EXAMPLES FOR PRACTICE.

1. What will an annuity of $378 amount to in 5 years, at 6 per cent. compound interest? Ans. $2130.821+.

2. What will an annuity of $1728 amount to in 4 years, at 5 per cent. compound interest?

3. What will an annuity of $87 per cent. compound interest?

4. What will an annuity of $500 per cent. compound interest?

5. What will an annuity of $96 per cent. compound interest?

Ans. $7447.89,6+. amount to in 7 years, at 6 Ans. $730.26,3+.

amount to in 6 years, at 6 Ans. $3487.65,9+. amount to in 10 years, at 6 Ans. $1265.35,6+.

6. What will an annuity of $1000 amount to in 3 years, at 6 per cent, compound interest? Ans. $3183.60.

7. July 4, 1842, H. Piper deposited in an annuity office, for his daughter, the sum of $56, and continued his deposits each. year, until July 4, 1848. Required the sum in the office July 4, 1848, allowing 6 per cent. compound interest.

Ans. $470.05,4+.

8. C. Greenleaf has two sons, Samuel and William. On Samuel's birth-day, when he was 15 years old, he deposited for him, in an annuity office, which paid 5 per cent. compound interest, the sum of $25, and this he continued yearly, until he was 21 years of age. On William's birth-day, when he was 12 years old, he deposited for him, in an office which paid 6 per cent. compound interest, the sum of $20, and continued this until he was 21 years of age. Which will receive the larger sum, when 21 years of age?

Ans. $60.06,5+ William receives more than Samuel. 9. I gave my daughter Lydia $10 when she was 8 years old, and the same sum on her birth-day each year, until she was 21 years old. This sum was deposited in the savings bank, which pays 5 per cent. annually. Now, supposing each deposit to remain on interest until she is 21 years of age, required the amount in the bank. Ans. $195.98,6+.

ART. 316. ALLIGATION is a rule employed in the solution of questions relating to the compounding or mixing of several ingredients. The term signifies the act of connecting or tying together. It is of two kinds: Alligation Medial and Alligation Alternate.

ALLIGATION MEDIAL.

ART. 317. Alligation Medial is the method of finding the mean price of a mixture composed of articles of different values, the quantity and price of each being given.

ART. 318. To find the mean price of several articles or ingredients, at different prices, or of different qualities.

RULE. Find the value of each of the ingredients, and divide the amount of their values by the sum of the ingredients. The quotient will be the price of the mixture.

EXAMPLES FOR PRACTICE.

Ex. 1. A grocer mixed 20lb. 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? Ans. $0.55.

Proof, $0.5 5 × 20 lb.

$5 5.0 0, value.

Then, $55.00÷100 = $0.5 5 per pound.

$0.55 x 30 lb.

$1 1.0 0

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. $0.31.

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

QUESTIONS.

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

corn 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.

ART. 319. Alligation Alternate is the method of finding what quantity of ingredients or articles, whose prices or qualities are given, must be taken, to compose a mixture of any given price or quality.

ART. 320. To find what quantity of each ingredient must be taken to form a mixture of a given price.

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 value 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 llb. more of spice at 28 cents. We take the 1lb. at 28 cents, and thus have a mixture of the required average value, by having taken, in all, llb. at 20 cents, 1lb. at 23 cents, Îlb. 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.

QUESTIONS.

Having arranged in a column the prices of the ingredients, with the given mean price on

Ans. the left, we connect together the terms denoting the price of each ingredient, so that a price less than the given

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

We then proceed

mean is united with one that is greater. to find what quantity of each of the two kinds, 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 ingredients 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 515 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 ingredients, 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.

We obtain, as a result, 31b. 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 ingredients in a column, with the mean price on the left, and connect the price of each ingredient which is less than the given mean price with one that is greater.

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

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 ?

QUESTIONS. What is 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.

ART. 321. When the quantity of one ingredient 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 51b., which is but of the given quantity. Therefore, if we increase the 5lb. to 20, the other differences must be increased in the same proportion. Hence the propriety of the following

RULE. Find the difference between each price and the mean price; then say, As the difference of that ingredient 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. ?

QUESTION.

Ans. 3831lb. at 10 cents, and 100lb. at 12 cents.

Art. 321. What is the rule for finding the quantity of each of

the other ingredients when one is given?