$1,02 per and also their prices per pound, 75cts. and $1,02, are per - 6,00 1616. 16,32 24 sum of simples Total cost $22,32 Now if the entire cost 24)$22,32(93cts. of the mixture, which is 216 $22,32, be divided by 24 the number of pounds, or 72 sum of the simples, the 93cts. quotient 93cts. will be the price per pound. Hence we have the following 72 RULE. at 3s C Divide the entire cost of the whole mixture by the sum of the simples : the quotient will be the price of the mixture. Ex. 1. A farmer mixes 30 bushels of wheat worth 58 per bushel, with 72 bushels of rye at 3s per bushel, and with 60 bushels of barley worth 2s per bushel: what is the value of a bushel of the mixture ? 30 bushels of wheat at 58 150s of 216s 120s 162)486(3s 486 Ans. 3s. 2. A wine merchant mixes 15 gallons of wine at $1 per gallon, with 25 gallons of brandy worth 75 cts. per gallon : what is the value of a gallon of the compound? Ans. 84cts. + 3. A grocer mixes 40 gallons of whiskey worth 31cts. per gallon, with 3 gallons of water, which cost. nothing: what is the value of a galloa of the mir ture? Ans. 21 cts. 4. A goldsmith melts together 26. of gold of 2 carats fine, 6oc. of 20 carats fine, and 602. of 16 ta: rats foe: what is the fineness of the mixture ? Ans. 20 carats. 5. On a certain day the mereary in the thermometer was observed to average the following heights : from 6 in the morning to 9, 64o ; from 9 to 12, 74; from 12 to 3, 84°; and from 3 to 6, 70°: what was the mean temperature of the day? Ans. 73o. ALLEGATION ALTERNATE. 0 206. A farmer would mix oats worth 35 per bushel with wheat worth 9s per bushel, so that the mixture shall be worth 58 per bushel : what proportion must be taken of each sort? The method of find ing how much of each sort must be taken, is called Adlegation Alternate. Hence, ALLEGATION ALTERNATE teaches the method of find ing what proportion must be taken of several sim. ples, whose prices are known, to form a compound of a given price. Allegation Alternate is the reverse of Allegation Medial and may be proved by it. For the first example, let us take the one above, If oats worth 3s per bushel be mixed with wheat worth 9s, how much must be taken of each sort that the compound may be worth 5s per bushel ? If the price of the mixture 3 4 Oats half the sum of the 5 simples, it is 2 Wheat, plain that it would be necessary to take just as much oats as wheat. But since the price of the mixture is nearer to the price of the oats than to that of the wheat, less wheat will be required in the mixture than oats. Having set down the prices of the simples under each other, and linked them together, we next set 5s, the price of the mixture, on the left. We then take the difference between 9 and 5 and place it opposite 3, the price of the oats, and also the difference between 5 and 3 and place it opposite 9, the price of the wheat. The difference standing opposite each kind shows how much of that kind is to be taken. In the present example, the mixture will consist of 4 bushels of oats and 2 of wheat; and any other quantities, bearing the same proportion to each other, as 8 and 4, 20 and 10, &c. will give a mixture of the same value. PROOF BY ALLEGATION MEDIAL. 4. bushels of oats at 3s 12s 18s 6)30 5s. Ans. CASE I. § 207. To find the proportion in which several simples of given prices must be mixed together, that the compound may be worth a given price. RULE. 1. Set down the prices of the simples under each other, in the order of their values, beginning with the lowest. II. Link the least price with the greatest, an next least with the next greatest, and so on, until the price of each simple which is less than the price of the mixture is linked with one or more that is greater ; and every one that is greater with one or more that is less. III. Write the difference between the price of the mixture and that of each of the simples, opposite that price with which the particular simple is linked: then the difference standing opposite any one price, or the sum of the differences when there is more than one, will express the quantity to be taken of that price. Ex. 1. A merchant would mix wines worth 16s, 185 and 22s per gallon, in such a way that the mis. ture be worth 20s per gallon: how much must be taken of each sort ? 162 at 163. 20. 181 2 at 18s. 224 4+2=6 at 22s. Ans. 2gal. at 16s, 2 at 18s, and 6 at 22s : Or any other quantities bearing the proportion of 2, 2 and 6. 2. What proportions of coffee at 16cts., 20cts. and 28cts. per pound, must be mixed together, so that the compound shall be worth 24cts. per pound? Ans. { 10 the proportion of 416, at 16cts., 416. at 3. A goldsmith has gold of 16, of 18, of 23 and 24 carats fine : what part must be taken of each so that the mixture shall be 21 carats fine? Ans. 3 of 16, 2 of 18, 3 of 23, and 5 of 24. 4. What portion of brandy at 14s per gallon, of ld Madeira at 24s per gallon, of new Madeira at on, and of brandy at 10s per gallon, must . s per be mixed together, so that the mixture shall be worth 18s per gallon ? Ans. 6gal. at 10s, 3 at 14s, 4 at 21s, and 8gal. at 24s. CASE II. $ 208. When a given quantity of one of the simples is to be taken. RULE, I. Find the proportional quantities of the simples as in Case I. II. Then say, as the number opposite the simple whose quantity is given, is to the given quantity, so is either proportional quantity, to the part of its simple to be taken. Ex. 1. How much wine at 5s, at 5s 6d and 6s per gallon, must be mixed with 4 gallons at 4s per galIon, so that the mixture shall be worth 58 4d per gallon ? 48 8. - simple whose quantity is known 60 2 64 66 proportional quantities. 72 16 : : Then 4 2 : 1 8 4 4 : 2 8 4 16: 8 Ans. Igal. at 5s, 2 at 5s 6d, and 8 at 6s. : |