How to add, subtract, multiply and divide fractions
Each of the four operations on fractions has one rule. This guide gives all four with worked examples, shows how to handle mixed numbers and negatives, and explains how to simplify an answer.
The short version
- Add or subtract: rewrite both fractions over a common denominator, then add or subtract the numerators.
- Multiply: multiply the numerators and multiply the denominators.
- Divide: flip the second fraction (take its reciprocal) and multiply.
- Simplify: divide the top and bottom by their greatest common factor.
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The four rules with examples
| Operation | Rule | Example |
|---|---|---|
| Add | common denominator, add tops | 1/2 + 1/3 = 3/6 + 2/6 = 5/6 |
| Subtract | common denominator, subtract tops | 5/6 − 1/3 = 5/6 − 2/6 = 3/6 = 1/2 |
| Multiply | top × top, bottom × bottom | 3/2 × 2/3 = 6/6 = 1 |
| Divide | flip the second, then multiply | 3/4 ÷ 3/8 = 3/4 × 8/3 = 24/12 = 2 |
Finding a common denominator
The easiest common denominator is the least common multiple (LCM) of the two denominators. For 1/4 + 1/6 the LCM of 4 and 6 is 12, so rewrite as 3/12 + 2/12 = 5/12. Multiplying the denominators together (4 × 6 = 24) also works, but you get 10/24 and must simplify afterwards.
Mixed numbers
Convert a mixed number to an improper fraction before multiplying or dividing: 1 1/2 = (1 × 2 + 1) ÷ 2 = 3/2. Then 1 1/2 × 2/3 = 3/2 × 2/3 = 6/6 = 1. To turn an improper fraction back into a mixed number, divide: 7/4 = 1 with a remainder of 3, so 1 3/4.
Simplifying
A fraction is in lowest terms when its numerator and denominator share no factor except 1. Divide both by their greatest common factor (GCF): 18/24 has GCF 6, so 18/24 = 3/4. Simplifying at the end is enough; you do not need to simplify at every step.
Mistakes to avoid
- Adding denominators. 1/2 + 1/3 is 5/6, not 2/5. Denominators only stay the same, they are never added.
- Using a common denominator for multiplication. You do not need one; only for adding and subtracting.
- Forgetting to flip when dividing, or flipping the wrong fraction. Only the second fraction flips.
- Dividing by zero. A fraction cannot have a denominator of 0, and you cannot divide by a fraction equal to 0.
Negative fractions follow the usual sign rules, and the minus sign is normally written on the top: −3/4, not 3/−4.