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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 cent. ?
Ans. 6 per cent. 8. If & 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 4fd. when corn is 4s. 2d. per bushel, what weight of it may be bought for ls. 2d. when the price per bushel is 5s. 6d. ?
Ans. 1612 oz. 13. If 496 men, in 54 days of 11 hours each, dig a trench of 7 degrees of hardness 465 feet long, 3 wide, 24 deep, in how many days of 9 hours long will 24 men dig a trench of 4 degrees of hardness 3371 feet long, 5% wide, and 3] deep ?
Ans. 132 days.
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 2016. at Boston make 23lb. at Antwerp, and 1551b. at Antwerp make 180lb. at Leghorn, how many pounds at Boston are equal to 1441b. at Leghorn ?
It will be perceived 2015. of Boston = *23 of Antwerp, in the operation that 155lb. of Antwerp = 180 of Leghorn, the continued product 1441b of Leghorn.
of the antecedents is
divided by the con044 = 4148 = 1073;lb. Ans.
tinued product of the
16 gø x 155 x 844
23 X 180 = = 107131b. Ans.
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 righthand 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 121b. at Boston make 10lb. at Amsterdam, and 10lb. at Amsterdam make 121b. at Paris, how many pounds at Boston are equal to 80lb. at Paris ?
Ans. 80lb. CANCELLING. 12 X 10 x 80
NOTE. - The pupil may cancel all the following questions in a similar manner.
4. If 24 shillings in Massachusetts are equal to 32 shillings in New York, and if 48 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. 831 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. 364 men. 6. If 10 barrels of apples will pay for 5 cords of wood, and 20 cords of wood for 4 tons of hay, how many barrels of apples will it take to purchase 50 tons of hay ? Ans. 500bbl.
7. If 100 acres in Bradford be worth 120 in Haverhill, and 50 in Haverhill be worth 65 in Methuen, how many acres in Bradford are equal to 150 in Methuen ? Ans. 964 acres.
8. If 10lb. of cheese are equal in value to 71b. of butter, and 11lb. 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 lb.
SECTION LV. PARTNERSHIP, OR COMPANY BUSINESS.
PARTNERSHIP is the association of two or more persons in business, with an agreement to share the profits and losses in proportion to the amount of the capital stock contributed by each.
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 = ļof $ 54, which is $18, is B’s gain. C's stock is $ 480 ; therefore his part of the whole stock is too = $; consequently C's share of the gain is 4 of $ 54, which is $ 24. Hence, to find any man's gain or loss in trade we have the fol. lowing
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 answers 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.
C's loss, $27.50. 3. Two merchants, C and D, engaged in trade ; C put in $ 6780, and D put in $12,000; they gain $1000. What is each man's share ?
C's share, $361.02,241.
D's share, $ 638.97,7}is. 4. M, P, and 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 were A's and B's shares of the gain, and what was 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.
C's gain, $ 2700. 7. A merchant, failing in trade, owes A $ 600, B $ 760, C $ 840, and D $ 800. His effects are sold for $ 2275. What will each receive of the dividend ?
A, $ 455.00.
D, $ 606.664. 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 ?
When merchants in partnership employ their stock for unequal times, it is called Partnership on Time.
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 5 x $ 420 = $ 2100 for 1