Q. What is the RULE for alligation medial? A. Find the value of each quantity to be mixed, at its given price, and add the several quantities into one sum, and their several amounts into another; then, by the rule of three, say, as the whole quantity to be mixed: is to the whole value of the mixture:: so is any part, or quantity of the mixture: to its proper value. EXAMPLES. 1. A grocer mixed 25 gallons of wine, at 80 cents a gallon, with 30 gallons at 70 cts. 45 gallons at 90 cts. and 50 gallons at 45 cents a gallon; what was the mixture worth per gallon? Operation. 25 gallons, 80-$20 X70= 21 ×90= 40,50 X45= 22,50 30 66 45 66 50 66 [mixture. Whole quantity, 150 galls. worth $104,00 whole value of the Then, as 150 gallons, $104: 1 gallon, 693 cents, Ans. 2. A grocer had 5 cwt. of sugar, at $8 the cwt., 4 cwt. at $6, 3 cwt. at $10, and 3 cwt. at $12, which he wished to mix, in order to find a more ready sale; what will 4 cwt. of the mixture be worth? 3. A farmer mixed an equal quantity of oats, at 37 cents, and buckwheat, at 62 what is the mixture worth per bushel ? Ans. $34,663. corn, at 75 cents, cents, together; Ans. 58 cents. 4. A vintner mixed 120 gallons of wine, worth $1,25 a gallon, with 130 gallons at 90 cents, 150 gallons at 80 cents, and 275 gallons at 37 cents, into which he put 75 gallons of water; what must he sell it for per gallon, to make $150 on the whole quantity ? Ans. 85 cents, 3 mills. 5. A refiner mixed 6 ounces of gold, of 22 carats fine, 8 ounces of 18 carats, 9 ounces of 16 carats, with 12 ounces of alloy; of how many carats fine was the whole compound? Ans. 12 carats. 6. A butcher bought 15 sheep, at $1,25 a head, 12 at $1,124, 25 at $1,374, 30 at 90 cents, and 18 at $1,20; on driving them to market, he sold them all at the same price, by which he made $62; at how much did he sell them per head? Ans. $1,77 cents, 74 mills. 7. A merchant bought four different kinds of tea, at 37 cts. 62 cts. 87 cts. and $1,12 per lb.; he was afterwards offered 85 cents a lb. for the whole together; I wish to know whether he would make or lose by selling it at that rate, and how much per pound? Ans. He would gain 10 cents per pound. ALLIGATION ALTERNATE. -Q. What is Alligation Alternate? A. It is a rule which teaches, by having the prices of several articles given, to find how much of each must be taken, to make a mixture which shall bear a certain given price. NOTE. This rule is exactly the reverse of alligation medial, and each may be proved by the other. CASE FIRST. Q. What is the first case in alligation alternate ? A. It is when the particular prices of several ingredients are given, to find what quantity of each ingredient will make a mixture, that shall bear a given price. Q. What is the RULE in this case? A. First, bring the prices of the several ingredients to the same denomination, and place them in a column under each other, the least being at the top, and increasing downwards in regular order; then place the mean or given rate of the mixture, (in the same denomination,) at the left hand of this column. Secondly, connect by a continued line, the price of each ingredient which is less than the mean rate, with one, or any number of those, which are greater than the price of the compound; and each greater price, with one, or any number of those, which are less than the mean rate, or mixture price. Thirdly, find the difference between the mean price, or mixture rate, and that of each ingredient, and place this difference opposite to that, with which it is tied or linked; then if one difference only stand against any rate, that difference, or number, will show the proper quantity belonging to that par ticular rate; but if two, or more differences, stand against any rate, their sum will show the proper quantity. EXAMPLES. 1. A wine merchant would mix wines of different prices, viz.: at 80 cents, 75 cents, $1,45, and $1,50, in such quantities that the mixture may be worth $1,35 a gallon; how much of each kind must he use in the mixture? -55 galls. at $1,50 cents. 2. A goldsmith would mix gold of 19 carats fine, with some of 16, 18, 23, and 24 carats fine, so that the compound may be 21 carats fine; what quantity of each must he take? NOTE 1.