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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. 2 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 184000 lbs. of provisions, of which they consume each man 5 lbs. 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 3150 lbs. 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 take to build a wall 80 feet long, 20 feet high and 24 feet thick. Ans. 17280 stones.

59. Bought 3 score pieces of Holland 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 after 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.

61. If I sell coffee at 2s. 3d. per lb. and gain 35 per cent., what did I give per lb. ? Ans. 20d.

62. Suppose 2000 soldiers had been supplied with bread, sufficient to last them 12 weeks, allowing each man 14 ounces a day, but, on examining, find 105 barrels, containing 200 lbs. each, wholly spoiled; how much a day may each man eat, that the remainder may supply them 12 weeks?

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

The whole, 147000 lbs.
Spoiled, 21000 lbs.

64. If 2000 soldiers, having lost 105 barrels of bread, weighing 200 lbs. 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, the same time. What was the whole weight, including what was lost; and how much had they left to subsist on?

The whole weight, 147000 lbs.
Left to subsist on, 126000 lbs.

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? The weight was 147000 lbs. The loss,

66. If .85 of a gallon of wine of a gallon cost?

67. If 61.3 pounds of tea cost

21000 lbs.

Had none been lost, they
might have had 14 ounces
per day.

cost $2.72, how much will .25
Ans. $0.80.
$44.9942, what is the price per
Ans. $0.73.4.

lb. ? 68. What is the value of .15 of a hogshead of lime, at $2.39 per hhd. ? 69. If .75 of a ton of hay cost $15, what is it per ton?

Ans. $0.35.85.

Ans. $20.00. 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 will he do it when the day is but 10 hours? 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 them who remain 9 months? Ans. 200 men. 73. The hour and minute hand of a watch are together at 12 o'clock, when are they next together?

Ans. 1 h. 5 m. 27 sec. 74. A and B can perform a piece of work in 5 days, B and C in 63 days; and  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?

A would do the work in 10 days, B would do the work in 12 days, C would do the work 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.

E

SECTION XLV.

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 20

= 78 miles.

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.

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

9 hours 12 hours :: 78 miles : 12378 = 104 miles, the answer to the question.

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

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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 W miles; and, travelling 9 hours a day, he will in one hour travel of miles; which is miles; and in a day of 12 hours, he will travel 12 times miles, which is 15 miles; and in 20 days he will travel 20 times 156 miles, which is 104 miles, the answer as before.

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= }

936

Thus, 117x=117÷=47x="-104 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, the length of the days, length of the wall, and its height and breadth. As the answer to the question is to be in days, we make the days the third term.

Were all the circumstances of the question alike, except the number of men and the number of days, the question would consist in finding in how many days 24 men would perform the same labor, that 6 men had done in 16 days; that is, if 6 men had built a certain wall in 16 days, how many days would it take 24 men to perform the same labor. This would furnish the following proportion.

24

=

4 days.

24 men : 6 men :: 16 days: 6116. Or, if this were the question. If a certain number of men, by laboring 9 hours a day, perform a piece of work in 16 days, how many days would it take the same men to do the labor, by working 8 hours a day? The following would be the propor tion.

=

8 hours 9 hours: 16 days: 916 18 days. Or, if this were the question. If a certain number of men build a wall 20 feet long in 16 days, how long would it take the same men to build a wall 200 feet long; the following would be the statement.

20 feet 200 feet: 16 days: 16 1200

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160 days.

Or, if only the days and height of the wall were considered, this would be the statement.

6 feet 8 feet: 16 days: 816-21 days.

:

Lastly, were we to consider only the days and the thickness of the wall, it would furnish the following statement.

4 feet 6 feet :: 16 days: 616

=

24 days.

We see, by this mode of resolving the question, that 16 days must have, to the true answer, the ratio compounded of the ratios, That 24 men have to 6 men; That 8 hours have to 9 hours; That 20 feet have to 200 feet; That 6 feet have to 8 feet; and That 4 feet have to 6 feet.

Stating the above in Compound Proportion, we have

24 men : 6 men

8 hours: 9 hours

20 feet

: 200 feet

:: 16 days: 90 days= Answer.

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The continued product of all the second terms by the third term, and this divided by the continued product of the first terms, will produce the answer.

Thus 6200 x 8x € 116-829440
24 1 8 120 Lỗ14

Hence the following

RULE.

90 days= Answer.

Make that number which is of the same kind as the answer required, the third term; and of the remaining numbers, take any two, that are of the same kind, and consider whether an answer, depending upon these alone, would be greater or less than the third term, and place them as directed in Simple Proportion. Then take any other two, and consider whether an answer depending only upon them would be greater or less than · the third term, and arrange them accordingly; and so on, until all are used. Multiply the continued product of the second terms by the third, and divide by the continued product of the first, and you produce the answer.

3. If $100 gain $6 in one year, how much would $500 gain in 4 months? Ans. $10.

4. If $100 gain $6 in one year, what must be the surn to gain $10 in 4 months? Ans. $500.

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