ALLIGATION.

316. Alligation is a process employed in the solution of ques. tions relating to the compounding or mixing of articles of different qualities or values.

It is of two kinds: Alligation Medial and Alligation Alternate.

ALLIGATION MEDIAL.

317. Alligation Medial is the process of finding the mean or average rate of a mixture composed of articles of different qualities or values, the quantity and rate of each being given.

318. To find the AVERAGE VALUE of several articles mixed, the quantity and rate being given.

RULE. - Find the value of each of the articles, and divide the sum of their values by the sum of the quantities of the articles. The quotient will be the average value of the mixture.

EXAMPLES FOR PRACTICE.

Ex. 1. A grocer mixed 201b. of tea worth $0.50 a pound, with 30lb. worth $0.75 a pound, and 50lb. worth $0.45 a pound; what is 1 pound of the mixture worth?

OPERATION.

$0.50 X 20 = $10.00

Ans. $0.55.

Proof.

20lb., at 50 cts. per lb., is worth $10.00; 30lb., at 75 cts. per lb., is worth $22.50; and 50lb., at 45 cts. per lb., is worth $22.50. Then, 20lb. +30lb. + 50lb. 100lb., is worth $10.00 +$22.50 +$ 22.50 $55.00; and 1lb. is worth as many dollars as 100 is contained times in 55.00, or $0.55.

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2. I have four kinds of molasses, and a different quantity of each, as follows: 30 gal. at 20 cents, 40 gal. at 25 cents, 70 gal. at 30 cents, and 80 gal. at 40 cents; what is a gallon of the mixture worth? Ans. $0.314.

3. A farmer mixed 4 bush. of oats at 40 cents, 8 bush. of corn

316. What is alligation? What two kinds are there?-317. What is alligation medial ?-318. The rule for finding the mean value of several articles at different rates? How does it appear that this process will give the mean value of a mixture?

at 85 cents, 12 bush. rye at $1.00, and 10 bush. of wheat at $1.50 per bushel. What will one bushel of the mixture be worth? Ans. $1.04.

ALLIGATION ALTERNATE.

319. Alligation Alternate is a process of finding what quantity of articles, whose rate or qualities are given, must be taken, to compose a mixture of any given rate or quality.

320. To find what quantity of each article must be taken to form a mixture of a given rate.

Ex. 1. I wish to mix spice, at 20 cents, 23 cents, 26 cents, and 28 cents per pound, so that the mixture may be worth 25 cents per pound. How many pounds of each must I take?

Compared with the mean or average price given, by taking 1lb. at 20 cents there is a gain of 5 cents, by taking 1lb. at 23 cents a gain of 2 cents, by taking 1lb. at 26 cents a loss of 1 cent, and by taking 1lb. at 28 cents a loss of 3 cents; making an excess of gain over loss of 3 cents. Now, it is evident that the mixture, to be of the average rate named, should have the several items of gain and loss in the aggregate exactly offset one another. This balance we can effect, in the present case, either by taking 3lb. more of the spice at 26 cents, or 1lb. more of spice at 28 cents. We take the 1lb. at 28 cents, and thus have a mixture of the average rate, by having taken, in all, 1lb. at 20 cents, 1lb. at 23 cents, 1lb. at 26 cents, and 2lb. at 28 cents. We prove the correctness of the result by dividing the value of the whole mixture, or $1.25, by the number of pounds taken, or 5, which gives 25 cents, or the given mean price per pound.

SECOND OPERATION.

20cts.- 3lb.

25cts.

23cts. 1lb.
26cts. 21b.
28cts. 5lb.

Having arranged in a column the prices of the articles with the given mean price on the left, wẹ Ans. connect together the terms denoting the price of each article, so that a price less than the given mean is united with one that is greater.

We then proceed to find what quantity of each of the two kinds,

319. What is alligation alternate? Explain the first operation. How is it proved to be correct? How do you connect the prices?

whose prices have been connected, can be taken, in making a mixture, so that what shall be gained on the one kind shall be balanced by the loss on the other. By taking 1lb. of spice at 20 cents, the gain will be 5 cents; and by taking 1lb. at 28 cents, the loss will be 3 cents. To equalize the gain and loss in this case, it is evident we should take as many more pounds of that at 28 cents as the loss on 1lb. of it is less than the gain on 1lb. of that at 20 cents; or, in other words, the quantity of the articles taken should be in the inverse ratio (Art. 236) of the difference between their respective prices and the given mean price. Therefore, we take 5lbs. at 28 cents, and 3lbs. of that at 20 cents, and the loss, 3cts. X 5: 15 cents, on the former, exactly offsets the gain, 5cts. X 3 = 15 cents, on the latter. We write the 3lb. against its price, 20 cents; and the 5lb. against its price, 28 cents. In like manner we determine the quantity that may be taken of the other two articles, whose prices are connected, by finding the difference between each price and the mean price; and, as before, write the quantity taken against its price.

