# Reduction of Fractions

142. From the principles which have been stated, are de rived the rules for the *reduction* of fractions, which are sub. stantially the same in algebra, as in arithmetic.

**A FRACTION MAT BE REDUCED TO LOWER TERMS, BY DIVI DING BOTH THE NUMERATOR AND DENOMINATOR, BY ANY QUAN TITY WHICH WILL DIVIDE THEM WITHOUT A REMAINDER.**

According to Art. 137, this will not alter the value of the fraction.

Thus $\frac{ab}{cb} = \frac{a}{c}$. And $\frac{6dm}{8dy} = \frac{3m}{4y}$. And $\frac{7m}{7mr} = \frac{1}{r}$.

In the last example, both parts of the fraction are divided by the numerator.

If a letter is in every term, both of the numerator and de nominator, it may be cancelled, for this is dividing by that letter. (Art. 117.)

Thus, $\frac{3am + ay}{ad + ah} = \frac{3m + y}{d + h}$.

If the numerator and denominator be divided by the *greatest common measure*, it is evident that the fraction wdl be reduced to the lowest terms. For the method of finding the greatest common measure.

143. **Fractions of different denominators may be reduced to reduce to a common denominator, by multypliing each numerator into all the denominators except its own, fo a new numetrator; and all the denominators together, for a common denominator.**

Ex. 1. Reduce $\frac{a}{b}$ and $\frac{c}{d}$, and $\frac{m}{y}$ to a common denominator.

$a\cdot d\cdot y = ady$

$c\cdot b\cdot y = cby$

$m\cdot b\cdot d = mbd$

The three numerators.

$b\cdot d\cdot y = bdy$

the common denominator.

The fractions reduced are $\frac{ady}{bdy}, \frac{bcy}{bdy}, \frac{bdm}{bdy}$

Here it will be seen, that the reduction consists in multiplying the numerator and denominator of each fraction, into all the other denominators. This does not alter the value. (Art. 137.)

2. Reduce $\frac{dr}{3m}$, and $\frac{2h}{g}$, and $\frac{6c}{y}$.

3. Reduce $\frac{1}{a + b}$, and $\frac{1}{a - b}$.

After the fractions have been reduced to a common de* nominator, they may be brought to lower terms, by the rule in the last article, if there is any quantity which will divide the denominator, and *all* the numerators without a remainder.

An *integer* and a fraction, are easily reduced to a common denojptunator. (Art. 138.)

Thus $a$ and $\frac{b}{c}$ are equal to $\frac{a}{1}$ and $\frac{b}{c}$, or $\frac{ac}{c}$ and $\frac{b}{c}$.

And $a, b, \frac{h}{m}, \frac{d}{y}$ are equal to $\frac{amy}{my}, \frac{bmy}{my}, \frac{hy}{my}, \frac{dm}{my}$.

147. ** TO REDUCE AN IMPROPER FRACTION TO A MIXED QUANTITY, DIVIDE THE NUMERATOR BT THE DENOMINATOR, as in Art. 124.**

Thus $\frac{ab + bm + d}{b} = a + m + \frac{d}{b}$.

Reduce $\frac{am - a + ady - hr}{a}$, to a mixed quantity.

For the reduction of a *mixed quantity* to an improper fraction, see Art. 147. And for the reduction of a *compound fraction* to a simple one, see Art. 157.