GCSE Maths Practice: inverse-proportion

Question 1 of 10

This question tests inverse proportion using machines and time.

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\( \begin{array}{l}\text{3 machines take 10 hours to complete a job.} \\ \text{How long would 5 machines take to complete the same job?}\end{array} \)

Choose one option:

Understanding Inverse Proportion in GCSE Maths

Inverse proportion is used when two quantities are linked in a way that one increases while the other decreases, but the total amount of work stays the same. This type of relationship appears frequently in GCSE Maths, especially in questions involving machines, workers, time, speed, or rates.

In problems like this one, the key idea is that the total amount of work does not change. Whether fewer machines work for a longer time or more machines work for a shorter time, the job being completed is identical.

The Core Rule

For inverse proportion problems involving work and time, we use the rule:

machines × time = constant

This means that if one value goes up, the other must go down so that their product stays the same.

Step-by-Step Method

  1. Identify the two quantities involved (for example, machines and time).
  2. Work out the total amount of work using the first set of values.
  3. Keep this total the same for the new situation.
  4. Form a simple equation and solve it.

Following this structure helps prevent mistakes and keeps your working clear for examiners.

Worked Example (Different Numbers)

Example: 4 workers can complete a task in 12 hours. How long would it take 6 workers to complete the same task?

  • Total work = 4 × 12 = 48
  • Let the new time be t
  • 6 × t = 48
  • t = 8 hours

This example shows how increasing the number of workers reduces the time needed.

Another Example Using Machines

Example: 2 machines take 15 hours to complete a job. How long would 5 machines take?

  • Total work = 2 × 15 = 30
  • 5 × t = 30
  • t = 6 hours

The method is always the same, regardless of the numbers used.

Common Mistakes to Avoid

  • Multiplying when you should divide, or dividing when you should multiply.
  • Forgetting that inverse proportion means one value goes up while the other goes down.
  • Adding or subtracting instead of multiplying to find total work.
  • Using direct proportion methods for inverse proportion questions.

Real-Life Applications

Inverse proportion appears in many everyday situations. For example, if more people help clean a room, the job finishes faster. In factories, adding more machines can reduce production time. Understanding this relationship is useful not only for exams but also for practical problem-solving.

Frequently Asked Questions

Is inverse proportion always about time?
No. It can also involve speed, number of workers, machines, or other rates.

How do I know if a question is inverse proportion?
Look for wording like “more means less” or “fewer means longer”.

Study Tip

If the question mentions work being completed, always check whether the total amount stays constant. Writing down quantity × quantity = constant before calculating can help you avoid errors.

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