SECTION XIV.

ALLIGATION.

164. Definitions and Explanations.

(.) MERCHANTS and others often find it convenient to mix articles of different kinds together, so as to obtain a compound differing in value from any of its ingredients. The various problems connected with the subject are called PROBLEMS IN ALLIGATION.

(b.) Questions in alligation are usually divided into two classes, viz. First, ALLIGATION MEDIAL, in which the quantities of the several ingredients and their prices are given, and we are required to find the price of the mixture per pound, per gallon, or per bushel.

Second, ALLIGATION ALTERNATE, in which the prices of the various ingredients are given, and we are required to find what quantities of each must be taken to make a mixture having a given value per pound, per bushel, or per gallon.

165. Problems.

1. A trader mixed together 6 lb. of coffee worth 10 cents per pound, 4 lb. worth 8 cents per lb., and 7 lb. worth 16 cents per lb. How much was the mixture worth per lb. ?

Solution. 6 lb. at 10 cents are worth 60 cents.

2. A silversmith melted together 9 oz. of silver, 14 carats fine; 6 oz., 18 carats fine; 12 oz., 22 carats fine; and 23 oz., 24 carats fine. What was the fineness of the mixture?

3. A dry goods dealer sold 25 yd. of sheeting at 9 cents per yd.; 30 yd. of shirting at 10 cents per yd.; 11 yd. of

delaine at 18 cents per yd.; 9 yd. of gingham at 22 cent; per yd.; and 25 yd. of linen at 40 cents per yd. What was the average price of the whole per yard?

4. A merchant has sugars at 6, 7, 10, and 13 cents per pound, of which he wishes to make a mixture such that, by selling it for 9 cents per pound, he will neither gain nor lose. How many pounds of each kind must he take?

Solution. - It is obvious that by selling the mixture for 9 cents per pound, he will gain 3 cents on each pound which he puts in of the first kind, and 2 cents on each pound of the second kind; that he will lose 1 cent on each pound of the third kind, and 4 cents on each pound of the fourth; and further, that to make the mixture worth just cents per pound, he must take such proportions of the several kinds as will make his gains equal his losses. Moreover, he may take as many pounds as he chooses of the kinds which cost less than 9 cents, provided he takes enough of the others to counterbalance the gain on them.

Suppose that he takes 8 lb. of the first kind, and 11 lb. of the second. Then, since on 1 lb. of the first he gains 3 cents, on 8 lb. he will gain 8 times 3 cents, or 24 cents; and since on 1 lb. of the second he gains 2 cents, on 11 lb. he will gain 11 times 2 cents, or 22 cents, which, added to the 24 cents, gives 46 cents as the sum of his gains. He must, therefore, take enough of the other kinds to lose 46 cents. Suppose he takes 10 lb. of that at 10 cents. Then, since on 1 lb. he loses 1 cent, on 10 lb. he will lose 10 times 1 cent, or 10 cents, and he must take enough of that at 13 cents to lose 36 cents. Since on 1 lb. he loses 4 cents, he must take as many pounds to lose 36 cents as there are times 4 in 36 which are 9 times. Hence, he may take 8 lb. at 6 cents, 11 lb. at 7 cents, 10 lb. at 10 cents, and 9 lb. at 13 cents.

The following is a convenient form of writing the work:

But 38 lb., at 9 cents per lb., would bring just $3.42, which shows that the merchant would get the same sum by selling the mixture at 9 cents per lb. that he would by selling the ingredients separately, at their respective prices.

(c.) There is no limit to the number of answers which may be obtained to such questions as the above; for however many or few pounds of any kind we take, we may take enough of other kinds to counterbalance the gain or loss. In the solution, we may as well consider first the number of pounds to be taken of the kinds which cost less than the mean rate, as of those which cost more.

(d.) Annexed is a part of the written work of two other solutions to the above example. Let the student complete and explain it, and also prove the correctness of his results.

(e.) If it should be necessary to make a mixture containing a given number of pounds, we should first get an answer to the question, as though no limit had been specified, and then find how many times as much should be taken to give the required quantity.

