GCSE Maths Practice: direct-proportion

Question 6 of 10

This question tests your understanding of inverse proportion using workers and time.

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\( \begin{array}{l}\text{If 3 workers paint a wall in 12 hours,} \\ \text{how long will it take 6 workers working at the same rate?}\end{array} \)

Choose one option:

For inverse proportion, multiply first to find the total work, then divide.

Understanding Inverse Proportion

Inverse proportion describes a relationship where one quantity increases while the other decreases in such a way that their product stays the same. In GCSE Maths, inverse proportion is often tested using work-rate problems, such as workers completing a task or machines producing items.

When quantities are inversely proportional, doubling one quantity causes the other quantity to halve. This is the opposite of direct proportion, where both quantities increase together. Recognising whether a situation involves direct or inverse proportion is a key exam skill.

Workers and Time

A very common example of inverse proportion involves workers and the time taken to complete a job. If more workers are added and everyone works at the same rate, the job will be completed faster. The total amount of work stays the same, but it is shared between more people.

The key idea is that:

number of workers × time taken = constant

This means that if you know the number of workers and the time taken in one situation, you can find the time taken for a different number of workers.

Using the Constant Product Method

Example: If 4 workers take 10 hours to complete a task, the total amount of work is 4 × 10 = 40 worker-hours. If 8 workers do the same task, divide 40 by 8 to find the new time, which is 5 hours.

This method works for any inverse proportion problem involving workers, machines, or shared workloads.

Scaling Using Simple Factors

Sometimes inverse proportion problems can be solved quickly by spotting simple relationships.

Example: If the number of workers triples, the time taken will be divided by 3. If the number of workers halves, the time taken will double. This approach is especially useful in Foundation-level questions where numbers are chosen to make the relationship clear.

Common Mistakes to Avoid

  • Multiplying time by the number of workers instead of dividing.
  • Assuming the relationship is direct instead of inverse.
  • Forgetting that workers must be working at the same rate.
  • Not checking whether the time should increase or decrease.

A good sense check is to ask: if more people help, should the job take longer or less time?

Real-Life Applications

Inverse proportion appears frequently in real life. Construction projects, factory production, cleaning jobs, and homework group tasks all use this relationship. For example, if one machine takes 8 hours to produce a batch of items, two identical machines working together will take 4 hours.

Frequently Asked Questions

Is inverse proportion always exact?
In exam questions, it is assumed that all workers work at the same rate. In real life, this may not always be perfectly true.

How do I know if a question is inverse proportion?
If increasing one quantity causes the other to decrease, it is likely to be inverse proportion.

Study Tip

For GCSE Maths exams, write down the relationship “workers × time = constant” before calculating. This helps you choose the correct method and avoid treating the problem as direct proportion.

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