Arithmetic with Fractions

Topic Overview

For your GCSE maths exam, you will need to add, subtract, multiple, divide, simplify and convert fractions. This guide will introduce you to the basic principles of fractions as well as providing several worked examples of how to answer examination questions.

A fraction is defined as the part of a whole. Within a fraction you have two values; the numerator and the denominator.

The numerator is the top number of the fraction, which tells you how many parts of a whole you have.

The denominator is the bottom number, which tells you how many parts the whole is divided into.

For example, when looking at the fraction:

1/2

1 is the numerator and 2 is the denominator.

Key Concepts

In the new linear GCSE Maths paper, you will be required to recognise certain properties of fractions and solve related calculations. According to the Edexcel Revision Checklist for the linear GCSE Maths paper, you will be required to:

  • Multiply and divide fractions
  • Understand equivalent fractions
  • Simplify a fraction by cancelling all common factors
  • Add and subtract fractions
  • Recognise that each terminating decimal is a fraction
  • Recognise that recurring decimals are exact fractions and that some exact fractions are recurring decimals
  • Interpret fractions, decimals and percentages as one another

Listed below are a series of summaries and worked examples to help you solidify your knowledge about fractions.

Worked Examples

1 - Adding and Subtracting Fractions
When adding and subtracting fractions, there is one simple rule you must remember:

Find the lowest common denominator

The lowest common denominator is the lowest number which the denominators will both go into. After you have worked out this value, multiply the number and denominator of each fraction by the same number.

After doing this, add or subtract the numerator values whilst keeping the denominator value the same.

Example
(a) - Work out

\[\frac{4}{6} + \frac{2}{8}\]

Solution
(a) - The lowest common denominator for 6 and 8 is 24

\[6*4 = 24\]
\[8*3 = 24\]

Therefore, you must multiply the numerators of each fraction by these values.

4/6 becomes 16/24 and 2/8 becomes 6/24

\[\frac{{16}}{{24}} + \frac{6}{{24}} = \frac{{22}}{{24}}\]

This value can be simplified to,

\[\frac{{11}}{{12}}\]

Therefore,

\[\frac{4}{6} + \frac{2}{8} = \frac{{11}}{{12}}\]

2 - Fractions of a quantity
During your exam you will be asked to find the fraction of a quantity. There are two methods of working this out, which are displayed as follows:

Example
(a) - What is 5/6 of 30?

Solution
(a) - Method One

  • Work out 1/6 of 30
  • Multiply this value by 5

1/6 of 30 is 30 รท 6 = 5

Therefore to work out 5/6 of 30 you multiply 5 by 5

5/6 of 30 = 5x5 = 25

(a) - Method 2

  • Multiply 5/6 by 30
\[\frac{5}{6}*30 = \frac{5}{6}*\frac{{30}}{1} = \frac{{150}}{6} = 25\]

3 - Equivalent Fractions
A fraction can be written in several ways and still have the same value. These are referred to as equivalent fractions.

You can produce lots of equivalent fractions by multiplying or dividing the top and bottom by the same number.

For example:

4/8 has the same value as 1/2

These values are the same because when you multiply or divide both the top and bottom by the same number, the fraction keeps the same value.

4 - Simplified Fractions
Through this process of recognising equivalent fractions, you can reduce large fractions into simplified values of themselves. These are referred to as simplified fractions.

During your exam, you may be asked to simplify a fraction, or fill in a missing number. When doing so, you must remember the following rules:

  • You must multiply or divide both the numerator and denominator by the same value
  • Only divide by a value which will ensure the numerator and denominator remain whole numbers
  • Only multiply or divide, never add or subtract, to calculate an equivalent fraction

Example
(a) - Simplify the fraction 20/35

Solution
(a) - 20 and 35 are both dividable by 5

\[20/5 = 4\]
\[35/5 = 7\]

Therefore, 20/35 can be simplified to 4/7

5 - Fractions and decimals
During your exam, you may be asked to convert a fraction into a decimal. To do this, you divide the numerator by the denominator.

