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 I gallon of the mixture be worth? G. 1: 1562: 86.77+ Answer. 2. If 19 bushels of wheat at 6 s. per bushel; 40 bushels of rye at 4 s. per bushel, and 12 bushels of barley at 3 s. per bushel, be mixed together, what will a bushel of the mixture be worth? Ans. 4 s. 44 d. 3. If a grocer mix 2 cwt. of sugar at 56 s. per cwt.; 1 cwt. at 43 s. per cwt.; and 2 cwt. at 50 s. per cwt.; what will be the value of 1 cwt. of the mixture? Ans. 2 L. 11 s. 4. A farmer mingled 20 bushels of wheat at 5 s. per bushel, and 36 bushels of rye at 3 s. per bushel, with 40 bushels of barley at 2 s. per bushel; I desire to know the worth of a bushel of this mixture? Ans. 3 s. 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. 65 cts. 6. A wine merchant mixes 12 gallons of wine at 4 s. 10 d. per gallon, with 24 gallons at 5 s. 6 d., and 16 guns at 6 s. 34 d.; what is a gallon of the mixture Worth? Ans. 5 s. 7 d. 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 4 s. per bushel, barley at 3 s. per bushel, and oats at 2 s. per bushel, will make a mixture worth 2 s. 6 d. per bushel? bu. S. 18+6=24 at 2 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? Ans. { 50 gals. at 160 cts., 50 gals. at 180 cts., and 40 gals. at 240 cts. 3. How much sugar at 4 d. at 6 d. and at 11 d. per lb. mus': be mixed together to make a composition worth 7 d. per lb.? Ans. an equal quantity of each kind. 4. It is required to mix several sorts of wine, viz. at 9 s. 15 s. and 21 s. per gallon, with water, that the mixture may be worth 12 s. per gallon; how much of each soit must be taken ? Ans. {3 gals. 9 s., 3 gals. 15s., and 12 gals. at 21 s. with 9 gals. 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. 1. A grocer would mix 30 lb. of sugar at 14 cents per lb. with some at 9 cents, 10 cents, and 13 cents per lb.; 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 5 s., at 5 s. 6 d., and at 6 s. per gallon, must be mixed with 3 gallons at 4 s. per gallon, so that the mixture may be worth 5 s. 4 d. per gallon? Ans. 3 gals. at 5 s., 6 at 5 s. 6 d., and 6 at 6 s. 4. How much tea at 12 s., 10 s., and at 6 s. per lb. must be mixed with 20 pounds at 4 s. per lb. to make a mixture worth 8 s. per lb.? Ans. 10 lb. at 6 s., 10 lb. at 10 s., and 20 lb. at 12 s. 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.? 15 2 As 8:2::20: 5lb. at 10 cts. 2 S:4:20: 10lb. at 15 cts. 3+1=4 8:2::20: 5lb. at 12 cts. 8 Sum of differences. Ans. 2. A brewer has three sorts of beer, viz. at 10 d., 8 d., and 6 d. per gallon; how much of each sort must he take to make a mixture of 30 gallons, worth 7 d. per gallon? Ans. 5 gals. at 10d., 5 gals. at 8d., and 20 gals. at 6 d. 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 40 oz. of 18 carats fine; how much of each sort is necessary? 16 oz. of 15 carats, 8 oz. of 17 carats, 4 oz. Ans. {of 20 carats, and 12 oz. of 22 carats fine. 4. How many gallons of water must be mixed with wine, at 4 s. per gallon, so as to fill a vessel of 80 gallons, that may be afforded at 2 s. 9 d. per gallon? Ans. 25 gallons of water, with 55 of wine. POSITION. Position is a rule for nading an unknown number, by one or more supposed numbers. It is divided into two parts, single and double. 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 he 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 с C pa 51 } 340 Proof. As 51 340 :: 36 240 sum paid by A. 2. A person after spending and of his money, had 60 L. left; how much had he at first? Ans. 144 L. 3. What number of dollars is that, of which the 4, , and, make 74? Ans. 120. |