3. Fellowship. ANALYSIS. 203. 1. Two men, A and B, trade in company; A puts in $100, and B $200, and they gain $30. What is each man's share of the gain? Each man's gain must evidently have the same relation to the whole gain, that the money which he puts in, has to the whole amount put in. In other words, the whole amount put in, will be to the whole gain as each man's share of the amount put in, is to his share of the gain, i. e. $300: $30 :: {$100} $10 A's share. Ans. 204. 2. A and B hired a pasture for 12 dollars; A put in 3 cows for 8 weeks, and B put in 4 cows for 9 weeks; what part of the rent ought each to pay? Three cows 8 weeks are equal to 1 cow (3X8) 24 weeks, and 4 cows 9 weeks are equal to 1 cow (4X9) 36 weeks; their shares, then, of the pasturage are 24 weeks and 36 weeks, equal to 60 weeks' pasturage. Then, as the whole pasturage is to the whole rent, so is each man's share of the pasturage to his share of the rent; that is, (3×8 24w. $4.80 A's share. 4X9-36w:: 87.20 B's share. } Ans. To prove the correctness of the work, we add together the shares, and find them to amount to (4.80+7.20) $12, the whole rent (54). DEFINITIONS. 205. Money, or property employed in trade, is called capital, or stock,-gain to be divided, the dividend. Fellowship is a general rule, by which merchants, or others, trading in company with a joint stock, compute each person's particular share of the gain or loss. RULE. 206. When the stocks are employed for equal times, say: As the whole stock is to the whole gain or loss: so is each man's share of the stock to his share of the gain or loss (203). When the times are unequal, multiply each man's stock by the time of its continuance in trade; then say, As the sum of the products is to the whole gain, or loss: so is each man's product to his share of the gain, or loss (204). QUESTIONS FOR PRACTICE. 3. A and B made a joint $ Ans. 75.00 pr'f. 4. Three persons make a joint stock, of which each puts in an equal share; A continues his stock in trade 4 months, B his 6 months, and C his 10 months, and they gained $480 what was each man's share? $96 A's. 144 B's. Ans. 240 C's. 207. 1. If I mix 6 quarts of currants, which are worth 8 cents a quart, with 2 quarts worth 12 cents a quart, what will a quart of the mixture be worth? (60) Six quarts at 8 cents are worth (8×6=) 48 cents, and 2 quarts at 12 oents are worth (12×2) 24 cents; then 48+24-72 cents, the worth of the whole mixture, and 72-8 (6+2, the whole mixture) 9 cents, the worth of 1 quart of the mixture. When the prices and quantities of the simples are given, and it is required to find the price of a given quantity of the mixture, as in the preceding example, it is called ALLIGATION MEDIAL. RULE. 208. Multiply each quantity by its price, and divide the sum of the products by the sum of the quantities, the quotient will be the rate of the compound required. QUESTIONS FOR PRACTICE. lons of wine at 4s. 10d. a gal- 4. If 5lb. of tea at 6s. per lb., 8lb. at 5s., and 4lb. at 4s. 6d., be mixed together, what is a pound of the mixture worth? Ans. 58. 2,2d. 5. A goldsmith melted together 10 oz. of gold 20 carats fine, 8 oz. 22 carats fine, and 1 lb. 8 oz. 21 carats fine; what is the fineness of the mixture? Ans. 2018 carats fine. ALLIGATION ALTERNATE. 209. When the prices of the simples, and also the price, or rate of the mixture, are given, the method of finding the proportion, or quantities of the several simples, is called Alligation Alternate. 1. A person has tea worth 40 cents a pound, which he wishes to mix with tea worth 60 cents a pound, in such manner that the mixture shall be worth 50 cents a pound; in what proportion must it be mixed? Ans. Equal quantities of each; for the price of one kind exceeds the mean just as much as the price of the other falls short of it, the difference between the given rate and the mean being 5 in each case. 2. In what proportion must I mix currants worth 9 cents a pound, with currants worth 12 cents a pound, in order that the mixture may be worth 10 cents a pound? Here a pound at 9 cents falls one cent short of the mean, and a pound at 12 cents exceeds the mean 2 cents; hence, 2 lb. at 9 cents will fall short of the mean by the same quantity that one lb. at 12 cents exceeds it; we must therefore take twice as many of the 9 cent currants as we do of those worth 12 cents, in order that the mixture may be worth 10 cents. From the foregoing examples it appears, that the less the price of any simple differs from that of the mixture, the quantity required of that simple to form the mixture will be proportionately greater, and the greater the dif ference the less the quantity; and that the differences between the values of the simples and the given value of a mixture of those simples, mutually exchanged, express the relative quantities of those simples necessary to make a mixture of the given value. Exchanging these differences in the above examples, we have in the first, 5 lb. at 40 cents, with 5 lb. at 60 cts., or equal quantities of each; and in the second, we have 2 lb. at 9 cts. with 1 lb. at 12. RULE. 210. Reduce the rates of all the simples to the same de nomination, and write them in a column with the rate of the required compound at the left hand. Connect each rate which is less than the rate of the compound, with one that is greater, and each that is greater with one that is less. Write the difference between each rate and that of the compound against the number with which it is connected. Then if only one difference stand against any rate, it will express the relative quantity to be taken of that rate; but if there be more than one, their sum will express the relative quantity to be taken of that rate in making up the coinpound. QUESTIONS FOR PRACTICE. 3. A farmer wishes to mix rye worth 4s., corn worth 3s., barley worth 2s. 6d., and oats worth 2s., so that the mixture may be worth 2s. 10d. per bushel; what proportion must he take o each sort? 4. A merchant would mix 5. How must barley at 40 wines at 14s., 15s., 19s. and 22s. a gallon, so that the mixture may be worth 18s. a gallon; how much must he take of each sort? cents, rye at 60 cents, and wheat at 80 cents a bushel, be mixed together, that the compound may be worth 624 cents a bushel? Alligation Alternate is the reverse of Alligation Medial, and may be proved by it Questions under this rule admit of as many different answers as there are different ways of linking. 211. When the whole composition is limited to a certain quantity. RULE.-Find the differences by linking as before; then say, As the sum of the quantities or differences, thus determined is to the given quantity :: so is each of the differences: to the required quantity of that rate. 212. When one of the simples is limited to a certain quantity. RULE. Find the differences as before; then, As the difference standing against the given quantity is to the given quantity :: so are the other differences, severally,: to the several quantities required. |