How do you multiply fractions?
Short answer. Multiply straight across: numerator times numerator, denominator times denominator. 2/3 × 3/4 = 6/12 = 1/2. No common denominator is needed, which makes multiplying easier than adding. Turn mixed numbers into improper fractions first, and cancel common factors before you multiply to keep the numbers small.
Multiplication is the one fraction operation that behaves. There is no hunting for a least common denominator, no rewriting both numbers before you start. Tops times tops, bottoms times bottoms, done.
2/3 × 3/4 = (2 × 3) / (3 × 4) = 6/12 = 1/2
That is the whole method. The rest of this page is about why it works, how to keep the numbers small, and the two places people still trip.
Why it does not need a common denominator
Adding fractions needs matching denominators because you are counting pieces, and you can only count things together when they are the same size. Multiplying is a different question entirely. Times means "of." When you multiply, you are taking a fraction of a fraction, so the pieces are supposed to change size.
Take 1/2 × 1/3. Cut a bar into thirds, then take half of one of those thirds. You get a piece that fits into the whole bar six times:
1/2 × 1/3 = 1/6 half of a third is a sixth The denominators multiply because you cut twice: 3 cuts, then each piece halved, gives 3 × 2 = 6 pieces.
The numerators multiply for the same reason. 2/3 × 3/4 is "three quarters of two thirds," and three quarters of two pieces is 2 × 3 = 6 quarter-pieces, each of them a twelfth of the whole. Hence 6/12, which is 1/2.
This is also why multiplying by a proper fraction always makes a number smaller, which surprises people who learned that multiplication makes things bigger. Half of something is less than the something.
Cancel first, multiply second
Multiplying straight across works every time, but it can hand you ugly numbers to reduce at the end. You are allowed to cancel any top against any bottom before multiplying, including across the multiplication sign:
4/9 × 3/8
Straight across: 12/72 then reduce by 12 = 1/6
Cancel first: 4 and 8 both divide by 4 -> 1 and 2
3 and 9 both divide by 3 -> 1 and 3
1/3 × 1/2 = 1/6
Same answer, and you never had to notice that 72 divided by 12 is 6. On bigger numbers the saving is real:
6/25 × 15/8
Cancel: 6 and 8 share 2 -> 3 and 4
15 and 25 share 5 -> 3 and 5
3/5 × 3/4 = 9/20
Mixed numbers: convert first
This is the mistake that costs the most marks. You cannot multiply the whole parts and the fraction parts separately. Convert each mixed number to an improper fraction, then multiply straight across:
2 1/2 × 1 3/5
Convert 2 1/2 = (2 × 2 + 1)/2 = 5/2
1 3/5 = (1 × 5 + 3)/5 = 8/5
Multiply 5/2 × 8/5 = 40/10 = 4
The tempting wrong way:
2 × 1 = 2, 1/2 × 3/5 = 3/10 -> 2 3/10
2 3/10 is 2.3. The real answer is 4. Not close.
The reason the shortcut fails is that 2 1/2 means 2 + 1/2, and multiplying two sums means every part has to meet every other part. Dropping the cross terms drops most of the answer.
Whole numbers
A whole number is a fraction with 1 underneath. Write it that way and nothing special happens:
3 × 2/7 = 3/1 × 2/7 = 6/7
In practice you can skip the middle step: multiply the numerator by the whole number and leave the denominator alone.
Where this actually comes up
Scaling a recipe is the everyday case. A recipe calls for 2/3 cup of sugar and you want three quarters of a batch:
3/4 × 2/3 = 6/12 = 1/2 cup
Cutting timber, splitting a bill by unequal shares, and working out how much of a tank of fuel a trip uses are all the same operation wearing different clothes.
The decimal trap
You could convert everything to decimals and multiply those instead. Sometimes that is fine, because 3/4 is exactly 0.75. But plenty of fractions have no exact decimal form, and multiplication does not forgive the rounding, it scales it:
1/3 × 3 exactly: 3/3 = 1
as decimals: 0.333 × 3 = 0.999
1/3 × 6/7 exactly: 6/21 = 2/7 = 0.285714...
as decimals: 0.333 × 0.857 = 0.285381
The first one is the famous case: three thirds is one, but three lots of 0.333 is not. The error was created the moment 1/3 was written as a decimal, and multiplying by 3 tripled it. That is the same failure as 0.1 + 0.2 = 0.30000000000000004, which is what happens when a calculator stores numbers in binary floating point instead of keeping the fraction.
The fix on paper is to stay in fractions until the last step. A calculator should do the same, and hand you 2/7 rather than a decimal that is already wrong in the fourth place. If you do want the decimal at the end, converting a fraction to a decimal is one division.
Common mistakes, quickly
| Mistake | What happens | Fix |
|---|---|---|
| Finding a common denominator first2/3 × 3/4 = 8/12 × 9/12 | Wasted work, and the answer comes out as 72/144 | Only addition needs matching denominators |
| Multiplying mixed parts separately2 1/2 × 1 3/5 = 2 3/10 | 2.3 instead of 4 | Convert to 5/2 and 8/5 first |
| Cancelling top against top2/3 × 3/4, cancelling the 2 and 3 on top | Changes the value of the answer | Cancel a numerator only against a denominator |
| Rounding to decimals first0.333 × 3 | 0.999 instead of 1 | Stay in fractions until the end |
Every number on this page was computed exactly and checked before publishing.
Or type 2/3 × 3/4 and get 1/2 back
Prism has a fraction mode that multiplies exactly and reduces for you, so 1/3 × 3 is 1 and not 0.999. Fractions are free forever, no subscription. See the app.