A complex fraction is a fraction whose numerator, denominator, or both contain fractions.
A nested fraction has fractions inside fractions, sometimes across several layers.
1. What is a complex fraction?
A simple example is:
\[
\begin{aligned}
\frac{\frac{3}{4}}{\frac{2}{5}}.
\end{aligned}
\]
This means:
\[
\begin{aligned}
\frac{3}{4}\div\frac{2}{5}.
\end{aligned}
\]
2. Main rule: divide by multiplying by the reciprocal
Dividing by a fraction is the same as multiplying by its reciprocal:
\[
\begin{aligned}
\frac{a}{b}\div\frac{c}{d}
&=
\frac{a}{b}\times\frac{d}{c},
\qquad c\ne0.
\end{aligned}
\]
The divisor cannot be zero.
3. Simplifying a complex fraction
For example:
\[
\begin{aligned}
\frac{\frac{3}{4}}{\frac{2}{5}}
&=
\frac{3}{4}\div\frac{2}{5}\\
&=
\frac{3}{4}\times\frac{5}{2}\\
&=
\frac{15}{8}.
\end{aligned}
\]
In mixed-number form:
\[
\begin{aligned}
\frac{15}{8}
&=
1\frac{7}{8}.
\end{aligned}
\]
4. Nested expression example
Consider:
\[
\begin{aligned}
\frac{\left(\frac{3}{4}\div\frac{2}{5}\right)}
{\left(\frac{1}{2}+\frac{3}{7}\right)}.
\end{aligned}
\]
First simplify the numerator:
\[
\begin{aligned}
\frac{3}{4}\div\frac{2}{5}
&=
\frac{3}{4}\times\frac{5}{2}\\
&=
\frac{15}{8}.
\end{aligned}
\]
Then simplify the denominator:
\[
\begin{aligned}
\frac{1}{2}+\frac{3}{7}
&=
\frac{7}{14}+\frac{6}{14}\\
&=
\frac{13}{14}.
\end{aligned}
\]
Now divide:
\[
\begin{aligned}
\frac{15}{8}\div\frac{13}{14}
&=
\frac{15}{8}\times\frac{14}{13}\\
&=
\frac{105}{52}.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{
\frac{\left(\frac{3}{4}\div\frac{2}{5}\right)}
{\left(\frac{1}{2}+\frac{3}{7}\right)}
=
\frac{105}{52}
=
2\frac{1}{52}
}.
\end{aligned}
\]
5. Parentheses matter
Parentheses tell you which fraction layer should be simplified first.
For example:
\[
\begin{aligned}
\left(\frac{1}{2}+\frac{1}{3}\right)\div\frac{5}{6}
\end{aligned}
\]
first combines \(1/2\) and \(1/3\), then divides by \(5/6\).
6. Common denominator inside a layer
When a layer contains addition or subtraction, use a common denominator:
\[
\begin{aligned}
\frac{a}{b}+\frac{c}{d}
&=
\frac{ad+bc}{bd}.
\end{aligned}
\]
A least common denominator is often cleaner because it keeps the numbers smaller.
7. Multiplication and cancellation
When a layer contains multiplication, multiply numerators and denominators:
\[
\begin{aligned}
\frac{a}{b}\times\frac{c}{d}
&=
\frac{ac}{bd}.
\end{aligned}
\]
Before multiplying, you may cross-cancel common factors between a numerator and the opposite denominator.
8. Multi-level nested fractions
A multi-level expression may look like:
\[
\begin{aligned}
\frac{\left(\frac{\frac{2}{3}}{\frac{4}{5}}\right)}
{\frac{7}{9}}.
\end{aligned}
\]
Work from the innermost layer outward:
\[
\begin{aligned}
\frac{2}{3}\div\frac{4}{5}
&=
\frac{2}{3}\times\frac{5}{4}\\
&=
\frac{5}{6}.
\end{aligned}
\]
Then:
\[
\begin{aligned}
\frac{5}{6}\div\frac{7}{9}
&=
\frac{5}{6}\times\frac{9}{7}\\
&=
\frac{15}{14}.
\end{aligned}
\]
9. Restrictions
A denominator cannot be zero. This means no layer can require division by zero:
\[
\begin{aligned}
\frac{a}{0}
\quad\text{is undefined.}
\end{aligned}
\]
Also:
\[
\begin{aligned}
a\div0
\quad\text{is undefined.}
\end{aligned}
\]
10. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
11. Common mistakes
- Forgetting to solve the numerator and denominator layers separately.
- Dividing fractions without using the reciprocal of the second fraction.
- Flipping both fractions instead of only the divisor.
- Ignoring parentheses in nested expressions.
- Adding fractions without first using a common denominator.
- Canceling across addition or subtraction instead of multiplication.
- Allowing a denominator or divisor to become zero.
Key idea: simplify the innermost fraction layers first, then rewrite every division as multiplication by the reciprocal.
