GCSE Maths Practice: direct-proportion

Direct Proportion and Rates of Pay

At Higher GCSE level, direct proportion questions often involve rates such as pay per hour, cost per unit, or speed per hour. These questions test whether you can identify the constant rate and then apply it correctly to a new situation. The key idea is that when pay is directly proportional to time worked, the hourly rate stays constant.

If a job pays the same amount for each hour worked, increasing the number of hours will increase the total pay by the same factor. This relationship only holds when there are no bonuses, overtime rates, or deductions involved.

Finding the Hourly Rate

The first step in solving pay-related proportion problems is to find the hourly rate. This is done by dividing the total pay by the number of hours worked. Once the hourly rate is known, it can be multiplied by any number of hours to calculate total pay.

Example: A worker earns £48 for 6 hours of work. Dividing £48 by 6 gives an hourly rate of £8 per hour. If the worker completes 9 hours at the same rate, the total pay can be found by multiplying £8 by 9.

Why Direct Proportion Applies Here

In this type of problem, the total amount of work completed is measured by time. As long as each hour of work is paid at the same rate, the relationship between time and pay is directly proportional. Doubling the number of hours will double the pay, and tripling the hours will triple the pay.

This proportional relationship can be written algebraically as:

pay = (hourly rate) × (time worked)

Common Mistakes to Avoid

• Multiplying the original pay by the new number of hours instead of finding the hourly rate first.
• Dividing by the wrong number of hours.
• Assuming the relationship is direct when overtime or bonuses apply.
• Forgetting to check whether the final value is reasonable.

A quick estimate can help. If the pay for 7 hours is a little over £60, then the pay for 10 hours should be somewhat higher but not dramatically larger.

Real-Life Applications

Understanding proportional pay calculations is essential in everyday life. Employees estimate wages, freelancers calculate invoices, and employers plan staffing costs. Being confident with unit rates helps avoid underpayment or budgeting errors.

For example, if a tutor charges a fixed hourly rate, increasing the number of sessions increases the total cost proportionally.

Frequently Asked Questions

Is pay always directly proportional to time?
No. Overtime rates, shift bonuses, or unpaid breaks can change the relationship.

Why is this Higher tier?
Higher-tier questions often test whether students can recognise when direct proportion applies and when it does not.

Study Tip

For GCSE Higher exams, always write down the unit rate clearly before calculating the final amount. This reduces mistakes and helps secure method marks.