mathsGrade 6-82 min read

Fractions: every operation on one page

Add, subtract, multiply, divide — the four things you can do to a fraction, the one rule each needs, and the trap that catches everyone with division.

Fractions feel like four unrelated procedures, so students mix them up — they find a common denominator when multiplying, or "flip" when they should add. Here is all of it on one page, with the single thing each operation actually needs.

The golden rule that decides everything

Adding and subtracting needs a common denominator. Multiplying and dividing does not. Almost every fraction mistake comes from forgetting which camp you're in. Decide that first, every time.

Multiply — the easy one

Straight across, top times top and bottom times bottom. No common denominator.

\frac{2}{3}\times\frac{4}{5}=\frac{2\times4}{3\times5}=\frac{8}{15}

Divide — multiply by the flip

Keep the first fraction, flip the second, and multiply. "Keep, flip, multiply."

\frac{2}{3}\div\frac{4}{5}=\frac{2}{3}\times\frac{5}{4}=\frac{10}{12}=\frac{5}{6}

The trap: students flip the wrong fraction, or flip both. Only the one you're dividing by turns over.

Add and subtract — make the bottoms match first

You cannot add quarters to thirds until they're the same kind of piece. Rewrite both over a common denominator, then add the tops only — the bottom stays put.

\frac{1}{3}+\frac{1}{4}=\frac{4}{12}+\frac{3}{12}=\frac{7}{12}

The trap: adding the denominators too. $\frac{1}{3}+\frac{1}{4}$ is not $\frac{2}{7}$ . The denominator names the size of the piece; it doesn't get added.

Quick reference

Operation	Common denominator?	What you do
$\times$	No	Multiply straight across
$\div$	No	Keep, flip the second, multiply
$+$ and $-$	Yes	Match bottoms, then add/subtract the tops

Always simplify the answer. $\frac{10}{12}$ and $\frac{5}{6}$ are the same number, but only one of them gets full marks. Divide top and bottom by their biggest common factor at the end.

Last revised 20 September 2024.