# Adding and subtracting fractions and mixed numbers

## Introduction

Before you start working with fractions you need to remember that the **numerator** is the top number in a fraction and the **denominator** is the bottom number.

The common mistake people make when adding fractions is to think you simply add the numerators and denominators together.
*This is wrong!*
For example, we know a half plus a quarter is 3 quarters but using that logic \( \frac{1}{2} + \frac{1}{4} = \frac{2}{6} \) which is obviously wrong!

A **mixed number** is a combination of a whole number and a fractional part. For example \(3 {1 \over 2} \) is a mixed number. When adding or subtracting fractions and mixed numbers the most important part is to find a
**common denominator**. This means *getting the denominators for both fractions the same*.

## How to add and subtract fractions

### Example 1 - Adding and subtracting with fractions

Work out \({3 \over 4} + {1 \over 6}\)

Before we can add any fractions we must convert to a common denominator first. This is found by finding the first common multiple of 4 and 6 which is 12.

If you don't find the first common multiple but instead use 24 or 36 etc you will still get to the same answer but have to do harder calculations along the way!

Now we have to convert both fractions to the common denominator of 12 by multiplying the numerators and denominators of both fractions by 3 and 2 respectively: \( \frac{3× 3}{4× 3} = \frac{9}{12} \) and \(\frac{1 × 2}{6 × 2} = \frac{2}{12} \).

Now the fractions have the same denominator we simply add the numerators together to get: \( \frac{9}{12} + \frac{2}{12} = \frac{11}{12} \)

## How to add and subtract mixed numbers

### Example 2 - Adding and subtracting with mixed numbers

Work out 2\( \frac{3}{5} + 3\frac{1}{2} \)

When adding mixed numbers you can make life easier for yourself by first adding the whole numbers together. In this case 2 + 3 = 5. Now we just have to work out the remaining fraction sum \( \frac{3}{5} + \frac{1}{2} \).

The common denominator of \( \frac{3}{5} + \frac{1}{2} \) is 10. So we multiply the numerators and denominators of both fractions by 2 and 5 respectively. \( \frac{3× 2}{5× 2} = \frac{6}{10} \) and \(\frac{1 × 5}{2 × 5} = \frac{5}{10} \). Adding these together gives us \( \frac{11}{10} \).

Finally, remember to add on the 5 from the beginning of the questions to get \( 5 \frac{11}{10} = 6 \frac{1}{10} \).

## Test yourself!

Now try these worksheets to practise your skills.