Hence to find the share of either, multiply the whole sum to be divided, by the proportion of the stock which he furnished, and divide the product by the sum of their proportions.

The propriety of this rule is easily seen. numbers instead of the letters, A's share is

$800, B's share is

3

5
8+5+3

For, putting in the

8

8+5+3

or of

or of it, and C's share is

or of it. That is, the sum of all their propor

8+5+3
tions is 16, and of these A furnished 8; B, 5; and C, 3.

13. Let it be required to find what sum, put at interest at a given rate, will amount to a given sum in a given time; that is, to find a rule, by which the principal may be found, when the rate, time, and amount are given.

First take a particular case.

A man lent some money for 3 years, interest at 6 per cent, and received for interest and principal $472. What was the sum lent?

It is a custom established among mathematicians to use the first letters of the alphabet for known quantities, and some of the last letters for unknown quantities. It is, however, frequently convenient to choose letters, that are the initials of the words for which they stand, whether the quantities be known or unknown.

To generalize the above example,

Let

p

the principal, or sum lent.

r

the rate per annum, which in the above case

t = the time for which it was lent,

That is, multiply the rate by the time, add 1 to the product, and divide the amount by this, and it will give the principal.

In the above example the rate is .06, which, multiplied by 3 (the time), gives .18, and one added to this makes 1.18; 472 divided by 1.18 gives 400, as before.

Apply this rule to the following example.

A man owes $275, due two years and three months hence, without interest." What ought he to pay now, supposing money to be worth 1 per cent. per annum ?

N. B. 2 years and 3 months is 21 years.

See Arithmetic, page 84.

78875

Ans. $2497125

The learner may now make rules for the following purposes: 14. The interest, time, and rate being given, to find the principal.

15. The amount, time, and principal being given, to find the rate.

16. The amount, principal, and rate given, to find the time

17. A man agreed to carry 20 (or a) earthen vessels to a certain place, on this condition; that for every one delivered safe he should receive S (or b) cents, and for every one he broke, he should forfeit 12 (or c) cents; he received 100 (crd). cents. How many did he break?

Let x

Then 20

the number unbroken.

xor a ― x = the number broken.

For every one unbroken he was to receive 8 or b cents, these will amount to 8 x or b x; and for every one broken he was to pay back 12 or c cents, these will amount to 240 -12 x cents, or a c -cx; this must be subtracted from the former. 12 x, subtracted from 8 x, is

240

Also ca

Sx

- 240 + 12 x, or 20 x cx subtracted from b is b X,

240.

ca+cx; for the quantity ca―c x is not so large as ca, by the quantity ex, therefore when we subtract ca from bx, we subtract too much by c x, and in order to obtain a correct result, it is necessary to add cx.

Particular Ans. 17 unbroken, and 3 broken.
General Ans. Unbroken d+ac
b+c

Putting numbers into the general answer,

The propriety of this answer may be shown as follows: If he had broken the whole 20 (or a) he must have paid 12 × 20 =240 (or a c) cents; but instead of paying this, he received 100 (or d) cents. Now the difference to him between paying 240 and receiving 100 is evidently 340, (or da c) cents. The difference for each vessel between paying 12 and receiving 8 is 20 (or b+c) cents; 340 divided by 20 gives 17, the an

swer.

The

The above is a good illustration of positive and negative quantities, or quantities affected with the signs and — signis placed before the quantities, which he is to receive, and the sign before his losses. We observed that the difference between receiving 100 and losing 240 is 340, that is, the difference between + 100 and 240 is 340, or their sum. Also the difference between + d and

difference between 8 and 12 is 20, or between + b and

cis b+c.

Hence it follows, that to subtract a quantity which has the sign -, we must give it the opposite sign, that is, it must be added.

X. The learner, by this time, must have some idea of the use of letters, or general symbols, in algebraic reasoning. It has been already observed that, strictly speaking, we cannot actually perform the four fundamental operations on these quantities, as we do in arithmetic; yet in expressing these operations, it is frequently necessary to perform operations so analogous to them, that they may with propriety be called by the same names. Most of these have already been explained; but in order to impress them more firmly on the mind of the learner, they will be briefly recapitulated, and some others explained which could not be introduced before.

Note. Algebraic quantities, which consist of only one term, are called simple quantities, as + 2 a,—3 ab, &c.; quantities which consist of two terms are called binomials, as a+b, a — b, 3b+2c, &c.; those which consist of three terms are called trinomials; and in general those which consist of many terms are called polynomials.

Simple Quantities.

The addition of simple quantities is performed by writing them after each other with the sign + between them. To express that a is added to b, we write a + b. To express that α, b, c, d, and e are added together, we write a+b+c+d+e. It is evidently unimportant which term is written first, for 358 is the same as 5 + 3 + 8, or as 8 + 5 + 3. So a+b+c has the same value as b + a + c.

It has been remarked (Art. I.) that x + x + x may be written 3 x. This is multiplication; and it arises, as was observed in Arithmetic, Art. III., from the successive addition of the same quantity. 3x, it appears, signifies 3 times the quantity x, that is, a multiplied by 3. Sob+b+b+b+b may be written 5 b. In the same manner, if x is to be repeated, any number of times, for instance as many times as there are units in a, we write a x, which signifies a times x, or x multiplied by a.

N. B. The learner should constantly bear in mind that the letters, a, b, c, &c. may be used to represent any known number; or they may be used indefinitely, and any number may afterwards be substituted in their place.

Again, ab+ab+ab may be written 3 a b, that is, 3 times the product a b; also c times the product a b may be written cab. It may be remarked that a times b is the same as b times a; for a times 1 is a, and a times 6 must be b times as much, that is, b times a. Hence the product of a and b may be written either ab or ba. In the same manner it may be shown that the product cab is the same as ab c. Suppose a = 3, b = 5, and c = 2, then a b c = 3 × 5 × 2, and c a b = 2 × 3 × 5. In fact it has been shown, in Arith. Art. IV., that when a product is to consist of several factors, it is not important in what order those factors are multiplied together. The product of a, b, c, d, e, and f, is written abcdef. They may be written in any other order, as a cd bef, or fbedca, but it is generally more convenient to write them in the order they stand in the alphabet.

Let it be required to multiply 3ab by 2 cd. The product is 6abcd; for d times 3 a b is 3 ab d, but c d times 3 a b is c times as much, or 3 a b c d, and 2 c d times 3 a b must be twice as much as the latter, that is, 6 a b c d.

Hence, the product of any two or more simple quantities must consist of all the letters of each quantity, and the product of the coefficients of the quantities.

N. B. Though the product of literal quantities is expressed by writing them together without the sign of multiplication, the same cannot be done with figures, because their value depends upon the place in which they stand. 3ab multiplied by 2 cd, for instance, cannot be written 32 a b c d. If it is required to express the multiplication of the figures as well as of the letters, they must be written 3 a b 2 dc, or 3 × 2 abcd, or 3.2 a b cd. That is, the figures must either be separated by the letters or by the sign of multiplication.