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

Simplify complex fractions that contain fractions in the numerator or denominator.

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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.

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Simplify Complex Fractions: what to notice and how to work it

Complex skill

Complex numbers have a real part and an imaginary part. Keeping those parts organized makes operations feel much more predictable.

What to notice first

Group real terms with real terms and imaginary terms with imaginary terms. The special fact \(i^2=-1\) drives multiplication and division.

Common student mistake

Do not leave \(i^2\) unchanged. Replacing it with -1 is the key simplification step.

Key formulas and cues

\(i^2=-1\)

\((a+bi)+(c+di)=(a+c)+(b+d)i\)

\((a+bi)(c+di)=ac+adi+bci+bd i^2\)

\(|a+bi|=\sqrt{a^2+b^2}\)

A reliable path

Separate partsKeep real and imaginary terms in their own lanes.
Use i squaredReplace \(i^2\) with -1 whenever it appears.
Use conjugatesFor division, multiply by the conjugate to make the denominator real.

Worked examples

Add complex numbers

Example: \((4+3i)+(2-5i)\)

Add real parts: 4 + 2.
Add imaginary parts: 3i – 5i.
Write both parts together.

Answer: \(6-2i\)

Use i squared

Example: \(i(5i)\)

Multiply coefficients to get 5.
i times i is i squared.
Replace i squared with -1.

Answer: \(-5\)

Open pop-up practice

Try one before moving on

Try: Simplify \((5-2i)+(1+6i)\).

Answer: \(6+4i\).

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

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 \(\color{blue}{x = 2}\): \(\color{blue}{\frac{(2 + 0.5)}{(3 – 0.5)} = \frac{2.5}{2.5} = 1}\); \(\color{blue}{\frac{(4+1)}{(6-1)} = \frac{5}{5} = 1}\) &checkmark;

Step-by-Step Summary

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).
Method 1: Keep the top fraction, \(\color{blue}{\text{ change } \div \text{ to }}\) ×, flip the bottom fraction (write its reciprocal). Simplify.
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.
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.

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

Answers

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

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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:

Frequently Asked Questions

How do you simplify complex fractions step by step?

Step one: simplify the top and bottom into single fractions if needed. Step two: rewrite the main fraction bar as division. Step three: multiply by the reciprocal of the bottom fraction and reduce. For (3/4)/(5/8), compute 3/4 × 8/5 = 24/20, which simplifies to 6/5. A second method is to multiply the top and bottom by the LCD of all the small fractions.

How do you simplify complex fractions with variables?

The LCD method is usually fastest with variables. Find the least common denominator of every small fraction, then multiply the whole numerator and the whole denominator by it. For (1/x + 1/y) ÷ (1/(xy)), multiply top and bottom by xy: the top becomes y + x and the bottom becomes 1, so the result is x + y, with x ≠ 0 and y ≠ 0.

How do you simplify complex fractions with exponents?

When a complex fraction has negative exponents, rewrite them as fractions first, then clear them with the LCD. For (1 + x⁻¹)/(1 − x⁻²), write it as (1 + 1/x)/(1 − 1/x²) and multiply top and bottom by x². That gives (x² + x)/(x² − 1) = x(x + 1)/((x + 1)(x − 1)), which simplifies to x/(x − 1) for x ≠ 0, 1 or −1.

How do you simplify complex fractions with mixed numbers?

Change each mixed number to an improper fraction, then divide as usual. For 2½ over 1¼, rewrite as (5/2)/(5/4). Multiply the top by the reciprocal of the bottom: 5/2 × 4/5 = 20/10 = 2. Converting first matters, because a mixed number like 2½ means 2 + ½, not 2 × ½.

How do you simplify complex fractions with square roots?

Treat square roots like any other factor: divide by multiplying by the reciprocal, cancel matching roots, and rationalize any root left in the denominator. For (√3/2)/(√3/4), compute √3/2 × 4/√3; the √3 cancels and 4/2 = 2. If you end up with something like 1/√2, multiply top and bottom by √2 to get √2/2.

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