2. Compound Proportion.

ANALYSIS.

199. 1. If a person can travel 96 miles in 4 days, when the days are 8 hours long, how far can he travel in 2 days, when the days are 12 hours long?

I. If a person can travel 96 miles in 4 days, he can travel (96÷4=) 24 miles in 1 day, and, if he can travel 24 in a day, which is 8 hours long. he can travel (24-8) 3 miles in 1 hour, and if he can travel 3 miles in an hour, he can travel, when the days are 12 hours long, (12 X3=) 36 miles in 1 day. or (36 × 2 =) 72 miles in 2 days, which is the answer.

II. It must be evident that the distances travelled by a person going all the time at the same rate will be in proportion to the times in which they are travelled. In this case, 4 days, which are 8 hours long, are equal to (84) 32 hours, and 2d. 12 hours long equal (12×2) 24h. and hence we have this proportion, 32h.:: 96m.:: 24h. x, or the distance travelled in the 2 days, which we find to be 72 miles as before.

III. It will be obvious, in the above question, that the distance travelled, depends upon two circumstances, viz. the number of days and the length of the days. Now, supposing the days had all been of the same length, we should have had this proportion, viz. 4d.: 96m. : : 2d. : x, or the distance travelled in 2 days; or, supposing the number of days had been the same in both cases, the proportion would stand, 8h.: 96m. :: 12h. x, or the distance travelled, when the days are 12 hours long. Uniting these proportions together, we have

4d. 2 8h. S

: 96m. ::

21 125

: x,

by which it appears that 96 is to be multiplied by 2 and 12, or (2 × 12—) 24, and divided by 4 and 3, or (4 × 8=) 32, which is the same as the second method of solving the question.

200. 2. If 12 men can make 9 1od of fence in 6 days, when the days are 10 hours long, how many men will be required to make 18 rods of fence in 4 days, when the days are 8 hours long?

In this question, the number of days and their length being supposed to be the same in both cases, we should have this proportion, 9rds.: 12 men :: 18 a, or the number of men required to build the 18 rods-supposing the number of rods to be the same in both cases, and the days to be of equal length, we should have this proportion, 4d. : 12 men :: 6d. : x, or the number required to build the fence in 4 days, and supposing the number of rods and also the number of days to be the same in both cases, we should have this proportion, 8 hours: 12 men :: 10h. : æ, or the number required, when the days are 8 hours long. These three proportions combined, we have

9rds.

4d.

ab.

18rds.

by which it appears that 9 x 4×8:12:: 16×6×10 : 2, and multiplying. the product, the third terms by the second, and dividing by the product of the first terms, we find the value of x to Le 45 men, which is the answer,

DOUBLE RULE OF THREE.

201. A proportion, which is formed by the combination of two, or more, simple proportions, as in the preceding exam

ples, is called a Compound Proportion. The rule by which the fourth term of a compound proportion is found, is called the Double Rule of Three, and may be understood from the preceding analysis.

RULE.

202. Make that number, which is of the same kind as the Take any two of the rerequired answer, the second term. maining terms which are of the same kind, and place one for a first, and the other for a third term, as directed in the Single Rule of Three, (198): then take any other two of the same kind, and place them in the same way, and so on till all are used. Multiply the product of the third terms by the second term, and divide the result by the product of the first terms; the quotient will be the required answer.

QUESTIONS FOR PRACTICE.

8 Ans. $12. 6. If the transportation of 20cwt. 37 miles cost 16 dolls. what will the transportation of 12cwt. 50 miles cost?

Ans. $12.972.

7. If the interest of 45 dolls. for 6 months be $1.80, what is the rate per annum?

Ans. 8 per cent.

8. If 8 men spend 48 dolls. in 24 weeks, how much will 40 men spend in 48 weeks at Ans. $480. the same rate?

9. If the freight of 5 tierces of salt, each weighing 53 cwt. 80 miles, cost $80, what will be the freight of 75 sacks of salt, each weighing 24 cwt. 150 miles ?

Ans. $322.159.

10. A man lent $350 to receive interest, and when it had continued 9 months, he received principal and interest together, $360.50; at what rate per cent did he lend his money? Ans. 4 per cent.

11. With how many pounds sterling could I gain £5 per annum, if with £450 I gain in 16 months, £30? Ans. £100.

3. Fellowship.

ANALYSIS.

203. 1. Two men, A and B, trade in company; A puts in $100, and B $200, and they gain $30. What is each man's share of the gain?

Each man's gain must evidently have the same relation to the whole gain, that the inoney which he puts in, has to the whole amount put in. In other words, the whole amount put in, will be to the whole gain as each man's share of the amount put in, is to his share of the gain, i. e.

$100 :

$300: $30 :: $200

$10 A's share,
$20 B's share,

Ans.

204. 2. A and B hired a pasture for 12 dollars; A put in 3 cows for 3 weeks, and B put in 4 cows for 9 weeks; what part of the rent ought each to pay?

Three cows 8 weeks are equal to 1 cow (38) 24 weeks, and 4 cows 9 weeks are equal to 1 cow (49) 36 weeks; their shares, then, of the pasturage are 24 weeks and 36 weeks, equal to 60 week's pasturage. Then as the whole pasturage is to the whole rent, so is each man's share of the pasturage to his share of the rent; that is,

60 w. $12 ::

S38=24w. $4.80 A's share.
(4×9-36w.: $7.20 B's share.

}

Ans.

To prove the correctness of the work, we add together the shares, and find them to amount to (4.807.20=) $12, the whole rent. (54)

DEFINITIONS.

205. Money, or property employed in trade, is called capital, or stock,-gain to be divided, the dividend. Fellowship is a general rule, by which merchants, or others, trading in company with a joint stock, compute each person's particular share of the gain or loss.

RULE.

206. When the stocks are employed for equal times, say; As the whole stock is to the whole gain or loss :: so is each man's share of the stock to his share of the gain or loss. (203.) When the times are unequal, multiply each man's stock by the time of its continuance in trade; then say, As the sum of the products is to the whole gain, or loss: so is each man's product : to his share of the gain, or loss.(204.)

5. A, B and C companied; | end of the 18 months they had A put in £480, B £680, C gained £263; what is each £840, and they gained £1010; | man's share? what is each man's share?

42 A's.

£96 9 642
166 10 558 B's. (

£263 00

Ans.

D. Alligation.

ANALYSIS.

Ans. $375.

207. 1. LfI mix 6 quarts of currants,which are worth 8 cents a quart, with 2 quarts worth 12 cents a quart, what will a quart of the mixture be worth ?(60)

Six quarts at 8 cents are worth (86) 48 cents, and 2 quarts at 12 cents are worth (12><2) 24 cents, then 48+24-72 cents, the worth of