Fractions

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Arithmetic Place value

This topic is relevant for:

GCSE Maths Number Fractions, Decimals and Percentages

Fractions

Here we will learn about fractions, including equivalent fractions and how to convert between improper fractions and mixed numbers. You will learn how to order fractions, how to calculate a fractions of an amount and how to add, subtract, multiply and divide fractions.

There are also fractions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are fractions?

Fractions are equal parts of a whole.

The denominator of a fraction (number below the line) shows how many equal parts the whole has been divided into. The numerator of a fraction (number above the line) shows how many of the equal parts we have.

E.g.

2 equal parts

One-half is shaded

4 equal parts

Three-quarters are shaded

12 equal parts

Seven-twelfths are shaded

Here we will learn about all the different ways we can use fractions.

Step-by-step guide: Numerator and denominator

Fractions worksheet

Get your free fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Fractions worksheet

Get your free fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Fractions less than 1

Lots of fractions you will come across are fractions less than 1. These are recognisable if the numerator is less than the denominator.

For example, here are some fractions which are less than 1,

\frac{1}{4}, \frac{3}{5}, \frac{11}{12}, \frac{99}{100}

Fractions which are greater than 1 can be written as improper fractions, where the numerator is greater than the denominator. Or they may be written as a mixed number, with an integer part and a fraction part.

For example, here are some fractions which are greater than 1,

\frac{5}{4}, \frac{27}{5}, 3\frac{1}{2}, 7\frac{3}{10}

Fraction arithmetic

Fraction arithmetic involves adding, subtracting, multiplying and dividing with fractions. There are techniques and skills you should learn and practise to help you with fraction arithmetic.

In order to add or subtract fractions they must have the same denominator (bottom number). If the denominators are the same then you can perform the addition or the subtraction to the numerators (top numbers).

For example,

$\frac{2}{5} + \frac{1}{5} = \frac{2+1}{5} = \frac{3}{5}$

$\frac{4}{7} - \frac{3}{7} = \frac{4-3}{7} = \frac{1}{7}$

If the denominators are not the same then you must first use equivalent fractions to give the fractions a common denominator.

For example,

$\frac{2}{3} + \frac{1}{6}$

$\frac{2}{3}$ is equivalent to $\frac{4}{6}$ therefore,

$\frac{2}{3} + \frac{1}{6} = \frac{4}{6} + \frac{1}{6} = \frac{4+1}{6} = \frac{5}{6}$

Step-by-step guide: Adding and subtracting fractions

To multiply two fractions together all you need to do is multiply the numerators (top numbers) together, and then multiply the denominators (bottom numbers) together.

For example,

$\frac{4}{5} \times \frac{2}{3} = \frac{4 \times 2}{5 \times 3} = \frac{8}{15}$

To divide by a fraction you should remember the following equivalent calculation rule: dividing by fraction ab \frac{a}{b} is the same as multiplying by fraction ba\frac{b}{a}

For example,

$\frac{2}{7} \div \frac{3}{4}$

Note here that $\div \frac{3}{4}$ is the same as $\times \frac{3}{4}$ , therefore

$\frac{2}{7} \div \frac{3}{4} = \frac{2}{7} \times \frac{4}{3} = \frac{2 \times 4}{7 \times 3} = \frac{8}{21}$

Step-by-step guide: Multiplying and dividing fractions

Adding fractions

To add fractions they need to have the same denominator.

Step-by-step guide: Adding fractions

Example 1: adding fractions

Work out:

\[\frac{7}{8}+\frac{3}{5}\]

The fractions have different denominators and in order to add fractions they need to have the same denominators.
We need to find a common denominator.

The LCM (the Lowest Common Multiple) of 8 and 5 is 40, so we change the fractions into equivalent fractions with a common denominator of 40.
(This is also known as the least common denominator).

\[\frac{7}{8}=\frac{7\times5}{8\times5}=\frac{35}{40}\]

\[\frac{3}{5}=\frac{3\times8}{5\times8}=\frac{24}{40}\]

We can now add the fractions as they have a common denominator.

\[\frac{35}{40}+\frac{24}{40}=\frac{35+24}{40}=\frac{59}{40}\]

The answer is

\[\frac{59}{40}\]

The answer is an improper fraction as the numerator is larger than the denominator.

