Simplify (1/5-1/y)/(1/y-1/5)

$\frac{\frac{1}{5} - \frac{1}{y}}{\frac{1}{y} - \frac{1}{5}}$

Step 1

Multiply the numerator and denominator of the fraction by $5 y$ .

Multiply $\frac{\frac{1}{5} - \frac{1}{y}}{\frac{1}{y} - \frac{1}{5}}$ by $\frac{5 y}{5 y}$ .

$\frac{5 y}{5 y} \cdot \frac{\frac{1}{5} - \frac{1}{y}}{\frac{1}{y} - \frac{1}{5}}$

Combine.

$\frac{5 y (\frac{1}{5} - \frac{1}{y})}{5 y (\frac{1}{y} - \frac{1}{5})}$

Step 2

Apply the distributive property.

$\frac{5 y \frac{1}{5} + 5 y (- \frac{1}{y})}{5 y \frac{1}{y} + 5 y (- \frac{1}{5})}$

Step 3

Simplify by cancelling.

Cancel the common factor of $5$ .

Factor $5$ out of $5 y$ .

$\frac{5 (y) \frac{1}{5} + 5 y (- \frac{1}{y})}{5 y \frac{1}{y} + 5 y (- \frac{1}{5})}$

Cancel the common factor.

$\frac{5 y \frac{1}{5} + 5 y (- \frac{1}{y})}{5 y \frac{1}{y} + 5 y (- \frac{1}{5})}$

Rewrite the expression.

$\frac{y + 5 y (- \frac{1}{y})}{5 y \frac{1}{y} + 5 y (- \frac{1}{5})}$

Cancel the common factor of $y$ .

Move the leading negative in $- \frac{1}{y}$ into the numerator.

$\frac{y + 5 y \frac{- 1}{y}}{5 y \frac{1}{y} + 5 y (- \frac{1}{5})}$

Factor $y$ out of $5 y$ .

$\frac{y + y \cdot 5 \frac{- 1}{y}}{5 y \frac{1}{y} + 5 y (- \frac{1}{5})}$

Cancel the common factor.

$\frac{y + y \cdot 5 \frac{- 1}{y}}{5 y \frac{1}{y} + 5 y (- \frac{1}{5})}$

Rewrite the expression.

$\frac{y + 5 \cdot - 1}{5 y \frac{1}{y} + 5 y (- \frac{1}{5})}$

Multiply $5$ by $- 1$ .

$\frac{y - 5}{5 y \frac{1}{y} + 5 y (- \frac{1}{5})}$

Cancel the common factor of $y$ .

Factor $y$ out of $5 y$ .

$\frac{y - 5}{y \cdot 5 \frac{1}{y} + 5 y (- \frac{1}{5})}$

Cancel the common factor.

$\frac{y - 5}{y \cdot 5 \frac{1}{y} + 5 y (- \frac{1}{5})}$

Rewrite the expression.

$\frac{y - 5}{5 + 5 y (- \frac{1}{5})}$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$\frac{y - 5}{5 + 5 y \frac{- 1}{5}}$

Factor $5$ out of $5 y$ .

$\frac{y - 5}{5 + 5 (y) \frac{- 1}{5}}$

Cancel the common factor.

$\frac{y - 5}{5 + 5 y \frac{- 1}{5}}$

Rewrite the expression.

$\frac{y - 5}{5 + y \cdot - 1}$

Step 4

Cancel the common factor of $y - 5$ and $5 + y \cdot - 1$ .

Factor $- 1$ out of $y$ .

$\frac{- 1 (- y) - 5}{5 + y \cdot - 1}$

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{- 1 (- y) - 1 (5)}{5 + y \cdot - 1}$

Factor $- 1$ out of $- 1 (- y) - 1 (5)$ .

$\frac{- 1 (- y + 5)}{5 + y \cdot - 1}$

Reorder terms.

$\frac{- 1 (- y + 5)}{- y + 5}$

Cancel the common factor.

$\frac{- 1 (- y + 5)}{- y + 5}$

Divide $- 1$ by $1$ .

$- 1$

Simplify (1/5-1/y)/(1/y-1/5)

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