THE CONDITION of the above obligation is such, that. if the above bounden James Wilson and Thomas Ash, their heirs, executors, or administrators, shall well and truly pay or cause to be paid, unto the above-named John Pickens, his executors, administrators, or assigns, the just and full sum of

[Here insert the condition.]

then the above obligation to be void, otherwise to remain in full force and virtue.

NOTE. The part in Italic to be filled up according to cir

cumstance.

If there is no condition to the bond, then all to be omitted after and including the words "THE CONDITION, &c."

ALLIGATION MEDIAL.

248. A merchant mixes 876. of tea, worth 75cts. per pound, with 167. worth $1,02 per lb: what is the value of the mixture per pound?

The manner of finding the price of this mixture is called Alligation Medial. Hence,

ALLIGATION MEDIAL teaches the method of finding the price of a mixture when the simples of which it is composed, and their prices, are known.

In the example above, the simples 87b. and 167b., and also their prices per pound, 75cts. and $1,02, are known. 8lb. of tea at 75cts. per гъ. 1676. 66 66° $1,02 per 16. 24 sum of simples.

6,00 16,32

Total cost $22,32

248. What is Alligation Medial? How do you find the price of the mixture?

Now, if the entire cost of the mixture, which is $22,32, be divided by 24, the number of lbs. or sum of the simples, the quotient 93cts. will be the price per pound. Hence, to find the price of the mixture,

OPERATION.

24)22,32(93cts.

216

72

72

Divide the entire cost of the whole mixture by the sum of the simples, and the quotient will be the price of the mixture.

EXAMPLES.

1. A farmer mixes 30 bushels of wheat worth 5s. 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?

:

2. A wine merchant mixes 15 gallons of wine at $1 per gallon with 25 gallons of brandy worth 75 cents 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 costs nothing: what is the value of a gallon of the mixture?

Ans. 283cts.

4. A goldsmith melts together 27b. of gold of 22 carats fine, 6oz. of 20 carats fine, and 6oz. of 16 carats fine: what is the fineness of the mixture? Ans. 20 carats.

5. On a certain day the mercury in the thermometer was observed to average the following heights: from 6 in the morning to 9, 640; 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. 73°.

ALLIGATION ALTERNATE.

bushel

249. A farmer would mix oats worth 3s. per with wheat worth 9s. per bushel, so that the mixture shall be worth 5s. per bushel: what proportion must be taken of each sort?

The method of finding how much of each sort must be taken is called Alligation Alternate. Hence,

ALLIGATION ALTERNATE teaches the method of finding what proportion must be taken of several simples, whose prices are known, to form a compound of a given price.

Alligation Alternate is the reverse of Alligation Medial, and may be proved by it.

For a first example, let us take the one above stated. 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

were 6s., half the sum of the

prices of the simples, it is plain that it would be necessary to take just as much oats as wheat.

3

4 Oats.

5

9

2. 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

249. What is Alligation Alternate? How do you prove Alligation Alternate?

each other, such as 8 and 4, 20 and 10, &c., will give a mixture of the same value,

PROOF BY ALLIGATION MEDIAL.

250. To find the proportion in which several simples of given prices must be mixed together, that the compound may be worth a given price.

I. 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, and the one next to the least with the one next to the 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.

EXAMPLES.

1. A merchant would mix wines worth 16s., 18s., and 22s. per gallon in such a way, that the mixture may be worth 20s. per gallon: how much must be taken of each sort?

Ans.

2gal. at 16s., 2 at 18s., and 6 at 22s.: or any other quantities bearing the proportion of 2, 2, and 6.

250. How do you find the proportions so that the compound may be of a given price?

2. What proportions of coffee at 16cts., 20cts., and 23cts. per lb. must be mixed together so that the compound shall be worth 24cts. per lb. ?

Ans.

In the proportion of 47b. at 16cts., 47b, at 20cts., and 127b. at 28cts.

3. A goldsmith has gold of 16, of 18, of 23, and of 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 old Madeira at 24s. per gallon, of new Madeira at 21s. per gallon, and of brandy at 10s. per gallon, must 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.

251. When a given quantity of one of the simples is to be taken.

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.

EXAMPLES.

1. How much wine at 5s., at 5s. 6d., and 6s. per gallon must be mixed with 4 gallons at 4s. per gallon, so that the mixture shall be worth 5s. 4d. per gallon?

4 : 16 :

8

Ans. 1gal. at 5s., 2 at 5s. 6d., and 8 at 6s.

251. How do you find the proportion when the quantity of one of the simples is given?