Adding and subtracting fractions
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How to add and subtract common fractions.

This topic is part of the TCS FREE high school mathematics 'How-to Library'. It shows you how to add and subtract common fractions.
(See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Algematics software. Click here for instructions.


Common fractions can only be added or subtracted only if they have they have common denominators.

The 'denominator' of a common fraction is the bottom number. Fractions having the same denominator, are said to have common denominators.

For example, these fractions have a common denominator of '4', so it is possible to add them:

When adding fractions with common denominators, add the numerators.

The 'numerator' of a common fraction is the top number.

In the same way, you can subtract fractions with common denominators by subtracting the numerators from each other:

If two fractions do not have a common denominator, you can still add and subtract them, but you must first change them so that they do have a common denominator.

There is a simple rule that works for any two fractions. This rule is based on the fact that if you multiply both the numerator and the denominator of a fraction by the same number, then the value of the fraction has not changed.

Here is the rule:

To add fractions using this rule you:

1. Multiply the numerator of each fraction by the denominator of the other fraction then add the two answers. (This is the numerator of your answer.)

2. Multiply the denominators together. (This is the denominator of your answer.)

3. Simplify

Because , the fractions now have a common denominator and can be added.

For example, to add these fractions:

We first apply the rule, like this:

Simplifying, we have:

For subtraction, the rule is exactly the same, but you subtract the numbers in the numerator instead of add.

There are often short cuts for adding and subtracting fractions.

You may have realised already that 3/4 is the same as 6/8,

so that 6/8 + 5/8 = 11/8

The 'Method' section below shows you how you can use 'Algematics' to help with adding and subtracting fractions. It will also correct your work.

Download the free support file... We have created an Algematics document containing the completed example from this topic. It also includes practice exercises to improve your skills.

File name:  'Adding and subtracting fractions.alg'   File size: 4kb
Click here
to download the file.

If you choose 'Open this file from its current location', then Algematics should open the document immediately. If not, try the other option: 'Save this file to disk', then run Algematics and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Algematics installed on your computer, click here for instructions.


In Algematics, you type common fractions with both numbers on the same line,

like this: 2/3 for 

Type mixed numbers  as the sum of the whole number and the fraction in brackets,

eg (2+7/9) for 

Use the  (simplify) command to convert mixed numbers to improper fractions in their lowest terms.

Use the  (common denominator) command to display the working for adding or subtracting two common fractions.

The steps below demonstrate how to add or subtract common fractions showing working, and also how to add or subtract mixed numbers.

Step 1  Set display options

Click  to display the ‘Set Colours And Fonts’ Dialog Box.

Make sure that this check box option is not selected:

o Use ¸ signs, eg: 2¸a

and then click

Step 2  Enter the data

Click  and type your expression into the maths box in the data entry dialog box.

If the ‘EMPTY’ message is not displayed between the blue buttons, click the  button until the message: ‘INSERT’ appears.

Use the ‘ * ’ symbol for multiply, and the ‘ / ’ symbol for divide.



   3/4 + 5/8




and then click

Step 3 Only if you have mixed numbers, for example: 

 Type this:

in the maths box like this:  (2+7/9) – (1+4/3)

Click  (simplify) repeatedly until you have converted all of the mixed numbers to common fractions:


Step 4  Put over a common denominator

Click  (common denominator).

This creates the following intermediate working step:

Step 5  Simplify

Click  (simplify) until a single answer is obtained.

In this case:  


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