Fractions

What you'll learn

How to add and subtract fractions using a common denominator.
How to multiply and divide fractions accurately.
How to change between mixed numbers and improper fractions.
How to simplify your final answer and choose the right form.

Fraction basics

A fraction tells you how many equal parts of a whole you have.

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

A fraction is in simplest form when the numerator and denominator have no common factor apart from 1. For example, $610\frac{6}{10}$ simplifies to $35\frac{3}{5}$ because both 6 and 10 divide by 2.

Example

Simplifying a fraction

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

Find a common factor of 12 and 18. Both divide by 6.
Divide the numerator and denominator by 6.
$1218=12÷618÷6=23\frac{12}{18}=\frac{12\div 6}{18\div 6}=\frac{2}{3}$
The answer is $23\frac{2}{3}$ because 2 and 3 have no common factor apart from 1.

Tip

Quick simplification check

If both numbers are even, divide by 2 first. If the digits add to a multiple of 3, the number divides by 3.

Adding and subtracting fractions

To add or subtract fractions, the denominators must be the same. This is called using a common denominator.

Key Idea

Adding and subtracting fractions

Make the denominators the same, then add or subtract the numerators. Do not add or subtract the denominators.

A good common denominator is often the lowest common multiple of the denominators. The diagram shows how different-sized fraction pieces can be changed into twelfths before adding.

Fraction strips showing one quarter plus one sixth as twelfths

Adding fractions with different denominators

Example

Adding fractions

Work out $18+34\frac{1}{8}+\frac{3}{4}$ .

The denominators are 8 and 4. Use 8 as the common denominator because fourths can be changed into eighths.
Rewrite $34\frac{3}{4}$ as eighths.
$34=68\frac{3}{4}=\frac{6}{8}$
Add the numerators and keep the denominator the same.
$18+68=78\frac{1}{8}+\frac{6}{8}=\frac{7}{8}$
The answer is $78\frac{7}{8}$ .

Subtracting fractions with different denominators

Example

Subtracting fractions

Work out $56−14\frac{5}{6}-\frac{1}{4}$ .

Find a common denominator for 6 and 4. The lowest common denominator is 12.
Change both fractions into twelfths.
$56=101214=312\frac{5}{6}=\frac{10}{12} \qquad \frac{1}{4}=\frac{3}{12}$
Subtract the numerators.
$1012−312=712\frac{10}{12}-\frac{3}{12}=\frac{7}{12}$
The answer is $712\frac{7}{12}$ .

Common Mistake

Adding the denominators

A very common mistake is writing $15+25=310\frac{1}{5}+\frac{2}{5}=\frac{3}{10}$ . The denominator stays as 5, so the correct answer is $35\frac{3}{5}$ .

Multiplying fractions

Multiplying fractions is usually more direct than adding or subtracting. You multiply the numerators together, then multiply the denominators together.

Key Idea

Multiplying fractions

For multiplication: numerator times numerator, denominator times denominator.

Example

Multiplying two fractions

Work out $23×910\frac{2}{3}\times\frac{9}{10}$ .

Multiply the numerators.
$2×9=182\times 9=18$
Multiply the denominators.
$3×10=303\times 10=30$
Write the fraction and simplify.
$1830=35\frac{18}{30}=\frac{3}{5}$
The answer is $35\frac{3}{5}$ .

You can sometimes simplify before multiplying. This is called cancelling. It works because multiplication can be done in any order.

Tip

Cancel before multiplying

In $34×89\frac{3}{4}\times\frac{8}{9}$ , you can cancel 8 with 4 first, and cancel 3 with 9 first. This keeps the numbers smaller.

Common Mistake

Cancelling across addition

You may cancel when fractions are being multiplied, but not when they are being added or subtracted. For example, do not cancel in $34+89\frac{3}{4}+\frac{8}{9}$ .

Dividing fractions

To divide by a fraction, multiply by its reciprocal.

Definition

Reciprocal

The reciprocal of a fraction is made by swapping its numerator and denominator. The reciprocal of $25\frac{2}{5}$ is $52\frac{5}{2}$ .

The method is often remembered as keep, change, flip: keep the first fraction, change divide to multiply, flip the second fraction.

$Diagram showing keep change flip for fraction division$

Example

Dividing fractions

Work out $47÷23\frac{4}{7}\div\frac{2}{3}$ .

