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)
❓ 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.
✅ 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 ✓
🎯 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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