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OPERATION.

12:825: Ans.

Ans. 163 days.

In this case the number of days decreases in the same ratio as the number of men increases.

NOTE 1. When more requires more, as in Ex. 25, or less requires less, the proportion is called direct. When more requires less, as in Ex. 26, or less requires more, the proportion is called inverse. Thus, the more men employed the more work, and the less men the less work, in a given time; that is, the amount of work varies directly as the number of men. But the more men the less time, and the less men the more time, to do a given piece of work; that is, the time varies inversely as the number of men.

422. To solve examples by proportion,

Rule.

Make the number that is of the same kind as the required answer the third term.

If from the nature of the question the answer ought to be greater than this third term, of the two remaining terms make the greater the second, and the less the first; but if the answer ought to be less than the third term, make the less the second, and the greater the first.

Divide the product of the second and third terms by the first.

NOTE 2. If possible always cancel factors from the first with equal factors from either the second or third terms (Art. 80, b).

27. If a man earns $ 48 in 4 months, how much will he earn in 9 months?

OPERATION.

4:9 $48: Ans.

=

12

9

$108, Ans.

Since we are seeking for dollars, we make $48 the third term; and then, as a man will earn more in 9 months than he will in 4 months, we make 9 the second term and 4 the first. Cancelling the equal factor 4 from the first and third terms, we have $12 × 9 = $ 108, Ans.

28. If 20 bushels of wheat make 4 barrels of flour, how many bushels of wheat will be required to make 7 barrels of flour?

Ans. 35.

29. If a man can walk 120 kilometers in 3 days, how far can he walk in 10 days?

30. If 30 bushels of wheat make 6 barrels of flour, how many barrels of flour will 85 bushels of wheat make?

31. If a man travel 74 miles in 2 days, how long will it take him to travel 259 miles ?

32. If 24 men perform a piece of work in 6 days, in how many days will 8 men perform the same work?

33. If a locomotive run 65000 kilometers in 8 weeks, how far, at that rate, would it run in 52 weeks?

34. If $30 pay for 20 sters of wood, how many dollars will pay for 48 sters?

35. If 12 cords of wood cost $ 60, what will 28 cords cost? 36. If 5 cords of wood cost $25, how many cords may be bought for $95?

37. What will 8 tons of coal cost if 9 tons cost $54?

38. If 5 horses eat 16 hektoliters of oats in 6 weeks, how many hektoliters will 17 horses eat in the same time?

39. In how many days can 8 men build a house, if 14 men can build it in 82 days?

40. The interest of $200 for 1 year being $12, what is the interest of $950 for the same time?

41. How many tons of coal can be bought for $84, when 6 tons cost $28?

42. How many tons of hay will 9 horses eat in 27 weeks, if 6 horses eat 15 tons in the same time?

43. If 8 horses eat a ton of hay in 22 days, how many horses will eat a ton in 32 days?

44. If of a ship cost $5679, what are of her worth?

45. If of a barrel of flour cost $3.75, what will 9 barrels cost?

46. If a man walks 135 miles in 5 days of 9 hours each, in how many days of 10 hours each will he walk 135 miles?

47. How many meters of cloth of a meter wide are equal to 30 meters 14 meters wide?

48. Lent a friend $600 for 8 months; afterwards he lent me $500. How long may I keep it to balance the favor?

49. If, when flour is $9 a barrel, a 4-cent loaf weighs 2 hektograms, what ought it to weigh when flour is $8 a barrel?

50. If 8 men can do a piece of work in 15 days, how many men must be added to the number to do it in 5 days?

51. A certain house was built by 40 workmen in 48 days, but, being burned, it is required to rebuild it in 30 days; how many men must be employed?

52. If 18 horses eat 85 bushels of oats in 4 weeks, how many bushels will 16 horses eat in the same time?

53. A garrison of 1200 men has provisions for 12 months; how long will the same provisions last if the garrison is reinforced by 150 men?

54. A ship's crew of 16 men has food for 56 days; how many men must be discharged that it may last 8 days longer?

55. If 0.95 of a ship cost $5000, what will 0.675 of it cost? 56. At $14 for a hundred pounds, what is the cost of 621 pounds?

57. If a steeple 150 feet high casts a shadow 210 feet, what is the length of the shadow cast, at the same time, by a staff 5 feet high?

NOTE 3. Since each of the three terms in the above example is in feet, the learner may be uncertain which number to place as the third term; but he has only to notice that he is required to find the length of a shadow: hence, the third term should be the number expressing the length of shadow in the given example, that is, 210 feet.

58. If a staff 5 feet long casts a shadow 7 feet, what is the height of a steeple which, at the same time, casts a shadow 210 feet?

59. If a staff 5 feet long casts a shadow 7 feet, how long is the shadow, cast at the same time, by a steeple which is 150 feet high?

60. If a steeple 150 feet high casts a shadow 210 feet, what is the height of a staff which, at the same time, casts a shadow 7 feet?

61. If 75 kilos of cheese are worth as much as 30 kilos of butter, how many kilos of cheese will pay for 24 kilos of butter?

62. The interest of $600-for 6 months being $15, what principal will gain $ 64 in the same time?

63. If a man's salary amounts to $7200 in 4 years, what will it amount to in 15 years?

64. If a man's salary amounts to $18900 in 15 years, in how many years will it amount to $7560?

65. If 15 meters of silk that is of a meter wide will make a dress, how many meters of muslin that is 1 meters wide will be required to line it?

66. If 3 men can mow 10 a. 136 sq. rd. of grass in 4 days, by working 6 hours a day, how many days will it take them to mow the same area if they work only 4 hours a day?

67. If of an acre of land is worth $22.50, what is the value of 25 acres, at the same price?

NOTE 4. All the examples given above can be solved without the aid of Proportion, by what is sometimes called Analysis in distinction from Proportion.

COMPOUND PROPORTION.

423. Compound Proportion is an equality of two ratios, one of which is compound and the other simple; thus,

3:12
2

16:

}

18: 9, is a compound proportion;

and 48: 24 = 189, is the same reduced to a simple form.

NOTE. The compound ratio may consist of any number of couplets.

424. Written Exercises.

68. If 6 men in 8 hours thresh 30 bushels of wheat, in how many hours will 2 men thresh 5 bushels?

OPERATION.

2:6 30:5

= 8: Ans.

As the required answer is to be in hours, we make 8 hours the third term. To do the same work it will take 2 men longer than it will 6 men ;

hence, we make 2 the first and 6 the second term. To thresh 5 bushels will take less time than to thresh 30 bushels; hence, we make 30 the first and 5 the second term. The proportion then stands 2 × 30 : 6 × 5 = 8: 4, Ans. Hence,

425. To solve questions in Compound Proportion,

Rule.

Write that given number which is of the same kind as the required answer for the third term. Take any two of the remaining terms of the same kind, and, considering the question as depending on these alone, arrange them as in simple proportion; arrange each pair of like terms by the same principles. Divide the product of the second and third terms by the product of the first terms.

NOTE 1. If possible always cancel factors in the second and third terms with equal factors in the first terms.

69. If 6 men in 15 days earn $135, how many dollars will 9 men earn in 18 days?

6: 9 13: 18 5 3

=

OPERATION.

$135: 9 x $ 27 = $243, Ans.

27

70. If 4 men, in 24 days of 9 hours each, build a wall 40 ft. long, 9 ft. high, and 4 ft. thick, in how many days of 6 hours each can 8 men build a wall 60 ft. long, 12 ft. high, and 5 ft. thick?

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