Krista King Math | Online math help

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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!

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}???


You can do the same thing with variables.


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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