AQA GCSE Mathematics Paper 1 (Foundation), June 2025: Question 14

Estimation & Significant Figures

📋 What this question tests

This question assesses fundamental number and estimation skills from GCSE Mathematics:

Rounding decimal numbers accurately to 1 significant figure (1 s.f.).
Applying the correct order of operations (BIDMAS) to evaluate an expression involving powers, multiplication, and addition.
Critical evaluation: determining and clearly justifying whether an estimate is an overestimate or an underestimate based on the direction each value was rounded.

Question 14 (a) • 3 Marks

Estimating the Value of an Expression

Expression: 1.98 × 3.82 + 6.75²

📐 Step-by-Step Calculation

Round each value to 1 s.f.:
• 1.98 rounds up to 2
• 3.82 rounds up to 4
• 6.75 rounds up to 7
Substitute into expression:
2 × 4 + 7²
Apply order of operations (BIDMAS):
Indices: 7² = 49
Multiply: 2 × 4 = 8
Add: 8 + 49 = 57

✅ Final Answer & Mark Breakdown

Final Answer: 57

M1: Rounding at least one value correctly to 1 s.f. (seeing 2, 4, 7, or 49).
M1: Rounding all three values correctly (seeing 2, 4, and 7 or 49).
A1: Correct answer of 57 with values 2, 4, and 7 shown in working. (Note: the mark scheme condones 60 if rounded to 1 s.f. at the very end).

💡 Key Knowledge

1 Significant Figure: Keep the first non-zero digit. Look at the next digit to decide whether to round up (≥ 5) or stay the same (< 5).
Order of Operations: Powers (indices) and multiplication come before addition. Do not add 4 + 7 first!

❌ Common Errors

Calculating exact values first: Working out 1.98 × 3.82 + 6.75² directly with a calculator or long hand gains 0 marks. The question explicitly says "By rounding each number to 1 significant figure".
Incorrect BIDMAS: Multiplying first to get 8, then writing (8 + 7)² = 15² = 225.
Squaring before rounding: Squaring 6.75 to get 45.5625 and then rounding leads to errors. Round first.

Question 14 (b) • 1 Mark

Overestimate or Underestimate?

Deciding and giving a valid mathematical reason

✅ Correct Answer

Tick: [✔] Overestimate

Acceptable Reason:

"Every number was rounded up."
"I rounded them all up."
"All the numbers were increased when rounded."

🧠 Exam Technique & Examiner Insight

Look at every operation: Both operations here are addition and multiplication with positive terms. Because every single component was rounded up (1.98 → 2, 3.82 → 4, 6.75 → 7), the final calculated answer must be strictly larger than the true answer.
Reference ALL inputs: Examiners strictly require you to state that all / every number was rounded up, not just one.

❌ Common Misconceptions (B0)

"I rounded up" – Lost the mark for ambiguity (does not specify that all inputs were rounded up).
"The answer is bigger than the real answer" – Scored 0. This merely restates the definition of an overestimate without giving a reason why.
Giving exact calculation comparisons: It is a 1-mark question; do not waste time working out the exact answer to compare values. Just explain the direction of rounding.