# How To Multiply Fractions: Step By Step Guide For Primary School Teachers & Pupils

Learning how to multiply fractions is quite a complex procedure and requires children to have a solid understanding of fractions. For this reason, it isn’t introduced until Upper Key Stage 2.

Here we include a step-by-step guide on how to multiply fractions, including worked examples and practice questions. You can use these in the classroom to test your pupils’ knowledge of fractions so you have everything you need when it comes to comparing fractions, decimals, and percentages.

### When do pupils learn how to multiply fractions in the national curriculum?

Children begin learning this in Year 5, where they are expected to “multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams”.

The non-statutory guidance in Year 5 advises that “pupils connect multiplication by a fraction to using fractions as operators (fractions of), and to division, building on work from previous years. This relates to scaling by simple fractions, including fractions > 1.”

In Year 6, pupils should “multiply simple pairs of proper fractions, writing the answer in its simplest form [for example \frac{1}{4} × \frac{1}{2} = \frac{1}{8} ]”.

The non-statutory guidance in Year 6 advises that “pupils should use a variety of images to support their understanding of multiplication with fractions. This follows earlier work about fractions as operators (fractions of), as numbers, and as equal parts of objects, for example as parts of a rectangle.”

### How to multiply fractions: step by step

When multiplying two fractions, we need to multiply the numerators (top numbers) and the denominators (bottom numbers). We therefore need to be secure with our times tables (multiplication facts). You may also wish to take a look at our article on simplifying fractions as this can aid your pupils with the process.

Let’s begin by multiplying unit fractions.

Step 1

\frac{1}{2} × \frac{1}{4} is the same as saying \frac{1}{2} of \frac{1}{4} (‘of’ can mean multiply). This bar model shows \frac{1}{4} .

Step 2

When the quarter is made half the size, the fraction of the whole is \frac{1}{8} . When a fraction is multiplied by a proper fraction, it makes it smaller.

The bar model now shows \frac{1}{4} halved. This is written as \frac{1}{4} × \frac{1}{2} = \frac{1}{8} (when multiplying fractions, multiply the numerators and the denominators).

The same method applies when multiplying non-unit fractions.

Step 1

When multiplying \frac{2}{3} × \frac{3}{5} , we want to find two-thirds of three-fifths. The bar model below shows \frac{3}{5} .

Step 2

We then find \frac{2}{3} of \frac{3}{5} by dividing it into 3 equal parts and shading 2 of them. This is shown on the diagram below in darker blue.

The new shaded section ( \frac{2}{3} of \frac{3}{5} ) is clearly shown here as \frac{6}{15} of the whole (which we could simplify to \frac{2}{5} ) - so the final answer to \frac{2}{3} × \frac{3}{5} is \frac{6}{15} (as previously mentioned, when multiplying fractions, multiply the numerator and denominator).

### How to multiply fractions with whole numbers

When multiplying fractions by whole numbers, we can use repeated addition, just as we would when multiplying whole numbers by whole numbers.

Step 1

3 × \frac{2}{9} is the same as 3 groups of \frac{2}{9} , or \frac{2}{9} + \frac{2}{9} + \frac{2}{9} , as shown below.

So 3 × \frac{2}{9} = \frac{6}{9} . To find the answer without a diagram, multiply the numerator by the whole number.

Step 2
Sometimes the answer is greater than one whole. The diagram below shows 3 × \frac{3}{5}, or \frac{3}{5} + \frac{3}{5} + \frac{3}{5} .

The diagram shows the answer as 1\frac{4}{5} . As previously discussed, we can also multiply the numerator by the whole number - 3 × \frac{3}{5} is \frac{9}{5} (which is equivalent to 1\frac{4}{5} but presented as an improper fraction instead of a mixed number).

If the denominator of the fraction is a factor of the whole number, the multiplication can be completed as a fraction of an amount. For example, 6 × \frac{2}{3} is the same as \frac{2}{3} of 6 which is 4 . If we use the method of multiplying the numerator by the whole number, we’d get the same answer as 6 × \frac{2}{3} = \frac{12}{3} = 4 .

### How to multiply improper fractions or mixed numbers

To multiply mixed numbers by whole numbers, you can either:

A) Partition the mixed number and multiply each part by the whole number.

B) Convert it into an improper fraction first and then multiply it.

For example, let’s calculate 1\frac{2}{3} × 2 using each method:

1. 1 × 2 = 2 and \frac{2}{3} × 2 = \frac{4}{3} , or 1\frac{1}{3} . Add each part to make 3\frac{1}{3} .
1. 1\frac{2}{3} = \frac{5}{3} and \frac{5}{3} × 2 = \frac{10}{3} , or 3\frac{1}{3} as a mixed number.