It can be converted from an improper fraction to a mixed number.

\[\frac{59}{40}=\frac{40+19}{40}=1\frac{19}{40}\]

The final answer is

\[1\frac{19}{40}\]

Subtracting fractions

To subtract fractions they need to have the same denominator.

Step-by-step guide: Subtracting fractions

Example 2: subtracting fractions

Work out:

\[\frac{7}{9}+\frac{1}{2}\]

The fractions have different denominators and in order to add fractions they need to have the same denominators.
We need to find a common denominator.

The LCM (the Lowest Common Multiple) of 9 and 2 is 18, so we change the fractions into equivalent fractions with a common denominator of 18.
(This is also known as the least common denominator).

\[\frac{7}{9}=\frac{7\times2}{9\times2}=\frac{14}{18}\]

\[\frac{1}{2}=\frac{1\times9}{2\times9}=\frac{9}{18}\]

We can now subtract the fractions as they have a common denominator.

\[\frac{14}{18}-\frac{9}{18}=\frac{14-9}{18}=\frac{5}{18}\]

The final answer is

\[\frac{5}{18}\]

The final answer can not be simplified further. This is because 5 and 18 have no common factors other than 1.

Multiplying fractions

To multiply fractions we multiply the numerators and multiply the denominators.

Step-by-step guide: Multiplying fractions

Example 3: multiplying fractions

Work out:

\[1\frac{2}{3}+\frac{3}{10}\]

So that we can multiply the fractions they both need to be either proper fractions or improper fractions.
We need to convert the first number from a mixed number into an improper fraction.

\[1\frac{2}{3}=\frac{3}{3}+\frac{2}{3}=\frac{5}{3}\]

We can now multiply the fractions.

\[\frac{5}{3}\times\frac{3}{10}=\frac{5\times3}{3\times10}=\frac{15}{30}\]

The answer is

\[\frac{15}{30}\]

The answer can be simplified.
This is because 15 and 30 have common factor 15.

Since 15 is the HCF (Highest Common Factor) of the numerator and the denominator they can be cancelled by the common factor of 15.

\[\frac{15}{30}=\frac{1\times15}{2\times15}=\frac{1}{2}\]

The final answer is

\[\frac{1}{2}\]

Dividing fractions

To divide fractions we change the division to a multiplication and use the reciprocal of the second fraction.

Step-by-step guide: Dividing fractions

Example 4: dividing fractions

Work out:

\[\frac{2}{5}\div\frac{7}{10}\]

We can divide the fractions by changing the division to a multiplication and finding the reciprocal of the second fraction. When we find the reciprocal of a fraction we turn it upside down.

\[\frac{2}{5}\div\frac{7}{10}=\frac{2}{5}\times\frac{10}{7}=\frac{20}{35}\]

The answer is

\[\frac{20}{35}\]

The answer can be simplified.
This is because 20 and 35 have common factor 5.

Since 5 is the HCF (Highest Common Factor) of the numerator and the denominator they can be cancelled by the common factor of 5.

\[\frac{20}{35}=\frac{4\times5}{7\times5}=\frac{4}{7}\]

The final answer is

\[\frac{4}{7}\]

Equivalent fractions

Equivalent fractions are fractions that are the same size.
We use equivalent fractions to simplify a fraction by cancelling both the numerator and the denominator by the HCF (Highest Common Factor).

We can also use equivalent fractions to find a common denominator by multiplying both the numerator and the denominator by the same number. This is very useful for adding fractions, subtracting fractions and ordering fractions.

