ratio of the quantity required to the given quantity of the same kind. A ratio resulting in this manner, from the multiplication of several ratios, is called a compound ratio.

From the examples and illustrations now given we deduce the following general

RULE

for solving questions in compound proportion, or double rule of three, viz.-Make that number which is of the same kind with the required answer, the third term; and, of the remaining numbers, take away two that are of the same kind, and arrange them according to the directions given in simple proportion; then, any other two of the same 'kind, and so on till all are used.

Lastly, multiply the third term by the continued product of the second terms, and divide the result by the continued product of the first terms, and the quotient will be the fourth term, or answer required.

EXAMPLES FOR PRACTICE.

1. If 6 men build a wall 20 ft. long, 6 ft. high, and 4 ft. thick, in 16 days, in what time will 24 men build one 200 ft. long, 8 ft. high, and 6 ft. thick? Ans. 80 days.

2. If the freight of 9 hhds. of sugar, each weighing 12 ewt, 20 leagues, cost 16 £., what must be paid for the freight of 50 tierces, each weighing 24 cwt., 100 leagues? Ans. 92 £. 11 s. 10 d.

3. If 56 lbs. of bread be sufficient for 7 men 14 days, how much bread will serve 21 men 3 days? Ans. 36 lbs.

The same by analysis. If 7 men consume 56 lbs. of bread, 1 man, in the same time, would consume of 56 lbs. = 56 lbs.; and if he consume 56 lbs. in 14 days, he would consume of 56: = 8 lb. in 1 day. 21 men would consume 21 times so much as 1 man; that is, 21 times §§ = 7 lbs. in 1 day, and in 3 days they would consume 3 times as much; that is, 36 lbs., as before.

Ans. 36 lbs.

Note. Having wrought the following examples by the rule of proportion, let the pupil be required to do the same by analysis.

4. If 4 reapers receive $11'04 for 3 days' work, how many men may be hired 16 days for $103'04?

Ans. 7 men.

5. If 7 oz. 5 pwt. of bread be bought for 42 d. when corn is 4 s. 2 d. per bushel, what weight of it may be bought for 1 s. 2 d. when the price per bushel is 5 s. 6 d. ?

Ans. 1 lb. 4 oz. 347 pwts 6. If $100 gain $6 in 1 year, what will $400 gain in 9 months?

Note.

This and the three following examples reciprocally prove each other.

7. If $100 gain $6 in 1 year, in what time will $400 gain $18?

8. If $400 gain $18 in 9 months, what is the rate per cent. per annum ?

9. What principal, at 6 per cent. per. ann., will gain $18 in 9 months?

10. A usurer put out $75 at interest, and, at the end of 8 months, received, for principal and interest, $79; I demand at what rate per cent. he received interest.

Ans. 8 per cent. 11. If 3 men receive 8 £. for 19 days' work, how much must 20 men receive for 100 days'? Ans. 305£. 0 s. 8 d.

SUPPLEMENT TO THE SINGLE RULE OF THREE.

QUESTIONS.

1. What is proportion? 2. How many numbers are required to form a ratio? 3. How many to form a proportion?

4. What is the first term of a ratio called? 5.

the second

term? 6. Which is taken for the numerator, and which for the denominator of the fraction expressing the ratio? 7. How may it be known when four numbers are in proportion? 8. Having three terms in a proportion given, how may the fourth term be found? 9. What is the operation, by which the fourth term is found, called? 10. How does a ratio become inverted ? 11. What is the rule in proportion? 12. In what denomination will the fourth term, or answer, be found? 13. If the first and second terms contain different aenominations, what is to be done? 14. What is compound proportion, or double rule of three? 15. Rule?

EXERCISES.

1. If I buy 76 yds. of cloth for $113'17, what does it cost per ell English? Ans. $1'861. 2. Bought 4 pieces of Holland, each containing 24 ells English, for $96; how much was that per yard?

Ans. $0'80. 3. A garrison had provision for 8 months, at the rate of 15 ounces to each person per day; how much must be allowed per day, in order that the provision may last 94 months? Ans. 1213 oz 4. How much land, at $250 per acre, must be given in exchange for 360 acres, at $375 per acre?

Ans. 540 acres. 5. Borrowed 185 quarters of corn when the price was 19 s.; how much must I pay when the price is

17 s. 4 d.? Ans. 2024. of his share Ans. 380 £ costs of a Ans. $140. 8. At 1. per cwt., what cost 3 lbs. ? Ans. 10 d. 9. If 4 cwt. can be carried 36 miles for 35 shillings, how many pounds can be carried 20 miles for the same money? Ans. 907 lbs. 10. If the sun appears to move from east to west 360 degrees in 24 hours, how much is that in each hour? each minute?

