Multiplying and dividing fractions step by step, with worked examples

To multiply two fractions, multiply the numerators together and the denominators together, then simplify: 3/4 × 2/5 = 6/20 = 3/10.

Division uses the same rule once the second fraction is turned upside down. This guide works through each case in turn: a fraction times a whole number, a fraction times a fraction, cancelling before multiplying, mixed numbers and division, then sets out what pupils learn in Year 5 and Year 6 in England and in grades 5 and 6 in the United States.

The rule for multiplying two fractions

A fraction times a fraction is the simplest of the four operations with fractions, because there is no common denominator to find. Multiply the top numbers to get the new numerator, multiply the bottom numbers to get the new denominator, and simplify the result if you can.

Take 3/4 × 2/5. The numerators give 3 × 2 = 6 and the denominators 4 × 5 = 20, so the product is 6/20. Both 6 and 20 can be divided by 2, which leaves 3/10. The US Common Core standards for fractions in grade 5 write the general rule as (a/b) × (c/d) = (ac)/(bd), with (2/3) × (4/5) = 8/15 as an example.

A picture shows why it works. 3/4 × 2/5 means three quarters of two fifths. Cut a rectangle into fifths one way and into quarters the other, and you get 20 small pieces; three quarters of two fifths of the rectangle covers 6 of them, so 6/20. The grade 5 standards use exactly this model: a rectangle with fractional side lengths, whose area is the product of the two sides.

When both fractions are less than 1, the answer is smaller than either of them: taking part of a part always leaves less than you started with. That gives a quick check on any result.

Multiplying whole numbers and fractions

A whole number can be written as a fraction over 1: 6 is 6/1. So 2/3 × 6 becomes 2/3 × 6/1 = 12/3, which is 4. In practice you only multiply the numerator, 2 × 6 = 12, and the denominator stays 3.

This is the same as finding a fraction of an amount. Two thirds of 6 is 6 ÷ 3 × 2 = 4, and the Common Core standards describe the product of a/b and a whole number q in those terms, as a × q ÷ b. Dividing first keeps the numbers small: 3/8 of 40 is 40 ÷ 8 = 5, then 5 × 3 = 15. Knowing your multiplication facts by heart does most of the work here.

When the result is an improper fraction, turn it into a mixed number. 3/4 × 5 = 15/4, and 15 ÷ 4 is 3 remainder 3, so the answer is 3 3/4. In England this case comes first: the national curriculum asks Year 5 pupils to multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams.

Simplify before multiplying to keep the numbers small

With bigger numbers, multiplying first and simplifying afterwards gets heavy. 8/9 × 15/16 gives 120/144, and it is not obvious that both numbers can be divided by 24. The quicker route is to cancel common factors before you multiply: a numerator and a denominator can be divided by the same number, even when they sit in different fractions.

• 8 and 16 share the factor 8: the 8 becomes 1 and the 16 becomes 2.
• 15 and 9 share the factor 3: the 15 becomes 5 and the 9 becomes 3.
• What is left is 1/3 × 5/2 = 5/6, already in its simplest form.

This works because the order of a multiplication does not matter, so 8/9 × 15/16 is the same as 8/16 × 15/9, and each of those can be simplified on its own. Cancelling only works across a multiplication sign. In 8/9 + 15/16 nothing can be cancelled, because adding fractions needs a common denominator first.

Cancelling also matches what the curriculum asks for at the end of primary school: Year 6 pupils multiply simple pairs of proper fractions and write the answer in its simplest form.

Mixed numbers, turned into improper fractions first

A mixed number such as 2 1/2 cannot be multiplied as it stands. The usual mistake is to multiply the whole parts and the fractions separately, which turns 2 1/2 × 1 1/3 into 2 1/6. Convert each mixed number into an improper fraction first.

1. Multiply the whole number by the denominator and add the numerator: 2 1/2 = (2 × 2 + 1)/2 = 5/2, and 1 1/3 = (1 × 3 + 1)/3 = 4/3.
2. Multiply as usual: 5/2 × 4/3 = 20/6.
3. Simplify and convert back: 20/6 = 10/3 = 3 1/3.

