B's interest in the partnership is 2400 3, and he will suffer of the loss, or $140 × 3 = $84. We may also solve by proportion, the causes being compounded of the two elements, capital and time thus: 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; 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, as in Case I EXAMPLES FOR PRACTICE. 1. A, B, and C enter into partnership. A puts in $357 for 5 months, B $371 for 7 months, and C $154 for 11 months, and they gain $347.20; how much is each one's share? Ans. A's $102; B's $148.40; C's $96.80. 2. Three men hire a pasture for $55.50. A put in 5 cows, 12 weeks; B, 4 cows, 10 weeks; and C, 6 cows, 8 weeks; how much ought each to pay? Ans. A $22.50; B $15; C $18. 3. B commenced business with a capital of $15000. Three months afterward C entered into partnership with him, and put in 125 acres of iand. At the close of the year their profits were $4500, of which C was entitled to $1800; what was the value of the land per acre? $106 4. A and B engaged in trade. months afterward $200 more. B put in at first $1500, and at the end of 6 months took out $500. At the end of 16 months their gain was $772.20; how much is the share of each? 5. Four companies of men worked on a railroad. In the first company there were 30 men who worked 12 days, 9 hours a day; in the second, there were 32 men who worked 15 days, 10 hours a day; in the third, there were 28 men who worked 18 days, 11 hours a day; and in the fourth, there were 20 men who worked 15 days, 12 hours a day. The entire amount paid to all the companies was $1500; how much were the wages of each company? 6. A and B are partners. A's capital is to B's as 5 to 8; at the end of 4 months A withdraws of his capital, and B of his; at the end of the year their whole gain is $1000; how much belongs to each? Ans. A, $17142; B, $2285§. 7. B, C, and D form a manufacturing company, with capitals of $15800, $25000, and $30000 respectively. After 4 months B draws out $1200, and in 2 months more he draws out $1500 more, and 4 months afterward puts in $1000. C draws out $2000 at the end of 6 months, and $1500 more 4 months afterward, and a month later puts in $800. D puts in $1800 at the end of 7 months, and 3 months after draws out $5000. If their gain at the end of 18 months be $15000, how much should each receive? Ans. B, $3228.07; C, $5258.15; D, $6513.78. 8. The joint stock of a company was $5400, which was doubled at the end of the year. A put for of a year, B. for a year, and C the remainder for one year. How much is each one's share of the entire stock at the end of the year? 9. Three men engage in merchandising. A's money was in 10 months, for which he received $456 of the profits; B's was in 8 months, for which he received $343.20 of the profits; and C's was in 12 months, for which he received $750 of the profits. Their whole capital invested was $14345; how much was the capital of each? Ans. A's, $4332; B's, $4075.50; C's, $5937.50. 10. Three men take an interest in a coal mine. B invests his capital for 4 months, and claims of the profits; C's capital is in 8 months; and D invests $6000 for 6 months, and claims of the profits; how much did B and C put in? 11. A, B, and C engage in manufacturing shoes. A puts in $1920 for 6 months; B, a sum not specified for 12 months; and C, $1280 for a time not specified. A received $2400 for his stock and profits, B $4800 for his, and C $2080 for his. Required, B's stock, and C's time? ALLIGATION. 631. Alligation treats of mixing or compounding two or more ingredients of different values or qualities. 632. 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. CASE I. 633. To find the mean price or quality of a mixture, when the quantity and price of the several ingredients are given. NOTE.―The process of finding the mean or average price of several ingredients is called Alligation Medial. 1. A produce dealer mixed together 84 bushels of oats worth $.30 a bushel, 60 bushels of oats worth $.38 a bushel, and 56 bushels of oats worth $40 a bushel; required, the mean price. $22.80+$22.40 $70.40. One bushel of the mixture is therefore worth $70.40200 = $.352. Hence the following RULE. Find the entire cost or value of the ingredients, and divide it by the sum of the simples. EXAMPLES FOR PRACTICE. 1. A grocer mixed 4 lb. of tea at $.60 with 3 lb. at $.70, I IK. at $1.10, and 2 lb. at $1.20; how much is 1 lb. of the mixture worth? Ans. $.80. 