5. How long will it take $500 to gain $10, if $100 gain $6 in one year? Ans. 4 months. 6. If $500 gain $10 in 4 months, what is the rate per cent. ? Ans. 6 per cent.

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

8. If 12 men, in 15 days, can build a wall 30 feet long, 6 feet high, and 3 feet thick, 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. 9. If 16 horses consume 84 bushels of grain in 24 days, how many bushels will suffice 32 horses 48 days?

Ans. 336 bushels.

10. If the carriage of 5 cwt. 3. qrs., 150 miles, cost $24.58, what must be paid for the carriage of 7 cwt. miles, at the same rate?

2 qrs. 25 lbs., 64 Ans. $14.08.6. 11. If 7 oz. 5 dwt. of bread be bought at 44d., 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. 1 lb. 4 oz. 31 dwt.

SECTION XLVI.

CHAIN RULE.

THE CHAIN RULE consists in joining many proportions together; and by the relation which the several antecedents have to their consequents, the proportion between the first antecedent and the last consequent is discovered.

This rule may often be abridged by cancelling equal quantities on both sides, and abbreviating commensurables.

NOTE. The first numbers in each part of the question are called antecedents, and the following consequents.

1. If 20 lbs. at Boston, make 23 lbs. at Antwerp, and 155 lbs. at Antwerp make 180 lbs. at Leghorn, how many pounds at Boston are equal to 144 lbs. at Leghorn ?

RULE

Write the numbers alternately, that is, the antecedents at the left hand, and the consequents at the right; and, if the last number stands at the left hand, multiply the numbers of the left hand column continually together for a dividend, and those at the right for a divisor; but, if the last number stands at the right hand, multiply the numbers of the right hand column continually together for a dividend, and those at the left for a divisor; and the quotient will be the answer.

NOTE. The demonstration for this rule is the same as for Compound Proportion.

2. If 12 lbs. at Boston make 10 lbs. at Amsterdam, and 10 lbs. at Amsterdam make 12 lbs. at Paris, how many pounds at Boston are equal to 80 lbs. at Paris? Ans. 80 lbs.

3. If 25 lbs. at Boston are equal to 22 lbs. at Nuremburg, and 88 lbs. at Nuremburg are equal to 92 lbs. at Hamburg, and 46 lbs. at Hamburg are equal to 49 lbs. at Lyons, how many pounds at Boston are equal to 98 lbs. at Lyons? Ans. 100 lbs.

4. If 24 shillings in Massachusetts are equal to 32 shillings in New York; and if 43 shillings in New York are equal to 45 shillings in Pennsylvania; and if 15 shillings in Pennsylvania are equal to 10 shillings in Canada; how many shillings in Canada are equal to 100 shillings in Massachusetts ?

Ans. 83 shillings.

5. If 17 men can do as much work as 25 women, and 5 women do as much as 7 boys, how many men would it take to do the work of 75 boys? Ans. 363 men.

6. If 10 barrels of cider will pay for 5 cords of wood, and 20 cords of wood for 4 tons of hay, how many barrels of cider will it take to purchase 50 tons of hay? Ans. 500 bbls.

7. If 100 acres in Bradford be worth 120 in Haverhill, and 50 in Haverhill worth 65 in Methuen, how many acres in Bradford are equal to 150 in Methuen ? Ans. 962 acres.

8. If 10 pounds of cheese are equal in value to 7 lbs. of butter, and 11 lbs. of butter to 2 bushels of corn, and 11 bushels of corn to 8 bushels of rye, and 4 bushels of rye to one cord of wood, how many pounds of cheese are equal in value to 10 cords of wood? Ans. 432 lbs.

SECTION XLVII.

FELLOWSHIP, OR COMPANY BUSINESS.

FELLOWSHIP is a rule by which merchants, and others in partnership, estimate their gain or loss in trade. It is of two kinds, single and double.

Single Fellowship is, when merchants in partnership, employ their stock for equal times.

NOTE. - Partnership is the union of two or more persons in the same trade.

EXAMPLES.

1. Three men, A, B, and C, enter into partnership for two years, with a capital of $1080. A puts in $240; B $360; and C $480. They gain $54. What is each man's share of the gain?

