GCSE Maths Practice: inverse-proportion

Question 1 of 10

This question tests inverse proportion using machines and production time.

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\( \begin{array}{l} \text{10 machines take 18 minutes to produce 300 items.} \\ \text{How long would 15 machines take to produce the same number of items?} \end{array} \)

Choose one option:

Inverse Proportion with Machines and Production Time (Higher Tier)

This question tests your understanding of inverse proportion in a production setting, where machines are used to produce a fixed number of items. At Higher GCSE level, questions often include extra context such as output quantities, rates, or efficiency, but the underlying proportional relationship remains the same.

The Key Idea

Inverse proportion occurs when one quantity increases while another decreases, but an important overall value stays constant. In this question, the number of machines increases, the time decreases, and the total amount of work required to produce the items stays the same.

For machine-based problems, the constant quantity is often described using machine-minutes:

number of machines × time = total work

Why This Works

If each machine works at the same rate, doubling the number of machines doubles the production rate. As a result, the time required is reduced proportionally. This assumption is standard in GCSE exam questions unless stated otherwise.

Step-by-Step Strategy

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

This structured approach helps you avoid common errors in multi-step Higher-tier questions.

Worked Example (Different Numbers)

Example: 6 machines take 25 minutes to produce a batch of components. How long would 15 machines take to produce the same batch?

  • Total work = 6 × 25 = 150 machine-minutes
  • 15 × t = 150
  • t = 10 minutes

Increasing the number of machines reduces the time taken.

Another Worked Example

Example: 20 identical machines take 9 minutes to complete a job. How long would 12 machines take?

  • Total work = 20 × 9 = 180 machine-minutes
  • 12 × t = 180
  • t = 15 minutes

Fewer machines means more time is needed.

Common Higher-Tier Pitfalls

  • Forgetting that the output quantity is fixed.
  • Using direct proportion instead of inverse proportion.
  • Not writing down the constant total work.
  • Making arithmetic errors when multiplying or dividing.

Real-World Relevance

This type of inverse proportion is widely used in manufacturing, engineering, and scheduling. Factories often calculate how adding or removing machines affects production time, assuming each machine works at the same speed.

Exam Tip

When you see phrases like “the same number of items” or “identical machines”, immediately think inverse proportion and write down the constant relationship before calculating.

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