How to Simplify Complex Fractions

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.

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.

1. $$\color{blue}{\frac{\frac{1}{4}}{\frac{6}{7}}}$$

2. $$\color{blue}{\frac{5}{\frac{1}{4}+\frac{3}{4}}}$$

3. $$\color{blue}{\frac{1+\frac{4}{x}}{\frac{2}{7}}}$$

4. $$\color{blue}{\frac{x}{\frac{3}{x+4}}}$$

5. $$\color{blue}{\frac{\frac{x}{x+1}}{\frac{x-1}{x+3}}}$$

6. $$\color{blue}{\frac{\frac{1}{x}-\frac{9}{x}}{20}}$$
1. $$\color{blue}{\frac{7}{24}}$$

2. $$\color{blue}{5}$$

3. $$\color{blue}{\frac{7x+28}{2x}}$$

4. $$\color{blue}{\frac{x^2+4x}{3}}$$

5. $$\color{blue}{\frac{x^2+3x}{x^2-1}}$$

6. $$\color{blue}{-\frac{2}{5x}}$$

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