N14: Estimation & Approximation
Estimate calculations; check results using approximation
📋 Key Concepts
Estimation: Finding an approximate answer by rounding numbers to make calculations easier. Useful for checking if an answer is sensible.
Approximation: Using rounded values to find a reasonable estimate. The symbol ≈ means "approximately equal to".
Estimation Process:
1. Round each number to 1 significant figure
2. Calculate with the rounded numbers
3. Show your estimate using ≈
4. Compare with exact answer to check
1. Round each number to 1 significant figure
2. Calculate with the rounded numbers
3. Show your estimate using ≈
4. Compare with exact answer to check
📝 Rounding to 1 Significant Figure
Method: Keep the first non-zero digit and replace the rest with zeros.
| Number | 1 sig fig |
|---|---|
| 47 | 50 |
| 382 | 400 |
| 0.043 | 0.04 |
| 0.78 | 0.8 |
| 5630 | 6000 |
| 0.0062 | 0.006 |
Example 1
Estimate: 47 × 32
Solution:
Round to 1 sig fig: 50 × 30
5 × 3 = 15, so 50 × 30 = 1500
Estimate: ≈ 1500
Exact answer: 1504
📝 Estimating Multiplication
Example 2
Estimate: 382 × 24
Solution:
Round: 400 × 20
4 × 2 = 8, so 400 × 20 = 8000
Estimate: ≈ 8000
Example 3
Estimate: 6.8 × 4.3
Solution:
Round: 7 × 4 = 28
Estimate: ≈ 28
Exact: 29.24
📝 Estimating Division
Example 4
Estimate: 847 ÷ 32
Solution:
Round: 800 ÷ 30
800 ÷ 30 = 80 ÷ 3 = 26.67
Estimate: ≈ 27
Exact: 26.47
Example 5
Estimate: 59.2 ÷ 1.8
Solution:
Round: 60 ÷ 2 = 30
Estimate: ≈ 30
Exact: 32.89
📝 Estimating with Decimals
Tip: When rounding decimals, think about what makes the calculation easy. Sometimes 2 sig figs gives a better estimate.
Example 6
Estimate: 0.047 × 3800
Solution:
Round: 0.05 × 4000
5 × 4000 = 20000, so 0.05 × 4000 = 200
Estimate: ≈ 200
Exact: 178.6
📝 Checking Answers
Why estimate? To check if your calculated answer is sensible. If your exact answer is very different from your estimate, check your calculation.
Example 7
A student calculates 34 × 28 = 952. Is this answer sensible?
Solution:
Estimate: 30 × 30 = 900
952 is close to 900, so the answer is sensible ✓
Example 8
A student calculates 47 × 23 = 108. Is this answer sensible?
Solution:
Estimate: 50 × 20 = 1000
108 is NOT close to 1000 - this answer is wrong ✗
Correct answer: 1081
📝 Real-World Estimation
Example 9
A garden is 23.6 m by 7.8 m. Estimate the area.
Solution:
Round: 24 × 8 ≈ 25 × 8 = 200 m²
Or: 20 × 8 = 160 m² (also acceptable)
🧮 Worked Examples — Exam Style
Estimation means rounding EVERY number to 1 significant figure, then computing exactly with the rounded values. State the rounded calculation — the method mark lives there.
Estimate: 41.2 × 5.8
Round to 1 s.f.: 40 × 6
Estimate: 240
M1 for 40 × 6, A1 for 240. Writing '≈ 40 × 6 = 240' is the full-mark presentation.
Use estimation to check whether 79 × 21 = 1659 is sensible.
Round: 80 × 20 = 1600. The claimed 1659 is close to 1600 → sensible ✓
M1 for the rounded product, A1 for the verdict. Checking questions expect a verdict sentence, not just the estimate.
Exam wording to expect: "Estimate …", "by rounding each number to 1 significant figure", "check whether … is sensible".
❓ Practice Questions
Q1: Estimate: 73 × 48
Q2: Estimate: 592 ÷ 28
Q3: Estimate: 0.68 × 412
Q4: Estimate: 8.9 × 3.1
Q5: A student says 62 × 89 = 5518. Use estimation to check.
