# How do you simplify \frac{y}{y-\frac{y}{y+\frac{1}{y}}}?

Nov 9, 2017

$\text{The Exp.=} \frac{{y}^{2} + 1}{{y}^{2} - y + 1} .$

#### Explanation:

For ease of writing, let, $x = \frac{y}{y + \frac{1}{y}} ,$ so that, the given

Expression (Exp.) becomes,

Exp.$= \frac{y}{y - x} \ldots \ldots \ldots \left(\star\right) .$

Now, $x = \frac{y}{y + \frac{1}{y}} = \frac{y}{\frac{{y}^{2} + 1}{y}} = {y}^{2} / \left({y}^{2} + 1\right) .$

$\therefore y - x = y - {y}^{2} / \left({y}^{2} + 1\right) = \frac{y \left({y}^{2} + 1\right) - {y}^{2}}{{y}^{2} + 1} .$

$\Rightarrow y - x = \frac{y \left({y}^{2} + 1 - y\right)}{{y}^{2} + 1} .$

Therefore, the Exp. = $\frac{y}{y - x} = y \div \frac{1}{y - x} ,$

$= y \div \frac{y \left({y}^{2} + 1 - y\right)}{{y}^{2} + 1} ,$

$= \cancel{y} \times \frac{{y}^{2} + 1}{\cancel{y} \left({y}^{2} + 1 - y\right)} .$

$\Rightarrow \text{ The Exp.=} \frac{{y}^{2} + 1}{{y}^{2} - y + 1} .$