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}???
You can do the same thing with variables.
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}???
