37. If the moon moves 13° 10′ 35′′ in one day, in what time does she perform one revolution? Ans. 27da. 7h. 43m.+ 38. If 7lb. of sugar cost of a dollar, what are 12lb. worth? Ans. $1.284. 39. If $1.75 will buy 7lb. of loaf-sugar, how much will $213.50 buy? Ans. 7cwt. 2qr. 14lb. 40. If 7 ounces of gold are worth 30£., what is the value of 7lb. 11oz.? Ans. 407£. 2s. 10 d.

41. My friend borrowed of me $500 for 6 months, promising me like favor; soon after, I had occasion for $ 600; how long should I keep it to receive full compensation for the kindness? Ans. 5 months.

42. If the penny loaf weighs 7oz. when flour is $8 per barrel, how much should it weigh when flour is $7.50 per barrel ? Ans. 77 ounces.

43. If a regiment of soldiers consisting of 1000 men were to be clothed, each suit containing 3 yards of cloth, that is 17 yards wide, and to be lined with flannel 14 yards wide, how many yards will it take to line the whole? Ans. 5625yd. 44. If 9 yards of broadcloth cost $113, what will 16 ells English cost?

Ans. $24. 45. A merchant failing in trade owes in all $17280; his effects are sold for $15120; what does he pay on a dollar, and what does A receive, to whom he owes $5670 ?

Ans. He pays $0.87 on the dollar; A receives $4961.25. 46. Bought in London 57 yards of broadcloth for 49 guineas, 28 shillings each; what did it cost per ell English?

Ans. 1£. 10s. 11⁄2d. 47. Bought a cask of wine, at $1.15 per gallon, for $100; how much did it contain ? Ans. 86gal. 3qt. 11pt. 48. A merchant bought 9 packages of cloth for $34560, each package containing 8 parcels, each parcel 12 pieces, and each piece 20 yards; what was the price per yard?

Ans. $2.00.

49. If 75 gallons of water fall into a cistern containing 500 gallons, and by a pipe in the cistern 40 gallons run out in an hour, in what time will it be filled? Ans. 14h. 17m. 8 sec.

50. How many dozen pairs of gloves, at $0.56 per pair, may be bought for $120.96 ? Ans. 18doz. 51. A certain cistern has three pipes; the first will empty it in 20 minutes, the second in 40 minutes, and the third in 75 minutes; in what time would they all empty it?

Ans. 11m. 19 sec.

52. A can mow a certain field in 5 days, and B can mow it in 6 days; in what time would they both mow it?

Ans. 28 days.

53. A wall, which was to be built 32 feet high, was raised 8 feet by 6 men in 12 days; how many men must be employed to finish the wall in 6 days? Ans. 36 men.

54. A can build a boat in 20 days, but with the assistance of C he can do it in 12 days; in what time would C do it himself? Ans. 30 days. 55. In a fort there are 700 men provided with 184000lb. of provisions, of which each man consumes 5lb. a week; how long can they subsist? Ans. 52 weeks 4 days.

56. If 25 men have of a pound of beef each three times in a week, how long will 3150lb. last them? Ans. 56 weeks. 57. How many tiles 8 inches square will lay a floor 20 feet long, and 16 feet wide?

Ans. 720.

58. How many stones 10 inches long, 9 inches broad, and 4 inches thick, would it require to build a wall 80 feet long, 20 feet high, and 24 feet thick? Ans. 17280 stones.

59. Bought threescore pieces of Hollands for three times as many dollars, and sold them again for four times as many dollars; but if they had cost me as much as I sold them for, for what should I have sold them to gain at the same rate?

Ans. $320. 60. A sets out on a journey, and travels 27 miles a day; 7 days after, B sets out and travels the same road 36 miles a day; in how many days will B overtake A ? Ans. 21 days. gain 35 per cent.,

61. If I sell coffee at 2s. 3d. per lb. and what did I give per lb. ?

Ans. 1s. 8d.

62. A detachment of 2000 soldiers were supplied with bread sufficient to last them 12 weeks, allowing each man 14 ounces a day; but on examination find 105 barrels, containing 200lb. each, wholly spoiled; how much a day may each man eat, that the remainder may supply them 12 weeks? Ans. 12oz.

63. In consequence of having a seventh part of their bread spoiled, 2000 soldiers were put on an allowance of 12 ounces of bread per day for 12 weeks; what was the whole weight of their bread (good and bad), and how much was spoiled?

Ans. The whole weight, 147000lb.; spoiled, 21000lb. 64. Two thousand soldiers, having lost 105 barrels of bread, weighing 200lb. each, were obliged to subsist on 12 ounces a day for 12 weeks; but had none been lost, they might have had 14 ounces a day for the same time. What was the whole

weight, including what was lost, and how much had they left to subsist on ?

Ans. The whole weight, 147000lb.; left to subsist on, 126000lb.

