ALLIGATION ALTERNATE.

ALLIGATION ALTERNATE, (from Lat. Alligo, to tie, and Alternus, by turns,) is that process by which ingredients of different values are mixed, so as to form a compound of a specified price, for which we have the following

RULE.

First. Link the GREATER price with the less, and then, first, subtract the mean price from the FORMER, placing the difference opposite the latter; second, subtract the LESS price from the MEAN, and place the difference opposite the greater, these differences, as they now stand, will show the quantity of each ingredient, which, being mixed, will give the compound required.

Second. If there are three or more quantities, link any two of them, one of which has a less, and the other a greater value than the mean price, and proceed as before.

Lastly. If one of the ingredients be greater in quantity than the rest, proceed as before, and take such a part of the quantity to be mixed of each, as is suggested by that ingredient whose quantity is specified, the results will be the

ANSWER.

EXAMPLES.

1. A grocer has three kinds of sugar, at 7 cents, 9 cents, and 10 cents, which he wishes to mix, so as to sell the mixture at 8 cents per pound; what quantity of each must he take?

mean price.

cts. lbs.

mean price.

cts. lb. =1

Thus: 1st, 8 cents, {11}2d, 8 cents, {]=

Hence, the answer,

9

Here we compare, first, 1lb., at 7 cents, with 1lb., at 10 cents, and the result is 2lbs., at 7 cents, with 1lb., at 10 cents.

Second. We compare 1lb., at 7 cents, with 1lb., at 9 cents, and the result is 1lb. of each; and,

Lastly, since the first comparison gives 2lbs., at 7 cents, and the second, 1lb., at 7 cents, it is plain, that the number

of pounds that must be taken at 7 cents, equals 2lbs.+1lb.= to 3lbs., and that of the rest, we must take 1lb. of each to form the mixture required.

2. A grocer has four kinds of tea, the first is worth 40 cents; the second, 50 cents; the third, 70 cents; and the fourth, 90 cents per lb., of which he wishes to form a mixture worth 60 cents per pound; how many pounds of each must he take?

Answer, 10lbs., at 40 cents; 20lbs., at 70 cents; 30lbs., at 50 cents; and 10lbs., at 90 cents. Or, 1lb., at 40 cents; 2lbs., at 70 cents; 3lbs., at 50 cents; and 1lb., at 90 cents.

Here we compare, first, that at 40 cents, and 70 cents; and, second, that at 50 cents, and 90 cents; but the result would have been equally true if we had compared that at 40 cents, with 90 cents, and that at 50 cents with 70 cents, i. e., it matters not what quantities we compare, so that the price of one is greater and the other less than the mean price.

3. A wine merchant has a hogshead of wine, worth 50 cents per gallon, which he wishes to mix with other wines, worth 75 cents, 80 cents, $1.25, and $1.50 per gallon, so as to sell the mixture at $1 per gallon; how much must he take of each?

cents.

Thus :
First.

cts. gals.
=25

=50

100,50 }

125

100 {150]=20

The mixture, therefore, is 25 gallons, at 50 cents; 50 gal. lons, at $1.25; 25 gallons, at 75 cents; 25 gallons, at $1.25; 50 gallons, at 80 cents; and 20 gallons, at $1.50. Or, 25 gallons, at 50 cents, and 75 cents; 50 gallons, at 80 cents; 75 gallons, at $1.25; and 20 gallons, at $1.50; but there are 63 gallons, (1hhd.), at 50 cents, which are 3 of 25 gallons, and hence, it will be necessary to take a like part of the other proportionate quantities, i. e., of each, making 63 gallons, at 50 cents, and 75 cents; 126 gallons, at 80 cents; 189 gallons, at $1.25; and 50 gallons, at $1.50. ANSWER.

25

Hence, we see that 4912491.40 gallons will cost $491.40, or 1 gallon, will cost $1, which is the PROOF REQuired.

4. A goldsmith wishes to mix 8ozs. of gold that is 20 carats fine, with 1oz. of each of the following values, viz. 16, 18, 22, and 24 carats fine, so that an ounce of the compound may be 21 carats fine; what quantity of each must be taken?

