How to multiply fractions

How to multiply fractions in 3 steps

Every fraction has two parts. The numerator is the top number and the denominator is the bottom number. In 3/4, the numerator is 3 and the denominator is 4.

1. Multiply the numerators. This gives the numerator of the answer.
2. Multiply the denominators. This gives the denominator of the answer.
3. Simplify. Divide the top and bottom by their greatest common factor, so the answer is in its lowest terms.

Example 1: two simple fractions

8 and 15 share no factor except 1, so 8/15 is already in its simplest form. As a check, 2/3 is about 0.667 and 4/5 is 0.8. Their product is about 0.533, and so is 8 ÷ 15.

Example 2: simplifying the answer

The greatest common factor of 6 and 36 is 6, so divide both by 6 to get 1/6. If you are not sure of the greatest common factor, our GCF Calculator finds it with working, or you can divide by small factors one at a time until nothing more divides.

Why multiplying fractions works

"Times" with fractions means "of". So 1/2 × 3/4 means half of three quarters. Picture a square split into 4 columns, with 3 of them shaded: that is 3/4. Now split the square in half across the middle and keep one half of the shaded part. The square is now cut into 2 × 4 = 8 small pieces, and 1 × 3 = 3 of them are in both. So half of three quarters is 3/8.

That is the whole rule. Multiplying the denominators counts how many small pieces the whole is cut into, and multiplying the numerators counts how many of those pieces you keep. The US Common Core standards teach it this way in grade 5, by tiling a rectangle with fractional side lengths.

It also explains something that surprises people: multiplying by a proper fraction (one less than 1) makes a number smaller. Half of something is less than the thing itself. Multiplying by a fraction bigger than 1, like 3/2, makes it larger.

Cancelling before you multiply

You can simplify before you multiply, which keeps the numbers small. Any numerator can cancel with any denominator, including the one in the other fraction. This is called cross cancelling.

Take example 2 again, 3/4 × 2/9. The 3 on top and the 9 below share a factor of 3, so they become 1 and 3. The 2 on top and the 4 below share a factor of 2, so they become 1 and 2.

Same answer, smaller numbers, and no simplifying at the end. Cancelling only works across a multiplication: you can't do it when adding or subtracting fractions.

How to multiply fractions with whole numbers

Write the whole number as a fraction over 1, then multiply as usual. This works because any number divided by 1 is itself, so 5 is the same as 5/1.

In practice, you just multiply the whole number by the numerator and keep the denominator. 15/8 is an improper fraction (the top is bigger than the bottom). 8 goes into 15 once with 7 left over, so it is 1 7/8 as a mixed number.

Finding a fraction of an amount

"2/5 of 30" means 2/5 × 30. Cancelling the 5 into the 30 first is the quickest route: 30 ÷ 5 = 6, and 6 × 2 = 12. Dividing by the denominator and then multiplying by the numerator is the same calculation done in a friendlier order.

How to multiply mixed numbers

A mixed number has a whole part and a fraction part, like 2 1/3. To multiply mixed numbers, turn each one into an improper fraction first. Multiply the whole number by the denominator, add the numerator, and put the result over the same denominator.

1. 2 1/3 becomes (2 × 3 + 1)/3 = 7/3.
2. 1 1/2 becomes (1 × 2 + 1)/2 = 3/2.
3. Multiply: 7/3 × 3/2. The 3s cancel, leaving 7/1 × 1/2 = 7/2.
4. Turn back into a mixed number: 2 goes into 7 three times with 1 left over, so the answer is 3 1/2.

More types of fraction multiplication

Three or more fractions

The rule doesn't change: multiply all the numerators and all the denominators. Cancelling first matters more here, because the numbers grow quickly. For 4/7 × 14/15 × 5/8, the 7 cancels into the 14 (leaving 2), the 5 into the 15 (leaving 3), and the 4 and the 2 into the 8 (leaving 1).

Without cancelling you get 280/840 and have to spot that 280 goes into 840 exactly three times. With cancelling, everything on top becomes 1 and the bottom becomes 3, so the answer is 1/3 straight away.

Negative fractions

Use the same sign rules as whole numbers: one negative gives a negative answer, and two negatives give a positive one. So −3/4 × 2/5 = −6/20 = −3/10, while −2/3 × −3/4 = 1/2.

Word problems

Look for "of", or for scaling something by a fraction. A recipe needs 3/4 of a cup of sugar and you want to make 2/3 of the recipe: 3/4 × 2/3 = 6/12 = 1/2 a cup. A rug is 5/6 of a metre long and 3/5 of a metre wide, so its area is 5/6 × 3/5 = 15/30 = 1/2 a square metre.

Multiplying fractions also turns up in probability. The chance of two independent events both happening is their probabilities multiplied, which our Probability course covers with tree diagrams.

How to divide fractions: keep, change, flip

Dividing fractions turns into multiplying. Keep the first fraction, change the ÷ to ×, and flip the second fraction upside down (its reciprocal). Then multiply as normal.

This makes sense: there are 6 eighths in three quarters. Only the second fraction flips, and only when you divide.

Multiplying vs adding fractions

Common mistakes

• Finding a common denominator first. It isn't wrong, but it isn't needed for multiplying and it makes the numbers bigger.
• Multiplying the whole number into the top and the bottom. 5 × 3/8 is 15/8, not 15/40. Writing 5 as 5/1 stops this.
• Splitting mixed numbers. Convert to improper fractions before multiplying.
• Cancelling two numbers on the same side. Cancel a numerator with a denominator, never two numerators with each other.
• Forgetting to simplify. Many teachers and exams expect the answer in its lowest terms.
• Flipping when multiplying. Keep, change, flip is only for dividing.

Fractions come up in almost every maths exam. In England, the national curriculum has pupils multiplying fractions by whole numbers in Year 5 and multiplying pairs of fractions in Year 6. For practice, try a short timed daily maths paper.

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Maths · AlgebraPartial fractions questions with worked answersPartial fractions questions with full answers: the two methods for finding constants, repeated factors, integrating with partial fractions, and practice.