A complex fraction has fractions inside its numerator, its denominator or both. Simplifying it means rewriting the whole thing as one ordinary fraction in lowest terms. This calculator accepts a numerator and a denominator that can each be a single fraction or two fractions combined with +, −, × or ÷, simplifies them, divides, and then double-checks the answer with the least-common-denominator method.
How to use the calculator
- For the numerator, choose whether it is a single fraction or two fractions combined with an operation, and enter the fractions.
- Do the same for the denominator.
- The tape shows the simplified result as a mixed number and improper fraction, the value of the numerator and denominator, and the decimal.
- The steps show the divide method, and when the parts are sums or differences, a check with the LCD method. The complex fraction you entered is drawn below the steps.
Method 1: simplify, then divide
- Combine the fractions in the numerator into one fraction.
- Combine the fractions in the denominator into one fraction.
- Divide: multiply the numerator by the reciprocal of the denominator.
- Simplify.
Method 2: multiply by the LCD
- Find the LCD of all the small denominators inside the complex fraction.
- Multiply every term in the numerator and in the denominator by that LCD. Each small fraction becomes a whole number.
- Simplify the resulting ordinary fraction.
Because you multiply the top and bottom by the same number, the value is unchanged — it is the same idea as writing equivalent fractions.
Worked example
Simplify (1/2 + 1/3) / (3/4 − 1/6).
Method 1.
- Numerator: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
- Denominator: 3/4 − 1/6 = 9/12 − 2/12 = 7/12.
- Divide: 5/6 ÷ 7/12 = 5/6 × 12/7 = 60/42.
- Simplify by 6: 10/7 = 1 3/7 (≈ 1.4286).
Method 2. The small denominators are 2, 3, 4 and 6, with LCD 12. Multiply each term by 12: the top becomes 6 + 4 = 10 and the bottom becomes 9 − 2 = 7. The result is 10/7 directly.
When to use which method
| Form of the complex fraction | Easier method |
|---|---|
| (a/b) / (c/d) — single fractions | Divide by multiplying by the reciprocal |
| Sums or differences on top and bottom | LCD method — one multiplication clears everything |
| Products or quotients inside | Simplify each part first, then divide |
| Algebraic fractions with variables | LCD method — the standard approach in algebra |
Common mistakes
- Flipping the wrong fraction. Only the denominator of the main fraction bar is inverted.
- Multiplying only some terms by the LCD. Every term in the top and bottom must be multiplied, including whole numbers.
- Stopping before lowest terms. 60/42 is correct but not simplified; the simplifying fractions calculator reduces any fraction.
To add, subtract, multiply or divide two complex fractions with each other, use the complex fractions calculator.
Frequently asked questions
How do you simplify a complex fraction?
Simplify the numerator and the denominator separately until each is a single fraction, then divide the top by the bottom by multiplying by the reciprocal. For (1/2 + 1/3)/(3/4 − 1/6), the top is 5/6, the bottom is 7/12, and 5/6 × 12/7 = 10/7.
What is the LCD method?
Multiply the numerator and the denominator of the complex fraction by the least common denominator of every small fraction inside it. All the small denominators cancel at once. In the example the LCD is 12: the top becomes 6 + 4 = 10 and the bottom 9 − 2 = 7, giving 10/7.
Which method is better?
Both give the same answer. The divide method is the natural one when the top and bottom are single fractions. The LCD method is usually faster when the top and bottom are sums or differences, and it is the standard technique in algebra with variables.
Can the denominator of a complex fraction be zero?
If the bottom part works out to 0 — for example 1/6 − 1/6 — the complex fraction is undefined. The calculator stops and tells you.