How to multiply fractions

Multiplying fractions is easier than adding them. The bottom numbers do not have to match, and there is nothing to change first. It is two times-table facts, one on top and one underneath.

The rule

Multiply the top numbers to get the top of the answer. Multiply the bottom numbers to get the bottom of the answer. Then simplify if you can.

How to do it

Multiply the two top numbers, the numerators. That is the top of the answer.
Multiply the two bottom numbers, the denominators. That is the bottom of the answer.
To multiply a whole number by a fraction, write the whole number as a fraction over 1 first: 3 is the same as 3/1.
Simplify the answer if its top and bottom divide by the same number. You can also do this before you multiply, by dividing a top number and a bottom number by the same thing, which keeps the numbers small.

Worked examples

What is 1/2 × 3/4?

1 × 3 = 3
2 × 4 = 8
1/2 × 3/4 = 3/8

1/2 × 3/4 means half of 3/4. Cut the bar into eighths and 3/4 is 6 of them. Half of 6 eighths is 3 eighths, which is what top times top and bottom times bottom gave.

What is 2/3 × 4/5?

2 × 4 = 8
3 × 5 = 15
2/3 × 4/5 = 8/15

8 and 15 share no factor but 1, so 8/15 is already in its simplest form.

What is 3 × 2/5?

3 × 2/5 = 3/1 × 2/5
3 × 2 = 6 and 1 × 5 = 5
3 × 2/5 = 6/5 = 1 1/5

Three lots of 2/5 is 2/5 + 2/5 + 2/5, which is 6 fifths. Writing 3 as 3/1 gives the same thing, because only the top is multiplied by 3 and the bottom stays 5. 6 fifths is one whole with 1 fifth over.

What is 2/3 × 3/4?

2/3 × 3/4 = 6/12 = 1/2
or first divide the 3s by 3: 2/1 × 1/4 = 2/4 = 1/2

Multiplying first gives 6/12, and dividing top and bottom by 6 makes it 1/2. Or simplify first: there is a 3 on the bottom of the first fraction and a 3 on the top of the second, so divide both by 3 before multiplying. The numbers stay small, and the answer is the same.

Why "of" means times, and why it works

Of has meant times since the times tables, where 3 lots of 4 is 3 × 4. It works the same with fractions. Half of 8 is 4, and 1/2 × 8 = 8/2 = 4.

Look at half of 3/4 on the bars. To take half of each quarter, cut every quarter into 2. Now the whole bar has 2 × 4 = 8 pieces, which is why the bottom numbers multiply. You keep 1 of the 2 halves of each of the 3 shaded quarters, so you keep 1 × 3 = 3 pieces, which is why the top numbers multiply. The answer is 3/8.

Multiplying by a fraction smaller than 1 makes a number smaller, because you are taking only part of it. 3/8 is less than 3/4. That surprises children who have learned that multiplying makes things bigger, and it is worth saying out loud.

Where mistakes happen

Making the bottoms the same first, out of habit from adding, and then keeping the bottom: 1/2 × 3/4 turns into 2/4 × 3/4 = 6/4. Multiplying needs no shared bottom, and the bottoms are multiplied too, so the answer is 3/8.
Multiplying a whole number into the top and the bottom. 3 × 2/5 is not 6/15, which is just 2/5 again. Only the top is multiplied, so the answer is 6/5.
Simplifying a top number against the other top number, or a bottom against a bottom. Simplifying before you multiply always pairs a top with a bottom.
Expecting a bigger answer, and changing a right one. Half of 3/4 has to be less than 3/4.

Questions people ask

Do the bottom numbers have to be the same?

No. That rule is for adding and taking away. For multiplying, any two fractions will do just as they are.

Should I simplify before or after multiplying?

Either way gives the same answer. Before keeps the numbers smaller, after is easier to remember. The Math Basics practice asks for the answer exactly as the multiplying gives it, 6/12 rather than 1/2, so that the two multiplications are what get practised.

What is cancelling?

Simplifying a top number and a bottom number before you multiply, as in 2/3 × 3/4 above. British teachers usually call it cancelling, and American books often say cross-canceling. It is ordinary simplifying, just done earlier.

How do I multiply mixed numbers?

Turn each one into an improper fraction first, then multiply as usual. 1 1/2 is 3/2, so 1 1/2 × 2/3 = 3/2 × 2/3 = 6/6, which is 1.