Subtraction of Fractions

148. The methods of performing subtraction in algebra, depend on the principle, that adding a negative quantity is equivalent to subtracting a positive one; and v. r. (Art. 81.) For the subtraction of fractions, then, we have the following simple rule. CHANGE THE FRACTION TO BE SUBTRACTED, FROM POSITIVE TO NEGATIVE, OR THE CONTRARY, AND THEN PROCEED AS IN ADDITION. (Art. 145.) In making the required change, it will be expedient to alter, in some instances, the signs of the numerator, and in others, the sign before the dividing line, (Art. 140,) so as to leave the latter always affirmative.

Ex. 1. From $\frac{a}{b}$ subtract $\frac{h}{m}$.

First change $\frac{h}{m}$, the fraction to be subtracted, to $-\frac{h}{m}$.

Secondly, reduce the two fractions to a common denominator, making, $\frac{am}{bm}$ and $-\frac{bh}{bm}$.

Thirdly, the sum of the numerators $am - bh$, placed over the common denominator, gives the answer, $\frac{am - bh}{bm}$.

2. From $\frac{a + y}{r}$ subtract $\frac{h}{d}$. Ans. $\frac{ad + dy - hr}{dr}$.

3. From $\frac{a}{m}$ subtract $\frac{d - b}{y}$. Ans. $\frac{ay - dm + bm}{my}$.

4. From $\frac{b - d}{m}$ subtract $-\frac{b}{y}$. Ans. $\frac{by - dy + bm}{my}$.

5. From $\frac{3}{a}$ subtract $\frac{4}{b}$.

149. Fractions may also be subtracted, like integers, by setting them down, after their signs are changed, without reducing them to a common denominator.

From h/m subtract $-\frac{h + d}{y}$. Ans. $\frac{h}{m} + \frac{h + d}{y}$.

In the same manner, an integer may be subtracted from a fraction, or a fraction from an integer.

From a subtract $\frac{b}{m}$. Ans. $a-\frac{b}{m}$.

150. Or the integer may be incorporated with the fraction, as in Art. 147.

Ex. 1. From $\frac{h}{y}$ subtract $m$. Ans. $\frac{h}{y} - m = \frac{h - my}{y}$.

2. From $1 + \frac{b - c}{d}$ subtract $\frac{c - b}{d}$. Ans. $\frac{d + 2b-2c}{d}$.

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