6. A man has in his possession 200 dollars, and

owes debts to the amount of 500 dollars.

much is he worth?

How

Here, the money the man has is a positive quantity, and the amount of his debts, which is to be subtracted, is a negative quantity; therefore, the expression 200 500 will represent the state of his property. Now, if he pay off his debts, as far as his 200 dollars will go, there will still be $ - 300 left; that is, he will be 300 dollars worse than nothing.

In the last question, the amount of the debts might be regarded as the positive quantity; and then the - opposite quantity, the money on hand, would be negative, and 500 - 200 would represent the amount of debts which the man could not pay.

It is evident, therefore, that positive and negative are merely relative terms, which are, in general, opposed to each other. In any calculation, whatever quantity is assumed as positive, all other quantities of a similar nature, or which tend to increase it, are also positive; and whatever quantities are opposed to it, in any way, or which serve to diminish it, are negative.

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CHAPTER II.

ADDITION.

SECTION I.

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 sum of the coefficients, 1 + 3 + 4, is also 8. Then, 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 stu

dent 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 120; and 5 xa will be

will be 5 × 4 × 6, or 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 abc, and 4 abc. 8. What is the sum of 5 arx, and 3arx, and arx; and 7arx, and 17 arx?

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 abc, and 5 abc, and 7 abc, and 12 abc?

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 - x, and - 3 x, and 5 х.

ANS.-9 x.

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

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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, -3 x = 300, 5 x = -500, and the answer - 9x-900 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 - 100, -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 - 6 х у,

x y, and 16 х у. 13. What is the amount of - 7авс, — 12 abc,

17. What is the sum of - 5 a b x, and - 3 abx,

and

18. What is the sum of - 12 abxy,
17 abxy, - a b xy, and - 2abxy?

19. Add together - 3 a, and

7 a, and

xy, and

2 х.

Tabxy,

3 x y, and

ANS. x.

21. Add together 3 x, and 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

and the sum due is evidently a dollars.

2x;

Suppose the value of x to be 10 dollars; then 3 x = 30, and

2 x = -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 + 2x, leaving + x due.

ANS. -x.

22. Add together 2 x, and 3 х. 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 2x, 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