Evaluate (1+1/(1-1/2))(1-1/(1+1/2))

$(1 + \frac{1}{1 - \frac{1}{2}}) (1 - \frac{1}{1 + \frac{1}{2}})$

Step 1

Simplify each term.

Simplify the denominator.

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

$(1 + \frac{1}{\frac{2}{2} - \frac{1}{2}}) (1 - \frac{1}{1 + \frac{1}{2}})$

Combine the numerators over the common denominator.

$(1 + \frac{1}{\frac{2 - 1}{2}}) (1 - \frac{1}{1 + \frac{1}{2}})$

Subtract $1$ from $2$ .

$(1 + \frac{1}{\frac{1}{2}}) (1 - \frac{1}{1 + \frac{1}{2}})$

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $2$ by $1$ .

$(1 + 2) (1 - \frac{1}{1 + \frac{1}{2}})$

Step 2

Add $1$ and $2$ .

$3 (1 - \frac{1}{1 + \frac{1}{2}})$

Step 3

Simplify each term.

Simplify the denominator.

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

$3 (1 - \frac{1}{\frac{2}{2} + \frac{1}{2}})$

Combine the numerators over the common denominator.

$3 (1 - \frac{1}{\frac{2 + 1}{2}})$

Add $2$ and $1$ .

$3 (1 - \frac{1}{\frac{3}{2}})$

Multiply the numerator by the reciprocal of the denominator.

$3 (1 - (1 (\frac{2}{3})))$

Multiply $\frac{2}{3}$ by $1$ .

$3 (1 - \frac{2}{3})$

Step 4

Reduce the expression by cancelling the common factors.

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

$3 (\frac{3}{3} - \frac{2}{3})$

Simplify the expression.

Combine the numerators over the common denominator.

$3 \frac{3 - 2}{3}$

Subtract $2$ from $3$ .

$3 (\frac{1}{3})$

Cancel the common factor of $3$ .

Cancel the common factor.

$3 (\frac{1}{3})$

Rewrite the expression.

$1$