Q6: Estimate: (5.8 × 21.7) ÷ 0.48
Q7: Estimate the cost of 38 items at £4.90 each
✅ Answers
- 70 × 50 = 3500
- 600 ÷ 30 = 20
- 0.7 × 400 = 280
- 9 × 3 = 27
- Estimate: 60 × 90 = 5400. 5518 is close to 5400, so sensible ✓
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. 70 × 50 = 3500
Q2. 600 ÷ 30 = 20
Q3. 0.7 × 400 = 280
Q4. 9 × 3 = 27
Q5. 60 × 90 = 5400; 5518 ≈ 5400 → sensible ✓
Q6. (6 × 20) ÷ 0.5 = 120 ÷ 0.5 = 240 — dividing by 0.5 doubles: the step students miss
Q7. 40 × 5 = £200
📌 Key Facts to Memorise
- Round EVERY value to 1 significant figure before calculating
- ÷ by 0.5 doubles; ÷ by 0.25 quadruples
- Show the rounded calculation — "≈ 70 × 50 = 3500" — for the method mark
- Estimates check sensibility: the real answer should be within ~20% of the estimate
- 1 s.f. examples: 73 → 70 · 48 → 50 · 592 → 600 · 0.68 → 0.7 · 8.9 → 9
🗺️ In the Exam
Estimation: 2–3 marks on Paper 1 (non-calculator). The check-the-answer version adds an AO3 mark for the verdict. Rounding to 1 s.f. itself is examined directly in N15.
Links to: significant figures and error intervals (N15), calculator checks (all Papers 2/3).
⚡ Targeted Drills
Cover the answers, work each one, then mark yourself. If any takes over a minute, revisit the matching section above.
- Estimate 88 × 21 → 90 × 20 = 1800
- Estimate 41.9 ÷ 0.21 → 40 ÷ 0.2 = 200
- Estimate 6.2 × 7.8 → 6 × 8 = 48
- Check: is 61 × 29 = 1769 sensible? → 60 × 30 = 1800 ≈ 1769 ✓
- Estimate √98 → √100 = 10
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always show your rounding before calculating
- Use ≈ to show it's an estimate, not an exact answer
- Your estimate should be close to the real answer
- If asked for estimation, don't calculate the exact answer
- Choose rounding that makes calculations easy (like 50 × 20 rather than 47 × 23)
🧠 Problem-Solving Strategies
Problem-Solving
For estimation, round each number to 1 significant figure first, then calculate. Show your rounded numbers before calculating. Use the ≈ symbol to show it's an estimate. If the estimate seems unreasonable, try rounding to 2 significant figures for a better approximation. Estimation is essential for checking calculator answers.
Multi-Step Problem
Estimate the cost of 28 tiles costing £3.85 each and 5 kg of adhesive costing £12.40 per kg.
Solution: Tiles: 30 × 4 = £120. Adhesive: 5 × 10 = £50. Total ≈ £170. (Actual: £107.80 + £62 = £169.80)
⚠️ Common Errors
Watch Out!
1. Wrong: Rounding 0.047 to 1 sig fig gives 0.05 (rounding the wrong digit) Correct: 0.047 rounded to 1 sig fig = 0.05 (the first significant figure is 4, next digit 7 rounds up to 5)
2. Wrong: Estimating 4.9 ÷ 0.51 by rounding to 5 ÷ 1 = 5 Correct: 0.51 rounds to 0.5 (1 sig fig), so 5 ÷ 0.5 = 10
3. Wrong: Calculating the exact answer when asked to estimate Correct: Estimation means rounding first, then calculating — show the rounded values
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A theatre has 38 rows of 22 seats. Tickets cost £18.50 each. (a) Estimate the maximum revenue if all seats are sold. (b) A student calculates the exact revenue as £15,546. Use estimation to determine whether this is a sensible answer. (c) The actual revenue was £14,790. Explain why this might be different from your estimate in part (a).
(a) Round: 40 × 20 = 800 seats. 800 × £20 = £16,000 estimated revenue.
(b) Student's answer: £15,546. Estimate: £16,000. £15,546 is reasonably close to £16,000, so the student's answer is sensible.
(c) The actual revenue (£14,790) is lower because not all seats were sold — some seats were empty. The estimate assumed a full theatre.
Mark scheme: 2 marks for (a) showing rounded values, 2 marks for (b) with comparison, 2 marks for (c) with valid reasoning
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A company needs to order paving slabs for a path that is 9.6 m long and 1.8 m wide. Each slab is 0.6 m × 0.6 m and costs £4.95.
(a) Estimate the number of slabs needed and the total cost.
(b) Would your estimate lead to ordering too many or too few slabs? Explain why.
(c) The company manager says "Estimation is useless for ordering — we need exact numbers." Do you agree? Give a reason for your answer.
Answers: (a) Area ≈ 10 × 2 = 20 m². Each slab = 0.36 m² ≈ 0.4 m². Slabs ≈ 20 ÷ 0.4 = 50. Cost ≈ 50 × £5 = £250. (b) Rounding the path dimensions up means overestimating the area, and rounding slab size down means overestimating the number needed — so we'd order slightly too many, which is safer than too few. (c) Disagree — estimation provides a useful check. An exact calculation of 50 slabs costing £247.50 is close to the estimate, confirming the answer is reasonable. Estimation catches major errors.
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🎬 Video Resources
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