GCSE Maths Number

Fractions

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 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

4 equal parts

12 equal parts

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

To add fractions they need to have the same denominator.

Work out:

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

The fractions have different denominators. In order to add fractions they need to have the same denominators.

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.

$\frac{35}{40}+\frac{24}{40}=\frac{35+24}{40}=\frac{59}{40}$

$\frac{59}{40}$

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.

$\frac{59}{40}=\frac{40+19}{40}=1\frac{19}{40}$

$1\frac{19}{40}$

Subtracting fractions

To subtract fractions they need to have the same denominator.

Example 2: subtracting fractions

Work out:

$\frac{7}{9}+\frac{1}{2}$

The fractions have different denominators.  In order to subtract fractions they need to be fractions with the same 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.

$\frac{14}{18}-\frac{9}{18}=\frac{14-9}{18}=\frac{5}{18}$

$\frac{5}{18}$

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.

Example 3: multiplying fractions

Work out:

$1\frac{2}{3}+\frac{3}{10}$

So that we can multiply fractions they 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.

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

$\frac{15}{30}$

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.

$\frac{15}{30}=\frac{1\times15}{2\times15}=\frac{1}{2}$

$\frac{1}{2}$

Dividing fractions

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

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.

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

$\frac{20}{35}$

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.

$\frac{20}{35}=\frac{4\times5}{7\times5}=\frac{4}{7}$

$\frac{4}{7}$

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 to add and subtract fractions.  We do this by multiplying both the numerator and the denominator by the same number.  This is very useful for adding fractions, subtracting fractions and ordering 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.

$\frac{12}{20}=\frac{3\times4}{5\times4}=\frac{3}{5}$

$\frac{3}{5}$

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

Improper fractions and mixed numbers

To 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.

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.

$3\frac{2}{5}$

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.

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 (bottom of the fraction).

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

$\frac{10}{30} \quad \quad \frac{12}{30} \quad \quad \frac{13}{30} \quad \quad \frac{14}{30}$

BUT - The original fractions should be used in the final answer.  The fractions have been simplified.

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

Fractions of amounts

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

Example 8: Fractions of amounts

Work out:

$\frac{3}{4} \text{ of £}48$

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

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

Alternatively you can think of it as 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:

$3\times12=36$

$£36$

Common misconceptions

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

• To multiply or divide mixed numbers they need to be fractions

If we try to multiply (or divide) mixed numbers it is easy not to do the full calculation.

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

$1\frac{1}{3}\times2\frac{1}{4}=1\times2+\frac{1}{2}\times{1}{4}=2\frac{1}{8}$ ✘

If you think about it the answer should be larger than one of the original numbers and it is not.

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

• Whole numbers 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

1. Write down these fractions in order of size from smallest to largest:

2. Work out:

\frac{5}{7} of 42

30

35

32

24
\frac{5}{7}\times42=\frac{5\times42}{7}=42\div7\times5=30

3. Work out:

\frac{3}{5}+\frac{2}{7}

\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}

4. Work out:

\frac{3}{4}-\frac{2}{9}

\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}

5. Work out:

\frac{1}{5}\times\frac{3}{8}

\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}

6. Work out the following, giving your answer as a fraction in its simplest form:

\frac{5}{6}\div\frac{2}{3}

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}\\\\ &=1\frac{1}{4}\end{aligned}

Fractions GCSE questions

1.  Without a calculator.

Work out

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

(3 marks)

\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}

(4 marks)

\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)

\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:

• Subtract fractions
• Multiply fractions
• Divide fractions
• Find equivalent fractions
• Convert between improper fractions and mixed numbers
• Order fractions
• Work out fractions of amounts

• Decimals
• Percentages
• Ratios

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Find out more about our GCSE maths revision programme.