-By the above operations, it will be seen, that the answers depend on the manner of connecting the different rates with each other. In some questions a number of different answers may be given, yet each of them will make the required compound. Any person, therefore, wishing to make a compound, after finding the requisite quantities by different operations, can make choice of those quantities which are found to be most profitable and convenient. NOTE 2.-When one of the simples or ingredients of the compound possesses no value, as when water is mixed with spirits, or alloy with the precious metals, its rate is nothing, or 00, and must be placed at the top of the column of rates, and connected in the same manner as all the other prices are connected, and its difference taken and placed in the same manner. 3. A vintner mixed wines, at $2,50, $2,25, $1,75, and 90 cents a gall. with enough water to make it worth $1 a gall.; how many galls. of each kind, and how much water did he use? 255 gs. of water. 125 gs. at 90 cts. 100 gs. at 175 cts. Ans. 1. 10 gs. at 255 cts. 100 gs. at 250 cts. 272 gs. of water. 10 gs. at 175 cts. Ans. 2. 4. How much barley, at 40 cents, rye, at 60 cents, and wheat, at 80 cents per bushel, must be mixed together, to make a compound worth 62 cents the bushel? Ans. 17 bushels of barley, 174 of ryc, and 25 of wheat. 5. It is required to mix wine at 60 cents, 90 cents, and $1,15 a gallon, with water, that the mixture may be worth 75 cents per gallon; how much of each sort must be used to make this compound? Ans. 40 gallons of water; 15 gallons of wine, at 60 cents; 15 gallons at 90 cents; and 75 gallons at $1,15. 6. A merchant has sugars, at $6, $8, $12, and $14 per cwt.; how much of each must he mix, that the compound may be worth $11 per cwt. Q. What is the second case in alligation alternate ? A. It is when the price of each ingredient, the quantity of one of them, and the mean rate of the whole compound, are given, to find the several quantities of the other ingredients which will make the desired mixture. Q. What is the RULE in this case? A. Take the difference between the several prices and the mean rate, and place them as in case first. Then by the rule of three, say, as the difference, (standing against that price whose quantity is given,) is to that quantity, so is each of the other differences, severally, to the several quantities required. EXAMPLES. 1. A wine merchant has 40 gallons of wine, at $1 a gallon, which he wishes to mix with some at $1,25, some at 90 cents, and some at 75 cents the gallon; how much of each kind must be put to the 40 gallons, to make a mixture worth 95 cents the gallon? -30 gallons. 75. 90 Operation. 95 5 66 in case first, we here find the number of gallons of each ne, that will make the desired mixture, without re riven quantity of either kind; and as 5 galls. stands rice, whose quantity is given, therefore, say As 5:40:: { 30: 240 gallons, at 75 cts. 5: 40 gallons, at 90 cts. Ans. 20 160 gallons, at 125 cts. 2. How much gold, of 16, 20, and 24 carats fine, and how much alloy, must be mixed with 10 ounces, of 18 carats fine, that the composition may be 22 carats fine? Ans. 10 oz. of 16, 10 oz. of 20, 170 of 24 carats fine, and 10 oz. of alloy. 3. A grocer has 14 cwt. of sugar, that cost him $14 per cwt.; but being too dear to sell readily, he wishes to mix it with some at $10, some at $8, and some at $6 the cwt., and to make a composition that he can sell at $9 the cwt. and make 50 cents the cwt.; how much of each kind must he put with his 14 cwt.? Ans. 30 cwt. at $6. 8 cwt. at $8. 2 cwt. at $10. 4. A tobacconist mixed 20 lbs. of tobacco, worth 15 cts. a lb., with others, at 16 cts., 18 cts., and 22 cts. a lb.; how many lbs. of each, must he take, that the mixture may be worth 17 cts. Ans. 4 lbs. at 16 cts. 4 lbs. at 18 cts. 8 lbs. at 22 cts. CASE THIRD. Q. What is the third case in Alligation Alternate? A. It is when the prices of the several ingredients, the quantity to be compounded, and the mean rate of the whole mix. ture, are given, to find how much of each sort will make the desired compound. Q. What is the RULE in this case? A. Find the difference between the mean rate and the several prices, and place them as in case first. Then say, as the sum of all these differences is to the given quantity: so is each difference to its own particular quantity. EXAMPLES. 1. A person wishes to ship 2000 lbs. of cheese, that may average 9 cts. per lb.; he has four sorts on hand, some at 12 cts., some at 10 cts., 8 cts., and 6 cts. the lb.; how much of each sort will make up the required lot? |