How to add fractions
Short answer. If the denominators match, add the numerators: 1/5 + 2/5 = 3/5. If they do not, rewrite both fractions over a common denominator first: 1/2 + 1/3 becomes 3/6 + 2/6, which is 5/6. Never add the denominators themselves.
The rule fits in one sentence, and yet fractions are where most people quietly gave up on math. The trouble is never the adding. It is knowing what you are allowed to add.
Why you cannot just add straight across
A fraction is two different kinds of number stapled together. The bottom one, the denominator, names the size of the pieces. The top one, the numerator, counts how many pieces you have. You can only count things together when they are the same size, so the denominators have to match before any adding happens.
When they already match, the whole job is one step:
1/5 + 2/5 = 3/5 count the fifths: 1 + 2 = 3 not 3/10. The pieces did not change size, so the denominator does not change either.
Different denominators: the full method
Take 2/3 + 3/4. Thirds and quarters are different sizes, so first find a piece size that fits both. The smallest number both 3 and 4 divide into is 12, the least common denominator:
2/3 = 8/12 multiply top and bottom by 4 3/4 = 9/12 multiply top and bottom by 3 8/12 + 9/12 = 17/12 = 1 5/12
Multiplying the top and bottom of a fraction by the same number does not change its value, only its name. That is the entire trick. 2/3 and 8/12 are the same amount of pizza.
The cross-multiply shortcut
If you do not feel like hunting for the least common denominator, multiply the two denominators together and cross-multiply the tops:
a/b + c/d = (a×d + c×b) / (b×d) 1/4 + 1/6 = (1×6 + 1×4) / 24 = 10/24 = 5/12
It always works, and the price is a bigger number to simplify at the end. The least common denominator would have given 3/12 + 2/12 = 5/12 directly. Same answer, smaller numbers, your choice.
Mixed numbers
For something like 2 3/4 + 1 2/3, add the whole parts and the fraction parts separately:
Wholes 2 + 1 = 3 Fractions 3/4 + 2/3 = 9/12 + 8/12 = 17/12 = 1 5/12 Total 3 + 1 5/12 = 4 5/12
If the fraction parts add up past one, as they do here, carry the whole into the whole-number column, exactly like carrying in ordinary addition.
The decimal trap
The tempting shortcut is to convert everything to decimals and add those. Sometimes that is fine: 3/4 is exactly 0.75. But many fractions have no exact decimal form. 1/3 is 0.3333 repeating forever, so whatever you type into a calculator is already rounded before the addition even starts:
1/3 + 1/4 as decimals: 0.33 + 0.25 = 0.58
exactly: 4/12 + 3/12 = 7/12 = 0.58333...
Off by a third of a percent before you have done anything. In a recipe nobody notices. Across a cut list for a woodworking project, or repeated in a spreadsheet a few hundred times, that rounding drift is real. It is the same disease as the famous 0.1 + 0.2 = 0.30000000000000004 problem, just wearing different clothes.
The fix is to keep fractions as fractions until the final answer. That is how the method above works on paper, and it is how a calculator should work too.
Common mistakes, quickly
| Mistake | What happens | Fix |
|---|---|---|
| Adding denominators1/5 + 2/5 = 3/10 | Answer is roughly half the true value | Denominator stays put when piece sizes match |
| Scaling only the bottom2/3 = 2/12 | The fraction silently shrinks | Whatever multiplies the bottom multiplies the top |
| Forgetting to carry3 + 17/12 | An improper fraction hides in the answer | 17/12 = 1 5/12, so carry the 1 |
| Rounding to decimals first0.33 + 0.25 | Error is baked in before the addition | Stay in fractions until the end |
Or type 2 3/4 and let it stay 2 3/4
Prism has a fraction mode that computes exactly, so 1/3 + 1/4 is 7/12, not 0.58. Fractions are free forever, no subscription. See the app.