How to add fractions with different denominators
Quick answer
Find a common denominator, usually the least common multiple of the two bottoms. Rewrite each fraction with that denominator, add the numerators, keep the denominator, and simplify. For example, 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
Step by step
- Find a common denominatorList multiples of each denominator until you find one they share. For 3 and 4, the smallest shared multiple is 12.
- Rewrite each fractionMultiply the top and bottom of each fraction by whatever turns its denominator into the common one. 1/3 becomes 4/12, and 1/4 becomes 3/12.
- Add the numerators4 + 3 = 7. Keep the denominator: 7/12.
- SimplifyReduce the answer if the top and bottom share a factor, and turn improper fractions into mixed numbers if needed.
Think of it like adding thirds of a pizza to quarters of a pizza. You can’t count them together until you cut both into slices of the same size, in this case twelfths.
Example: 2/3 + 5/6
Here, 6 is already a multiple of 3, so 6 is the common denominator.
2/3 = 4/6, so 4/6 + 5/6 = 9/6
9/6 simplifies to 3/2, or 1½
Adding mixed numbers
Add the whole numbers and the fractions separately, then combine.
1½ + 2⅓
Whole numbers: 1 + 2 = 3
Fractions: 1/2 + 1/3 = 3/6 + 2/6 = 5/6
Answer: 3⅚
If the fractions add up to more than 1, carry the extra whole into the whole-number part.
A shortcut for two fractions
Multiply diagonally and add for the top, then multiply the bottoms:
a/b + c/d = (a × d + b × c) / (b × d)
1/3 + 1/4 = (1 × 4 + 3 × 1) / (3 × 4) = 7/12
It always works, but with bigger denominators it can give a large answer you’ll need to simplify.
What about subtracting?
Exactly the same steps, but subtract the numerators. For example, 3/4 − 1/6 = 9/12 − 2/12 = 7/12.
Multiplying is easier, since you don’t need a common denominator at all: see how to multiply fractions.
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