ALLIGATION.

Art. 260.-ALLIGATION is the method of mixing two or more simples, of different qualities, so that the composition may be of a mean, or middle quality.

When the quantities and prices of the simples are given, to find the mean price of the mixture compounded of them, the process is called

ALLIGATION MEDIAL.

Art. 261.-1. If I mix 8 lbs. of sugar, worth 10 cents a pound, with 10 lbs., worth 15 cents a pound, what is 1 lb. of the mixture worth?

Eight pounds, at 10 cents a pound, are worth 10×8=80 cents, and 10 pounds, at 15 cents, are worth 15 × 10=150 cents; then, 80+150=230 cents, the price of the whole mixture, and 8+10=18 pounds, the whole mixture; then $2.30÷18 lbs.=123 cts., the worth of 1 pound of the mixture. Hence the

RULE.

Multiply each quantity by its price, and divide the sum of the products by the sum of the quantities. The quotient will be the rate of the compound required.

EXAMPLES.

2. A grocer mixes sugar, 5 lbs. at 6 cts., 8 lbs. at 5 cts., and 7 lbs. at 10 cts. a lb. What is 1 lb. of the mixture worth? Ans. 7 cts.

3. A farmer mixes 12 bushels of wheat at $1.75 a bushel, 8 bushels of rye at $1, and 6 bushels of corn at 80 cts. a bushel. What is a bushel of the mixture worth? Ans. $1.30.

4. A goldsmith melted together 12 lbs. of gold, 21 carats fine, 8 lbs. 20 carats fine, 9 lbs. 22 carats fine, and 7 lbs. 18 carats fine. Of what fineness is the mixture?

Ans. 204 carats fine. 5. A merchant mixed 8 gallons of wine, at 4s. 2d. per gal

lon, with 10 gallons at 6s. 5d., and 12 gallons at 8s. 4d. per gallon. What is a gallon of the mixture worth?

Ans. 6s. 7d.

6. If 4 lbs. of tea, at 6s. per lb., 8 lbs. at 5s., and 6 lbs. at 38., be mixed together, what is 1 lb. of the mixture worth? Ans. 45 shillings.

ALLIGATION ALTERNATE.

Art. 262.-ALLIGATION ALTERNATE is when the prices of the simples to be mixed, and the mean rate, are given, to find what quantity of each is to be taken at a given rate.

1. I have corn at 50 cents a bushel, and oats at 30 cents a bushel, which I would mix, so that the mixture may be worth 40 cents a bushel. What quantity of each must be taken ?

It is evident that equal quantities of each must be taken, for the price of the corn exceeds the mean rate as much as the price of the oats falls short of it, which is 10 cents in each case. We find, also, that the whole mixture, which is 20 bushels, at the mean rate, 40 cents a bushel, equals the price of 10 bushels of oats at 30, and 10 bushels of corn at 50 cents a bushel.

RULE.

I. Reduce the rates of all the simples to the same denomination, and write them in a column under each other, and the mean rate on the left hand.

II. Connect the rate of each simple, which is less than the rate of the compound, with one that is greater, and each that is greater with one that is less.

III. Write the difference between each rate, and that of the compound against the number with which it is connected. Then, if only one difference stand against any rate, it will express the relative quantity to be taken of that rate; but if more than one, their sum will express that quantity.

EXAMPLES.

Art. 263.-2. A farmer has wheat at $1.50, rye at $1.00, corn at 90, and oats at 40 cents a bushel, which he mixes so

that the mixture is worth 95 cents a bushel.

of each does he take?

What quantity

By linking the price of the different simples, as above, their quantities are mutually mixed, and the portion taken of each depends upon the manner of linking them. In the first operation, the price of the wheat, which is greater than the mean price, is linked with the price of the oats, which is less. The price of the wheat is found to be as much greater than the mean rate, as the price of the oats is less; therefore an equal quantity of each is taken. The same is true of the corn and oats. In the second operation, the price of the wheat is linked with the price of the corn. The difference between the price of the wheat and the mean rate, is 55, and the difference between the price of the corn and the mean rate, is 5. Hence, it appears that the less the difference between the price of a simple and the mean rate, the greater will be the quantity taken of that simple; and the greater the difference the less the quantity.

3. A merchant has teas at 72 cents, at 62 cents, and 57 cents a pound, which he would mix, so that the mixture may be worth 67 cents per lb. What quantity of each must be taken?

The correctness of the above operation may be ascertained thus: The cost of 15 lbs. at 72 cents, is $10.80, and the cost of 5 lbs. at 62 cents, is $3.10, and the cost of 5 lbs. at 57 cents, is $2.85. Then the whole cost is $10.80+$3.10+ $2.85 $16.75, which, divided by 25 lbs., gives the mean price, $16.75 25-67 cents. Hence, it appears that Alligation Alternate is the reverse of Alligation Medial, and may be proved by it.

4. A grocer mixes wines at 29s., 24s., 22s., and 17s. a gallon, so that the mixture is worth 23s. per gallon. How much of each sort does he take?

As many different answers may be obtained to questions in this rule as there are modes of linking the prices of the simples. Let the number of simples be what it may, and with how many soever each one is linked, since the price of one that is less than the mean rate, is always linked with one that is greater, there will always be an equal balance of loss and gain between the two, and consequently an equal balance on the whole. 5. It is required to mix brandy at 12s., wine at 9s., cider at 2s., beer at is., and water at Os., per gallon, so that the mixture may be worth 7s. per gallon. What quantity of each must be taken?

Art. 264.-When the composition is limited in quantity.

RULE.

Find the proportion of each quantity as before; then say, As the sum of the quantities is to the given quantity, so is each of the differences to the required quantity.

EXAMPLES.

6. Suppose a mass of pure gold, a mass of pure silver, and a mass which is a mixture of gold and silver, each weighing 9 oz.; by immersing them in water, it is found that the quantity of water displaced by the gold is 5; by the silver 8, and by

the mixture 7. What part of the mixture is gold, and what part silver?

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By a similar problem, Archimedes detected the fraud of the artist employed by Hiero, king of Syracuse, to make him a crown of pure gold.

7. A druggist has medicines at 6d., 3d., 9d., and 4d. per oz., and would form a compound of 15 oz., worth 5d. per oz. How much of each sort must he take?

8. A goldsmith would melt together gold of 13, of 14, of 15, and of 21 carats fine, to form a composition of 35 oz. 18 carats fine. What proportion of each must he take?

9. How many gallons of water, worth Os. per gal., must be mixed with wine worth 12s. per gal., so as to fill a cask of 20 gallons, and that a gallon of the mixture may be afforded at 98. per gallon?

Ans.

5 gal. water. 15 gal. wine.

Art. 265.—When one of the simples is limited to a certain quantity.

RULE.

Find the proportional quantities, or differences, as before; then say, As the difference standing against the given quantity is to the given quantity, so are the other differences severally to the several quantities required.

EXAMPLES.

10. A grocer mixes sugar at 9 cts., 12 cts., and 14 cts., with 16 lbs. at 15 cts. How much of each sort must he take, that the mixture may be worth 13 cts. per

lb.?