AQA GCSE Mathematics Paper 1 (Foundation), June 2025: Question 16

4 marks · Easy difficulty · Multi-step Problem

Use an inverse proportion formula relating days and workers to calculate the number of workers needed, and determine the effect on the number of workers if fewer days are required.

Practise this question

Question

Question 16(a) gives a formula in a box: d = 800/w, where d represents the number of days to complete a job and w represents the number of workers needed. It asks: 'Assume the job needs completing in 20 days. How many workers are needed?' for 3 marks. Question 16(b) states: 'In fact, the job needs completing in fewer than 20 days. What does this mean about the number of workers that are needed? Tick one box.' with options: 'It is less than the answer to part (a)', 'It is the same as the answer to part (a)', and 'It is greater than the answer to part (a)', for 1 mark.

Mark scheme

Show the mark scheme Mark scheme for question 16(a) awards M1 for 20 = 800/w or (w =) 800/d (oe), M1dep for 800 ÷ 20 (oe), and A1 for 40. For question 16(b), B1 is awarded for ticking 'It is greater than the answer to part (a)'.

How to answer it

Inverse Proportion: Formula Rearrangement & Practical Reasoning

📌 What this question tests
  • Formula substitution: Replacing variables with given numerical values ( d = 20 ).
  • Solving fractional equations: Rearranging an inverse proportion formula of the form y = k / x to find the denominator.
  • Qualitative reasoning about inverse proportion: Understanding that as one variable decreases (days needed), the related variable must increase (workers needed).

Question 16 (a)

Calculate the number of workers needed for 20 days [3 marks]

📐 Step-by-Step Calculation

  1. Substitute the given value:
    Substitute d = 20 into the given equation:
    20 = 800 / w
    awarded M1 for setting up the equation or rearranging to w = 800 / d
  2. Rearrange to make w the subject:
    Multiply both sides by w :
    20 × w = 800
    Divide both sides by 20:
    w = 800 ÷ 20
    awarded M1dep (dependent on first method mark) for the division step
  3. Evaluate the final answer:
    800 ÷ 20 = 40
    awarded A1 for the final answer of 40

✅ Correct Answer

40 workers

Total: 3 marks (M1, M1dep, A1)

💡 Key Knowledge

  • The formula d = 800 / w models inverse proportion.
  • In this relationship, the total "person-days" needed is constant: d × w = 800 .
  • To solve for a denominator variable: swap the subject with the denominator, so d = 800 / w becomes w = 800 / d .

❌ Common Errors

  • Multiplying instead of dividing: Calculating 800 × 20 = 16 000 .
  • Dividing the wrong way: Calculating 20 ÷ 800 = 0.025 .
  • Arithmetic slips: Miscounting zeroes when dividing 800 ÷ 20 , leading to 4 or 400.

Question 16 (b)

Interpreting changes in the number of days [1 mark]

✅ Correct Answer

Tick the third box:

☑ It is greater than the answer to part (a)

Awarded B1 (standalone mark)

🧠 Exam Technique & Logic

  • Use real-life logic: If a job needs to be finished faster (fewer days), you need more people working on it.
  • Test with numbers: Pick a number fewer than 20 days, e.g., 10 days:
    w = 800 ÷ 10 = 80 workers .
    Since 80 > 40, the number of workers must be greater.

❌ Common Pitfalls

  • Direct proportion bias: Assuming that "fewer days" automatically means "fewer workers" without thinking about inverse variation.
  • Ticking more than one box: The question clearly asks to "Tick one box". Selecting multiple options automatically loses the mark.

Topics

Algebra · Ratio, proportion and rates of change · 3.2.1 Notation, vocabulary and manipulation · 3.2.3 Solving equations and inequalities · 3.3 Ratio, proportion and rates of change

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 1 (Foundation), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.