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Again, we say, If $ 800 will gain $ 48 in 12 months, what will the same sum gain in 8 months ? Ans. $ 32.

OPERATION.

ner.

12 months : 8 months :: $ 48 : $ 32 Ans, as before. This question may be analyzed in the following man

We say, If $ 100 gain $ 6, $ 800 will gain 8 times as much, = $ 48. Again, we say, If 12 months gain $ 48, 1 month will gain It of $ 48, = $4, and, if 1 month gain $4, 8 months will gain 8 times $ 4, = $ 32 Answer, as before. Note. The pupil should analyze each question. From the above illustrations, we deduce the following

RULE.

Make that number, which is of the same kind as the answer required, the third term; and, of the remaining num. bers, take any two, that are of the same kind, and consider, whether an answer, depending upon these alone, would be greater or less than the third term, and place them as directed in Simple Proportion. Then take any other two, and consider, whether an answer, depending only upon them, would be greater or less than the third term, and arrange them accordingly; and so on 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.

per cent.

2. If $ 100 gain $ 6 in 12 months, in how many months will $ 800 gain $ 32.

Ans. 8 months. 3. If $ 100 gain $ 6 in 12 months, how large a sum will

it require to gain $ 32 in 8 months ? Ans. $ 800. 4. If $ 800 gain $ 32 in 8 months, what is the per cent. ?

Ans. 6 5. If 15 carpenters can build a bridge in 60 days, when the days are 15 hours long, how long will it take 20 men to build the bridge, when the days are 10 hours long?

Ans. 671 days. 6. If a regiment of soldiers, consisting of 939 men can eat 351 bushels of wheat in 3 weeks, how

many

soldiers will it require to eat 1404 bushels in 2 weeks ?

Ans. 5634 soldiers.

7. If 248 men, in 54 days of 11 hours each, dig a trench of 7 degrees of hardness, and 2324 feet long, 3 feet wide, and 21 feet deep; in how many days of 9 hours each, will 24 men dig a trench of 4 degrees of hardness, and 3374 feet long, 5% feet wide, and 31 feet deep?

Ans. 132 days.

Section 44.

COMPANY BUSINESS.

COMPANY Business, or 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. 1. John Smith and Henry Grey enter into partnership for three years, with a capital of $ 6000, of which Smith puts in $ 4000, and Grey $ 2000. They gain $ 570. What is each man's share of the gain ?

$ .

.

Proof. $ 570. As the whole stock is $ 6000, of which $ 4000 belongs to Smith, it is evident, that his share of the stock is 4:::=*; and, as each m&n's gain is in proportion to his stock, of $ 570 = $ 380 is Smith's share of the gain. Grey's stock is $ 2000, therefore, 2888 = of $ 570 = $ 190 is Grey's share of the gain.

Hence, to find any man's gain or loss in trade, we have the following

Ans. { Grey's gain $ 190

RULE,

Multiply the whole gain or loss by each man's FRACTIONAL PART of the stock.

2. Three merchants, A., B., and C., engage in trade. A. put in $ 6000, B. put in $ 9000, and C. put in

M

$5000. They gain $ 840. What is each man's share of the gain?

A.'s gain $ 252. Ans. B.'s gain $ 378.

C.'s gain $ 210.

Proof. $ 840. 3. A bankrupt owes Peter Parker $8750, James Dole $ 3610, and James Gage $ 7000. His effects sold at auction, amount to $ 6875 ; of this sum $ 375 are to be deducted for expenses, &c. What will each receive of the dividend ?

(Parker $ 2937.75191. Ans. { Dole $ 1212.03 21.

Gage $ 2350.20.30 4. A merchant, failing in trade, owes A. $ 500, B. $ 386, C. $ 988, and D. $ 126. His effects are sold for $ 100. What will each man receive ?

Ans. A. receives $ 25.00, B. $ 19.30, C. $ 49.40, D. $ 6.30.

12T:

Section 45.

DOUBLE FELLOWSHIP.

