How to add fractions (and why the rule works)

Fraction addition is where school math loses the most people, and it loses them at one specific spot: the rule "you can't just add the bottoms" gets taught as law rather than explained as sense. So here is the sense first, then the method, then the shortcuts. (The fraction calculator shows its work in the same steps, so you can check anything below against it.)

Why 1/2 + 1/3 isn't 2/5

A denominator isn't a normal number — it's a unit. "1/2" means one piece of the kind you get cutting a pizza in two; "1/3" is one piece of a three-way cut. Adding 1/2 + 1/3 = 2/5 is the same mistake as 1 metre + 1 foot = 2 something — you added counts of different-sized things. And the answer 2/5 fails a sanity check instantly: 2/5 is less than the 1/2 you started with, so "adding" made it shrink.

The fix is the same as with metres and feet: convert to a common unit first. That's all a common denominator is.

The method, step by step

For 1/2 + 1/3:

• 1. Find a common denominator — a size of piece both fractions can be re-cut into. Sixths work: cut every half into three and every third into two.
• 2. Convert both fractions, multiplying top and bottom by the same number (which is multiplying by 1, so nothing changes but the units): 1/2 = 3/6, and 1/3 = 2/6.
• 3. Add the tops: 3/6 + 2/6 = 5/6. Same-sized pieces now, so counting them is legal.
• 4. Simplify if possible. 5/6 already is.

Subtraction is the identical procedure with a minus in step 3 — every idea on this page transfers unchanged.

Finding the least common denominator

Any common multiple of the denominators works, but the least one keeps numbers small. Three tactics, in order of speed: check whether one denominator already divides the other (4 and 8 → use 8); otherwise walk the multiples of the larger one until the smaller divides it (6 and 8: 8, 16, 24 ✓); and when in doubt, denominator × denominator always works — it just may need more simplifying at the end. For 5/12 + 7/18, the walk gives 36: 15/36 + 14/36 = 29/36.

The butterfly shortcut

For any two fractions, cross-multiply for the new tops and multiply the bottoms for the new bottom: a/b + c/d = (ad + bc)/bd. So 2/3 + 1/4 = (2×4 + 3×1)/(3×4) = 11/12. It's the "denominator × denominator" tactic dressed as a mnemonic — always correct, no LCD hunting, at the price of bigger numbers and a likely simplification step (2/6 + 3/9 gives 36ths before it gives the real answer, 2/3). Use it for two fractions in a hurry; use the LCD when denominators are friendly or there are three-plus fractions.

Mixed numbers

For 2¾ + 1⅔, two honest routes. Convert to improper fractions — 11/4 + 5/3 = 33/12 + 20/12 = 53/12 = 4 5/12 — which is mechanical and never surprises you. Or add parts separately — wholes: 3; fractions: ¾ + ⅔ = 17/12 = 1 5/12; total 4 5/12 — which is faster mentally but requires noticing when the fraction part carries past 1. Both are fine; the errors live in half-remembered mixtures of the two.

Checking your answer

• Size check: the sum must exceed the larger addend; ½ + ⅓ must land between ½ and 1. This one glance catches the classic added-the-bottoms error every time.
• Decimal check: 0.5 + 0.333 ≈ 0.833 = 5/6 ✓ — thirty seconds and near-unfoolable.
• Tool check: the fraction calculator shows the LCD, the converted fractions and the simplified result, so you can find which step went wrong rather than just that one did. The same convert-to-common-units instinct, pointed at real life, is unit conversion — fractions are just the first place school makes you do it.