · 42?

2)7391 1

SOLUTION. Since 42 = 2 X 3 X 7, we may find the

required quotient by dividing by 2, then by 3, and then 3)3695 2

by 7. 7)1231 6

Dividing by 2 gives 3695 for a quotient and 1 for a

remainder, i. c. 3695 twos and 1 unit remaining. 175

Dividing 3695 by 3 gives 1231 for a quotient, and 2 for a remainder, i. e. 1231 units each equal to 3 times 2 or to 6, and 2 twos remaining.

Dividing 1231 by 7 gives 175 for a quotient and 6 for a remainder, i. e. 175 units, each equal to 7 times 3 times 2, or to 42, and 6 sixes remaining.

The quotient, then, is 175, and the remainder is 1 + 2 times 2 + 6 times 3 times 2 =

41.

(c.) From the above considerations, we infer that the true remainder, after any division by factors, may be found by multiplying the remainder of each division after the first by the divisors of all the preceding divisions, and then adding these products to the first remainder.

(d.) Find the quotient and remainder in the following examples:

11. 5855 • 36 ?
12. 8794 • 28?
13, 3795 ; 42 ?
14. 2767 • 14?
15. 3859 : 81 ?

16. 2987 • 51 ?
17. 5321 • 16 ?
18. 6743 • 25 ?
19. 5425 · 36 ?
20. 3891 • 27 ?

(e.) One of the most convenient applications of the division by factors is made when the divisor contains 10, 100, 1000, or some higher power of 10, as a factor.

21. What is the quotient of 4573 = 800 ?

SOLUTION, - Since 800 8 X 100, we may find the required quotient by dividing by 100 and then by 8, i. e. by removing the point two places to the left (see 17) and dividing by 8.

Removing the point gives 45 for a quotient and 73 for a remainder. Dividing 45 by 8 gives 5 for a quotient and 5 for a remainder. The quotient, then, is 5, and the remainder is 5 times 100 + 73, i. e. 500 + 73 573.

(f.) Perform the foļlowing examples :

22. 5975 • 600 ?
23. 3794 = 20 ?
24. 48697 = 5000
25. 32697 • 50 ?
26. 67328 – 900 ?
27. 69487 • 1400 ?

28. 31754 • 2100 ?
29. 168597 • 1200 ?
30. 32754 • 240 ?
31. 59867 1807
32. 37642 • 360 ?
33. 69483 • 800 ?

(g.) Had we wished for the complete quotient in the 21st example, we should have considered the first quotient to be 45.73, instead of 45 and 73 remainder. We should then have divided 45.73 by 8, giving 5.715, or 5.71625 for the full quotient.

(h.) Find the complete quotient in each of the examples under f.

55. Divisor a Large Number,

(a.) When the divisor is large, it is not always easy to determine the true quotient figure directly. In such cases, the number expressed by one or two of the left-hand figures of the divisor, may be selected as a sort of TRIAL DIVISOR ; but to test the cor. rectness of the quotient figure thus obtained, it will be necessary to find the product of the divisor by the quotient, and the remainder after subtracting that product from the dividend.

Note. — The product should be either equal to or less than the dividend, and the remainder should be less than the divisor.

1. What is the quotient of 537 · 82?
SOLUTION. — Writing the divisor on the left of the divi-

82)537(6 dend, as opposite, we make 8 the trial divisor.* Since 8 is

492 contained 6 times with a remainder in 53, we write 6 as the

45 probable quotient figure. To see whether it be correct or not, we multiply the divisor by it. The product is 492, which, being less than the dividend, shows that the quotient figure is not too large. Subtracting 492 from 537 leaves a remainder of 45, which, being less than the divisor, shows that the quotient figure is not too small. Being neither too large nor too small, it must be correct. Hence, the quotient is 6 and the remainder is 45, or the complete quotient is 65.

* For 82 is contained in 537 about the same number of times that 80 is, or that 8 is contained in'53.

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