How to multiply fractions

Fractions can be multiplied together, which means we multiply numerator by numerator and denominator by denominator. It is the simplest operation there is for fractions, simpler than adding.

If we have, say, $\tfrac{3}{8}$ and $\tfrac{2}{6}$, we multiply the two numerators and the two denominators directly:

$$\frac{3}{8} \times \frac{2}{6} = \frac{3 \times 2}{8 \times 6} = \frac{6}{48}$$

That is the right answer, but it does not look neat. The first thing we do is simplify. $6$ divides both $6$ and $48$, so we divide both the numerator and the denominator by $6$:

$$\frac{6}{48} = \frac{1}{8}$$

So: $\tfrac{3}{8} \times \tfrac{2}{6} = \tfrac{1}{8}$.

I always try to find a number that divides both the numerator and the denominator, so we can simplify. If you do not see one straight away, look for $2$, $3$ or $5$. Most fractions that come out of a multiplication can be simplified by one of those.

Why is multiplying simpler than adding?

It is worth noticing that the rule is different from the one we use when we add fractions. To add, we need a common denominator. To multiply, we just multiply numerator by numerator and denominator by denominator, in the same spirit as the rule "plus times plus gives plus".

The reason is that multiplying is about "a fraction of a fraction". If we take $\tfrac{2}{6}$ and ask "how much is $\tfrac{3}{8}$ of that?", we need both the numerators and the denominators multiplied together. It is a completely different move from putting two portions together.

Simplify before you multiply, if you can

If the numbers are big, you often save work by simplifying first. Take $\tfrac{4}{9} \times \tfrac{3}{8}$. If we multiply directly, we get:

$$\frac{4}{9} \times \frac{3}{8} = \frac{12}{72}$$

and then we have to simplify $\tfrac{12}{72}$. But if we had spotted that $4$ and $8$ share the factor $4$, and $3$ and $9$ share the factor $3$, we could have simplified crosswise before multiplying:

$$\frac{4}{9} \times \frac{3}{8} = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$$

It is the same answer, just with smaller numbers to juggle. This is not a must, but it is a good habit to look out for.

Common mistakes

• "I need to find a common denominator too, like when I add." No. That is only for adding, not for multiplying. When you multiply, you just multiply numerator by numerator and denominator by denominator.
• "I multiply the numerators, but not the denominators." Both have to be multiplied. Numerator by numerator, denominator by denominator. If you forget one of them, you have actually scaled up a fraction instead of multiplying.
• "I forget to simplify at the end." An answer like $\tfrac{6}{48}$ is correct, but $\tfrac{1}{8}$ is neater. It is always a good idea to simplify as much as possible, so the answer is easy to compare with everyone else's.

Where to next

If you need to divide fractions, it sits very close to what we did here. You flip one of the fractions over and then multiply, in exactly the same way as in this guide. That walkthrough is in how to divide fractions. Adding works differently and needs a common denominator, see how to add fractions, and the simplifying step at the end is covered in equivalent fractions. The foundation underneath it all is in What is a fraction?.