GCSE Maths Practice: inverse-proportion

Question 4 of 10

This question tests inverse proportion using people and time for a fixed task.

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\( \begin{array}{l} \text{If 3 people complete a task in 24 hours,} \\ \text{how long would 8 people take to complete the same task?} \end{array} \)

Choose one option:

Inverse Proportion with People and Time (Higher Tier)

This question tests your understanding of inverse proportion using people and time, a common Higher GCSE Maths topic. In these problems, the number of people working changes, but the task itself remains exactly the same.

The Constant Quantity

In work-related inverse proportion questions, the quantity that stays constant is the total amount of work. This is often measured in person-hours, which combines the number of people and the time they work:

number of people × time = total work

If more people work on the same task, the time required must decrease proportionally.

Why This Is Inverse Proportion

Inverse proportion occurs when one quantity increases while another decreases, with a fixed overall outcome. In this case, increasing the number of people means the job is shared between more workers, so it is completed faster.

Step-by-Step Strategy

  1. Identify the two changing quantities (people and time).
  2. Calculate the total amount of work using the given values.
  3. Keep this total constant.
  4. Form an equation using the new number of people.
  5. Solve for the unknown time.

This structured approach is particularly important in Higher-tier questions, where numbers may not divide as neatly.

Worked Example (Different Numbers)

Example: 4 people complete a job in 18 hours. How long would it take 6 people?

  • Total work = 4 × 18 = 72 person-hours
  • 6 × t = 72
  • t = 12 hours

Increasing the workforce reduces the time taken.

Another Worked Example

Example: 10 workers take 6 hours to complete a task. How long would 5 workers take?

  • Total work = 10 × 6 = 60 person-hours
  • 5 × t = 60
  • t = 12 hours

Fewer workers means the task takes longer.

Common Higher-Tier Mistakes

  • Using direct proportion instead of inverse proportion.
  • Forgetting to calculate total work first.
  • Dividing when multiplication is needed, or vice versa.
  • Not checking whether the answer makes sense.

Real-World Context

This type of calculation is used in project planning, construction, and teamwork. Managers often estimate how adding or removing staff affects the time needed to complete a task.

Exam Tip

Whenever a question involves the same task with a different number of people, immediately write down people × time = constant before starting your calculations.

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