whose rates are given, will compose a mixture of a given rate.

EXPLANATIONS.

The same question, in this rule, often admits of different answers, and it is, therefore, called Alligation Alternate. It is the reverse of Alligation Medial.

RULE.

Q. When the prices of the several ingredients or simples are given, how do you find how much of each, at their respective rates, must be taken to make a mixture or compound, at any proposed price?

A. The rates of the ingredients or simples must be placed in a column under each other, with the mean price at the left hand; then each rate which is less than the mean rate must be connected with one or more that is greater, and the difference between each rate and the mean price must be taken and placed directly opposite that rate with which it is connected. If only one difference stand against either rate, it will be the quantity required at that rate; but if there be two or more, their sum will express the quantity.

EXAMPLES.

1. A merchant has oats at 30c. a bushel, barley at 44c. a bushel, corn at 48c. a bushel, and rye at 56c. a bushel; how many bushels of each sort must he mix, that the mixture may be worth 46c. a bushel?

and 48, as 30 and 44 are less than 46, the mean rate; and you set down 10 opposite 30, 16 opposite 56, 2 opposite 44, and opposite 48, as the difference between 46, the mean rate, and the several separate simples. You will readily perceive, that, by this operation, you connect a rate which is less than the mean rate with one that is greater than the mean rate, and set down the difference between them and the mean rate alternately, in such a manner, that there is precisely as much gained by one quantity as there is lost by the other; and that, therefore, the gain and loss on the whole are equal.

2. A grocer has three kinds of sugar; at 26c., 23c., and 20c. a pound, how many pounds of each kind must he mix, that the mixture may be worth 22c. a pound? Ans. 2lb. at 26c., 2lb. at 23c., and 5lb. at 20c.

3. A grocer wishes to mix four kinds of tea, at 60c., 70c., 80c., and 90c. a pound in such a manner that the mixture will be worth 75c. a pound; how many pounds of each kind must he mix? Ans. 15lb. at 60c., 5lb. at 70c., 5lb. at 80c., and 15lb. at 90c.

RULE.

Q. When one of the ingredients is limited to a certain quantity, how do you find the several quantities of the rest, in proportion to the given quantity?

A. First take the differences between each price and the mean rate, and set them down alternately, as in the preceding rule; then, as the difference standing against that simple whose quantity is given, is to that quantity, so is each of the other differences, severally, to the several quantities required.

EXAMPLES.

1. A merchant wishes to mix 10gal. of brandy at 70c. a gallon, with one kind at 48c. a gallon, another at 36c. a gallon, and another at 30c. a gallon, so that a gallon of the mixture may be sold for 38c.; what quantity of each must be taken ?

As 8: 10:

{

2: 2gal. 2qt. at 48c. a gallon.
10: 12gal. 2qt. at 36c.
32: 40gal. Oqt. at 30c.

2. A grocer wishes to mix 3lb. of sugar at 7c. a pound, with one kind at 4c., another at 5c., and another at 8c. a pound, so that a pound of the mixture may be sold for 6c. a pound; how many pounds of each must be taken? Ans. 3lb. at 7c., 6lb. at 4c., 3lb. at 5c., and 67b. at 8c. a pound.

RULE.

Q. How do you state and work the terms when the whole composition is limited to a given or certain quantity?

A. First find the differences between each price and the mean rate, and set them down, as directed in the preceding rule: then, as the sum of the quantities, or differences thus found, is to the given quantity, or whole composition, so is each ingredient, thus found, to the required quantity of each.

EXAMPLES.

1. A grocer has four kinds of sugar, at lc. 2c. 6c. and 9c. a pound, which he wishes to mix together to make a composition of 961b. at 3c. a pound; how many pounds of each must he take?

2. A grocer wishes to mix water, at O a gallon, with brandy, at $1,60c. a gallon, to make a mixture of 120gal., at $1,20c. a gallon; how much of each must he take? Ans. 30gal. of water, and 90gal. of brandy.

DOUBLE RULE OF THREE.

Q. What is DOUBLE RULE OF THREE?

A. The Double Rule of Three teaches to resolve, by one statement, such questions as require two or more statements in single proportion, or Single Rule of Three. In this rule there are always five terms given to find a sixth; the first three terms of which are a supposition, and the last two a demand.

RULE.

Q. How do you state and work the terms in the Double Rule of Three?

A. First reduce the terms, if in different denominations, as in Single Rule of Three. Then, in stating the questions, place the terms of supposition so that the principal cause of loss, gain, or action, possess the first place; that which expresses time, distance of place, the second place; and that which expresses the gain, loss, or action, the third place. Then place the terms of demand under those of the same kind in the supposition. If the blank place or term sought fall under the third term, the proportion is direct; then multiply the first and second terms together for a divisor, and the other three for a dividend, and the quotient will be the answer in the same denomination of the term directly above the blank. But if the blank fall under the first or second term, the proportion is inverse; then multiply the third and fourth terms together for a divisor, and the other three for a dividend, and the quotient will be the answer.

EXAMPLES.

1. If $100, in 12mo., gain $6, what will $200 gain in 8mo.?

Ans. $8.

2. If 20 men spend $18 in 24 weeks, how much will 40 men spend in 48 weeks? Ans. $72.

3. If the wages of 6 persons, for 21 weeks, be $288, what must 14 persons receive for 46 weeks? Ans. $1472.

4. If the carriage of 8cwt. 128 miles cost $12,80c, what will it cost for the carriage of 12cwt. 160 miles? Ans. $24.

5. If 10bu. of oats be sufficient for 18 horses 20 days, how many bushels will serve 60 horses 36 days? Ans. 60bu.

6. If 12bu. of wheat are sufficient for a family of 4 persons 9mo., how many bushels will be sufficient for a family of 8 persons 12mo.? Ans. 32bu.

7. If 8 men can make 72 rods of wall in 6 days, how many men can make 54 rods in 3 days? Ans 12 men.

8. If $700, in 6mo., gain $14 interest, how much will $400 gain in 5 years? Ans. $80.

9. What principal, at 7 per cent. per annum, will gain $42 interest in 9mo.? Ans. $800.

10. If 20 cows, for $80, can be kept 40 weeks, how many cows can be kept 12 weeks for $30? Ans. 25 cows.

11. If 7 men can reap 84 acres of wheat in 12 days, how many days will it require 20 men to reap 100 acres? Ans. 5da. 12. If a footman travel 720 miles in 27 days, in travelling. 16 hours each day, how many days will he require to travel 240 miles if he travel 12 hours in each day? Ans. 12da.