See 2 lots of 1\frac{2}{3} (or \frac{5}{3} ) represented in a diagram, where each row represents one whole.

To multiply mixed numbers by fractions, convert the mixed number to an improper fraction and then multiply the fractions as usual.

For example, 2\frac{1}{2} × \frac{5}{6} = \frac{5}{2} × \frac{5}{6} = \frac{25}{12} or 2 and \frac{1}{12} .

To multiply mixed numbers by mixed numbers, convert them both to improper fractions and then multiply as usual.

For example, 1\frac{4}{5} × 1\frac{2}{3} = \frac{9}{5} × \frac{5}{3} = \frac{45}{15} or 3

### How to multiply fractions with the same denominator

Multiplying fractions with the same denominators works exactly the same as the examples above. Let’s take \frac{3}{5} × \frac{2}{5} as an example.

Step 1: Multiply the numerators: 3 × 2 = 6 .

Step 2: Multiply the denominators: 5 × 5 = 25 .

### How to multiply fractions with different denominators

Multiplying fractions with the same denominators works exactly the same as the examples above. Let’s take \frac{3}{4} × \frac{5}{7} as an example:

Step 1: Multiply the numerators: - 3 × 5 = 15 .

Step 2: Multiply the denominators: 4 × 7 = 28 .

### Rules on how to multiply fractions

Multiply the numerators together and the denominators together. If one of the fractions is a mixed number, convert it to an improper fraction then follow the same method. If one number is an integer, place that digit over 1 and follow the same method (e.g. \frac{3}{4} × 9 = \frac{3}{4} × \frac{9}{1} = \frac{27}{4} ).

### Teaching ideas and activities for multiplying fractions

Find our teaching resources with fractions examples and multiplying fractions worksheets to test your pupils with:

You may also wish to check our articles on fraction word problems and fraction questions.

### How to multiply fractions: tips for teachers

Of all the calculations children are required to learn with fractions, multiplication is procedurally the simplest, as all it requires is multiplication of the numerators and denominators respectively. However, if you only approach this procedurally - without the children understanding why they’re doing what they’re doing - it is likely they will forget which procedure they need for which calculation. Always introduce this concept pictorially first.

### How to multiply fractions: worked examples and answers

1. \frac{3}{4} × \, \_ = \frac{9}{20}
We know we need to multiply the numerators together and the denominators together when multiplying fractions, so firstly we can look at the numerators: 3 × \, \_ = 9 ? The missing digit here is 3 . The denominators show 4 × \, \_ = 20 , so the missing digit here is 5 . The final answer is therefore \frac{3}{5} .
1. The length of a day on Earth is 24 hours. The length of a day on Planet X is 27\frac{3}{4} times the length of a day on Earth. What is the length of a day on Planet X in hours?
To calculate 24 × 27\frac{3}{4} , let’s partition the mixed number, multiply the parts, and then add them back together. 24 × 27 = 648 , and 24 × \frac{3}{4} (or \frac{3}{4} of 24) is 18, then 648 + 18 = 666
1. Calculate 16 × 1\frac{3}{8}
This can be solved in two ways - either by a) converting to an improper fraction first and then multiplying or b) partitioning the mixed number.
a) 16 × \frac{11}{8} = \frac{176}{8} = 22
b) 16 × 1 = 16 and 16 × \frac{3}{8} = \frac{3}{8} of 16 = 6 , then 16 + 6 = 22

### How to multiply fractions: practice questions and answers

1. \frac{5}{6} × 4
Answer: \frac{20}{6} or 3\frac{1}{3} or equivalent
1. Rafe had a piece of rope that was \frac{3}{5} of a metre long. He cut it and gave half to Sammi. What fraction of a metre did he give to Sammi?
1. What is the missing fraction? \, \_ × \frac{3}{4} = \frac{1}{4}
1. Leo has \frac{1}{2} a pizza. He cuts it into 4 equal slices and eats 3 of them. What fraction of the original pizza has he eaten?
1. Multiply \frac{3}{4} by \frac{5}{6}

What are the steps to multiply fractions with whole numbers?

The steps to multiply fractions with whole numbers are:
1. Multiply the numerator by the whole number.
2. Convert the improper fraction to a mixed number if necessary.

How can fractions with different denominators be multiplied?

Fractions with different denominators can be multiplied by:
1. Multiplying the numerators together.
2. Multiplying the denominators together.

What is the process for multiplying mixed fractions?

The process for multiplying mixed fractions is the same as multiplying proper fractions, but it would be easier to convert the mixed number into an improper fraction first.

Is there a specific method for multiplying fractions with the same denominator?

The method for multiplying fractions with the same denominator is the same as it would be for any other pairs of fractions - multiply the numerators together and multiply the denominators together.

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