When merchants in partnership employ their stock for unequal times, it is called Double Fellowship. 1. Josiah Brown and George Dole trade in company Brown put in $ 600 for 8 months, and Dole put in $ 400 for 6 months. They gain $ 60. What is each man's share of the gain ?

Operation by analysis. We say, $ 600 for 8 months is the same as 8 x $ 600 = $ 4800 for 1 month ; and $ 400 for 6 months is the same as 6 x $ 400 = $ 2400 for 1 month. The question is, therefore, the same, as if Brown had put in $ 4800 and Dole $ 2400 for 1 month each. The whole stock would then be $ 4800 + $ 2400 = $ 7200, and Brown's share of the gain would be 488 =of $ 60 = $ 40. Dole's share will be 400 = f of $ 60= $ 20. Hence the propriety of 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 sum, 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.

2. A., B., and C. trade in company. A. put in $ 700 for 5 months ; B. put in $ 800 for 6 months ; and C put in $ 500 for 10 months. They gain $ 399. What is each man's share of the gain ?

Ans. A.'s gain $ 105, B.'s gain $ 144, C.'s gain $ 150. 3. Leverett Johnson, William Hyde, and William Tyler, formed a connexion in business, under the firm of Johnson, Hyde, and Co.; Johnson at first put in $ 1000, and, at the end of 6 months, he put in $ 500 more. Hyde at first put in $ 800, and, at the end of 4 months, he put in $ 400 more, but, at the end of 10 months, he withdrew $ 500 from the firm. Tyler at first put in $ 1200, and, at the end of 7 months, he put in $ 300 more, and, at the end of 10 months, he put in $ 200. At the end of the year they found their net gain to be $ 1000. What is each man's share ?

Ans. Johnson's gain $ 348.02131, Hyde's $273.7843, Tyler's 8 378.19 13 4. George Morse hired of William Hale, of Haverhill, his best horse and chaise for a ride to Newburyport, for $3.00, with the privilege of one person's having a seat with him. Having rode 4 miles, he took in John Jones and carried him to Newbury port, and brought him back to the place from which he took him. What share of the expense should each pay, the distance from Haverhill to Newburyport being 15 miles ?

Ans. Morse pays $ 1.90, Jones pays $ 1.10. 5. J. Jones and L. Cotton enter into partnership for one year. January 1, Jones put in $ 1000, but Cotton did not put in any until the first of April. What did he then put in to have an equal share with Jones at the end of the year?

Ans. $ 1333.331.

Section 46.

DUODECIMALS.

Duodecimals are so called because they decrease by twelves, from the place of feet towards the right.

Inches are called primes, and are marked thus'; the next division after is called seconds, marked thus "; and

so on.

OPERATION.

1. Multiply 8 feet 6 inches by 3 feet 7 inches.

As feet are the integers of units, it 8 6 is evident, that feet multiplied by feet

3 7 will produce feet; and, as inches are 25

6' twelfths of a foot, the product of inches 4 11 6" by feet will be twelfths of a foot. For 30 5' 6"

the same reason, inches multiplied by

inches will produce twelfths of an inch, or one hundred and forty-fourths of a foot. Hence we deduce the following

RULE.

Under the multiplicand write the same names or denominations of the multiplier; that is, feet under feet, inches under inches, fc. Multiply each term in the multiplicand, beginning at the lowest, by the feet of the multiplier, and write each result under its respective term, observing to carry a unit for every 12 from each denomination to its next superior. In the same manner the multiplicand by the inches of the multiplier, and write the result of each term one place further towards the right of those in the multiplicand. Proceed in the same manner with the seconds, and all the rest of the denominations, and the sum of all the lines will be the product required.

2. Multiply Sít. 3in. by 7ft. Sin. Ans. 63ft. 11' 3". 3. Multiply 12ft. 9' by 9ft. 11. Ans. 126ft. 5' 3". 4. Multiply 14ft. 9' 11" by 6ft. 11' 8".

Ans. 103ft. 4' 5" 8" 4".

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