Fractions

What you'll learn

What the top and bottom numbers in a fraction mean.
How to add and subtract fractions using a common denominator.
How to multiply and divide fractions.
How to work with mixed numbers and give answers in simplest form.

The basics of fractions

A fraction shows part of a whole. For example, $35\frac{3}{5}$ means 3 parts out of 5 equal parts.

Definition

Numerator and denominator

The numerator is the top number in a fraction. It tells you how many parts you have.
The denominator is the bottom number. It tells you how many equal parts the whole has been split into.

For $35\frac{3}{5}$ , the numerator is 3 and the denominator is 5.

$The fraction \frac{3}{5} means 3 of 5 equal parts are selected.$

Simplest form

A fraction is in simplest form when the numerator and denominator cannot both be divided by the same whole number bigger than 1.

For example, $68\frac{6}{8}$ can be simplified because 6 and 8 can both be divided by 2:

$The diagram shows why \frac{6}{8} has the same value as \frac{3}{4} when pairs of eighths are grouped.$

68=34\frac{6}{8} = \frac{3}{4}

Key Idea

Simplest form

At GCSE, you should usually leave a fraction in simplest form unless the question says otherwise.

Example

Simplifying a fraction

Simplify $1218\frac{12}{18}$ .

Look for a number that divides both 12 and 18. They can both be divided by 6.
Divide the numerator and denominator by 6:
$1218=12÷618÷6\frac{12}{18} = \frac{12 \div 6}{18 \div 6}$
Write the simplified answer:
$1218=23\frac{12}{18} = \frac{2}{3}$

Equivalent fractions

Equivalent fractions are fractions with the same value, even though they look different.

For example:

$The three fraction bars show that \frac{1}{2}, \frac{2}{4} and \frac{3}{6} cover the same amount of the whole.$

12=24=36\frac{1}{2} = \frac{2}{4} = \frac{3}{6}

You make equivalent fractions by multiplying or dividing the numerator and denominator by the same number.

Adding fractions

To add fractions with the same denominator, just add the numerators.

For example:

27+37=57\frac{2}{7} + \frac{3}{7} = \frac{5}{7}

If the denominators are different, first change the fractions so they have a common denominator.

Definition

Common denominator

A common denominator is a shared bottom number for two or more fractions. It lets you add or subtract the fractions.

Example

Adding fractions with different denominators

Work out $16+23\frac{1}{6} + \frac{2}{3}$ .

$Changing thirds into sixths makes it clear that \frac{1}{6}+\frac{2}{3} is adding sixth-sized parts.$

The denominators are 6 and 3, so choose 6 as a common denominator.
Change $23\frac{2}{3}$ into sixths by multiplying the top and bottom by 2:
$23=46\frac{2}{3} = \frac{4}{6}$
Add the fractions:
$16+46=56\frac{1}{6} + \frac{4}{6} = \frac{5}{6}$
The answer is $56\frac{5}{6}$ .

Common Mistake

Adding the denominators

Do not add the bottom numbers. For example, $16+23\frac{1}{6} + \frac{2}{3}$ is not $39\frac{3}{9}$ . Make a common denominator first.

Subtracting fractions

Subtracting fractions works almost the same as adding fractions.

If the denominators are different, make them the same first. Then subtract the numerators.

Example

Subtracting fractions

Work out $58−14\frac{5}{8} - \frac{1}{4}$ .

The quarter is first shown as two eighths so it can be subtracted from five eighths.

The denominators are 8 and 4, so use 8 as the common denominator.
Change $14\frac{1}{4}$ into eighths:
$14=28\frac{1}{4} = \frac{2}{8}$
Subtract the numerators:
$58−28=38\frac{5}{8} - \frac{2}{8} = \frac{3}{8}$
The answer is $38\frac{3}{8}$ .

Tip

Common denominator shortcut

If one denominator is already a multiple of the other, use the bigger one. For example, with 4 and 12, use 12.

Multiplying fractions

Multiplying fractions is often the most straightforward fraction operation.

Key Idea

Multiply straight across

To multiply fractions, multiply the numerators together and multiply the denominators together.

So:

ab×cd=a×cb×d\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

Example

Multiplying two fractions

Work out $25×34\frac{2}{5} \times \frac{3}{4}$ .

$An area model shows that taking \frac{2}{5} of \frac{3}{4} gives 6 small parts out of 20.$

Multiply the numerators:
$\times 3 = 6$
Multiply the denominators:
$\times 4 = 20$
Write the fraction:
$25×34=620\frac{2}{5} \times \frac{3}{4} = \frac{6}{20}$
Simplify by dividing the top and bottom by 2:
$620=310\frac{6}{20} = \frac{3}{10}$

Dividing fractions

To divide by a fraction, multiply by its reciprocal.

