Operations on fractions Revision Notes for AQA GCSE Maths

Operations on fractions

This topic covers the four main operations you can perform with fractions: adding, subtracting, multiplying and dividing. Understanding these operations is essential for GCSE maths, and you need to be able to perform them accurately without a calculator.

Adding or subtracting fractions

When adding or subtracting fractions, the key principle is that both fractions must have the same denominator before you can combine them.

infoNote

The denominator is the bottom number of a fraction, while the numerator is the top number. Having the same denominator means the fractions represent parts of the same-sized whole, making them possible to add or subtract directly.

Method for adding or subtracting fractions

Write both fractions as equivalent fractions with the same denominator
- Find the lowest common multiple (LCM) of the denominators
- This becomes your common denominator
Add or subtract the numerators only
- The numerators are the numbers on top of the fractions
- Keep the common denominator unchanged
Simplify your answer if possible
- Check if your answer can be written in its simplest form

lightbulbExample

Worked Example: Adding fractions

To work out $\frac{1}{2} + \frac{3}{10}$ :

Step 1: Find the LCM of 2 and 10 The LCM of 2 and 10 is 10

Step 2: Convert to equivalent fractions $\frac{1}{2} = \frac{5}{10}$

Step 3: Add the numerators $\frac{5}{10} + \frac{3}{10} = \frac{8}{10}$

Step 4: Simplify $\frac{8}{10} = \frac{4}{5}$

lightbulbExample

Worked Example: Subtracting fractions

To work out $\frac{8}{9} - \frac{1}{6}$ :

Step 1: Find the LCM of 9 and 6 The LCM of 9 and 6 is 18

Step 2: Convert to equivalent fractions $\frac{8}{9} = \frac{16}{18}$ and $\frac{1}{6} = \frac{3}{18}$

Step 3: Subtract the numerators $\frac{16}{18} - \frac{3}{18} = \frac{13}{18}$

Multiplying fractions

Multiplying fractions is more straightforward than adding or subtracting because you don't need to find a common denominator.

infoNote

Unlike addition and subtraction, multiplication of fractions doesn't require equivalent fractions with the same denominator. This makes the process much simpler once you understand the method.

Method for multiplying fractions

Write any whole numbers as fractions
- Put the whole number over 1 (e.g., 3 = 3/1)
Multiply the numerators together
- Multiply all the top numbers
Multiply the denominators together
- Multiply all the bottom numbers
Simplify the result
- Cancel down to the simplest form if possible

lightbulbExample

Worked Example: Multiplying fractions

To work out $\frac{2}{5} \times \frac{7}{10}$ :

Step 1: Multiply numerators $2 \times 7 = 14$

Step 2: Multiply denominators $5 \times 10 = 50$

Step 3: Form the result $\frac{14}{50}$

Step 4: Simplify $\frac{14}{50} = \frac{7}{25}$ (dividing both by 2)

lightbulbExample

Worked Example: Multiplying with whole numbers

To work out $3 \times \frac{2}{11}$ :

Step 1: Write 3 as a fraction $3 = \frac{3}{1}$

Step 2: Multiply $\frac{3}{1} \times \frac{2}{11} = \frac{6}{11}$

Dividing fractions

Division of fractions uses a special technique that transforms the division into multiplication.

infoNote

The key to dividing fractions is remembering the phrase "flip and multiply". You flip the second fraction (find its reciprocal) and then multiply instead of dividing.

Method for dividing fractions

Write any whole numbers as fractions
- Put the whole number over 1
Turn the second fraction 'upside down'
- This means swap the numerator and denominator
- This is called finding the reciprocal
Change the division sign (÷) to multiplication (×)
Multiply the numerators and multiply the denominators
- Follow the same method as multiplying fractions

lightbulbExample

Worked Example: Dividing fractions

To work out $\frac{2}{5} \div \frac{4}{3}$ :

Step 1: Turn $\frac{4}{3}$ upside down Reciprocal of $\frac{4}{3}$ is $\frac{3}{4}$

Step 2: Change ÷ to × $\frac{2}{5} \times \frac{3}{4}$

Step 3: Multiply $\frac{2 \times 3}{5 \times 4} = \frac{6}{20}$

Step 4: Simplify $\frac{6}{20} = \frac{3}{10}$

lightbulbExample

Worked Example: Dividing with whole numbers

To work out $6 \div \frac{2}{3}$ :

Step 1: Write 6 as a fraction $6 = \frac{6}{1}$

Step 2: Flip $\frac{2}{3}$ to get $\frac{3}{2}$

Step 3: Multiply $\frac{6}{1} \times \frac{3}{2} = \frac{18}{2} = 9$

Important exam tips

chatImportant

Watch out for common mistakes!

You don't have to simplify your final answer unless the question specifically asks for the answer in its simplest form
You can compare and order fractions by converting them to equivalent fractions with the same denominator
Always check your working - it's easy to make arithmetic errors with fractions

bookmarkSummary

Key Points to Remember:

For addition/subtraction: find a common denominator first
For multiplication: multiply straight across
For division: flip the second fraction and multiply
Always look for opportunities to simplify your answer

Remember!

bookmarkSummary

Essential Fraction Operations Summary:

Adding and subtracting fractions requires finding a common denominator using the lowest common multiple (LCM) of the denominators
Multiplying fractions is done by multiplying numerators together and denominators together - no common denominator needed
Dividing fractions involves flipping the second fraction upside down and then multiplying
Equivalent fractions with the same denominator make it easier to compare and order fractions
Practice these operations regularly as they form the foundation for more complex fraction work in GCSE maths

Operations on fractions (AQA GCSE Maths): Revision Notes