How to Simplify Complex Fractions? (+FREE Worksheet!)

A complex fraction is a fraction that contains one or more fractions in its numerator, its denominator, or both. Simplifying complex fractions is a key algebra skill because it turns a “fraction of fractions” into a single, clean fraction. With two reliable methods, any complex fraction can be simplified quickly and accurately.

What Is a Complex Fraction?

A complex fraction looks like a fraction stacked inside another fraction. Examples:

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• \(\color{blue}{\frac{(\frac{1}{2})}{(\frac{3}{4})}}\) — a simple complex fraction with numeric values
• \(\color{blue}{\frac{(\frac{x}{3})}{(\frac{x^{2}}{9})}}\) — a complex fraction with variable expressions
• \(\color{blue}{\frac{(2 + \frac{1}{x})}{(3 – \frac{1}{x})}}\) — a complex fraction with sums in numerator and denominator

Two Methods for Simplifying Complex Fractions

Method 1 — Divide (multiply by the reciprocal)

Treat the complex fraction as division of the numerator by the denominator. Rewrite as multiplication by the reciprocal of the denominator, then simplify.

Example: \(\color{blue}{\frac{(\frac{1}{2})}{(\frac{3}{4})} = (\frac{1}{2}) \times (\frac{4}{3}) = \frac{4}{6}}\) = \(\color{blue}{\frac{2}{3}}\)

Example with variables: \(\color{blue}{\frac{(\frac{x}{3})}{(\frac{x^{2}}{9})} = (\frac{x}{3}) \times (\frac{9}{x^{2}}) = \frac{9x}{(3x^{2})} = \frac{3}{x}}\)

Method 2 — Multiply by the LCD

Find the LCD of all the small fractions (top and bottom). Multiply every term of the main numerator and denominator by that LCD, which clears all the small fractions at once.

Example: \(\color{blue}{\frac{(2 + \frac{1}{x})}{(3 – \frac{1}{x})}}\). LCD of all fractions is x.
Multiply top and bottom by x:
Numerator: \(\color{blue}{x(2 + \frac{1}{x}) = 2x + 1}\)
Denominator: \(\color{blue}{x(3 – \frac{1}{x}) = 3x – 1}\)
Result: \(\color{blue}{\frac{(2x + 1)}{(3x – 1)}}\)
Check at \(\color{blue}{x = 2}\): \(\color{blue}{\frac{(2 + 0.5)}{(3 – 0.5)} = \frac{2.5}{2.5} = 1}\); \(\color{blue}{\frac{(4+1)}{(6-1)} = \frac{5}{5} = 1}\) ✓

Step-by-Step Summary

1. Identify whether the complex fraction has a single fraction over a single fraction (use Method 1) or sums/differences in the numerator or denominator (use Method 2).
2. Method 1: Keep the top fraction, \(\color{blue}{\text{ change } \div \text{ to }}\) ×, flip the bottom fraction (write its reciprocal). Simplify.
3. Method 2: Find the LCD of all denominators inside the complex fraction. Multiply the top and bottom of the complex fraction by that LCD. Simplify the result.
4. Factor and cancel any common factors in the final fraction.

Watch: Simplifying Complex Fractions in Algebra (Video Lesson)

This lesson walks through the theory and mechanics of simplifying complex fractions with multiple worked examples:

Simplifying Complex Fractions – Worked Examples

Example 1: Simplify \(\color{blue}{\frac{(\frac{1}{2})}{(\frac{3}{4})}}\).

Method 1: Multiply by reciprocal of denominator.
\(\color{blue}{(\frac{1}{2}) \times (\frac{4}{3}) = \frac{4}{6}}\) = \(\color{blue}{\frac{2}{3}}\)

Example 2: Simplify \(\color{blue}{\frac{(\frac{3}{8})}{(\frac{9}{16})}}\).

\(\color{blue}{(\frac{3}{8}) \times (\frac{16}{9}) = \frac{48}{72}}\) = \(\color{blue}{\frac{2}{3}}\)

Example 3: Simplify \(\color{blue}{\frac{(\frac{x}{3})}{(\frac{x^{2}}{9})}}\).

\(\color{blue}{(\frac{x}{3}) \times (\frac{9}{x^{2}}) = \frac{9x}{(3x^{2})}}\) = \(\color{blue}{\frac{3}{x}}\)  (x ≠ 0)

Example 4: Simplify \(\color{blue}{\frac{(2 + \frac{1}{x})}{(3 – \frac{1}{x})}}\).

Method 2 with \(\color{blue}{\text{ LCD } = x}\):
Numerator: \(\color{blue}{2x + 1}\); Denominator: \(\color{blue}{3x – 1}\)
Result: \(\color{blue}{\frac{(2x + 1)}{(3x – 1)}}\)  (x ≠ 0, x ≠ \(\color{blue}{\frac{1}{3}}\))

More Practice: Two Methods Video

Mario’s Math Tutoring demonstrates both methods side by side on several examples so you can choose the approach that works best for each problem:

Exercises for Simplifying Complex Fractions

Simplify each complex fraction completely.

1. \(\color{blue}{\frac{(\frac{2}{3})}{(\frac{4}{9})}}\)
2. \(\color{blue}{\frac{(\frac{5}{x})}{(\frac{10}{x^{2}})}}\)
3. \(\color{blue}{\frac{(1 + \frac{1}{x})}{(1 – \frac{1}{x})}}\)
4. \(\color{blue}{\frac{(\frac{(x + 1)}{4})}{(\frac{(x + 1)}{8})}}\)
5. \(\color{blue}{\frac{(\frac{3}{a})}{(\frac{6}{a^{2}})}}\)

Answers

1. \(\color{blue}{(\frac{2}{3}) \times (\frac{9}{4}) = \frac{18}{12}}\) = \(\color{blue}{\frac{3}{2}}\)
2. \(\color{blue}{(\frac{5}{x}) \times (\frac{x^{2}}{10}) = 5x}\)\(\color{blue}{^{2}/(10x)}\) = \(\color{blue}{\frac{x}{2}}\)
3. Multiply by x: \(\color{blue}{\frac{(x + 1)}{(x – 1)}}\)
4. \(\color{blue}{(\frac{(x+1)}{4}) \times (\frac{8}{(x+1)}) = \frac{8}{4}}\) = 2
5. \(\color{blue}{(\frac{3}{a}) \times (\frac{a^{2}}{6}) = 3a}\)\(\color{blue}{^{2}/(6a)}\) = \(\color{blue}{\frac{a}{2}}\)

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Want More Practice?

We haven’t published a worksheet built specifically for Simplifying Complex Fractions just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work — they’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:

• Download Simplifying Algebraic Expressions Worksheet
• Download Properties of Exponents Worksheet

Frequently Asked Questions

Which method should I use to simplify a complex fraction?

Use Method 1 (reciprocal) when the numerator and denominator are each a single fraction. Use Method 2 (LCD) when the numerator or denominator has a sum or difference involving fractions, because it clears all the small fractions in one step.

Do I need to state domain restrictions?

Yes. Any variable value that makes a denominator equal to zero in the original expression is excluded from the domain, even after simplification. Always note these restrictions.

Are complex fractions the same as rational expressions?

Complex fractions are a special type of rational expression. A rational expression is any ratio of polynomials; a complex fraction is a rational expression whose numerator or denominator (or both) also contains a fraction.

Related Topics

• Adding and Subtracting Rational Expressions
• Multiplying Rational Expressions
• Dividing Rational Expressions
• Graphing Rational Functions
• Finding Domain and Range of a Function