A quick estimate catches the slip. Multiplying 2 1/2 by a number greater than 1 must give more than 2 1/2, so 2 1/6 cannot be right, while 3 1/3 can. The grade 5 standards make the same point: multiplying by a fraction greater than 1 gives a product greater than the number you started with, and multiplying by a fraction less than 1 gives a smaller one.

Cancelling works with mixed numbers too, once they are improper fractions. In 5/2 × 4/3, the 2 and the 4 share the factor 2, which gives 5/1 × 2/3 = 10/3 straight away.

Dividing fractions by multiplying by the reciprocal

To divide by a fraction, multiply by its reciprocal, the same fraction turned upside down. The reciprocal of 3/4 is 4/3, so 2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9. The Common Core standards for grade 6 use this example and write the general rule as (a/b) ÷ (c/d) = ad/bc.

You can check any division with a multiplication: 8/9 × 3/4 = 24/36 = 2/3, which takes you back to where you started. The same check explains why dividing by a unit fraction makes a number bigger. 4 ÷ 1/5 = 20, because there are 20 fifths in 4, and 20 × 1/5 = 4.

Dividing a fraction by a whole number follows the same rule, since the reciprocal of 2 is 1/2: 1/3 ÷ 2 = 1/3 × 1/2 = 1/6. Sharing a third between two people gives each of them a sixth. With mixed numbers, convert first, then turn over only the second fraction: 2 1/4 ÷ 1 1/2 = 9/4 × 2/3 = 18/12 = 3/2 = 1 1/2. If you need the answer as a decimal, divide the numerator by the denominator, here 3 ÷ 2 = 1.5, as in our guide to converting fractions, decimals and percentages.

When pupils learn to multiply and divide fractions

In England, the national curriculum for mathematics spreads the topic over the last two years of primary school and key stage 3. In the United States, the Common Core standards follow a similar order across grades 5 and 6.

In England, the Year 6 objectives stop at dividing a fraction by a whole number; dividing by another fraction, such as 2/3 ÷ 3/4, comes with the four operations at key stage 3. For what the end-of-primary tests look like, see our guide to the Year 6 maths SATs.

Checking your answer, and the mistakes to avoid

Most errors with fractions come from mixing up the rules of the different operations. Four checks catch nearly all of them:

• No common denominator to multiply. Finding one first only adds work: 1/2 × 1/3 is 1/6 directly.
• Whole numbers go over 1. In 3/4 × 5, the 5 multiplies the numerator only; multiplying the denominator as well would give 15/20, which is just 3/4 again.
• Only the second fraction is turned over. In 2/3 ÷ 3/4, flipping 2/3 instead of 3/4 gives 9/8, not 8/9.
• Mixed numbers become improper fractions first. Multiplying the parts separately gives an answer that is too small.

Then look at the size of the result. Multiplying by a fraction less than 1 makes a number smaller, dividing by a fraction less than 1 makes it bigger. If 3/4 × 2/5 comes out larger than 3/4, or 4 ÷ 1/5 comes out smaller than 4, go back over the working.

Frequently asked questions

How do you multiply two fractions?

Multiply the numerators together and the denominators together, then simplify: 3/4 × 2/5 = 6/20 = 3/10.

How do you multiply a fraction by a whole number?

Multiply the numerator by the whole number and keep the denominator: 2/3 × 6 = 12/3 = 4.

How do you divide one fraction by another?

Multiply the first fraction by the reciprocal of the second: 2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9.

How do you multiply mixed numbers?

Turn them into improper fractions, multiply, simplify, then convert back: 2 1/2 × 1 1/3 = 5/2 × 4/3 = 10/3 = 3 1/3.

When do children learn to multiply fractions in England?

In Year 5 they multiply fractions by whole numbers; in Year 6 they multiply pairs of proper fractions and divide fractions by whole numbers.