Definition

Reciprocal

The reciprocal of a fraction is the fraction turned upside down. For example, the reciprocal of $37\frac{3}{7}$ is $73\frac{7}{3}$ .

A useful phrase is: keep, change, flip.

Keep the first fraction.
Change divide to multiply.
Flip the second fraction.

Example

Dividing by a fraction

Work out $29÷45\frac{2}{9} \div \frac{4}{5}$ .

$The keep-change-flip diagram shows that only the second fraction is turned into its reciprocal.$

Keep the first fraction, change divide to multiply, and flip the second fraction:
$29÷45=29×54\frac{2}{9} \div \frac{4}{5} = \frac{2}{9} \times \frac{5}{4}$
Multiply the numerators and denominators:
$29×54=1036\frac{2}{9} \times \frac{5}{4} = \frac{10}{36}$
Simplify by dividing the top and bottom by 2:
$1036=518\frac{10}{36} = \frac{5}{18}$

Common Mistake

Flipping the wrong fraction

When dividing fractions, only flip the second fraction. Do not flip the first one.

Mixed numbers

A mixed number has a whole number and a fraction, such as $1231\frac{2}{3}$ .

An improper fraction has a numerator bigger than the denominator, such as $53\frac{5}{3}$ .

To calculate with mixed numbers, it is usually easiest to change them into improper fractions first.

Changing a mixed number into an improper fraction

For $2152\frac{1}{5}$ :

$The mixed number 2\frac{1}{5} is two wholes and one fifth, which makes 11 fifths altogether.$

Multiply the whole number by the denominator: 2 times 5 is 10.
Add the numerator: 10 + 1 = 11.
Keep the same denominator: $115\frac{11}{5}$ .

So:

215=1152\frac{1}{5} = \frac{11}{5}

Example

Multiplying mixed numbers

Work out $112×1231\frac{1}{2} \times 1\frac{2}{3}$ . Give your answer as a mixed number.

Change $1121\frac{1}{2}$ into an improper fraction:
$112=321\frac{1}{2} = \frac{3}{2}$
Change $1231\frac{2}{3}$ into an improper fraction:
$123=531\frac{2}{3} = \frac{5}{3}$
Multiply the fractions:
$32×53=156\frac{3}{2} \times \frac{5}{3} = \frac{15}{6}$
Simplify $156\frac{15}{6}$ to $52\frac{5}{2}$ .
Change $52\frac{5}{2}$ into a mixed number:
$52=212\frac{5}{2} = 2\frac{1}{2}$

Adding mixed numbers

When adding mixed numbers, you can add the whole numbers and fractions separately, or change everything into improper fractions.

For Grade 3 questions, adding separately is often quick.

Example

Adding mixed numbers

Work out $213+1162\frac{1}{3} + 1\frac{1}{6}$ .

Add the whole numbers:
$2 + 1 = 3$
Add the fractions:
$13+16\frac{1}{3} + \frac{1}{6}$
Change $13\frac{1}{3}$ into sixths:
$13=26\frac{1}{3} = \frac{2}{6}$
Add the fractions:
$26+16=36=12\frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$
Put the whole number and fraction together:
$3123\frac{1}{2}$

Exam technique

In the exam

Check the operation carefully: add, subtract, multiply, or divide.
For adding or subtracting, find a common denominator before combining the fractions.
For multiplying or dividing mixed numbers, change them into improper fractions first.
Always simplify your answer, and change to a mixed number if the question asks for one.

Self review

Check yourself

Can you explain why $23\frac{2}{3}$ is the same as $46\frac{4}{6}$ ?
When dividing by a fraction, which fraction do you flip?
Can you change $1341\frac{3}{4}$ into an improper fraction?

Fractions

What you'll learn

What the top and bottom numbers in a fraction mean.
How to add and subtract fractions using a common denominator.
How to multiply and divide fractions.
How to work with mixed numbers and give answers in simplest form.

The basics of fractions

A fraction shows part of a whole. For example, $35\frac{3}{5}$ means 3 parts out of 5 equal parts.

Definition

Numerator and denominator

The numerator is the top number in a fraction. It tells you how many parts you have.
The denominator is the bottom number. It tells you how many equal parts the whole has been split into.

For $35\frac{3}{5}$ , the numerator is 3 and the denominator is 5.

