Multiplying Fractions

Halving a recipe that calls for three-quarters of a cup, or taking half of the half-pizza that is left over — real life keeps asking for a fraction of a fraction. That is exactly what multiplying fractions does.

Here is a happy surprise. Adding and subtracting fractions is fiddly — you have to hunt for a common denominator first. But multiplying fractions is the easiest thing you can do to them. No common denominator, no lining anything up. Just multiply the tops together and multiply the bottoms together:

\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

For example, \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}. Top times top, bottom times bottom — that really is the whole rule.

The little word "of" is the secret to why it works: in the language of fractions, "of" means multiply. So \frac{1}{2} of \frac{3}{4} means \frac{1}{2} \times \frac{3}{4}. Take a chocolate bar, break it into quarters, and grab three of them — that's \frac{3}{4}. Now share those three quarters fairly between two friends: each friend gets half of the three quarters, which is \frac{3}{8} of the whole bar. "Half of three-quarters" and "one-half times three-quarters" are the very same question.

Three worked examples

1. A plain multiply. Work out \frac{1}{2} \times \frac{2}{3}. Tops: 1 \times 2 = 2. Bottoms: 2 \times 3 = 6. So it is \frac{2}{6}, and that simplifies to \frac{1}{3}.

\frac{1}{2} \times \frac{2}{3} = \frac{1 \times 2}{2 \times 3} = \frac{2}{6} = \frac{1}{3}

2. Cancel first to stay tidy. Work out \frac{4}{9} \times \frac{3}{8}. You could multiply straight across to get \frac{12}{72} and then wrestle it down — but those are big, ugly numbers. Instead, cancel before you multiply. The 3 on top and the 9 on the bottom share a factor of 3 (they become 1 and 3); the 4 on top and the 8 on the bottom share a factor of 4 (they become 1 and 2):

\frac{\cancel{4}^{\,1}}{\cancel{9}_{\,3}} \times \frac{\cancel{3}^{\,1}}{\cancel{8}_{\,2}} = \frac{1 \times 1}{3 \times 2} = \frac{1}{6}

Same answer, tiny numbers, no big simplification at the end. Cancelling early is a real time-saver.

3. A fraction times a whole number. A whole number is secretly a fraction with a 1 underneath: 7 = \frac{7}{1}. So \frac{3}{4} \times 7 = \frac{3}{4} \times \frac{7}{1} = \frac{21}{4} = 5\tfrac{1}{4}. The quick way: multiply the top by the whole number and keep the bottom.

"Of" a real amount

Because "of" means multiply, you can find a fraction of a real quantity — money, people, minutes — the same way.

Three-quarters of 20 sweets. \frac{3}{4} \text{ of } 20 = \frac{3}{4} \times \frac{20}{1} = \frac{60}{4} = 15. (Shortcut: 20 \div 4 = 5, then 5 \times 3 = 15 — split into quarters, take three of them.)

Two-thirds of a class of 30. Say \frac{2}{3} of a class of 30 came on the trip. \frac{2}{3} \text{ of } 30 = \frac{2}{3} \times \frac{30}{1} = \frac{60}{3} = 20 children went. First find one-third (30 \div 3 = 10), then take two of them (2 \times 10 = 20). Ten stayed behind.

Simplify at the end

After multiplying, the answer may not be in lowest terms, so simplify it. For instance \frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}. You can also cancel common factors first to keep the numbers small — the 3s cancel, leaving \frac{2}{4} = \frac{1}{2} directly.

See it as area

A unit square is a perfect picture of \frac{1}{2} \times \frac{1}{3}. Slice it into thirds one way and into halves the other way, and the overlap of "one third" and "one half" is a single small box — and the grid has six boxes in all, so that overlap is \frac{1}{6}. Step through it.

This is the trap that costs students the most time. When you add or subtract fractions you must first make the bottoms match. So people learn that rule so well that they try to use it for multiplying too — and waste minutes finding a common denominator they never needed.

To multiply, you just go straight across: top × top, bottom × bottom. Nothing to match up first. And beware the other slip — do not "add the tops": the answer to \frac{2}{3} \times \frac{4}{5} is \frac{8}{15} (multiply), never \frac{6}{8} (that would be adding). The single rule for multiplying is: multiply top × top and bottom × bottom.

Multiply two whole numbers and the answer grows: 3 \times 4 = 12. So it feels wrong that \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} — an answer smaller than either fraction you started with! But remember "of" means times: \frac{1}{2} \times \frac{1}{2} is asking for half of a half, and half of a half really is a quarter. Whenever both fractions are less than 1, the answer shrinks — "a third of a third" of a pizza is a tiny sliver, \frac{1}{9} of the whole.

This exact idea powers probability. Flip a coin: the chance of heads is \frac{1}{2}. Flip two coins — the chance they are both heads is \frac{1}{2} \text{ of } \frac{1}{2} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}. Three heads in a row? \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}. The more coins you ask about, the tinier the chance — the same shrinking you saw with the pizza.