GCSE Maths Practice: direct-proportion

Question 7 of 10

This question tests your ability to apply direct proportion using fractional time values.

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\( \begin{array}{l}\text{A worker earns £72 for 8 hours of work.} \\ \text{How much will they earn for 11.5 hours at the same rate?}\end{array} \)

Choose one option:

Keep decimal values exact and avoid rounding until the final step.

Direct Proportion with Fractional Time Values

At Higher GCSE level, direct proportion questions often involve non-integer values such as decimal hours or fractional quantities. These questions test whether you can apply proportional reasoning accurately while handling decimals confidently. The key idea remains unchanged: when two quantities are directly proportional, one increases at a constant rate with the other.

In pay-and-time problems, direct proportion applies when the hourly rate stays the same. This means that each additional hour of work adds the same amount to the total pay, regardless of whether the time worked is a whole number or a fraction.

Finding the Unit Rate

The most reliable method for solving Higher-tier proportion problems is to calculate the unit rate. In this context, the unit rate is the amount earned per hour. Once this rate is known, it can be multiplied by any number of hours to find the total pay.

Example: A worker earns £56 for 7 hours of work. Dividing £56 by 7 gives an hourly rate of £8 per hour. If the worker then works 9.5 hours at the same rate, the total pay is found by multiplying £8 by 9.5.

Working with Fractional Hours

Fractional hours such as 11.5 hours represent a combination of whole hours and part hours. For example, 11.5 hours means 11 hours plus half an hour. Treating this as a decimal makes calculations clearer and avoids unnecessary conversions.

Example: If the hourly rate is £10, working 2.25 hours means earning £10 for each full hour and a proportional amount for the remaining fraction of an hour. Multiplying £10 by 2.25 gives the total pay.

Common Mistakes to Avoid

  • Multiplying the original pay directly by the new number of hours.
  • Dividing by the wrong number of hours when finding the unit rate.
  • Rounding the hourly rate too early.
  • Ignoring the fractional part of the time worked.

A helpful sense check is to estimate roughly. If the hourly rate is £9, then working a little more than 11 hours should result in a total just over £99.

Real-Life Applications

This type of calculation is very common in everyday life. Freelancers bill by the hour, tutors charge for partial sessions, and employers calculate wages for shifts that do not end on the hour. Being comfortable with proportional reasoning ensures accurate pay calculations and fair budgeting.

Frequently Asked Questions

Is pay always directly proportional to time?
No. Direct proportion only applies when the hourly rate stays constant and there are no overtime rates or bonuses.

Should I convert fractions to decimals?
At Higher tier, decimals are usually clearer, but either method is acceptable if used consistently.

Study Tip

For GCSE Higher exams, always write down the unit rate first and keep decimal values exact until the final step. This reduces errors and helps secure method marks.