Simplify (1-y/x)/(1/y+1/x)

$\frac{1 - \frac{y}{x}}{\frac{1}{y} + \frac{1}{x}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $x y$ .

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

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

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

Factor $x$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Raise $y$ to the power of $1$ .

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

Raise $y$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Cancel the common factor of $y$ .

Factor $y$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

Factor $x$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

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

Factor $y$ out of $x y \cdot 1$ .

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

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

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

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

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

Multiply $x$ by $1$ .

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

Simplify (1-y/x)/(1/y+1/x)

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