2. A dealer in liquors would mix 14 gal. of water with 12 gal. of wine at $75, 24 gal. at $.90, and 16 gal. at $1.10; how much is a gallon of the mixture worth? Ans. $.7333. 3. If 3 lb. 6 oz. of gold 23 carats fine be compounded with 4 lb. 8 oz. 21 carats, 3 lb. 9 oz. 20 carats, and 2 lb. 2 oz. of alloy, what is the fineness of the composition? Ans. 18 carats. 4. A grain dealer mixes 15 bu. of wheat at $1.20 with 5 bu. at $1.10, 5 bu. at $.90, and 10 bu. at $.70; what will be his gain per bushel if he sell the compound at $1.25.25 5. A merchant sold 17 lb. of sugar at 5 cts. a pound, 51 lb. at 8 cts., 68 lb. at 10 cts., 17 lb. at 12 cts., and thereby gained on the whole 33 per cent.; how much was the average cost per pound? 0675 6. A drover bought 42 sheep at $2.70 per head, 48 at $2.85, and 65 at $3.24; at what average price per head must he sell them to gain 20 per cent.? Ans. $3.5671 7. A surveyor took 10 sets of observations with an instrument, for the measurement of an angle, with the following results: 1st, 36° 17′ 25.4"; 2d, 36° 17′ 24.5"; 3d, 36° 17′ 27.8"; 4th, 36° 17' 26.9"; 5th, 36° 17′ 25.4"; 6th, 36° 17′ 24.7"; 7th, 36° 17′ 24.2"; 8th, 36° 17′ 26.3"; 9th, 36° 17' 25.8"; 10th, 36° 17′ 26.7". What is the average of these measurements? Ans. 36° 17' 25.77". 8. Three trials were made with chronometers to determine the difference of time between two places; the first trial gave 37 min. 54.16 sec., the second 37 min. 55.56 sec., and the third 37 min. 54.82 sec. Owing to the favorable conditions of the third trial, it is entitled to twice the degree of reliance to be placed upon either of the others; what should be taken as the difference of longitude between the two places, according to these observations? Ans. 9° 28′ 42.6". CASE II. 634. To find the proportional quantity to be used of each ingredient, when the mean price and the prices of the several simples are given. NOTE. The process of finding the quantities to be used in any required mixture is commonly called Alligation Alternate. 1. A farmer would mix oats worth 3 shillings a bushel with peas worth 8 shillings a bushel, to make a compound worth 5 shillings a bushel; what quantities of each may he take? OPERATION. 5 (3 19 3 Ans. ANALYSIS. If a mixture, in any proportions, of oats worth 3 shillings a bushel and peas worth 8 shillings, be priced at 5 shillings, there will be a gain on the oats, the ingredient worth less than the mean price, and a loss on the peas, the ingredient worth more than the mean price; and if we take such quantities of each that the gain and loss shall each be 1 shilling, the unit of value, the result will be the required mixture. By selling 1 bushel of oats worth 3 shillings for 5 shillings, there will be a gain of 5—3=2 shillings, and to gain 1 shilling would require of a bushel; hence we place opposite the 3. By selling 1 bushel of peas worth 8 shillings for 5 shillings, there will be a loss of 8-5 = 3 shillings, and to lose 1 shilling will require of a bushel; hence we write opposite the 8. Therefore,bushel of oats to of a bushel of peas are the proportional quantities for the required mixture. It is evident that the gain and loss will be equal, if we take any number of times these proportional terms for the mixture. We may therefore multiply the fractions and by 6, the least common multiple of their denominators, and obtain the integers 3 and 2 for the proportional terms (418,III); that is, we may take, for the mixture, 3 bushels of oats to 2 bushels of peas. 2. What relative quantities of sugar at 7 cents, 8 cents, 11 cents, and 14 cents per pound, will produce a mixture worth 10 cents per pound? 10 OPERATION. 1|2|3|4 8 11 4 3 23 14 ¡Or, 7 8 10 111 3 1 12 ANALYSIS. To preserve the equality of gains and losses, we must compare two prices or simples, one greater and one less than the mean rate, and treat each pair or couplet as a separate example. Thus, comparing the simples whose prices are 7 cents and 14 cents, we find that, to gain 1 cent, of a pound at 7 cents must be taken, and, to lose 1 cent, of a pound at 14 cents must be taken; and comparing the simples the prices of which are 8 cents and 11 cents, we find that pound at 8 cents must be taken to gain 1 cent, and 1 pound at 11 cents must be taken to lose 1 cent. These proportional terms are |