Step-by-step guide: Equivalent fractions

See also: Simplifying fractions

Example 5: equivalent fractions

Write the following fraction in its simplest terms:

\[\frac{12}{20}\]

The numerator (top number) is 12 and the denominator (bottom number) is 20.
They have a HCF(Highest Common Factor) of 4. So we can cancel the numerator and the denominator by 4.

\[\frac{12}{20}=\frac{3\times4}{5\times4}=\frac{3}{5}\]

The answer is

\[\frac{3}{5}\]

The final answer cannot be simplified further.
This is because 3 and 5 have no common factors other than 1.

Improper fractions and mixed numbers

Improper fractions are fractions where the numerator is larger than the denominator. Fractions where the numerator is smaller than the denominator are known as proper fractions.

A mixed number has a whole number part and a fractional part.

Step-by-step guide: Improper fractions to mixed numbers

Example 6: improper fractions and mixed numbers

Write the following improper fraction as a mixed number:

\[\frac{17}{5}\]

The denominator can go into the numerator 3 times with a remainder of 2.

\[\frac{17}{5}=\frac{3\times5+2}{5}=3\frac{2}{5}\]

This means the whole number part is 3 and the fractional part is two-fifths.

The final answer is

\[3\frac{2}{5}\]

The final answer can not be simplified further.
This is because 2 and 5 have no common factors other than 1.

Ordering fractions

To be able to write fractions in order of size, usually from smallest to largest, we need to be able to compare them. To be able to compare fractions it is easier if the fractions have a common denominator. We can also convert the fractions to decimals to put them in order.

Step-by-step guide: Ordering fractions

See also: Comparing fractions

Example 7: ordering fractions

Write these fractions in order of size:

\[\frac{2}{5} \quad \quad \frac{1}{3} \quad \quad \frac{7}{15} \quad \quad \frac{13}{30} \]

The fractions have different denominators. So that we can compare them it is useful to convert them so that they all have the same denominator.

3, 5, 15 and 30 are factors of 30. We can use 30 as the common denominator.

\[\frac{2}{5}=\frac{2\times6}{5\times6}=\frac{12}{30}\]

\[\frac{1}{3}=\frac{1\times10}{3\times10}=\frac{10}{30}\]

\[\frac{7}{15}=\frac{7\times2}{15\times2}=\frac{14}{30}\]

The denominators are all the same. We can compare the numerators to put the fractions in size order.

\[\frac{12}{30} \quad \quad \frac{10}{30} \quad \quad \frac{14}{30} \quad \quad \frac{13}{30}\]

In order

\[\frac{10}{30} \quad \quad \frac{12}{30} \quad \quad \frac{13}{30} \quad \quad \frac{14}{30} \]

The original fractions should be used in the final answer:

\[\frac{1}{3} \quad \quad \frac{2}{5} \quad \quad \frac{13}{30} \quad \quad \frac{7}{15} \]

Fractions of amounts

To find a fraction of an amount we can multiply the fraction and the amount together.

Step-by-step guide: Fractions of amounts

Example 8: Fractions of amounts

Work out:

\[\frac{3}{4} \text{ of £}48\]

The “of” means that we multiply the fraction and the amount.

\[\frac{3}{4}\times48=\frac{3\times48}{4}=48\times3\div4=36\]

Alternatively you can think of it as first finding one quarter by dividing the amount by 4.

Then finding three quarters by multiplying by 3.

One quarter of 48:

\[48\div4=12\]

Three quarters of 48:

\[3\times12=36\]

The final answer is

\[£36\]

Common misconceptions

Common denominator
To add or subtract fractions they need to have a common denominator

Multiply or divide mixed numbers

To multiply or divide mixed numbers we should first convert them to proper of improper fractions.

We have to multiply ALL of the first number by ALL of the second number.

\[1\frac{1}{3}\times2\frac{1}{4}=\frac{4}{3}\times\frac{9}{4}=\frac{4\times 9}{3\times4}=\frac{36}{12}=3\] ✔

Whole numbers and fractions

Whole numbers can be written as fractions if needed.