Solution.-The whole may be done without separating the

NOTE. In this example, we find that loz. of 20 carats fine, is the quantity found by the first calculation; but by the conditions of the question, we must mix 8ozs. of 20 carat gold, which is 8 times 1oz., and since we are required to take 8 times as much of one of the ingredients, we must also take 8 times as much of all the rest, which gives the ANSWER Sought.

QUESTIONS FOR EXERCISE.

1. A farmer has two kinds of wheat, one at 80 cents, and the other at $1.20 per bushel, which he wishes to mix, so as to sell the mixture at $1 per bushel, how much of each sort must he take?

2. A grocer has sugars worth 8 cents, 9 cents, and 11 cents per lb., with which he wishes to form a mixture worth 10 cents per lb., what quantity of each must he take?

3. A tea dealer wishes to mix teas, at 50 cents, 60 cents,

80 cents, and 90 cents per pound, so as to sell the mixture for 75 cents per pound, what quantity of each must he take?

4. A wine merchant wishes to mix wines, at 60 cents, 65 cents, 80 cents, 95 cents, $1, and $1.25, so as to sell the mixture, at 70 cents, per gallon, how much of each sort must he take?

5. A jeweller has 10 ounces of 21 carat-gold, which he wishes to mix with other gold of 18, 20, and 24 carats per ounce, so as to form a mixture of 22 carats fine; what quantity, i. e., how many ounces of each must he take?

6. A grocer has four kinds of spices, one at 25 cents; one at 40 cents; one at 50 cents; and one at 60 cents, which he wishes to mix with 20lbs., at 30 cents, so as to make the compound worth 45 cents a pound, how many pounds of each must he take?

7. How can you mix five parcels of tea, worth 50, 60, 70, 80, and 90 cents per lb., so as to sell the mixture for 62 cents per pound?

8. A grocer has 3qrs. (84lbs.) of sugar, worth 8 cents per lb. which he wishes to mix with other sugars, at 5 cents, 6 cents, 9 cents, and 10 cents per pound, so as to form a compound worth 7 cents a pound; what quantity of each must he take?

9. A wine merchant has a tierce of wine, worth 68 cents per gallon, which he would mix with other wines worth 45 cents, 56 cents, 62 cents, 70 cents, and 75 cents, so as to sell the mixture at 65 cents per gallon; how much must he take of each sort to form this compound?

10. A merchant has sugars worth 64 cents, 7 cents, 83 cents, 9 cents, and 10 cents per pound, which he wishes to mix, so as to form a compound worth 8 cents a pound; how much of each sort must he take?

11. A goldsmith has 5 ounces of gold, 18 carats fine, which he wishes to mix with other gold 16, 21, and 23 carats fine, and thus form a mixture that shall be 20 carats fine; how much of each sort must he take? 21 *

ARTICLE XVIII.

MISCELLANEOUS EXERCISES.

INVOLUTION.

1. INVOLUTION is that PROCESS by which we commence at the UNIT, and form a succession, or series of NUMBERS, by multiplying continually by the same number.

2. The UNIT is called the base of the series.

3. The MULTIPLIER is called the PRIME ROOT. *

4. The several PRODUCTS thus formed are called POWERS. Thus: 1× 6-6, or the first power of 6, and 6×6=36, or the second power of 6, and 36×6=216, or the third power of 6, &c.; from which it appears, first, that the PRIME ROOT and FIRST POWER of any number, are the same as the number itself; second, that the SECOND POWER of any number is the PRODUCT of that number multiplied by itself, called its SQUARE; third, the THIRD POWER of any number is the PRODUCT of the SQUARE of that number, multiplied by itself, called its CUBE, &c.; hence, since the first power and prime root of a number are always identical with the number itself, it is only necessary to find the second, third, &c., powers which may be done by the following

RULE.

First. Multiply the number whose power is sought, by itself, and the PRODUCT will be the SECOND POWER.

Second. Multiply the second power by the number itself, and the PRODUCT will be the THIRD POWER, &c.

EXAMPLES.

1. What is the fourth power of 8? Thus: 8x8x8x8 =512. Answer.

2. What is the fifth power of 9? Thus: 9x9x9x9×9 59049. Answer.

NOTE.-When a number is to be raised to a certain power, * PRIME, (from Lat. PRIMUS,) means FIRST.