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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.

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.

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 ,

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 ,

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 . 

\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.

\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} 

\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

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).

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

The answer is

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.

The final answer is

Subtracting fractions

To subtract fractions they need to have the same denominator .

Step-by-step guide: Subtracting fractions

Example 2: subtracting fractions

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).

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

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

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.

We can now multiply the fractions.

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 .

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

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.

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 .

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:

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 .

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

See also: Mixed number to improper fraction

Example 6: improper fractions and mixed numbers

Write the following improper fraction as a mixed number:

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

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

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:

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.

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

The original fractions should be used in the final answer:

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

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

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 :

Three quarters of 48 :

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.

• 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 .

Practice fractions questions

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

2. Work out:

\frac{5}{7} of 42

3. Work out:

4. Work out:

5. Work out:

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

Fractions GCSE questions

1.  Without a calculator.

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

Give your answer as a mixed number.

2.  Without a calculator.

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

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?

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

• Percentages
• Comparing fractions, decimals and percentages

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Fraction Application Problem: GCSE Maths Question of the Week

Can your students solve a tricky GCSE question on fractions? There’s only one way to find out!

The latest edition of my GCSE Maths Question of the Week series is a lovely contextual question on fractions provided exclusively for my  Diagnostic Questions website by OCR, but suitable no matter which awarding body you are following.

Here is one suggestion for using this resource:

• Print out the worksheet for use as a starter in lessons, or a homework.
• Discuss your students answers and explanations, their reasons for the incorrect answers, and their choice of alternative wrong answers
• Use our data analytics to see how the rest of the country performed on this question, and view alternative student explanations
• Set your students the entire quiz that this question comes from as a follow-up activity
• Direct your students to the Topic section on my website, where they will find videos, worksheets and extension material on this topic.

All of this is free.

At our school we are using this once a week with our Year 11 classes. I really hope you and your students find it useful.

Try the  Question online View the Question Data and Explanations

Try the Quiz online View the Quiz Data

For all the questions in this series, please visit my  GCSE Maths Question of the Week   page.

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CPA - Problem solving: Bar model - Fraction of amounts - only Halves - Year 1

Subject: Mathematics

Age range: 5-7

Resource type: Worksheet/Activity

Last updated

26 August 2024

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Progression to one and two step word problems. Sentence stems provided to encourage children to verbalise value of the whole as well as the half.

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