A complex fraction is a fraction whose numerator, denominator, or both are fractions and in this article, we will teach you how to simplify this kind of fraction.

## Related Topics

- How to Add and Subtract Rational Expressions
- How to Multiply Rational Expressions
- How to Divide Rational Expressions
- How to Solve Rational Equations
- How to Graph Rational Expressions

## A step-by-step guide to Simplifying Complex Fractions

- Convert mixed numbers to improper fractions.
- Simplify all fractions.
- Write the fraction in the numerator of the main fraction line then write division sign (÷) and the fraction of the denominator.
- Use the normal methods for dividing fractions.
- Simplifly as needed.

### Simplifying Complex Fractions – Example 1:

Simplify: \(\frac{\frac{3}{5}}{\frac{2}{25}- \frac{5}{16}}\)

**Solution**:

First, simplify the denominator: \(\frac{2}{25} – \frac{5}{16}=-\frac{93}{400}\), Then: \(\frac{\frac{3}{5}}{\frac{2}{25} – \frac{5}{16} }=\frac{\frac{3}{5}}{-\frac{93}{400} }\)

Now, write the complex fraction using the division sign \((÷): \frac{\frac{3}{5}}{\frac{93}{400}}=\frac{3}{5}÷(-\frac{93}{400})\)

Use the dividing fractions rule: Keep, Change, Flip (keep the first fraction, change the division sign to multiplication, flip the second fraction)

\(\frac{ 3}{5}÷(-\frac{93}{400})=\frac{ 3}{5}×\frac{ 400}{93}=-\frac{ 240}{93}=-\frac{ 80}{31}=-2 \frac{ 18}{31}\)

### Simplifying Complex Fractions – Example 2:

Simplify: \(\frac{\frac{2}{3}}{\frac{7}{10}- \frac{1}{4}}\)

**Solution**:

First, simplify the denominator: \(\frac{7}{10} – \frac{1}{4}=\frac{9}{20}\), Then: \(\frac{\frac{2}{3}}{\frac{7}{10} – \frac{1}{4} }=\frac{\frac{2}{3}}{\frac{9}{20} }\)

Now, write the complex fraction using the division sign \((÷): \frac{\frac{2}{3}}{\frac{9}{20}}=\frac{2}{3}÷(\frac{9}{20})\)

Use the dividing fractions rule: Keep, Change, Flip (keep the first fraction, change the division sign to multiplication, flip the second fraction)

\(\frac{ 2}{3}÷\frac{9}{20}=\frac{ 2}{3}×\frac{ 20}{9}=\frac{ 40}{27}=1\frac{ 13}{27}\)

### Simplifying Complex Fractions – Example 3:

Simplify: \(\frac{\frac{3}{4}}{\frac{7}{15}- \frac{1}{5}}\)

**Solution**:

First, simplify the denominator: \(\frac{7}{15} – \frac{1}{5}=\frac{4}{15}\), Then: \(\frac{\frac{3}{4}}{\frac{7}{15} – \frac{1}{5} }=\frac{\frac{3}{4}}{\frac{4}{15} }\)

Now, write the complex fraction using the division sign \((÷): \frac{\frac{3}{4}}{\frac{4}{15}}=\frac{3}{4}÷(\frac{4}{15})\)

Use the dividing fractions rule: Keep, Change, Flip (keep the first fraction, change the division sign to multiplication, flip the second fraction)

\(\frac{ 3}{4}÷\frac{4}{15}=\frac{ 3}{4}×\frac{ 15}{4}=\frac{ 45}{16}=2\frac{ 13}{16}\)

### Simplifying Complex Fractions – Example 4:

Simplify: \(\frac{\frac{3}{5}}{\frac{3}{8}- \frac{2}{5}}\)

**Solution**:

First, simplify the denominator: \(\frac{3}{8} – \frac{2}{5}=-\frac{1}{40}\), Then: \(\frac{\frac{3}{5}}{\frac{3}{8} – \frac{2}{5} }=\frac{\frac{3}{5}}{-\frac{1}{40} }\)

Now, write the complex fraction using the division sign \((÷): \frac{\frac{3}{5}}{-\frac{1}{40}}=\frac{3}{5}÷(-\frac{1}{40})\)

Use the dividing fractions rule: Keep, Change, Flip (keep the first fraction, change the division sign to multiplication, flip the second fraction)

\(\frac{ 3}{5}÷(-\frac{1}{40})=\frac{ 3}{5}×(-\frac{ 40}{1})=\frac{ 120}{5}=24\)

## Exercises for Simplifying Complex Fractions

### Simplify Complex Fractions.

- \(\color{blue}{\frac{\frac{1}{4}}{\frac{6}{7}}}\)
- \(\color{blue}{\frac{5}{\frac{1}{4}+\frac{3}{4}}}\)
- \(\color{blue}{\frac{1+\frac{4}{x}}{\frac{2}{7}}}\)
- \(\color{blue}{\frac{x}{\frac{3}{x+4}}}\)
- \(\color{blue}{\frac{\frac{x}{x+1}}{\frac{x-1}{x+3}}}\)
- \(\color{blue}{\frac{\frac{1}{x}-\frac{9}{x}}{20}}\)

- \(\color{blue}{\frac{7}{24}}\)
- \(\color{blue}{5}\)
- \(\color{blue}{\frac{7x+28}{2x}}\)
- \(\color{blue}{\frac{x^2+4x}{3}}\)
- \(\color{blue}{\frac{x^2+3x}{x^2-1}}\)
- \(\color{blue}{-\frac{2}{5x}}\)