GCSE Maths Practice: direct-proportion

Direct Proportion and Cost Problems

Direct proportion is an important GCSE Maths topic that helps explain how two quantities change together at a constant rate. When quantities are directly proportional, increasing one quantity causes the other quantity to increase by the same factor. This relationship appears frequently in everyday situations involving prices, quantities, distance, and time.

In shopping scenarios, direct proportion usually means that the total cost depends on how many items are bought. If each item costs the same amount, the overall price increases steadily as more items are added. Understanding this relationship allows you to calculate costs accurately and confidently.

Using the Unit Cost Method

The unit cost method is the most reliable way to solve direct proportion problems involving money. This method involves finding the cost of one item first and then scaling that value up to find the cost of any number of items.

Example: Suppose 5 oranges cost £3.00. To find the cost of one orange, divide £3.00 by 5, giving £0.60 per orange. If someone buys 8 oranges, the total cost can be found by multiplying £0.60 by 8, resulting in £4.80.

Scaling Quantities Carefully

Another way to solve direct proportion problems is by scaling quantities up or down. This works best when the new quantity is a simple multiple of the original amount.

Example: If 3 bottles of water cost £2.10, then 6 bottles will cost twice as much because the quantity has doubled. This gives a total cost of £4.20. The key idea is that both quantities change in the same proportion.

Common Mistakes to Avoid

• Dividing by the wrong number when finding the unit cost.
• Multiplying before calculating the cost of one item.
• Assuming the price changes when the quantity changes.
• Forgetting to check whether the answer makes sense.

A good habit is to estimate roughly before finishing. If you are buying more items than before, the total cost should be higher.

Everyday Uses of Direct Proportion

Direct proportion is used in many real-life situations. Supermarkets display unit prices to help customers compare value, wages are often calculated based on hours worked, and recipes are adjusted depending on how many people are being served. For example, if one sandwich needs 2 slices of bread, then making 5 sandwiches will require 10 slices.

Frequently Asked Questions

Is direct proportion always solved the same way?
Most Foundation-level questions can be solved using the unit cost method, but scaling can also work when quantities are simple.

Why should I show my working?
Showing each step clearly helps avoid mistakes and can earn method marks even if the final answer is incorrect.

Study Tip

When practising GCSE Maths, always write down the unit cost first. This creates a clear pathway to the final answer and reduces confusion under exam pressure.