་ ALLIGATION. 328. ALLIGATION is the process of mixing substances in such a manner that the value of the compound shall be equal to the sum of the values of the several ingredients. It is divided into two parts: Alligation Medial and Alligation Alternate. ALLIGATION MEDIAL. 329. ALLIGATION MEDIAL is the method of finding the price or quality of a mixture of several simple ingredients whose prices and quantities are known. 1. A grocer would mix 200 pounds of lump sugar, worth 13 cents a pound, 400 pounds of Havana, worth 10 cents a pound, and 600 pounds New Orleans, worth 7 cents a pound: what should be the price of the mixture? ANALYSIS.-The quantity, 200 lb., at 13 cents a pound, costs $26; 400 pounds, at 10 cents a pound, costs $40; and 600 lb. at 7 cents a pound, costs $42: hence, the entire mixture, consisting of 1200 lb., costs OPERATION. 200 × 13 = 26.00 1200 $108. Now, the price of the mixture will be as many cents as 1200 is contained times in 10800 cents, viz., 9 times: hence, to find the price of the mixture, Rule.-I. Find the cost of the mixture: II. Divide the cost of the mixture by the sum of the simples, and the quotient will be the price of the mixture. Examples. 1. If 1 gallon of molasses, at 75 cents, and 3 gallons, at 50 cents, be mixed with 2 gallons, at 37, what is the mixture worth a gallon? 2. If teas at 371, 50, 621, 80, and 100 cents per pound, be mixed together, what will be the value of a pound of the mixture? 3. If 5 gallons of alcohol, worth 60 cents a gallon, and 3 gallons, worth 96 cents a gallon, be diluted by 4 gallons of water, what will be the price of one gallon of the mixture? 4. A farmer sold 50 bushels of wheat, at $2 a bushel; 60 bushels of rye, at 90 cents; 36 bushels of corn, at 62 cents; and 50 bushels of oats, at 39 cents a bushel: what was the average price per bushel of the whole? 5. During the seven days of the week, the thermometer stood as follows: 70°, 73°, 7310, 770, 700, 801°, and 81°: what was the average temperature for the week? 6. If gold 18, 21, 17, 19, and 20 carats fine, be melted together, what will be the fineness of the compound? sugar at 5 cents a pound, cents a pound, and 34 lb. at 7. A grocer bought 34 lb. of 102 lb., at 8 cents, 136 lb. at 10 12 cents a pound. He mixed it together, and sold the mixture so as to make 50 per cent. on the cost: what did he sell it for per pound? 8. A merchant sold 8 lb. of tea, 11 lb. of coffee, and 25 lb. of sugar, at an average of 15 cents a pound. The tea was worth 30 cents a pound; the coffee, 25 cents a pound; and the sugar 7 cents a pound: did he gain or lose, and how much? ALLIGATION ALTERNATE. 330. ALLIGATION ALTERNATE is the method of finding what proportion of several simples, whose prices or qualities are known, must be taken to form a mixture of any required price or quality. It is the reverse of Alligation Medial, and may be proved by it. The process of Alligation Alternate is founded on an equality of gain and loss. In selling a mixture at an average price, there is a gain on each simple below that price, and a loss on each simple above that price. The gain must be exactly equal. to the loss, otherwise the value of the compound would not be an average price. CASE I. 331. To find the proportional parts. 1. A miller would mix wheat, worth 12 shillings a bushel; corn, worth 8 shillings; and oats, worth 5 shillings, so as to make a mixture worth 7 shillings a bushel : what are the proportional parts of each? ANALYSIS.-On every bushel put into the mixture, whose price is less than the mean price, there will be a gain; on every bushel whose price is greater than the mean price, there will be a loss; and since the gains and losses must balance each other, we must connect an ingredient on which there is a gain with one on which there is a loss. A bushel of oats, when put into the mixture, will bring 7 shillings, giving a gain of 2 shillings; and to gain 1 shilling, we must take half as much, or a bushel, which we write opposite 5s. in column A. On 1 bushel of wheat there will be a loss of 5 shillings; and to make a loss of 1 shilling, we must take of a bushel, which we write in column A: and, are called proportional numbers. Again, comparing the oats and corn, there is a gain of 2 shillings on every bushel of oats, and a loss of 1 shilling on every bushel of corn: to gain 1 shilling on the oats, and lose 1 shilling on the corn, we must take a bushel of the oats, and 1 bushel of the corn: these numbers are written in column B Two simples, thus compared, are called a couplet: in one, the price of 1 is less than the mean price, and in the other it is greater. If every time we take a bushel of oats we take of a bushel of wheat, the gain and loss will balance; and if every time we take a bushel of oats we take 1 bushel of corn. the gain and loss will balance: hence, if the proportional numbere of a couplet be multiplied by any number, the gain. and loss denoted by the products will balance. When the proportional numbers, in any column, are fractional (as in columns A and B), multiply them by the least common multiple of their denominators, and write the products in new columns C and D. Then add the numbers in columns C and D standing opposite each simple, and if their, suns have a cominon factor, divide by it: the last result will be the proportional num bers. NOTE. The answers to the last, and to all similar questions, will be infinite in number, for two reasons: 1st. If the proportional numbers in column E be multiplied by any number, integral or fractional, the products will denote proportional parts of the simples. 2d. If the proportional numbers of any couplet be multiplied by any number, the gain and loss in that couplet will still balance, and the proportional numbers in the final result will be changed. Rule.-I. Write the prices or qualities of the simples in a column, beginning with the lowest, and the mean price or quality at the left: II. Opposite the first simple write the part which must be taken to gain 1 of the mean price, and opposite the other simple of the couplet write the part which must be taken to lose 1 of the mean price, and do the same for each simple : III. When the proportional numbers are fractional, reduce them to integral numbers, and then add those which stand opposite the same simple; if the sums have a common factor, reject it the resuit will denote the proportional parts. Examples. 1. What proportions of coffee, at 8 cents, 10 cents, and 14 cents per pound, must be mixed together so that the compound shall be worth 12 cents per pound? 2. A merchant has teas worth 40 cents, 65 cents, and 75 cents a pound, from which he wishes to make a mixture worth 60 cents a pound: what is the smallest quantity of each that he can take and express the parts by whole numbers? 3. A farmer sold a number of colts at $50 each, oxen at $40, cows at $25, calves at $10, and realized an average price of $30 per head: what was the smallest number he could sell of each? 4. What is the smallest quantity of water that must be mixed with wine worth 14s. and 15s. a gallon, to form a mixture worth 13s. a gallon, when all the parts are expressed by whole numbers? CASE II. 332. When the quantity of one of the simples is given. 1. A farmer would mix rye worth 80 cents a bushel, and corn worth 75 cents a bushel, with 66 bushels of oats worth 45 cents a bushel, so that the mixture shall be worth 50 cents a bushel: how much must be taken of each sort? ANALYSIS. Find the proportional parts as in Case I.: they are 11, 1 and 1. But we are to take 66 bushels of oats in the mixture; hence, each proportional number is to be taken 6 times; that is, as many times as there are units in the quotient of 6611. Rule.-I. Find the proportional numbers as in Case I., and write each opposite its simple: II. Find the ratio of the proportional number corresponding to the given simple, to the quantity of that simple to be taken, and multiply each proportional number by it. NOTE.-If we multiply the numbers in either or both of the columns C or D by any number, the proportion of the numbers in |