What are complex fractions (fractions of fractions)?

What are complex fractions, and how do you handle them?

A complex fraction might look intimidating at first, but don’t worry.

You just need to remember a few things and you will be good to go!

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What do you need to remember?

A reciprocal is a number “flipped upside down.”

The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$

The reciprocal of $\frac{x}{1}$ is $\frac{1}{x}$

A fraction bar can be thought of like a division sign.

$\frac{x}{y}=x\div y$

To divide by a fraction you can multiply by its reciprocal.

$\frac{x}{\frac{a}{b}}=x \div \frac{a}{b}=x \cdot \frac{b}{a}$

Any number or variable can be divided by $1$ and remain the same number.

$x = \frac{x}{1}$

How to rewrite complex fractions (fractions of fractions) by turning the division into multiplication

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Simplifying different complex fractions

Example

Simplify the expression.

$\frac{\frac{2}{3}}{\frac{3}{4}}$

Instead of dividing by the fraction in the denominator, we can multiply by its reciprocal. 

$\frac{\frac{2}{3}}{\frac{3}{4}}$

$\frac{2}{3} \div \frac{3}{4}$

Now that we have a fraction divided by another fraction, instead of dividing by the fraction in the denominator, we can multiply by its reciprocal.

$\frac{2}{3} \cdot \frac{4}{3}$

For fraction multiplication, multiply the numerators together, then multiply the denominators together.

$\frac{2 \cdot 4}{3 \cdot 3}$

$\frac{8}{9}$

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You can do the same thing with variables.

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To divide by a fraction you can multiply by its reciprocal.

Example

Simplify the expression.

$\frac{x}{\frac{a}{b}}$

We have to rewrite the given fraction.

$\frac{x}{\frac{a}{b}}$

$\frac{\frac{x}{1}}{\frac{a}{b}}$

$\frac{x}{1} \div \frac{a}{b}$

Now that we have a fraction divided by another fraction, instead of dividing by the fraction in the denominator, we can multiply by its reciprocal.

$\frac{x}{1} \cdot \frac{b}{a}$

For fraction multiplication, multiply the numerators together, then multiply the denominators together.

$\frac{xb}{1a}$

$\frac{xb}{a}$

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