AQA GCSE Mathematics Paper 3 (Foundation), June 2025: Question 4

4 marks · Easy difficulty · Short Answer

Use a number machine to find an output given an input, an input given an output, and determine a missing operation.

Practise this question

Question

Question 4 shows a number machine with an Input oval, followed by a box with 'times 3', then a box with 'minus 5', leading to an Output oval. Part (a) asks to work out the output when the input is 4 (1 mark). Part (b) asks to work out the input when the output is 19 (2 marks). Part (c) shows a second number machine with Input 16, followed by '+ 2', then an empty box, leading to Output 3, and asks to complete the number machine (1 mark).

Mark scheme

Show the mark scheme Mark scheme for Question 4: 4(a) awards 1 mark (B1) for an answer of 7. 4(b) awards M1 for '19 + 5' or 24, or 'n / 3' where 14 <= n <= 24, or '(x + 5)/3'; A1 for 8. 4(c) awards 1 mark (B1) for 'divide by 6' or '- 15' or 'x 1/6' or '+ -15'.

How to answer it

Working with Number Machines (Functions)

📌 What This Question Tests

This question assesses your foundational understanding of number machines (function machines):

  • Applying operations in the correct sequence moving forward (input to output).
  • Applying inverse operations in reverse order to backtrack from output to input.
  • Deducing a missing intermediate operation given an input, an initial step, and a final output.

Question 4 (a)

Finding the Output from a Given Input

1 Mark

📐 Step-by-Step Calculation

  1. Start with the input value: 4
  2. Apply the first box ( × 3 ):
    4 × 3 = 12
  3. Apply the second box ( − 5 ):
    12 − 5 = 7

✅ Correct Answer

7

Mark Scheme:
B1: Independent mark for the correct answer 7 .
Note: Examiners will check the output oval on the diagram if the answer line is blank.

🧠 Exam Technique

  • Write your intermediate number ( 12 ) between the boxes on the diagram to avoid simple mental arithmetic slips.

❌ Common Errors

  • Performing operations in reverse or mixing up signs (e.g., doing 4 + 5 first).

Question 4 (b)

Working Backwards (Inverse Operations)

2 Marks

📐 Step-by-Step Calculation

To go backwards from Output (19) to Input, reverse both the operations and the order:

  1. Reverse the second operation ( − 5 becomes + 5 ):
    19 + 5 = 24
  2. Reverse the first operation ( × 3 becomes ÷ 3 ):
    24 ÷ 3 = 8

✅ Correct Answer

8

Mark Scheme:
M1 (Method): 19 + 5 or 24 or n ÷ 3 (where 14 ≤ n ≤ 24) or (x + 5) ÷ 3 .
A1 (Accuracy): Final correct input of 8 .

💡 Key Knowledge: Inverse Pairs

  • Addition ( + ) reverses Subtraction ( − )
  • Division ( ÷ ) reverses Multiplication ( × )
  • Always reverse the order of steps: undo the last operation first!

❌ Common Errors & Traps

  • Forward Trap: Running 19 forward through the machine ( 19 × 3 − 5 = 52 ).
  • Wrong Inverse: Subtracting 5 instead of adding ( 19 − 5 = 14 , leading to 14/3 ). Note: The mark scheme awards M1 for 14 ÷ 3 as a partial method, but loses the accuracy mark!

Question 4 (c)

Deducing the Missing Operation

1 Mark

📐 Step-by-Step Deduction

  1. Find the intermediate value after the first box:
    16 + 2 = 18
  2. Find what single operation turns 18 into the output 3 :
    • Division: 18 ÷ 6 = 3
    • Subtraction: 18 − 15 = 3

✅ Acceptable Answers

÷ 6  or  − 15

Also accepted: × 1/6 or + −15

Mark Scheme:
B1: Any mathematically equivalent single operation.
Critical Condition: Writing more than one operation inside the box scores B0.

🧠 Exam Technique

  • Always check both multiplication/division and addition/subtraction. ÷ 6 and − 15 are equally valid!
  • Make sure to include both the operational sign ( ÷ or − ) and the number. Just writing 6 or 15 will lose the mark.

❌ Examiner Warning

  • Do not split the box into two steps (e.g. "÷ 2 then ÷ 3"). The scheme explicitly states: "More than one operation = B0".

Topics

Number · Algebra · 3.1.1 Structure and calculation · 3.2.3 Solving equations and inequalities

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 3 (Foundation), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.