Adding & Subtracting Fractions

A denominator tells you the size of each piece: in $\frac{1}{4}$ the whole is cut into four, so each piece is a "fourth." You can add $\frac{1}{4} + \frac{2}{4}$ directly — same-size pieces — to get $\frac{3}{4}$.

But $\frac{1}{2} + \frac{1}{4}$ mixes a half with a quarter: different-size pieces, so the tops can't just be added. The fix is to rewrite one (or both) fractions as equivalent fractions with a common denominator, so every piece is the same size again. Then add the tops and keep the shared bottom.

Same denominator? Add the tops. Take $\frac{1}{5} + \frac{2}{5}$ — both are fifths, so combine $1 + 2 = 3$ fifths:

$\frac{1}{5} + \frac{2}{5} = \frac{3}{5}$

Different denominators? Re-slice first. For $\frac{1}{2} + \frac{1}{4}$, rewrite $\frac{1}{2}$ as $\frac{2}{4}$ so both are quarters. The bars below show $\frac{1}{2}$ and $\frac{2}{4}$ covering the same amount:

Now both are quarters, so add the tops:

$\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}$

Subtraction works exactly the same way — match the denominators, then subtract the tops. Always simplify the result at the end.

Find the common denominator first, then combine. Predict before you check.