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24. A grocer has a false balance, by which 1 pound will weigh but 12 oz. ; what is the real value of a barrel of sugar that he sells for $28 ? Ans. $21.

25. A butcher in selling meat sells 1411 oz. for a pound; how much does he cheat a customer, who buys of him to the amount of $30 ? Ans. $2.46+.

26. If a man clear $750 by his business in 1 yr. 6 mo., how much would he gain in 3 yr. 9 mo. at the same rate? 27. If a certain business yield $350 net profits in 10 mo., in what time would the same business yield $1050 profits?

28. B and C have each a farm; B's farm is worth $25 an acre, and C's $30; if in trading B values his land at $28 an acre, what value should C put upon his? Ans. $34.16.

29. If I borrow $500, and keep it 1 yr. 4 mo., for how long a time should I lend $240 as an equivalent for the favor? Ans. 2 yr. 9 mo. 10 da.

COMPOUND PROPORTION.

400. Compound Proportion embraces that class of questions in which the causes, or the effects, or both, are compound.

The required term may be a cause, or a single element of a cause; or it may be an effect, or a single element of an effect. 1. If 16 horses consume 128 bushels of oats in 50 days, how many bushels will 5 horses consume in 90 days?

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If 16 horses in 50 days consume 128 bushels of oats, 5 horses in 90

days will consume how many, or (blank) bushels?

These questions are most readily performed by cancellation.

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2. If $480 gain $84 interest in 30 months, what sum will gain $21 in 15 months?

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the question may be read, If $480 in 30 months gain $84, what principal in 15 months will gain $21?

3. If 7 men dig a ditch 60 feet long, 8 feet wide, and 6 feet deep, in 12 days, what length of ditch can 21 men dig in 2 days, if it be 3 feet wide and 8 feet deep?

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Or, 7 x 12: 21 × :: 60 × 8 × 6: ( ) × 3 × 8.

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the causes, and those which constitute the effects, and arrange them in couplets, putting() in place of the required term.

II. Then, if the blank term () occur in either of the extremes, make the product of the means a dividend, and the product of the extremes a divisor; but if the blank term occur in either mean, make the product of the extremes a dividend, and the product of the means a divisor.

1. The causes must be exactly alike in the number and kind of their terms; the same is true of the effects.

2. The same preparation of the terms by reduction is to be observed as in simple proportion.

401. We will now solve an example according to the Second Method given in Simple Proportion.

1. If 18 men can build 42 rods of wall in 16 days, how many men can build 28 rods in 8 days?

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Since compound proportion is made up of two or more simple proportions, if this third or odd term be multiplied by the compound ratio, or by the simple ratio of each couplet successively, the product will be the required term.

By comparing the terms of each couplet with the third term we may readily determine whether the answer, or term sought, will be greater or less than the third term; if greater, then the ratio will be greater than 1, and the multiplier an improper fraction; if less, the ratio will be less than 1, and the multiplier a proper fraction.

First we will compare the terms composing the first couplet, 42 rods and 28 rods, with the third term, 18 men. If 42 rods require 18 men, how many men will 28 rods require? less men; hence the ratio is less than 1, and the multiplier a proper fraction, ; next, if 16 days require 18 men, how many men will 8 days require? more men; hence the ratio is greater than 1, and the multiplier an improper fraction, 1. Regarding the third term as the antecedent of a couplet, the consequent being the term sought, if we multiply this third term by the simple ratios, or by their product, we shall have the required term or answer, thus: 18 x x = 24, as shown in the operation.

2. 5 compositors, in 16 days, of 14 hours each, can compose 20 sheets of 24 pages in each sheet, 50 lines in a page,

and 40 letters in a line; in how many days, of 7 hours each, will 10 compositors compose a volume to be printed in the same letter, containing 40 sheets, 16 pages in a sheet, 60 Ans. 32 days. lines in a page, and 50 letters in a line?

OPERATION.

Days. Comp. Hours. Sheets. Pages. Lines. Letters.

16 × × 14 × 48 × 18 × 8 × 8 = 32 days.

BY CANCELLATION.

16
$

714

20 40

24 162

50 60 40 $0

32 days, Ans.

ANALYSIS. The required term or answer is to be in days; and we see that all the terms appear in pairs or couplets, except the 16 days, which is of the same kind as the answer sought.

We will proceed to compare the terms of each couplet with the 16 days. First, if 5 compositors require 16 days, how many days will 10 compositors require? less days; hence the multiplier is the proper fraction, and we have 16×5. Next, if 14 hours a day require 16 days, how many days will 7 hours a day require? more days; hence the multiplier is the improper fraction 4, and we have 16 × × 14. Next, if 20 sheets require 16 days, how many days will 40 sheets require? more days; hence the multiplier is the improper fraction 8, and we have 16××14×18. Pursuing the same method with the other couplets, we obtain the result as shown in the operation.

RULE. I. Of the terms composing each couplet form a ratio greater or less than 1, in the same manner as if the answer depended on those two and the third or odd term.

II. Multiply the third or odd term by these ratios successively, and the product will be the answer sought.

By the odd term is meant the one that is of the same kind as the answer.

The following examples may be solved by either of the given methods:

EXAMPLES FOR PRACTICE.

1. If 16 horses consume 128 bushels of oats in 50 days, how many bushels will 5 horses consume in 90 days?

2. If a man travel 120 miles in 3 days when the days are 12 hours long, in how many days of 10 hours each will he require to travel 360 miles ?. Ans. 10 days.

3. If 6 laborers dig a ditch 34 yards long in 10 days, how many yards can 20 laborers dig in 15 days? Ans. 170 yds. 4. If 450 tiles, each 12 inches square, will pave a cellar, how many tiles that are 9 inches long and 8 inches wide will pave the same?

Ans. 900. wide to clothe 500

5. If it require 1200 yards of cloth men, how many yards which is wide will it take to clothe 960 men? Ans. 3291 yds.

6. If 8 men will mow 36 acres of grass in 9 days, of 9 hours each day, how many men will be required to mow 48 acres in 12 days, working 12 hours each day? Ans. 6 men. 7. If 4 men, in 21 days, mow 6 acres of grass by working 84 hours a day, how many acres will 15 men mow in 3 days by working 9 hours a day? Ans. 401 acres.

8. If, by traveling 6 hours a day at the rate of 4 miles an hour, a man perform a journey of 540 miles in 20 days, in how many days, traveling 9 hours a day at the rate of 4 miles an hour, will he travel 600 miles?

Ans. 144 days.

9. If 2 yards of cloth 13 yards wide cost $3.371, what Ans. $52.79+.

cost 36 yards, 14 yards wide?

10. If 5 men reap 52.2 acres in 6 days, how many men Ans. 20 men.

will reap 417.6 acres in 12 days?

11. If 6 men dig a cellar 22.5 feet long, 17.3 feet wide, and 10.25 feet deep, in 2.5 days, of 12.3 hours, in how many days, of 8.2 hours, will 9 men take to dig another, measuring 45 feet long, 34.6 wide, and 12.3 deep? Ans. 12 days.

12. If 54 men can build a fort in 24 days, working 12 hours each day, in how many days will 75 men do the same, when they work but 10 hours each day? Ans. 21 days.

13. If 24 men dig a trench 334 yards long, 5 wide, and 3 deep, in 189 days, working 14 hours each day, how many hours per day must 217 men work, to dig a trench 231 yards long, 3 wide, and 24 deep, in 5 days? Ans. 16 hours.

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