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

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

Step 1

Simplify the denominator.

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{y}{y + z}$ by $1$ .

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

Write $1$ as a fraction with a common denominator.

$\frac{z}{\frac{y + z}{y + z} - \frac{y}{y + z}}$

Combine the numerators over the common denominator.

$\frac{z}{\frac{y + z - y}{y + z}}$

Rewrite $\frac{y + z - y}{y + z}$ in a factored form.

Subtract $y$ from $y$ .

$\frac{z}{\frac{z + 0}{y + z}}$

Add $z$ and $0$ .

$\frac{z}{\frac{z}{y + z}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$z \frac{y + z}{z}$

Step 4

Cancel the common factor of $z$ .

Cancel the common factor.

$z \frac{y + z}{z}$

Rewrite the expression.

$y + z$