Keep the first fraction the same.
Change the division sign to multiplication and flip the second fraction.
$47÷23=47×32\frac{4}{7}\div\frac{2}{3} = \frac{4}{7}\times\frac{3}{2}$
Multiply the numerators and denominators.
$4×37×2=1214\frac{4\times 3}{7\times 2}=\frac{12}{14}$
Simplify the fraction.
$1214=67\frac{12}{14}=\frac{6}{7}$
The answer is $67\frac{6}{7}$ .

Common Mistake

Never divide by zero

A denominator can never be zero, and you cannot divide by a fraction equal to zero. Division by zero is undefined.

Mixed numbers and improper fractions

A mixed number has a whole number part and a fraction part, such as $1341\frac{3}{4}$ .

An improper fraction has a numerator larger than or equal to its denominator, such as $74\frac{7}{4}$ .

Definition

Changing a mixed number to an improper fraction

To change a mixed number into an improper fraction: multiply the whole number by the denominator, add the numerator, then keep the same denominator.

For example:

134=1×4+34=741\frac{3}{4}=\frac{1\times 4+3}{4}=\frac{7}{4}

You should usually change mixed numbers into improper fractions before multiplying or dividing.

Example

Multiplying with mixed numbers

Work out $125×1131\frac{2}{5}\times1\frac{1}{3}$ . Give your answer as a mixed number.

Change each mixed number into an improper fraction.
$125=75113=431\frac{2}{5}=\frac{7}{5} \qquad 1\frac{1}{3}=\frac{4}{3}$
Multiply the fractions.
$75×43=2815\frac{7}{5}\times\frac{4}{3}=\frac{28}{15}$
Change $2815\frac{28}{15}$ into a mixed number. There is one whole 15 in 28, with 13 left over.
The answer is $113151\frac{13}{15}$ .

Dividing with a mixed number

Example

Dividing a mixed number by a fraction

Work out $112÷351\frac{1}{2}\div\frac{3}{5}$ .

Change the mixed number into an improper fraction.
$112=321\frac{1}{2}=\frac{3}{2}$
Keep, change, flip.
$32÷35=32×53\frac{3}{2}\div\frac{3}{5} = \frac{3}{2}\times\frac{5}{3}$
Multiply and simplify.
$3×52×3=156=52\frac{3\times 5}{2\times 3}=\frac{15}{6}=\frac{5}{2}$
Change $52\frac{5}{2}$ into a mixed number.
The answer is $2122\frac{1}{2}$ .

Adding mixed numbers

For addition and subtraction, you can either change to improper fractions or deal with the whole numbers and fractions separately. At this level, changing to improper fractions is often safest.

Example

Adding mixed numbers

Work out $214+1232\frac{1}{4}+1\frac{2}{3}$ .

Change both mixed numbers into improper fractions.
$214=94123=532\frac{1}{4}=\frac{9}{4} \qquad 1\frac{2}{3}=\frac{5}{3}$
Find a common denominator. For fourths and thirds, use 12.
Rewrite both fractions.
$94=271253=2012\frac{9}{4}=\frac{27}{12} \qquad \frac{5}{3}=\frac{20}{12}$
Add the numerators.
$2712+2012=4712\frac{27}{12}+\frac{20}{12}=\frac{47}{12}$
Change $4712\frac{47}{12}$ into a mixed number.
The answer is $311123\frac{11}{12}$ .

Exam technique

In the exam

Check the operation first: adding and subtracting need a common denominator; multiplying and dividing do not.
If you see a mixed number in a multiplication or division question, change it to an improper fraction before you start.
Always simplify your answer, and if the question asks for a mixed number, convert any improper fraction at the end.

Self review

Check yourself

Can you explain why $13+14\frac{1}{3}+\frac{1}{4}$ is not $27\frac{2}{7}$ ?
What is the reciprocal of $58\frac{5}{8}$ ?
How would you change $2352\frac{3}{5}$ into an improper fraction?

Fractions

What you'll learn

How to add and subtract fractions using a common denominator.
How to multiply and divide fractions accurately.
How to change between mixed numbers and improper fractions.
How to simplify your final answer and choose the right form.

Fraction basics

A fraction tells you how many equal parts of a whole you have.

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

Example

Simplifying a fraction

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

Find a common factor of 12 and 18. Both divide by 6.
Divide the numerator and denominator by 6.
$1218=12÷618÷6=23\frac{12}{18}=\frac{12\div 6}{18\div 6}=\frac{2}{3}$
The answer is $23\frac{2}{3}$ because 2 and 3 have no common factor apart from 1.

