Multiplying a Fraction by a Whole Number

A recipe needs \frac{3}{4} of a cup of flour, but you're baking three batches — so how much flour in total? Whenever you take the same fraction over and over — three quarter-cups, four half-slices, five thirds of a metre of ribbon — you're multiplying a fraction by a whole number. It's the everyday sum behind scaling up a recipe or a shopping list.

You already know that multiplication is a fast way to add the same amount many times — 3 \times 4 just means 4 + 4 + 4. The same trick works on a fraction: multiplying a fraction by a whole number means adding that fraction to itself, over and over.

So 3 \times \frac{2}{5} means three lots of two-fifths:

3 \times \frac{2}{5} = \frac{2}{5} + \frac{2}{5} + \frac{2}{5} = \frac{6}{5}

Each two-fifths adds two more shaded parts, and the parts are still fifths the whole time — the size of a part never changes, only how many you have. That is the one big idea on this page: the pieces stay the same size; you just collect more of them.

Imagine three hungry friends, and each one eats \frac{2}{5} of a pizza:

pizza pizza pizza

Two-fifths, plus two-fifths, plus two-fifths is \frac{6}{5} of a pizza eaten in all — that is one whole pizza and one extra fifth. The slices were always fifths; the friends just ate six of them between them.

The shortcut

Adding the same fraction k times just stacks up its top number k times, while the bottom number stays put. So you can skip the repeated addition: multiply the numerator (the top) by the whole number, and keep the denominator (the bottom).

k \times \frac{m}{n} = \frac{k \times m}{n}

For our example, 3 \times \frac{2}{5} = \frac{3 \times 2}{5} = \frac{6}{5} — exactly what the repeated addition gave. The denominator 5 is untouched because the pieces are still fifths; only the count of pieces grows.

The two traps everybody falls into the first time:

When the answer is more than one whole

When the new top number is bigger than the bottom, the fraction is improper — it is worth more than one whole. We usually tidy it into a mixed number: a whole number next to a fraction. To do it, see how many whole groups of the bottom fit into the top.

\frac{6}{5} = \frac{5}{5} + \frac{1}{5} = 1\tfrac{1}{5}

Five-fifths make one whole, and there is one fifth left over, so \frac{6}{5} = 1\tfrac{1}{5}. The pizza friends ate one whole pizza and one extra fifth — same thing, said two ways.

Now four friends each nibble \frac{1}{3} of a cookie:

cookie cookie cookie cookie

That is 4 \times \frac{1}{3} = \frac{4}{3} of a cookie. Three thirds make one whole cookie, with one third left over, so they ate 1\tfrac{1}{3} cookies altogether.

A few worked examples

Notice the recipe is always the same: top times the whole, bottom unchanged, then tidy the result if you can.

The same idea: a fraction of a quantity

Multiplying the other way round — a whole number by a unit fraction — is exactly how you take a fraction of a quantity. Because multiplication can be done in any order,

\tfrac{1}{4} \times 12 = 12 \times \tfrac{1}{4} = \frac{12}{4} = 3.

"One quarter of twelve" is just 12 lots of a quarter, collected up into \frac{12}{4}, which is 3. So the skill on this page is the skill of finding a fraction of an amount — the same multiply-the-top rule, looked at from the other side.

See it built

Each bar below is one whole, split into the same number of equal parts. Watch one lot of the fraction get shaded in every bar — then count up all the shaded parts to see the new numerator. The denominator never moves. Step through it.

Try a fresh one

Here is a random whole number times a fraction, drawn as repeated shaded bars. Each bar is split into the same number of parts, and the same lot is shaded in every bar — count all the shaded parts to get the new top number. Press Refresh for a new example.

See it explained

Sal Khan multiplies fractions by whole numbers two different ways and shows they agree.