As the whole stock in trade is $1080, of which $240 belongs to A, A's share of the stock therefore, will be 240, and as each man's gain is in proportion to his stock, ¦ of $54, which is $12, is A's gain. B's stock in trade is $360; therefore 360 10% of $54, which is $18, is B's gain. C's stock is $480; therefore his part of the whole stock is 1080; consequently C's share of the gain is of $54, which is $24. Hence to find any man's gain or loss in trade we have the following

RULE.

Multiply the whole gain or loss by each man's fractional part of the stock.

NOTE.

The pupil, who may be desirous of performing the questions of this rule in the "old way," will adopt the following

RULE.

As the whole stock is to the whole gain or loss, so is each man's particular stock to his particular share of the gain or loss.

The following is the statement of the first question with the nswers and proof.

As the stock $1080: $54:: $240: $12 A's gain.
$1080 $54: $360: $18 B's gain.
$1080 $54: $480: $24 C's gain.

Proof, $12+$18+ $24-$54 whole gain.

2. A, B, and C, enter into partnership with a capital of $1100, of which A put in $250, B put in $300, and C $550; they lost by trading 5 per cent. on their capital. What was each man's share of the loss?

A's loss, $12.50,
B's loss, $15.00,
C's loss, $27.50.

3. Two merchants, C and D, engaged in trade; C put in $6780, D put in $12000; they gain $1000. What is each man's share? C's share, $361.02.2113. D's share, $638.97.71.

4. M, P, Q, trade in company with a capital of $10.000; M put in $3000, P $2000, and Q $5000; they gain $500. What is each man's share of the gain?

M's gain, $150,
P's gain, $100,
Q's gain, $250.

5. A, B, and C, enter into partnership; A put in $500, B $350, and C put in 320 yards of broadcloth; they gain $332.50, of which C's share is $120. What was A's and B's shares of the gain, and the value of C's cloth per yard?

A's gain, $125.00, B's gain, $87.50, C's cloth per yd. $ 1.50.

6. A, B, and C, trade in company; A put in $5000, B put in $6500, and C put in $7500; they gain 40 per cent. on their capital, but receive the whole amount of their gains in bills, for which they are obliged to allow a discount of 10 per cent. How much was each man's net gain? A's gain, $1800, B's gain, $2340,

C's gain, $2700.

7. A merchant failing in trade, owes A $600, B $760, C $840, D $800. His effects are sold for $2275. What will each receive of the dividend?

A, $455.00
B, $576.33

C, $637.00

D, $606.663

8. A bankrupt owes $5000. His effects, sold at auction, amount to $4000. What will his creditors receive on a dollar? Ans. $0.80.

9. A merchant, having sustained many losses, is obliged to become a bankrupt. His effects are valued at $1728, with which he can pay only fifteen cents on the dollar. What did he owe? Ans. $11520.

SECTION XLVIII.

DOUBLE FELLOWSHIP.

WHEN merchants in partnership employ their stock for unequal times, it is called Double Fellowship.

EXAMPLES.

1. Two men, A, and B, trade in company. A puts in $420 for 5 months, and B puts in $350 for 8 months; they gain $84. What is each man's share of the gain?

Method of operation by analysis.

=

$420 for 5 months is the same as 5X $420=$2100 for 1 month; and $350 for 8 months, is the same as 8 X $350 $2800 for 1 month. The question therefore, is the same, as if A had put in $2100 and B $2800 for 1 month each. The whole stock would then be $4900. A's share of the gains therefore will be 2108 — of $84=$36. And B's share will be 2808 of $84 = $48.

4800

Hence we deduce the following

RULE.

Multiply each man's stock by the time it continued in trade, and consider each product a numerator to be written over their sums, as a common denominator; then multiply the whole gain or loss by each fraction, and the several products will be the gain or loss of each man.

NOTE.It might be well for the student to be acquainted with the "old way" of performing these questions. The following is the

RULE.

Multiply each man's stock by the time it was continued in trade; then as the sum of all the products is to the whole gain or loss, so is each man's particular product to his share of the gain or loss.

The first question would be performed thus,

$420X5-2100 4900: 84: 2100: $36 A's gain,
$350X8=2800 4900: 84:: 2800: $48 B's gain.

Proof, $36-$48-$84-A and B's gain.