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Compound Fraction

Compound Fraction
Complex Fraction

A fraction which has, as part of its numerator and/or denominator, at least one other fraction.

Examples: 1.  Compound fraction example: (1/4 + 3) divided by (2 - 2/7)
   
  2.  5 divided by (1/(x−1) + 1/(x+1)), a compound fraction with fractions in the denominator.

 

See also

Compound inequality

Worked Example

Problem: Simplify the compound fraction: the numerator is 3/4, and the denominator is 5/8.
Step 1: Write the compound fraction as a division problem. A fraction bar means division, so the compound fraction equals the numerator divided by the denominator.
  34    58  =34÷58\frac{\;\tfrac{3}{4}\;}{\;\tfrac{5}{8}\;} = \frac{3}{4} \div \frac{5}{8}
Step 2: To divide by a fraction, multiply by its reciprocal. Flip the denominator fraction and change the operation to multiplication.
34×85\frac{3}{4} \times \frac{8}{5}
Step 3: Multiply the numerators together and the denominators together.
3×84×5=2420\frac{3 \times 8}{4 \times 5} = \frac{24}{20}
Step 4: Simplify the resulting fraction by dividing the numerator and denominator by their greatest common factor, which is 4.
2420=65\frac{24}{20} = \frac{6}{5}
Answer: The simplified result is 65\frac{6}{5}, or equivalently 1151\frac{1}{5}.

Another Example

Problem: Simplify the compound fraction where the numerator is 2+132 + \frac{1}{3} and the denominator is 76\frac{7}{6}.
Step 1: First, simplify the numerator. Convert the mixed expression into a single fraction.
2+13=63+13=732 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}
Step 2: Now rewrite the compound fraction with the simplified numerator, and convert to a division problem.
  73    76  =73÷76\frac{\;\tfrac{7}{3}\;}{\;\tfrac{7}{6}\;} = \frac{7}{3} \div \frac{7}{6}
Step 3: Multiply by the reciprocal of the denominator fraction.
73×67=4221\frac{7}{3} \times \frac{6}{7} = \frac{42}{21}
Step 4: Simplify the result.
4221=2\frac{42}{21} = 2
Answer: The compound fraction simplifies to 22.

Frequently Asked Questions

What is the difference between a compound fraction and a simple fraction?
A simple fraction has integers in both the numerator and denominator, such as 35\frac{3}{5}. A compound fraction (also called a complex fraction) has at least one fraction within its numerator or denominator, such as  34 5\frac{\,\tfrac{3}{4}\,}{5}. You can always simplify a compound fraction down to a simple fraction.
How do you simplify a compound fraction?
The most common method is to rewrite the main fraction bar as division, then multiply by the reciprocal. If the numerator or denominator contains addition or subtraction, first combine those parts into a single fraction before dividing. An alternative method is to multiply both the numerator and denominator by the least common denominator of all the smaller fractions.

Compound (Complex) Fraction vs. Simple Fraction

A simple fraction has whole-number numerator and denominator, like 79\frac{7}{9}. A compound fraction contains at least one fraction inside the numerator or denominator, like  23  45 \frac{\,\tfrac{2}{3}\,}{\,\tfrac{4}{5}\,}. Every compound fraction can be simplified into a simple fraction (or a whole number) by dividing.

Why It Matters

Compound fractions appear frequently in algebra when you simplify rational expressions or solve equations involving multiple fractions. They also arise naturally in real-world rate problems—for instance, dividing 12\frac{1}{2} a pizza among 34\frac{3}{4} of a group. Knowing how to simplify them is an essential skill for working with ratios, proportions, and more advanced algebraic manipulation.

Common Mistakes

Mistake: Trying to simplify by canceling across the main fraction bar without first combining the numerator or denominator into a single fraction.
Correction: If the numerator or denominator involves addition or subtraction (e.g., 1+1234\frac{1 + \tfrac{1}{2}}{\tfrac{3}{4}}), you must first combine terms into one fraction before dividing. You cannot cancel individual pieces across a sum.
Mistake: Forgetting to flip the correct fraction when converting to multiplication.
Correction: When you rewrite a/bc/d\frac{a/b}{c/d} as a division, you flip the denominator fraction—the one on the bottom of the main fraction bar. You multiply by dc\frac{d}{c}, not ba\frac{b}{a}.

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