$The fraction \frac{3}{5} means 3 of 5 equal parts are selected.$

Simplest form

A fraction is in simplest form when the numerator and denominator cannot both be divided by the same whole number bigger than 1.

For example, $68\frac{6}{8}$ can be simplified because 6 and 8 can both be divided by 2:

$The diagram shows why \frac{6}{8} has the same value as \frac{3}{4} when pairs of eighths are grouped.$

68=34\frac{6}{8} = \frac{3}{4}

Key Idea

Simplest form

At GCSE, you should usually leave a fraction in simplest form unless the question says otherwise.

Example

Simplifying a fraction

Simplify $1218\frac{12}{18}$ .

Look for a number that divides both 12 and 18. They can both be divided by 6.
Divide the numerator and denominator by 6:
$1218=12÷618÷6\frac{12}{18} = \frac{12 \div 6}{18 \div 6}$
Write the simplified answer:
$1218=23\frac{12}{18} = \frac{2}{3}$

Equivalent fractions

Equivalent fractions are fractions with the same value, even though they look different.

For example:

$The three fraction bars show that \frac{1}{2}, \frac{2}{4} and \frac{3}{6} cover the same amount of the whole.$

12=24=36\frac{1}{2} = \frac{2}{4} = \frac{3}{6}

You make equivalent fractions by multiplying or dividing the numerator and denominator by the same number.

Adding fractions

To add fractions with the same denominator, just add the numerators.

For example:

27+37=57\frac{2}{7} + \frac{3}{7} = \frac{5}{7}

If the denominators are different, first change the fractions so they have a common denominator.

Definition

Common denominator

A common denominator is a shared bottom number for two or more fractions. It lets you add or subtract the fractions.

Example

Adding fractions with different denominators

Work out $16+23\frac{1}{6} + \frac{2}{3}$ .

$Changing thirds into sixths makes it clear that \frac{1}{6}+\frac{2}{3} is adding sixth-sized parts.$

The denominators are 6 and 3, so choose 6 as a common denominator.
Change $23\frac{2}{3}$ into sixths by multiplying the top and bottom by 2:
$23=46\frac{2}{3} = \frac{4}{6}$
Add the fractions:
$16+46=56\frac{1}{6} + \frac{4}{6} = \frac{5}{6}$
The answer is $56\frac{5}{6}$ .

Common Mistake

Adding the denominators

Do not add the bottom numbers. For example, $16+23\frac{1}{6} + \frac{2}{3}$ is not $39\frac{3}{9}$ . Make a common denominator first.

Subtracting fractions

Subtracting fractions works almost the same as adding fractions.

If the denominators are different, make them the same first. Then subtract the numerators.

Example

Subtracting fractions

Work out $58−14\frac{5}{8} - \frac{1}{4}$ .

The quarter is first shown as two eighths so it can be subtracted from five eighths.

The denominators are 8 and 4, so use 8 as the common denominator.
Change $14\frac{1}{4}$ into eighths:
$14=28\frac{1}{4} = \frac{2}{8}$
Subtract the numerators:
$58−28=38\frac{5}{8} - \frac{2}{8} = \frac{3}{8}$
The answer is $38\frac{3}{8}$ .

Tip

Common denominator shortcut

If one denominator is already a multiple of the other, use the bigger one. For example, with 4 and 12, use 12.

Multiplying fractions

Multiplying fractions is often the most straightforward fraction operation.

Key Idea

Multiply straight across

To multiply fractions, multiply the numerators together and multiply the denominators together.

So:

ab×cd=a×cb×d\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

Example

Multiplying two fractions

Work out $25×34\frac{2}{5} \times \frac{3}{4}$ .

$An area model shows that taking \frac{2}{5} of \frac{3}{4} gives 6 small parts out of 20.$

Multiply the numerators:
$\times 3 = 6$
Multiply the denominators:
$\times 4 = 20$
Write the fraction:
$25×34=620\frac{2}{5} \times \frac{3}{4} = \frac{6}{20}$
Simplify by dividing the top and bottom by 2:
$620=310\frac{6}{20} = \frac{3}{10}$

Dividing fractions

To divide by a fraction, multiply by its reciprocal.

Definition

Reciprocal

The reciprocal of a fraction is the fraction turned upside down. For example, the reciprocal of $37\frac{3}{7}$ is $73\frac{7}{3}$ .

A useful phrase is: keep, change, flip.

Keep the first fraction.
Change divide to multiply.
Flip the second fraction.

Example

Dividing by a fraction

Work out $29÷45\frac{2}{9} \div \frac{4}{5}$ .