Suppose, for instance, that the above question had read, "How many pounds of each kind must he take to make a mixture of 100 lb. worth 9 cents per lb.," we should have the following additional work:

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The first solution gives a mixture containing 38 lb.; and since 100 lb. = 213 times 38 lb., we must take 21 times as much of each of the former ingredients as before, which would give 21 times 8 lb., or 219 lb. of the first; 21 times 11 lb., or 2818 lb. of the second; 2 times 10 lb., or 26 lb. of the third; and 213 times 9 lb., or 231 lb. of the fourth. The proof is the same as though these quantities had been originally selected.

5. A trader has coffees at 8, 10, 13, and 15 cents per lb. How many pounds of each may he take to make a mixture worth 12 cents per lb.?

6. A trader has molasses at 22, 25, 29, and 33 cents per gallon.. How many gallons of each kind may he take to make a mixture worth 26 cents per gallon?

7. A trader has oils at $.95, $1.20, $1.42, and $1.60 per gallon, of which he wishes to make a mixture worth $1.25 per gallon. How many gallons of each kind may he take?

8. A trader wishes to mix 50 lb. of sugar at 7 cents per lb., and 30 lb. at 10 cents, with such quantities at 9 and 6 cents per lb. as will make a mixture worth 8 cents per lb. How many pounds of each may he take?

9. A trader wishes to mix 40 lb. of tea at 40 cents per lb., 30 lb. at 24 cents, and 50 lb. at 45 cents, with enough at 30 cents to make a mixture worth 35 cents per lb. How many pounds of the last must he take?

10. I have salt at 33, 37, and 50 cents per bushel. How many bushels of each kind may I take to make a mixture of 100 bushels worth 40 cents per bushel?

11. A farmer has oats worth 42 cents, barley worth 64 cents, rye worth 87 cents, and wheat worth $1.38 per bushel. How many bushels of each kind may he take to make a mixture of 200 bushels worth 75 cents per bushel?

SECTION XV.

INTEREST.

166. Introductory.

WHEN a person hires an article of property of another, it is evident that, at the expiration of the time for which he hires it, he ought to return it, and pay for its use. Moreover, the sum paid for the use of the borrowed article should be proportioned both to its value and the length of time it is kept.

For instance, if I hire two houses, one of which is worth twice as mucn as the other, I ought to pay twice as much per year for the first as for the second. If the values of the houses are alike, and one is kept one half as long as the other, only one half as much ought to be paid for the first as for the second.

To the Teacher. It will be well to illustrate the above important principles by questions similar in character to the following:

If one man hires a horse to go a certain distance, and another hiles one to go twice as far, how many times as much ought the second to pay for its use as the first pays ? What would have been the answer to the above question, provided the second man had gone 3 times as far as the first? 4 times as far? 3 times as far? as far as far? as far? &c. If the horses are hired by the hour, and the first man keeps his horse three times as many hours as the second keeps his, how many times as much ought he to pay for the use of it? What would have been the answer had he kept it 5 times as long as the second? times as long? 6 times as long as long as long as long? &c.

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Similar questions should be asked with reference to other objects hired, as houses, money, &c., till the principle is fully understood.

167. Definitions.

(a.) Money is very frequently hired, and the sum to be paid for its use is determined in accordance with the above principles. (See 177th page, Ex. 24, Note.)

(b.) Money thus paid for the use of money is called IN

TEREST.

(c.) The money used is called the PRINCIPAL.

(d.) The principal and interest added together form the AMOUNT, or entire sum due at any given time.

(e.) The interest of any principal is usually reckoned at a certain per cent, i. e., a certain number of one hundredths of that principal, for each year it is on interest. This per cent is called the RATE PER CENT, or simply the RATE.

NOTE.

The term per cent, from the Latin per centum, originally meant by the hundred; but as it is now used in arithmetic, it means one hundredths. Thus 6 per cent means ro, or .06; 4 per cent means