To make 3 into a fraction we can use a denominator of 1.

\[3=\frac{3}{1}\]

Practice fractions questions

\frac{1}{2} \quad \quad \frac{7}{12} \quad \quad \frac{2}{3} \quad \quad \frac{3}{4}

\frac{2}{3} \quad \quad \frac{7}{12} \quad \quad \frac{1}{2} \quad \quad \frac{3}{4}

\frac{1}{2} \quad \quad \frac{2}{3} \quad \quad \frac{3}{4} \quad \quad \frac{7}{12}

\frac{1}{2} \quad \quad \frac{3}{4} \quad \quad \frac{2}{3} \quad \quad \frac{7}{12}

\frac{3\times3}{4\times3}=\frac{9}{12} \quad \quad \frac{7}{12} \quad \quad \frac{1\times6}{2\times6}=\frac{6}{12} \quad \quad \frac{2\times4}{3\times4}=\frac{8}{12}

30

35

32

24

\frac{5}{7}\times42=\frac{5\times42}{7}=42\div7\times5=30

\frac{31}{35}

\frac{5}{11}

\frac{31}{70}

1\frac{3}{35}

\begin{aligned}&\frac{3}{5}+\frac{2}{7} \\\\ &=\frac{3\times7}{5\times7}+\frac{2\times5}{7\times5}\\\\ &=\frac{21}{35}+\frac{10}{35}\\\\ &=\frac{21+10}{35}\\\\ &=\frac{31}{35}\end{aligned}

\frac{19}{36}

\frac{35}{36}

\frac{21}{36}

\frac{1}{5}

\begin{aligned}&\frac{3}{4}-\frac{2}{9}\\\\ &=\frac{3\times9}{4\times9}-\frac{2\times4}{9\times4}\\\\ &=\frac{27}{36}-\frac{8}{36}\\\\ &=\frac{27-8}{36}\\\\ &=\frac{19}{36}\end{aligned}

\frac{3}{40}

\frac{3}{13}

\frac{4}{40}

\frac{1}{10}

\begin{aligned}&\frac{1}{5}\times\frac{3}{8}\\\\ &=\frac{1\times3}{5\times8}\\\\ &=\frac{3}{40}\end{aligned}

1\frac{1}{4}

1\frac{3}{12}

\frac{5}{9}

\frac{1}{4}

\begin{aligned}&\frac{5}{6}\div\frac{2}{3}\\\\ &=\frac{5}{6}\times\frac{3}{2}\\\\ &=\frac{5\times3}{6\times2}\\\\ &=\frac{15}{12}\\\\ &=\frac{3}{12}\\\\ &\text{So,} \quad \frac{5}{6}\div\frac{2}{3} =1\frac{1}{4} \end{aligned}

Fractions GCSE questions

1. Without a calculator.

Work out

$\frac{5}{7}+\frac{3}{8}$

Give your answer as a mixed number.

(3 marks)

Show answer

\frac{40}{56}+\frac{21}{56}

(1)

\frac{61}{56}

(1)

1\frac{5}{56}

(1)

2. Without a calculator.

Work out

$8\frac{1}{3}\div2\frac{3}{4}$

Give your answer as a mixed number.

(4 marks)

Show answer

\frac{25}{3}\div\frac{11}{4}

(1)

\frac{25}{3}\times\frac{4}{11}

(1)

\frac{100}{33}

(1)

3\frac{1}{33}

(1)

3. Lee has a bag containing only red apples and green apples.

$\frac{2}{9}$ of the apples are red.

If there are $6$ red apples, how many apples are green?

(3 marks)

Show answer

\frac{2}{9}=\frac{6}{27}

(1)

1-\frac{6}{27}=\frac{21}{27}

(1)

21

(1)

Learning checklist

You have now learned how to:

Add fractions

Subtract fractions

Multiply fractions

Divide fractions

Find equivalent fractions

Convert between improper fractions and mixed numbers

Order fractions

Work out fractions of amounts

The next lessons are

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