$The keep-change-flip diagram shows that only the second fraction is turned into its reciprocal.$

Keep the first fraction, change divide to multiply, and flip the second fraction:
$29÷45=29×54\frac{2}{9} \div \frac{4}{5} = \frac{2}{9} \times \frac{5}{4}$
Multiply the numerators and denominators:
$29×54=1036\frac{2}{9} \times \frac{5}{4} = \frac{10}{36}$
Simplify by dividing the top and bottom by 2:
$1036=518\frac{10}{36} = \frac{5}{18}$

Common Mistake

Flipping the wrong fraction

When dividing fractions, only flip the second fraction. Do not flip the first one.

Mixed numbers

A mixed number has a whole number and a fraction, such as $1231\frac{2}{3}$ .

An improper fraction has a numerator bigger than the denominator, such as $53\frac{5}{3}$ .

To calculate with mixed numbers, it is usually easiest to change them into improper fractions first.

Changing a mixed number into an improper fraction

For $2152\frac{1}{5}$ :

$The mixed number 2\frac{1}{5} is two wholes and one fifth, which makes 11 fifths altogether.$

Multiply the whole number by the denominator: 2 times 5 is 10.
Add the numerator: 10 + 1 = 11.
Keep the same denominator: $115\frac{11}{5}$ .

So:

215=1152\frac{1}{5} = \frac{11}{5}

Example

Multiplying mixed numbers

Work out $112×1231\frac{1}{2} \times 1\frac{2}{3}$ . Give your answer as a mixed number.

Change $1121\frac{1}{2}$ into an improper fraction:
$112=321\frac{1}{2} = \frac{3}{2}$
Change $1231\frac{2}{3}$ into an improper fraction:
$123=531\frac{2}{3} = \frac{5}{3}$
Multiply the fractions:
$32×53=156\frac{3}{2} \times \frac{5}{3} = \frac{15}{6}$
Simplify $156\frac{15}{6}$ to $52\frac{5}{2}$ .
Change $52\frac{5}{2}$ into a mixed number:
$52=212\frac{5}{2} = 2\frac{1}{2}$

Adding mixed numbers

When adding mixed numbers, you can add the whole numbers and fractions separately, or change everything into improper fractions.

For Grade 3 questions, adding separately is often quick.

Example

Adding mixed numbers

Work out $213+1162\frac{1}{3} + 1\frac{1}{6}$ .

Add the whole numbers:
$2 + 1 = 3$
Add the fractions:
$13+16\frac{1}{3} + \frac{1}{6}$
Change $13\frac{1}{3}$ into sixths:
$13=26\frac{1}{3} = \frac{2}{6}$
Add the fractions:
$26+16=36=12\frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$
Put the whole number and fraction together:
$3123\frac{1}{2}$

Exam technique

In the exam

Check the operation carefully: add, subtract, multiply, or divide.
For adding or subtracting, find a common denominator before combining the fractions.
For multiplying or dividing mixed numbers, change them into improper fractions first.
Always simplify your answer, and change to a mixed number if the question asks for one.

Self review

Check yourself

Can you explain why $23\frac{2}{3}$ is the same as $46\frac{4}{6}$ ?
When dividing by a fraction, which fraction do you flip?
Can you change $1341\frac{3}{4}$ into an improper fraction?

Fractions

What you'll learn

The basics of fractions

Simplest form

Equivalent fractions

Adding fractions

Subtracting fractions

Multiplying fractions

Dividing fractions

Mixed numbers

Changing a mixed number into an improper fraction

Adding mixed numbers

Recap questions

You've reached the end

Fractions

What you'll learn

The basics of fractions

Simplest form

Equivalent fractions

Adding fractions

Subtracting fractions

Multiplying fractions

Dividing fractions

Mixed numbers

Changing a mixed number into an improper fraction

Adding mixed numbers

Recap questions

You've reached the end

Fractions

What you'll learn

The basics of fractions

Simplest form

Equivalent fractions

Adding fractions

Subtracting fractions

Multiplying fractions

Dividing fractions

Mixed numbers

Changing a mixed number into an improper fraction

Adding mixed numbers

Recap questions

You've reached the end

Fractions Revision Guide

Fractions

What you'll learn

The basics of fractions

Simplest form

Equivalent fractions

Adding fractions

Subtracting fractions

Multiplying fractions

Dividing fractions

Mixed numbers

Changing a mixed number into an improper fraction

Adding mixed numbers

Recap questions

You've reached the end

Fractions Revision Guide