Complex Fractions

What’s a Complex Fraction?

A fraction that has fractions in its numerator, denominator, or both.

$\cfrac{\cfrac{1}{x+1}}{\cfrac{2}{x-1}}$

How to Simplify

Rewrite as division, then flip and multiply.

Example: Simplify the complex fraction above

Step 1: Rewrite as division

$\frac{1}{x+1} \div \frac{2}{x-1}$

Step 2: Flip and multiply

$\frac{1}{x+1} \times \frac{x-1}{2}$

Step 3: Multiply across

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

Another Example

$\cfrac{\cfrac{x}{2}}{\cfrac{x^2}{4}}$

Rewrite as division:

$\frac{x}{2} \div \frac{x^2}{4} = \frac{x}{2} \times \frac{4}{x^2} = \frac{4x}{2x^2} = \frac{2}{x}$

Alternative Method: Multiply by LCD

Instead of rewriting as division, multiply numerator and denominator by the LCD.

Example: Simplify $\cfrac{\frac{1}{x} + 1}{2}$

The LCD of all fractions is $x$. Multiply top and bottom by $x$:

$\frac{\left(\dfrac{1}{x} + 1\right) \cdot x}{2 \cdot x} = \frac{1 + x}{2x}$