Proportion is used to show how quantities and amounts are related to each other. The amount that quantities change in relation to each other is governed by proportion rules.

**Proportion calculations** can be used to calculate values when one relationship is known.

8 pens cost £2.16. Calculate how much 7 pens cost.

To find out the cost of one item use the **unitary method** - divide the cost by how many items have been bought. Any amount can be calculated when the value of 1 is known.

8 pens cost £2.16.

Dividing both numbers by 8:

\[\begin{array}{ccc} 8~\text{pens} & = & \pounds 2.16 \\ \div 8 && \div 8 \\ 1~\text{pen} & = & \pounds 0.27 \end{array}\]

Multiplying both numbers by 7:

\[\begin{array}{ccc} 1~\text{pen} & = & \pounds 0.27 \\ \times 7 && \times 7 \\ 7~\text{pens} & = & \pounds 1.89 \end{array}\]

So 7 pens cost £1.89.

Proportion calculations can be used to decide which items in a shop offer the best value. Many items sold in supermarkets have a price *per item* **and** a price *per 100 g or per kg*. This lets people compare products and get the best value for money.

Proportion calculations must be used to compare the cost of items. This makes either their cost or their size the same so comparisons can be made. It is generally easier to make the size of the items the same but how this is done can vary depending on the numbers that are used.

If a 2 kg bag of basmati rice costs £2.68 and a 500 g bag of basmati rice costs £0.95, which one represents the best value for money?

There are many ways to approach this problem:

- calculate the cost of 500 g in the 2 kg bag
- calculate the cost of 2 kg worth of the 500 g bags
- calculate the cost of 1 kg for each of the bags

These are just a small number of ways in which this problem could be attempted.

For this solution, the cost of 1 kg for each of the bags will be calculated.

Price per kg for 2 kg bag of basmati rice:

\[2.68 \div 2 = \pounds1.34\]

Price per kg for 500 g bag of basmati rice:

\[0.95 \times 2 = \pounds1.90\]

2 kg bag | 500 g bag | |
---|---|---|

Price | 2 kg costs £2.68 | 0.5 kg costs £0.95 |

Price per kg | £1.34 | £1.90 |

This shows that the 2 kg bag of basmati rice is cheaper per kg and better value for money than the smaller, 500 g bag.