Example
(a) - Convert 3/4 to a decimal

Solution
(a) -
To convert 3/4 to a decimal, you divide 3 by 4

Therefore,

\[3/4 = 0.75\]

In examples such as the one demonstrated above, some decimals will terminate. However, there are other fractions which, when converted into decimal form, will have a recurring value.

For example:

\[1/3 = 0.33333333.....\]

During your exam, you may also be asked to convert a decimal into a fraction. When doing so, it is important to remember that the denominator will be a value of either 10, 100 r 1000 depending on the amount of decimal places.

Example
(a) - Convert 0.4 into a fraction

Solution
(a) -
0.4 is four tenths of 1, therefore as a fraction it can be written as 4/10. You then simplify this fraction using the method explained above.

\[0.4 = 4/10 = 2/5\]

Therefore 0.4 can be converted into the fraction 2/5

Example
(b) - Convert 0.35 into a fraction

Solution
(b) -
0.35 is 35 hundredths of 1, therefore as a fraction it can be written as 35/100.

When simplified,

\[35/100 = 7/20\]

Therefore 0.35 can be converted into the fraction 7/20

Example
(c) - Convert 0. 240 into a fraction

Solution
(c) -
0.240 is 240 thousandths of 1, therefore as a fraction it can be written as 240/1000 . When simplified,

\[240/1000 = 6/25\]

Therefore 0.240 can be converted into the fraction 6/25

6 - Fractions and decimals: Higher Tier
If you are sitting the higher tier paper, you will be asked to convert fractions into decimals and where the value may recur. For example, 1/3 which when converted gives a recurring value of 0.333333...

When converting these values, it is necessary to find the prime factors of the denominator.

As a rule, when the prime factors of the denominator of a fraction in its simplest form are only 2 and/or 5, then its decimal will terminate.

Therefore, to work out whether a value will terminate, it is necessary to work out whether its prime factors are 2 and/or 5.

Example
(a) - Convert 3/20 into a decimal

Solution
(a) -

\[3/20 = 3/(2*2*5) = 0.15\]

Example
(b) - Convert 2/9 into a decimal

Solution
(b) -

\[2/9 = 2/(3*3) = 0.22222...\]

Therefore, this value will not terminate.

7 - Fractions and Percentages
During your exam, you may be asked to convert a fraction to a percentage. To do this, you must multiply the fraction by 100.

Example
(a) - Convert 4/10 into a percentage

Solution
(a) -
4/10 is equivalent to 4/10 x 100 = 40%

Therefore 4/10 can be converted into the percentage 40%

You may also be asked to convert a percentage to a fraction. To do this, you must present the percentage value as the numerator in a fraction with the denominator 100.

Example
(a) - Convert 12% into a fraction

Solution
(a) -
12% as a fraction can be displayed as 12/100

This fraction can then be simplified to 6/50

Therefore 12% can be converted into the fraction 6/50

Exam Tips

  1. Remember that percentages are simply fractions out of 100, and that decimals are simply tenths, hundredths and thousandths of 1
  2. When multiplying and dividing fractions, always search for the lowest common denominator
  3. Multiply and divide both the numerator and denominator by the same value
  4. Only divide by a value which will ensure the numerator and denominator will remain whole numbers
  5. Only multiple and divide, never add or subtract, to work out an equivalent fraction
  6. Write down ALL of your working out, no matter how simple it may seem!

Topic Summary

Ultimately, although fraction based questions may appear complicated at first, once you have learned the correct methods they are easy to understand and solve. Above all else, remember to find the lowest common denominator. Once you have this value, you can easily follow the steps displayed in the examples above in order to add, subtract, multiply, divide and convert fractions with ease!

Related Topics

  • Indices
  • Problem Solving with Decimals and Percentages
  • Accuracy of Measurement Problems
  • Arithmetic with Positive and Negative Integers
  • Standard Form
  • Rational and Irrational Numbers
  • Combinations