Simplify (1-x/y)/(1-(x^2)/(y^2))

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

Step 1

Multiply the numerator and denominator of the complex fraction by $y^{2}$ .

Multiply $\frac{1 - \frac{x}{y}}{1 - \frac{x^{2}}{y^{2}}}$ by $\frac{y^{2}}{y^{2}}$ .

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

Combine.

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

Step 2

Apply the distributive property.

$\frac{y^{2} \cdot 1 + y^{2} (- \frac{x}{y})}{y^{2} \cdot 1 + y^{2} (- \frac{x^{2}}{y^{2}})}$

Step 3

Simplify by cancelling.

Cancel the common factor of $y$ .

Move the leading negative in $- \frac{x}{y}$ into the numerator.

$\frac{y^{2} \cdot 1 + y^{2} \frac{- x}{y}}{y^{2} \cdot 1 + y^{2} (- \frac{x^{2}}{y^{2}})}$

Factor $y$ out of $y^{2}$ .

$\frac{y^{2} \cdot 1 + y \cdot y \frac{- x}{y}}{y^{2} \cdot 1 + y^{2} (- \frac{x^{2}}{y^{2}})}$

Cancel the common factor.

$\frac{y^{2} \cdot 1 + y \cdot y \frac{- x}{y}}{y^{2} \cdot 1 + y^{2} (- \frac{x^{2}}{y^{2}})}$

Rewrite the expression.

$\frac{y^{2} \cdot 1 + y (- x)}{y^{2} \cdot 1 + y^{2} (- \frac{x^{2}}{y^{2}})}$

Cancel the common factor of $y^{2}$ .

Move the leading negative in $- \frac{x^{2}}{y^{2}}$ into the numerator.

$\frac{y^{2} \cdot 1 + y (- x)}{y^{2} \cdot 1 + y^{2} \frac{- x^{2}}{y^{2}}}$

Cancel the common factor.

$\frac{y^{2} \cdot 1 + y (- x)}{y^{2} \cdot 1 + y^{2} \frac{- x^{2}}{y^{2}}}$

Rewrite the expression.

$\frac{y^{2} \cdot 1 + y (- x)}{y^{2} \cdot 1 - x^{2}}$

Step 4

Simplify the numerator.

Factor $y$ out of $y^{2} \cdot 1 + y (- x)$ .

Factor $y$ out of $y^{2} \cdot 1$ .

$\frac{y (y \cdot 1) + y (- x)}{y^{2} \cdot 1 - x^{2}}$

Factor $y$ out of $y (y \cdot 1) + y (- x)$ .

$\frac{y (y \cdot 1 - x)}{y^{2} \cdot 1 - x^{2}}$

Multiply $y$ by $1$ .

$\frac{y (y - x)}{y^{2} \cdot 1 - x^{2}}$

Step 5

Simplify the denominator.

Rewrite $y^{2} \cdot 1$ as ${(y \cdot 1)}^{2}$ .

$\frac{y (y - x)}{{(y \cdot 1)}^{2} - x^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y \cdot 1$ and $b = x$ .

$\frac{y (y - x)}{(y \cdot 1 + x) (y \cdot 1 - x)}$

Simplify.

Multiply $y$ by $1$ .

$\frac{y (y - x)}{(y + x) (y \cdot 1 - x)}$

Multiply $y$ by $1$ .

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

Step 6

Cancel the common factor of $y - x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (1-x/y)/(1-(x^2)/(y^2))

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