Adding and subtracting fractions made simple

Fractions have a reputation for being hard, but adding them really comes down to one rule: you can only add pieces that are the same size. Once you see why, the method stops feeling like a set of arbitrary steps.

When the bottoms already match

If the denominators are the same, you are counting pieces of one size. Three fifths plus one fifth is four fifths. The bottom stays the same because the pieces have not changed; only the count has. So 3/5 + 1/5 = 4/5.

When the bottoms are different

A quarter and a third are different sizes, so you cannot just add the tops. First, swap both for fractions of the same size. That size is the lowest common multiple (LCM) of the two denominators.

1. Find the common denominator. For 1/4 + 1/3, the LCM of 4 and 3 is 12.
2. Rewrite each fraction. 1/4 = 3/12 because you multiplied top and bottom by 3. 1/3 = 4/12 because you multiplied by 4.
3. Add the tops. 3 + 4 = 7, so the sum is 7/12.
4. Simplify if you can. 7/12 is already as simple as it gets.

Subtracting works the same way

For 5/6 - 1/4, the LCM of 6 and 4 is 12. That gives 10/12 - 3/12 = 7/12. Always subtract the second top from the first, and keep the common bottom.

Mixed numbers

For 2 1/2 + 1 3/4, add the whole numbers first (2 + 1 = 3), then the fractions (1/2 + 3/4 = 2/4 + 3/4 = 5/4 = 1 1/4). Combine: 3 + 1 1/4 = 4 1/4. Alternatively, convert both to improper fractions, add, and convert back. Pick whichever feels less error-prone.

Question	Common denominator	Answer
1/2 + 1/3	6	5/6
3/4 + 2/3	12	17/12 = 1 5/12
7/8 - 1/2	8	3/8
2/5 + 3/10	10	7/10

Three mistakes to avoid

• Adding the bottoms. 1/4 + 1/3 is not 2/7. The bottom tells you the piece size, and the size does not add up.
• Forgetting to change the top. If you multiply the bottom by 3, multiply the top by 3 too, or you have changed the value.
• Not simplifying. 6/8 is correct but 3/4 is what most teachers want to see.

The picture helps: think of a pizza cut into quarters and another cut into thirds. To combine slices you have to re-cut both into twelfths. That re-cutting is all a common denominator is.