RULE. As the sum of the several quantities, To the value of that part. PROOF. The value of the whole mixture at the mean price must agree with the total value of the several quantities at their respective prices. EXAMPLES. 1. If 6 gallons of wine at 67 cents per gallon; 7 at 80 cents, and 5 at 120 cents per gallon, be mixed together, what will 1 gallon of the mixture be worth? 2. If 19 bushels of wheat at 6s. per bushel; 40 bushels of rye at 4s. per bushel; and 12 bushels of barley, at 3s. per bushel, be mixed together, what will a bushel of the mixture be worth? Ans. 4s. 44d. 3. If a grocer mix 2cwt. of sugar at 56s. per cwt.; 1 cwt. at 43s. per cwt.; and 2cwt. at 50s. per cwt.; what will be the value of 1cwt. of the mixture? Ans. 2L. 11s. 4. A farmer mingled 20 bushels of wheat at 5s. per bushel, and 36 bushels of rye at 3s. per bushel, with 40 bushels of barley at 2s. per bushel; I desire to know the worth of a bushel of this mixture? Ans. 3s. 5. If 4 ounces of silver worth 75 cents per ounce, be melted with 8 ounces worth 60 cents per ounce, what will 1 ounce of the mixture be worth?' Ans. 65cts. 6. A wine merchant mixes 12 gallons of wine at 4s. 10d. per gallon, with 24 gallons at 5s. 6d. and 16 gallons at 6s. 34d.; what is a gallon of the mixture worth? Ans. 5s. 7d. CASE 2. When the prices of several simples are given, to find how much of each, at their respective rates, must be taken to make a compound or mixture at any proposed price. RULE. Set the prices of the simples one under another, and link every price which is not greater than the mean rate, to one or more that are greater than that rate; place the difference between each price and the mean rate opposite to the price or prices with which it is linked: then, if only one difference stand opposite to either particular price, it will be the quantity required at that price; but if there be several differences, their sum will be the quantity. Note.-Different modes of linking will produce different answers. EXAMPLES 1. How much rye at 4s. per bushel, barley at 3s. per bushel, and oats at 2s. per bushel, will make a mixture worth 2s. 6d. per bushel? 2. A vintner has three kinds of wine, viz. one kind at 160 cents per gallon, another at 180 cents, and another at 240 cents; how much of each kind must he take to make a mixture, worth 190 cents per gallon? lb. Ans. 50gals. at 160cts. 50gals. at 180cts. and 40gals. at 240cts. 3. How much sugar at 4d. at 6d. and at 11d. per must be mixed together, to make a composition worth 7d. per lb.? Ans. An equal quantity of each kind. 4. It is required to mix several sorts of wine, viz. at 9s. 15s. and 21s. per gallon, with water, that the mixture may be worth 12s. per gallon; how much of each sort must be taken? Ans. 3gals. 9s. 3gals. 15s. and 12gals. at 21s. with 9gals. of water. 5. A grocer has several sorts of sugar, viz. one sort at 12 cents per lb. another at 11 cents, a third at 9 cents, and a fourth at 8 cents per lb.; how much of each sort must he take to make a mixture worth 10 cents per lb. When the price of all the simples, the quantity of one of them, and the mean price of the whole mixture are given, to find the several quantities of the rest. RULE. Link the several prices, and place their differences as in case 2; then As the difference opposite to the price of the given quantity, Is to the differences respectively; So is the given quantity, To the several quantities required. EXAMPLES. per Ib.; 1. A grocer would mix 30lb. of sugar, at 14 cents per lb. with some at 9 cents, 10 cents, and 13 cents how much of each sort must he mix with the thirty lb. that the mixture may sell at 12 cents per lb.? 2. How much barley at 30 cents per bushel, rye at 36 cents, and wheat at 48 cents, must be mixed with 12 bushels of oats, at 18 cents, to make a mixture worth 22 cents per bushel? Ans. 1 bushel of each sort. 3. How much wine at 5s. at 5s. 6d. and at 6s. per gallon, must be mixed with 3 gallons at 4s. per gallon, so that the mixture may be worth 5s. 4d. per gallon? Ans. 3gals. at 5s. 6 at 5s. 6d. and 6 at 6s. 4. How much tea at 12s. 10s. and at 6s. per lb. must be mixed with 20 pounds at 4s. per lb. to make a mixture worth Ss. per lb.? Ans. 10lb. at 6s. 10lb. at 10s. and 20lb. at 12s. CASE 4. When the prices of the several simples, the quantity to be compounded, and the mean price are given, to find the quantity of each simple. RULE. Link the several prices, and place their differences as before; then, As the sum of the differences, Is to the difference opposite to each price; To the quantity required. EXAMPLES. 1. How much sugar at 10 cents, 12 cents, and 15 cents, per lb. will be required to make a mixture of 20 lb. worth 13 cents per lb.? 2 As 82: 20': '5lb. at 10cts. Ans. 2. A brewer has three sorts of beer, viz. at 10d. 8d. and 6d. per gallon; how much of each sort must he take to make a mixture of 30 gallons, worth 7d. per gallon? Ans. 5gals. at 10d. 5gals. at Sd. and 20gals. at 6d. 3. A goldsmith has gold of 15, 17, 20 and 22 carats fine, and would melt together of each of these so much as to make a mass of 40oz. of 18 carats fine; how much of each sort is necessary? Ans. S16oz. of 15 carats, 8oz. of 17 carats, 4oz. of 20 carats, and 12oz. of 22 carats fine. 4. How many gallons of water must be mixed with wine, at 4s. per gallon, so as to fill a vessel of 80 gallons, that may be afforded at 2s. 9d. per gallon? Ans. 25 gallons of water, with 55 of wine. POSITION. Position is a rule for finding an unknown number, by one or more supposed numbers. two parts, single and double. It is divided into SINGLE POSITION. Single Position teaches to resolve such questions as require only one supposition. RULE. Suppose any number to be the true one, and proceed with it agreeably to the tenor of the question; then, As the result of the operation, Is to the number given; PROOF. Work with the answer according to the tenor of the question, and the result must equal the given number. EXAMPLES. 1. A, B, and C, bought a quantity of wine for 340 dollars, of which sum A paid three times more than B, and B four times more than C; how much did each pay? $ A paid 240 340 Proof. As 51 340: 36 240 sum paid by A. 2. A person after spending and 4 of his money, had 60L. left; how much had he at first? Ans. 144L. 3. What number of dollars is that, of which the 4, , and, make 74. Ans. 120. |