How to Add Fractions
You can only add pieces that are the same size. Adding fractions is really just one idea — make the pieces match — followed by counting.
Watch the pieces match up
Pick a sum and step through it. Notice that re-slicing never changes how much of each bar is shaded — it only changes how many pieces it’s cut into.
2/3 and 1/4 are cut into different-sized pieces, so you can’t just add the tops. 2/3 + 1/4 is not 3/7.
The four steps
- Check the denominators. If they are already the same, skip to step 3.
- Find a common denominator — the least common multiple of the denominators — and rewrite each fraction over it. Multiply the top by whatever you multiplied the bottom by: 2/3 = 8/12 (both × 4).
- Add the numerators. Keep the denominator the same: 8/12 + 3/12 = 11/12. The denominator stays because the size of the pieces hasn’t changed.
- Simplify if the top and bottom share a factor, and turn improper fractions into mixed numbers if your teacher asks.
Example with mixed numbers
2 1/2 + 1 3/4: add the wholes (2 + 1 = 3), then the fractions (1/2 + 3/4 = 2/4 + 3/4 = 5/4 = 1 1/4). Put them together: 3 + 1 1/4 = 4 1/4.
Questions
Why can’t you just add the numerators and denominators?
Because the denominator names the size of the pieces. 1/2 + 1/3 would give 2/5, which is less than 1/2 on its own — clearly wrong. Pieces must be the same size before you count them.
Do I have to use the least common denominator?
No — any common denominator gives the right answer (multiplying the two denominators always works). The least one keeps the numbers smaller, so there is less to simplify.
How do I add mixed numbers?
Either add the whole numbers and the fractions separately (carrying if the fractions make more than 1), or turn both into improper fractions first and add those.
How is subtracting fractions different?
It isn’t, apart from the last step: find a common denominator, rewrite both fractions, then subtract the numerators.