# 3 Fractions

What is a fraction?

A fraction is defined as a part of a whole. So for example , or ‘one third’, is one part of three parts, all of equal size.

Fractions are an important feature of everyday life. For example, a healthy diet is based on eating appropriate fractions of each food group at each meal. The NHS recommends that more than of the food we eat each day should be fruit and vegetables. As you go through this section, you’ll see how fractions could be used to lead a healthier lifestyle or used within the workplace.

Fractions are related to decimals and percentages, which you’ll look at in the sections that follow this one.

This section will help you to:

- order and compare fractions
- identify equivalencies between fractions
- calculate parts of whole quantities and measurements (e.g. calculate discounts in sales).

Please look at the following example before you carry out the activity:

A **half** can be written as , i.e. one of two equal parts.

A **quarter** can be written as , i.e. one of four equal parts.

An **eighth** can be written as , i.e. one of eight equal parts.

**Hint:** The top of the fraction is called the numerator. The bottom of the fraction is called the denominator. Notice that is bigger than , even though the denominator 2 is smaller than the denominator 4. How would you explain one third? How would you write it as a fraction? Which is bigger: one third or two quarters?

## Example: Cooking a healthy meal

Sandra is cooking an evening meal. Half of her plate is made up of meat, one third is vegetables and one sixth is dairy products.

- Which food group is Sandra eating the most of?
- Which food group is Sandra eating the least of?

### Method

When numerators of fractions are all 1, the larger the denominator of the fraction, the smaller the fraction.

Looking at the example above, the fractions can be put in order of size starting from the smallest:

, ,

So:

- The food group Sandra is eating the most of () is meat.
- The food group Sandra is eating the least of () is dairy.

If you’re asked to arrange a group of fractions into size order, it’s sometimes helpful to change the denominators to the same number. This can be done by looking for the lowest common multiple – that is, the number that all of the denominators are multiples of.

## Example: Looking at equivalent fractions

Arrange the following fractions in order of size, starting with the smallest:

- , ,

### Method

The lowest common multiple is 12:

- 6 × 2 = 12
- 3 × 4 = 12
- 12 × 1 = 12

Whatever you do to the bottom of the fraction you must also do to the top of the fraction, so that it holds the equivalent value. The third fraction, , already has 12 as its denominator, so we don’t need to make any further calculations for this fraction. But what about and ?

- 2 × means calculating (2 × 3 = 6) and (2 × 6 = 12), so the equivalent fraction is
- 4 × means calculating (4 × 1 = 4) and (4 × 3 = 12), so the equivalent fraction is

Now you can now see the size order of the fractions clearly:

- , ,

So the answer is:

- , ,

Use the examples above to help you with the following activity. Remember to check your answers once you have completed the questions.

## Activity 8: Fractions in order of size

- Put these fractions in order of size, with the smallest first:

- , , , ,

### Answer

Remember that when the numerator of a fraction is 1, the larger the denominator, the smaller the fraction.

From smallest to largest, the order is:

- , , , ,

- What should you replace the question marks with to make these fractions equivalent?

- =
- =
- =
- =

### Answer

- =
- =
- =
- =

## Example: Drawing the fractions

If you need to compare one fraction with another, it can be useful to draw the fractional parts.

Look at the mixed numbers below. (A mixed number combines a whole number and a fraction.) Say you wanted to put these amounts in order of size, with the smallest first:

2 , 3 , 1

### Method

To answer this you could look at the whole numbers first and then the fractional parts. If you were to draw these, they could look like this:

So the correct order would be:

1 , 2 , 3

Use the example above to help you with the following activity. Remember to check your answers once you have completed the questions.

## Activity 9: Putting fractions in order

- Put these fractions in order of size, smallest first:

- 5 , 6 , 2

- Put these fractions in order of size, smallest first:

- 2 , 1 , 2

### Answer

- The correct order would be:

- 2 , 5 , 6

- In this case, even though is bigger than and is bigger than , you need to look at the whole numbers first and then the fractions. The diagram illustrates this more clearly:

- The correct order would be:

- 1 , 2 , 2