# Thread: Simplify the complex fraction

1. ## Simplify the complex fraction

I'm trying to simplify this complex fraction:

$\frac{1-\frac{x}{y}}{1-\frac{x^2}{y^2}}$

but I don't know what to do with the ones in front of the fractions. This is as far as I have gotten. The common denominator is $y^2$ so I need to multiply them by $y^2$, but how?

$\frac{(1-\frac{x}{y})\frac{y^2}{1}}{(1-\frac{x^2}{y^2})\frac{y^2}{1}}$
These are mixed numbers right? To multiply a mixed number by a fraction, you need to change the mixed number to an improper fraction, but how do I do it when I don't know how much x and y are worth?

2. Originally Posted by bryang
I'm trying to simplify this complex fraction:

$\frac{1-\frac{x}{y}}{1-\frac{x^2}{y^2}}$

but I don't know what to do with the ones in front of the fractions. This is as far as I have gotten. The common denominator is $y^2$ so I need to multiply them by $y^2$, but how?

$\frac{(1-\frac{x}{y})\frac{y^2}{1}}{(1-\frac{x^2}{y^2})\frac{y^2}{1}}$
These are mixed numbers right? To multiply a mixed number by a fraction, you need to change the mixed number to an improper fraction, but how do I do it when I don't know how much x and y are worth?
It'll help you if you turn the mixed number numerators and denominators into improper fractions.

Note that $\frac{y}{y} = 1$ and $\frac{y^2}{y^2} = 1$.

So $\frac{1-\frac{x}{y}}{1-\frac{x^2}{y^2}} = \frac{\frac{y}{y} - \frac{x}{y}}{\frac{y^2}{y^2} - \frac{x^2}{y^2}}$

$= \frac{\frac{y - x}{y}}{\frac{y^2 - x^2}{y^2}}$

$= \frac{y - x}{y} \div \frac{y^2 - x^2}{y^2}$

$= \frac{y - x}{y} \times \frac{y^2}{y^2 - x^2}$

$= \frac{y^2(y - x)}{y(y^2 - x^2)}$

$= \frac{y(y - x)}{y^2 - x^2}$

$= \frac{y(y - x)}{(y - x)(y + x)}$

$= \frac{y}{y + x}$.