Tip

Quick simplification check

If both numbers are even, divide by 2 first. If the digits add to a multiple of 3, the number divides by 3.

Adding and subtracting fractions

To add or subtract fractions, the denominators must be the same. This is called using a common denominator.

Key Idea

Adding and subtracting fractions

Make the denominators the same, then add or subtract the numerators. Do not add or subtract the denominators.

A good common denominator is often the lowest common multiple of the denominators. The diagram shows how different-sized fraction pieces can be changed into twelfths before adding.

Fraction strips showing one quarter plus one sixth as twelfths

Adding fractions with different denominators

Example

Adding fractions

Work out $18+34\frac{1}{8}+\frac{3}{4}$ .

The denominators are 8 and 4. Use 8 as the common denominator because fourths can be changed into eighths.
Rewrite $34\frac{3}{4}$ as eighths.
$34=68\frac{3}{4}=\frac{6}{8}$
Add the numerators and keep the denominator the same.
$18+68=78\frac{1}{8}+\frac{6}{8}=\frac{7}{8}$
The answer is $78\frac{7}{8}$ .

Subtracting fractions with different denominators

Example

Subtracting fractions

Work out $56−14\frac{5}{6}-\frac{1}{4}$ .

Find a common denominator for 6 and 4. The lowest common denominator is 12.
Change both fractions into twelfths.
$56=101214=312\frac{5}{6}=\frac{10}{12} \qquad \frac{1}{4}=\frac{3}{12}$
Subtract the numerators.
$1012−312=712\frac{10}{12}-\frac{3}{12}=\frac{7}{12}$
The answer is $712\frac{7}{12}$ .

Common Mistake

Adding the denominators

A very common mistake is writing $15+25=310\frac{1}{5}+\frac{2}{5}=\frac{3}{10}$ . The denominator stays as 5, so the correct answer is $35\frac{3}{5}$ .

Multiplying fractions

Multiplying fractions is usually more direct than adding or subtracting. You multiply the numerators together, then multiply the denominators together.

Key Idea

Multiplying fractions

For multiplication: numerator times numerator, denominator times denominator.

Example

Multiplying two fractions

Work out $23×910\frac{2}{3}\times\frac{9}{10}$ .

Multiply the numerators.
$2×9=182\times 9=18$
Multiply the denominators.
$3×10=303\times 10=30$
Write the fraction and simplify.
$1830=35\frac{18}{30}=\frac{3}{5}$
The answer is $35\frac{3}{5}$ .

You can sometimes simplify before multiplying. This is called cancelling. It works because multiplication can be done in any order.

Tip

Cancel before multiplying

In $34×89\frac{3}{4}\times\frac{8}{9}$ , you can cancel 8 with 4 first, and cancel 3 with 9 first. This keeps the numbers smaller.

Common Mistake

Cancelling across addition

You may cancel when fractions are being multiplied, but not when they are being added or subtracted. For example, do not cancel in $34+89\frac{3}{4}+\frac{8}{9}$ .

Dividing fractions

To divide by a fraction, multiply by its reciprocal.

Definition

Reciprocal

The reciprocal of a fraction is made by swapping its numerator and denominator. The reciprocal of $25\frac{2}{5}$ is $52\frac{5}{2}$ .

The method is often remembered as keep, change, flip: keep the first fraction, change divide to multiply, flip the second fraction.

$Diagram showing keep change flip for fraction division$

Example

Dividing fractions

Work out $47÷23\frac{4}{7}\div\frac{2}{3}$ .

Keep the first fraction the same.
Change the division sign to multiplication and flip the second fraction.
$47÷23=47×32\frac{4}{7}\div\frac{2}{3} = \frac{4}{7}\times\frac{3}{2}$
Multiply the numerators and denominators.
$4×37×2=1214\frac{4\times 3}{7\times 2}=\frac{12}{14}$
Simplify the fraction.
$1214=67\frac{12}{14}=\frac{6}{7}$
The answer is $67\frac{6}{7}$ .

Common Mistake

Never divide by zero

A denominator can never be zero, and you cannot divide by a fraction equal to zero. Division by zero is undefined.

Mixed numbers and improper fractions

A mixed number has a whole number part and a fraction part, such as $1341\frac{3}{4}$ .

An improper fraction has a numerator larger than or equal to its denominator, such as $74\frac{7}{4}$ .