65. If 2000 soldiers, after losing one seventh part of their bread, had each 12 ounces a day for 12 weeks, what was the whole weight of their bread, including that lost, and how much might they have had per day, each man, if none had been lost? Ans. The whole weight was 147000lb.; the loss, 21000lb.; had none been lost, they might have had 14 ounces per day. 66. If .85 of a gallon of wine cost $2.72, how much will .25 of a gallon cost? Ans. $0.80. 67. If 61.3 pounds of tea cost $44.9942, what is the price per lb. ? Ans. $0.73,4. 68. What is the value of .15 of a hogshead of lime, at $2.39 per hhd. ?

Ans. $0.35,85.

69. If .75 of a ton of hay cost $15, what is it

per ton? Ans. $20.

70. How many yards of carpeting that is half a yard wide. will cover a room that is 30 feet long and 18 feet wide? Ans. 120 yards.

71. If a man perform a journey in 15 days when the day is 12 hours long, in how many days will he do it when the day is but 10 hours long? Ans. 18 days.

72. If 450 men are in a garrison, and their provisions will last them but 5 months, how many must leave the garrison that the same provisions may be sufficient for those who remain 9 months? Ans. 200 men.

73. The hour and minute hands of a watch are together at 12 o'clock; when will they next be together?

Ans. 1h. 5m. 27 sec. 74. A and B can perform a piece of work in 5 days, B and C in 6 days, and A and C in 6 days; in what time would each of them perform the work alone, and how long would it take them to do the work together?

Ans. A would do the work in 10 days; B, in 12 days; C, in 15 days; A, B, and C, together, in 4 days.

75. A, B, and C can perform a piece of work in 4 days, B can do it in 12 days, C can do it in 15 days; in what time would A and B perform the labor? Ans. 5 days.

76. How many bricks 8 inches long, 4 inches wide, and 2 inches thick, will it require to build the walls of a house which is 46 feet long, 28 feet wide, and 25 feet high, and the walls to be 18 inches thick? Ans. 143,775 bricks.

77. Lent a friend $200 for 12 months, on condition of his returning the favor; how long ought he to lend me $150 to requite my kindness? Ans. 16 months. 78. If 5 oxen or 7 cows eat 3 tons of hay in 87 days, in what time will 2 oxen and 3 cows eat the same quantity of hay ? Ans. 105 days.

79. If 360 men be placed in a garrison, and have provisions for 6 months, how many men must be sent away at the end of 4 months that the remaining provision may last them 8 months longer? Ans. 270 men.

80. My tailor informs me it will take 10 square yards of cloth to make me a full suit of clothes. The cloth I am about to purchase is 17 yards wide, and on sponging it will shrink 5 per cent. in width and length. How many yards of the above cloth must I purchase for my Ans. 66yd. new suit"?

66

62

SECTION LIII.

COMPOUND PROPORTION,

OR

DOUBLE RULE OF THREE.

COMPOUND PROPORTION is the method of performing such operations in Proportion as require two or more statements.

EXAMPLES.

1. If a man travel 117 miles in 30 days, employing only 9 hours a day, how far would he go in 20 days, travelling 12 hours a day?

The distance to be travelled depends on two circumstances,

the number of days the man travels, and the number of hours he travels in each day.

We will first suppose the hours to be the same in each case; the question will then be,-If a man travel 117 miles in 30 days, how far will he travel in 20 days?

This will lead to the following proportion.

30 days 20: 117 miles :

117 X 20

78 miles.

30

That is, if we multiply 117 by 20, and divide the product by 30, we obtain the number of miles he will travel in 20 days, which is 78.

day for a certain far will he go in This will furnish

Now, if we take into consideration the number of hours, we must say, - If a man, travelling 9 hours a number of days, has travelled 78 miles, how the same time, if he travel 12 hours a day? the following proportion.

9 hours 12 hours: 78 miles : answer to the question.

12 X 78

X**

104 miles, the

By this mode of resolving the question, we see that 117 miles have, to the answer 104 miles, the proportion that 30 days have to 20 days, and that 9 hours have to 12 hours. Stating this in Compound Proportion, we have

Thus it appears that if 117 be multiplied by both 20 and 12, and the product be divided by 30 times 9, the quotient will be 104 miles; or if we multiply 117 by 20, and divide the product by 30, and then multiply this quotient by 12 and divide by 9, it will produce the same answer as before.

This question may be performed by analysis thus: - If he travel 117 miles in 30 days, in one day he will travel of 117 miles, which is 7 miles; and, travelling 9 hours a day, he will in one hour travel of miles, which is 18 miles; and in a day of 12 hours he will travel 12 times 18 miles, which is 156 miles; and in 20 days he will travel 20 times 156 miles, which is 104 miles, the answer, as before.

30

The answer to the above question might have been obtained by dividing the third term by the product of the two ratios which the first two terms have to the second terms; that is, by the ratio of 30 to 20, which is 38 = ; and of 9 to 12, which is. Thus,

117÷2×2=117÷=142x8936104

Ans.

2. If 6 men in 16 days of 9 hours each build a wall 20 feet long, 6 feet high, and 4 feet thick, in how many days of 8 hours each will 24 men build a wall 200 feet long, 8 feet high, and 6 feet thick?

In stating this question, there are several circumstances to be taken into consideration; the number of men employed,