CHAPTER II

ADDITION.

SECTION 1.

Simple Quantities that are Similar.

1. A man gave to one poor person a dollars, to a second 3 a dollars, and to a third 4 a dollars. How much did he give them all? ANS. 8 a dollars.

In this question, we have three simple quantities: they are all similar, and they are all positive; for it must be remembered that, when no sign is used, the sign is always understood. The first quantity is a ; the second, 3 a, which may be written a + a + a ; and the third, 4 a, or a + a + a + a. Now, by counting, we find there are eight a's or 8 a.

But the

Then,

sum of the coefficients, 1 + 3+ 4, is also 8. to perform questions of this kind, Add together all the coefficients, and place the sum before the common letters.. Suppose the value of a, in this question, to be 5; then we shall have

By assigning any other value to the letter a, the student will obtain a similar result.

6. Add together a x, and 3 a x, and 5 x a.

It is of no consequence in what order the letters are given; for 5 x a is evidently of the same value as 5 a x. Let x 6, and a = 4, for instance; then 5 a x will be 5 X 4 × 6, or 120; and 5 x a will be 5 × 6 × 4, which is also equal to 120. It is usual, however, to arrange the letters according to their order in the alphabet.

7. Add together 3 a b c, and 2 a b c, and 4 a b c. 8. What is the sum of 5 ar x, and 3 arx, and ar x; and 7 a r x, and 17 a rx?

9. A merchant is indebted 4 a dollars to A, 5 a dollars to B, 6 a dollars to C, and 8 a dollars to D. How much does he owe them all?

10. What is the amount of a b c, and 5 a b c, and Τα b C, and 12 a b c ?

The student will find it a useful exercise to prove his answers, by assigning definite values to the letters given in the questions, in the manner exhibited above. 11. Add together - 5 x.

x, and

3 x, and

ANS.

-9 x.

This example differs from the foregoing in only one particular; that is, the quantities are all negative :

the sign

must, therefore, be prefixed to the sum; for the whole must evidently be of the same character as the parts of which it is composed.

Let us suppose that x, in this example, stands for 100 dollars; then x= 100, 3x=- 300,

--

900

5x=- 500, and the answer 9x= dollars. But it may be asked, How can 100, or 300, or 500, or 900 dollars be subtracted from nothing? Such a subtraction may be represented, although it cannot be performed.

A merchant, for instance, wishes to ascertain the profits of his business. His gains are positive quantities; and his losses, because they must be subtracted from his gains before his clear profits can be known, must be negative. Now, it is evident, if he has lost 100 dollars by one speculation, 300 dollars by another, and 500 dollars by bad debts, that these sums should be written 300, and 500; and that they may be added together, as if they were positive quantities, their amount being written —- 900, to show that it is to be subtracted from some other quantity; that is, from the amount or sum total of

his profits.

12. Add together

100,

13. What is the amount of -abc,

C,

12 a b c,

Tabcd

12 a b c d

16 a b c d

3 x y z

4 x y z

13 x y z

18. What is the sum of-12 a b xy, - 7 ab xy,

This example contains both a positive and a negative quantity. If the Dr. column of a leger amount to 3 x dollars, and the Cr. column to 2 x dollars, the condition of the account may be expressed 3 x - 2x; and the sum due is evidently x dollars.

Suppose the value of x to be 10 dollars; then 3 x = 30, and 2x = 20, and the account will be Dr. 30 dollars, Cr. 20 dollars, due 10 dollars. The 20 dollars credited will cancel the same amount of the debt; that is, 2 x will cancel + 2 x, leaving +x due.

ANS.

X.

22. Add together 2 x, and 3 x. This example is like the last, only the negative is the larger quantity, and the answer must have the sign prefixed. For it is evident, if the Dr. side of an account be 2 x, or 20 dollars, and the Cr. side be 3 x. or 30 dollars, the debt has all been paid, and x, or 10 dollars more, which is to be paid back.

-

From these examples we derive the following RULE for adding together two similar quantities, when their signs are not alike: Subtract the less coefficient from

the greater. The answer must have the same sign as the greater quantity.

Hence it appears, that what is called Addition in Algebra, is sometimes performed by Subtraction. This is an apparent contradiction, which is very apt to perplex beginners. The whole difficulty will vanish, however, if they remember the object of Addition; which is, to express the value of two or more quantities, of any kind, in the most simple manner possible. 23. What is the sum of 5 x y, and

24.) + 4 ab

3 x y? ANS. 2x y.

Let us suppose the positive quantities in this question, namely, 5 x and 4 x, to represent sums of money received, and the negative quantities, 3 x and -2x, payments made. Then, 9 x dollars are received, and 5 x dollars paid away; and, of course, 4 x dollars. remain on hand.

If the value of x be 100 dollars, then 900 dollars are received, 500 dollars paid away, and a balance of 400 dollars remains.

30. Add together 4 x, and - 8 x, and and 6 x.

ANS.

7x,

.5 x.

This example is like the last; but here, the sum of the negative quantities is greater than that of the posi