Definition

Changing a mixed number to an improper fraction

To change a mixed number into an improper fraction: multiply the whole number by the denominator, add the numerator, then keep the same denominator.

For example:

134=1×4+34=741\frac{3}{4}=\frac{1\times 4+3}{4}=\frac{7}{4}

You should usually change mixed numbers into improper fractions before multiplying or dividing.

Example

Multiplying with mixed numbers

Work out $125×1131\frac{2}{5}\times1\frac{1}{3}$ . Give your answer as a mixed number.

Change each mixed number into an improper fraction.
$125=75113=431\frac{2}{5}=\frac{7}{5} \qquad 1\frac{1}{3}=\frac{4}{3}$
Multiply the fractions.
$75×43=2815\frac{7}{5}\times\frac{4}{3}=\frac{28}{15}$
Change $2815\frac{28}{15}$ into a mixed number. There is one whole 15 in 28, with 13 left over.
The answer is $113151\frac{13}{15}$ .

Dividing with a mixed number

Example

Dividing a mixed number by a fraction

Work out $112÷351\frac{1}{2}\div\frac{3}{5}$ .

Change the mixed number into an improper fraction.
$112=321\frac{1}{2}=\frac{3}{2}$
Keep, change, flip.
$32÷35=32×53\frac{3}{2}\div\frac{3}{5} = \frac{3}{2}\times\frac{5}{3}$
Multiply and simplify.
$3×52×3=156=52\frac{3\times 5}{2\times 3}=\frac{15}{6}=\frac{5}{2}$
Change $52\frac{5}{2}$ into a mixed number.
The answer is $2122\frac{1}{2}$ .

Adding mixed numbers

For addition and subtraction, you can either change to improper fractions or deal with the whole numbers and fractions separately. At this level, changing to improper fractions is often safest.

Example

Adding mixed numbers

Work out $214+1232\frac{1}{4}+1\frac{2}{3}$ .

Change both mixed numbers into improper fractions.
$214=94123=532\frac{1}{4}=\frac{9}{4} \qquad 1\frac{2}{3}=\frac{5}{3}$
Find a common denominator. For fourths and thirds, use 12.
Rewrite both fractions.
$94=271253=2012\frac{9}{4}=\frac{27}{12} \qquad \frac{5}{3}=\frac{20}{12}$
Add the numerators.
$2712+2012=4712\frac{27}{12}+\frac{20}{12}=\frac{47}{12}$
Change $4712\frac{47}{12}$ into a mixed number.
The answer is $311123\frac{11}{12}$ .

Exam technique

In the exam

Check the operation first: adding and subtracting need a common denominator; multiplying and dividing do not.
If you see a mixed number in a multiplication or division question, change it to an improper fraction before you start.
Always simplify your answer, and if the question asks for a mixed number, convert any improper fraction at the end.

Self review

Check yourself

Can you explain why $13+14\frac{1}{3}+\frac{1}{4}$ is not $27\frac{2}{7}$ ?
What is the reciprocal of $58\frac{5}{8}$ ?
How would you change $2352\frac{3}{5}$ into an improper fraction?

Fractions

What you'll learn

Fraction basics

Adding and subtracting fractions

Adding fractions with different denominators

Subtracting fractions with different denominators

Multiplying fractions

Dividing fractions

Mixed numbers and improper fractions

Dividing with a mixed number

Adding mixed numbers

Recap questions

You've reached the end

Fractions

What you'll learn

Fraction basics

Adding and subtracting fractions

Adding fractions with different denominators

Subtracting fractions with different denominators

Multiplying fractions

Dividing fractions

Mixed numbers and improper fractions

Dividing with a mixed number

Adding mixed numbers

Recap questions

You've reached the end

Fractions

What you'll learn

Fraction basics

Adding and subtracting fractions

Adding fractions with different denominators

Subtracting fractions with different denominators

Multiplying fractions

Dividing fractions

Mixed numbers and improper fractions

Dividing with a mixed number

Adding mixed numbers

Recap questions

You've reached the end

Fractions Revision Guide

Fractions

What you'll learn

Fraction basics

Adding and subtracting fractions

Adding fractions with different denominators

Subtracting fractions with different denominators

Multiplying fractions

Dividing fractions

Mixed numbers and improper fractions

Dividing with a mixed number

Adding mixed numbers

Recap questions

You've reached the end

Fractions Revision Guide