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411. To find each partner's share of the profit or loss, when the capital of each is employed for equal periods of time.
1. A and B engage in trade; A furnishes $300, and B $400 of the capital; they gain $182. What is each one's share of the profit?
1st cause. 2d cause. 1st effect. 2d effect.
$182 : ( )
$700 $300 ::
$182 : ( )
man's share of the profit or loss will have the same ratio to the whole profit or loss that his share of the stock has to the whole stock, A will have of the entire profit, and B ‡, as shown in the operation.
We may also regard the whole capital as the first cause, and each man's share of it as the second cause, the whole gain or loss as the first effect, and each man's share of it as the second effect, and solve by proportion, thus:
()= $78, A's profit. ( ) = $104, B's profit. RULE.-Multiply the whole profit or loss by the ratio of each man's share of the capital to the whole capital. Or,
The whole capital is to each man's share of the capital as the whole profit or loss is to each man's share of the profit or loss.
2. Three men trade in company; A furnishes $8000, B $12000, and C $20000 of the capital; their gain is $1680. What is each man's share?
Ans. A's, $336; B's, $504; C's, $840. 3. Three persons purchased a house for $2800, of which A paid $1200, B $1000, and C $600; they rented it for $224 a year. How much of the rent should each receive?
4. A man failed in business for $20000, and his available means amounted to only $13654. How much will two of his creditors respectively receive, to one of whom he owes $3060, and to the other $ 1530?
Ans. $2089.062; $1044.531. 5. Four men hired a coach for $13, to convey them to their respective homes, which were at distances from the place of starting as follows: A's 16 miles, B's 24 miles, C's 28 miles, and D's 36 miles. What ought each to pay? A, $2; C, $3.50; Ans. B, $3; D, $4.50.
6. A captain, mate, and 12 sailors took a prize of $2240, of which the captain took 14 shares, the mate 6 shares, and each sailor 1 share. What did each receive? 7. A cargo of corn, valued at $3475.60 was entirely lost; of it belonged to A, 4 of it to B, and the remainder to C. How much was the loss of each, there being an insurance of $2512 ?
Ans. $120.45, A's; $240.90, B's; $ 602.25, C's. 8. Three persons engaged in the lumber trade; two of the persons furnished the capital, and the third managed the business; they gained $2571.24, of which C received $6 as often as D $ 4, and E had as much as the other two for taking care of the business. How much was each one's share of the gain?
Ans. $1285.62, C's; $857.08, D's; $428.54, E's.
412. To find each partner's share of the profit or loss when the capital of each is employed for unequal periods of time.
It is evident that the respective shares of profit and loss will depend upon two conditions, viz.: the amount of capital invested by each, and the time it is employed.
1. Two persons form a partnership; A puts in $450 for 7 months, and B $300 for 9 months; they lose $156. How much is each man's share of the loss?
$450 × 7 = $3150, A's capital for 1 mo.
$5850, entire "
81587, A's share of the entire capital.
$156 × = $84, A's loss.
$156 × = $72, B's
5850 31507 ( ) 15612
( ) = $84, A's loss.
SOLUTION. The use of $450 capital for 7 months is the same as the use of 7 times $450, or $3150 for 1 month; and of $300 for 9 months the same as of 9 times $300, or $2700 for 1 month. The entire capital for 1 month is equivalent to $3150 + $2700 = $5850. If the loss, $156, is divided between the two partners, as in 411, the results will be the loss of each as shown in the operation. Examples of this kind may also be solved by proportion, the causes being compounded of capital and time; thus,
$5850 $3150 :: $156: ( )
() = $72, B's loss.
RULE.Multiply each man's capital by the time it is employed in trade, and add the products. Then multiply the entire profit or loss by the ratio of each product to the sum of the products, and the results will be the respective shares of profit or loss of each partner. Or,
Multiply each man's capital by the time it is employed in trade, and regard each product as his capital, and the sum of the products as the entire capital, and solve by proportion.
2. Three persons traded together; B put in $250 for 6 months, C $275 for 8 months, and D $450 for 4 months; they gained $825. What was each man's share of the gain?
3. Two merchants formed a partnership for 18 months. A at first put in $1000, and at the end of 8 months he put in $600 more; B at first put in $1500, but at the end of 4 months he drew out $300; at the expiration of the time they found that they had gained $1394.64. What was each man's share of the gain?
Ans. A's, $715.20; B's, $679.44.
4. Three men took a field of grain to harvest and thrash for of the crop; A furnished 4 hands 5 days, B 3 hands 6 days, and C 6 hands 4 days; the whole crop amounted to 372 bushels. What was each man's share?
5. William Gallup began trade January 1, 1890, with a capital of $3000, and, succeeding in business, took in M. H. Decker as a partner on the first day of the following March, with a capital of $2000; four months later, they admitted J. Newman as third partner, who put in $1800 capital; they continued their partnership until April 1, 1892, when they found that $4388.80 had been gained since January 1, 1890. What was each one's share? Ans. G.'s, $2106; D.'s, $1300; N.'s, $982.80.
413. Average is the method of finding the mean or average price of a mixture of two or more ingredients of different values or qualities.
An average or mean of any number, is a sum or quantity intermediate to a number of different sums or quantities, obtained by adding them together and dividing by the number of quantities added.
414. The Mean Price or Quality is the average price or quality of the ingredients, or the price or quality of a unit of the mixture.
415. Averaging consists of two distinct processes, as follows:
1. The process of finding the mean or average value of two or more things of different given values.
2. The process of finding the proportional quantities of articles at given prices or values, to be used to make a combination of a given average value.*
416. The Simples are the unmixed ingredients.
417. To find the average value of things of different given values.
1. A miller mixes 40 bushels of rye worth 80 cents a bushel, and 25 bushels of corn worth 70 cents a bushel, with 15 bushels of wheat worth $1.50 a bushel. What is the value of a bushel of the mixture?
*This process will be explained in the Higher Arithmetic under "Alligation Alternate."