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We obtain, as a result, 3lb. at 20 cents, 1lb. at 23 cents, 2lb. at 26 cents, 5lb. at 28 cents; this, in the same manner as the other answer, may be proved to satisfy the conditions of the question, since examples of this kind admit of several answers.

RULE.

Write the prices of the articles in a column, with the mean rate on the left, and connect the rate of each article which is less than the given mean rate with one that is greater.

Write the difference between the mean rate and that of each of the articles opposite to the rate with which it is connected; and the number set against each rate is the quantity of the article to be taken at that rate.

NOTE. There will be as many different answers as there are different ways of connecting the prices, and by multiplying and dividing these answers they may be varied indefinitely.

EXAMPLES FOR PRACTICE.

2. A farmer wishes to mix corn at 75 cents a bushel, with rye at 60 cents a bushel, and oats at 40 cents a bushel, and wheat at 95 cents a bushel; what quantity of each must he take to make a mixture worth 70 cents a bushel?

320.The rule for alligation alternate? How can you obtain different

answers? Are they all true?

3. I have 4 kinds of salt, worth 25, 30, 40, and 50 cents per bushel; how much of each kind must be taken, that a mixture might be sold at 42 cents per bushel?

Ans. 8 bushels at 25, 30, and 40 cents, and 31 bushels at 50

cents.

321. When the quantity of one article is given to find the quantity of each of the others.

Ex. 1. How much sugar, that is worth 6, 10, and 13 cents a pound, must be mixed with 20lb. worth 15 cents a pound, so that the mixture will be worth 11 cents a pound?

By the conditions of the question we are to take 20lb. at 15 cents a pound; but by the operation we find the difference at 15 cents a pound to be only 5lb., which is but of the given quantity. Therefore, if we increase the 5lb. to 20, the other differences must be increased in the same ratio. Hence the

RULE. Find the difference between the rate of each and the mean rate; then say, As the difference of that article whose quantity is given is to each of the differences separately, so is the quantity given to the several quantities required.

EXAMPLES FOR PRACTICE.

2. A farmer has oats at 50 cents per bushel, peas at 60 cents, and beans at $1.50. These he wishes to mix with 30 bushels of corn at $1.70 per bushel, that he may sell the whole at $1.25 per bushel; how much of each kind must he take?

Ans. 18 bushels of oats, 10 bushels of peas, and 26 bushels of beans.

3. A merchant has two kinds of sugar, one of which cost him 10 cents per lb., and the other 12 cents per lb.; he has also 100lb. of an excellent quality, which cost him 15 cents per lb. Now, as he ought to make 25 per cent. on his cost, how much of each quantity must be taken that he may sell the mixture at 14 cents per lb.?

Ans. 383 lb. at 10 cents, and 100 lb. at 12 cents.

321. What is the rule for finding the quantity of each of the other articles when one is given?

322. When the sum of the quantities of the articles and their mean rate are given, to find what quantity of each must be taken.

Ex. 1. I have teas at 25 cents, 35 cents, 50 cents, and 70 cents a pound, with which I wish to make a mixture of 180lb., that will be worth 45 cents a pound. How much of each kind must I take?

By the conditions of the question, the weight of the mixture is 180lb., but by the operation we find the sum of the differences to be only 60lb., which is but of the quantity required. Therefore, if we increase 60lb. to 180, each of the differences must be increased in the same ratio, in order to make a mixture of 180lb., the quantity required. Hence the

RULE. - Find the differences as before; then say, As the sum of the differences is to each of the differences separately, so is the given quantity to the required quantity of each article.

EXAMPLES FOR PRACTICE.

2. John Smith's "great box" will hold 100 bushels. He has wheat worth $2.50 per bushel, and rye worth $2.00 per bushel. How much chaff, of no value, must he mix with the wheat and rye, that, if he fill the box, a bushel of the mixture may be sold at $1.80 ?

Ans. 40bu. each of wheat and rye, and 20bu. of chaff.

3. I have two kinds of molasses, which cost me 20 and 30 cents per gallon; I wish to fill a hogshead, that will hold 80 gallons, with these two kinds. How much of each kind must be taken, that I may sell a gallon of the mixture at 25 cents per gallon, and make 10 per cent. on my purchase?

Ans. 58 of 20 cents, and 21 of 30 cents.

4. I have sugars at 10 cents and 15 cents per pound. How much of each must be taken, that a mixture containing 60 pounds shall be worth $7.20 ?

Ans. 36 pounds at 10 cents, and 24 pounds at 15 cents.

322. How do you find what quantity of each ingredient must be taken when the sum and mean price are given?