6. A person, owning of a coal mine, sells for 171.; what is the whole mine worth? 7. If of a gallon cost of a dollar, what tun?

in each second?

in

Ans. to last, 15" of a deg. 11. If a family of 9 persons spend $ 450 in 5 months, how much would be sufficient to maintain them 8 months if 5 persons more were added to the family? Ans. $1120.

Note. Exercises 14th, 15th, 16th, 17th, 18th, 19th, and 20th, "Supplement to Fractions," afford additional examples in single and double proportion, should more examples be thought necessary.

FELLOWSHIP.

T 98. 1. Two men own a ticket; the first owns 4, and the second owns of it; the ticket draws a prize of 40 dollars; what is each man's share of the money?

2. Two men purchase a ticket for 4 dollars, of which one pays 1 dollar, and the other 3 dollars; the ticket draws 40 dollars; what is each man's share of the money?

3. A and B bought a quantity of cotton; A paid 100 dollars, and B 200 dollars; they sold it so as to gain 30 dollars; what were their respective shares of the gain ?

The process of ascertaining the respective gains or losses of individuals, engaged in joint trade, is called the Rule of Fellowship.

The money, or value of the articles employed in trade, is called the Capital, or Stock; the gain or loss to be shared is called the Dividend.

It is plain, that each man's gain or loss ought to have the same relation to the whole gain or loss, as his share of the stock does to the whole stock.

Hence we have this RULE:-As the whole stock: to cach man's share of the stock: : the whole gain or loss: his share of the gain or loss.

4. Two persons have a joint stock in trade; A put in $250, and B $350; they gain $400; what is each man's share of the profit?

OPERATION.

Then,

$350

600: 250 :: 400: 166'666 dolls. A's gain. 600: 350 :: 400: 233′333

Whole stock, $600

dolls. B's gain. The pupil will perceive, that the process may be contracted by cutting off an equal number of ciphers from the first and second, or first and third terms; thus, 6: 250 :: 4· 166'666, &c.

It is obvious, the correctness of the work may be ascertained by finding whether the sums of the shares of the gains are equal to the whole gain; thus, $166'666 + $233′3331 $400, whole gain.

5. A, B and C trade in company; A's capital was $175, B's $200, and C's $500; by misfortune they lose $250; what loss must each sustain ? $ 50', A's loss. Ans. $ 57'1425, B's loss. $142'857, C's loss

6 Divide $600 among 3 persons, so that their shares may be to each other as 1, 2, 3, respectively.

Ans. $100, $200, and $300.

R

7. Two merchants, A and B, loaded a ship with 500 hhds. of rum; A loaded 350 hhds., and B the rest; in a storm, the seamen were obliged to throw overboard 100 hhds.; how much must each sustain of the loss?

Ans. A 70, and B 30 hhds. 8. A and B companied; A put in $45, and took out of the gain; how much did B put in ? Ans. $30. Note. They took out in the same proportion as they put in; if 3 fifths of the stock is $45, how much is 2 fifths of it?

9. A and B companied, and trade with a joint capital of $400; A receives, for his share of the gain, as much as B; what was the stock of each?

$133'333, A's stock.

Ans. {$266-666, B's stock.

10. A bankrupt is indebted to A $780, to B $460, and to C $760; his estate is worth only $600; how must it be divided?

Note. The question evidently involves the principles of fellowship, and may be wrought by it.

Ans. A $234, B $ 138, and C $228. 11. A and B venture equal stocks in trade, and clear $164; by agreement, A was to have 5 per cent. of the profits, because he managed the concerns; B was to have but 2 per cent.; what was each one's gain? and how much

did A receive for his trouble?

Ans. A's gain was $117'142, and B's $46,857, and A received $70'2854 for his trouble.

12. A cotton factory, valued at $12000, is divided into 100 shares; if the profits amount to 15 per cent. yearly, what will be the profit accruing to 1 share? to 2 shares ?

to 5 shares?

to 25 shares?

Ans. to the last. $450. 13. In the above-mentioned factory, repairs are to be made which will cost $340; what will be the tax, on each share, necessary to raise the sum ? shares?

on 10 shares ?

on 3

on 2 shares ? Ans. to the last, $34. 14. If a town raise a tax of $1850, and the whole town be valued at $37000, what will that be on $1? What will be the tax of a man whose property is valued at $1780 ? Ans. $'05 on a dollar, and $89 on $1780.