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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 new suit"? Ans. 66yd.

66

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.

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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

30

=

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 : answer to the question.

12 X 78

9

= 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

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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 miles; and, travelling 9 hours a day, he will in one hour travel of miles, which is 3 miles; and in a day of 12 hours he will travel 12 times 3 miles, which is 56 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 28 ; and of 9 to 12, which is. Thus,

117÷×=117÷8=112x=936 104 Ans.

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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.

6×16

24 men : 6 men :: 16 days : = 4 days.

24

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 proportion.

8 hours: 9 hours :: 16 days: 9x16 18 days.

= 8

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?

be the statement.

the following would

20 feet: 200 feet: 16 days:

16×200
20

= 160 days.

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

8X16
6

6 feet 8 feet :: 16 days: = 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:
:

6×16
4

= 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

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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.

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3. If 5 compositors in 16 days, 11 hours long, can compose 25 sheets of 24 pages in each sheet, and 44 lines in a page, and 40 letters in a line, in how many days 10 hours long may 9 compositors compose a volume, to be printed on the same letter, consisting of 36 sheets, 16 pages to a sheet, 50 lines to a page, and 45 letters in a line? Ans. 12 days.

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= 12 days, Ans.

RULE. 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.

NOTE. All the following questions are to be performed not only by the Rule, but by analysis. The pupil should also apply the cancelling rule.

4. If $100 gain $6 in one year, how much would $500 gain in four months ? Ans. $10. 5. If $100 gain $6 in one year, what must be the sum to gain $10 in 4 months? Ans. $500. 6. How long will it take $500 to gain $10, if $100 gain $6 in one year? Ans. 4 months. 7. If $500 gain $10 in 4 months, what is the rate per Ans. 6 per cent.

cent. ? 8. If 8 men spend $32 in 13 weeks, what will 24 men spend in 52 weeks? Ans. $384.

9. If 12 men can build a wall 30 feet long, 6 feet high, and 3 feet thick, in 15 days, when the days are 12 hours long, in what time will 60 men build a wall 300 feet long, 8 feet high, and 6 feet thick, when they work only 8 hours a day?

Ans. 120 days. 10. If 16 horses consume 84 bushels of grain in 24 days, how many bushels will suffice 32 horses 48 days?

Ans. 336 bushels. 11. If the carriage of 5cwt. 3qr. 150 miles cost $24.58, what must be paid for the carriage of 7cwt. 2qr. 25lb. 64 miles, at the same rate? Ans. $14.08,6. 12. If 74oz. of bread be bought for 43d. when corn is 4s. 2d. per bushel, what weight of it may be bought for 1s. 2d. when the price per bushel is 5s. 6d. ? Ans. 1611 oz.

13. If 496 men, in 5 days of 11 hours each, dig a trench of 7 degrees of hardness 465 feet long, 3 wide, 21 deep, in how many days of 9 hours long will 24 men dig a trench of 4 degrees of hardness 337 feet long, 5g wide, and 31 deep? Ans. 132 days.

SECTION LIV.

CHAIN RULE.

THE CHAIN RULE consists in joining many proportions together, and by the relation which the several antecedents have

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