To estimate a calculation, round every number to 1 significant figure, then work out the simpler calculation, using ≈ for “approximately equal to”. For powers and roots, round to a nearby square or cube number you know. Exam questions often ask whether your estimate is an over-estimate or an under-estimate, which depends on which way you rounded.
Estimation rules
- Round every number to 1 significant figure
- Use ≈ to show an approximate answer
- Roots: round to the nearest square or cube number you know
- Rounding up in a product gives an over-estimate; rounding up a divisor gives an under-estimate
Estimating a calculation
Round each number to 1 significant figure, so that the calculation can be done mentally: 8.7 becomes 9, 42 becomes 40 and 2.1 becomes 2. Then work out the simpler calculation. Show the rounded values in your working, because the method mark depends on them.
Estimating with squares and roots
For a root, round to the nearest square number you know: √65 ≈ √64 = 8. For a square, round the number first: 2.9² ≈ 3² = 9. Rounding inside a root to a perfect square is allowed even when it is not 1 significant figure.
Over-estimates and under-estimates
If you round up the numbers you multiply, your estimate is bigger than the true answer: an over-estimate. Rounding up a number you divide by makes the estimate smaller, an under-estimate. When some numbers are rounded up and some down, the effect has to be reasoned out, so explain which way each rounding pushes the answer.
Using estimates to check answers
An estimate is a quick way to check a calculator answer. If an estimate gives about 20 and your calculator shows 197, a key has probably been pressed twice or a decimal point lost.
Estimating in context
Real problems use the same method: estimate the cost of 29 items at £4.85 as 30 × £5 = £150. State whether the estimate is likely to be too high or too low when the question asks, giving a reason.
Worked examples
Example 1. Estimate (8.7 × 42) ÷ 2.1.
- Round to 1 s.f.: (9 × 40) ÷ 2.
- 9 × 40 = 360, and 360 ÷ 2 = 180.
Answer: ≈ 180.
Example 2. Estimate √65 + 2.9².
- √65 ≈ √64 = 8.
- 2.9² ≈ 3² = 9.
Answer: ≈ 17.
Example 3. Estimate 31.6 × 0.48.
- Round to 1 s.f.: 30 × 0.5.
- 30 × 0.5 = 15.
Answer: ≈ 15.
Example 4. A student estimates 6.7 × 4.3 as 7 × 4 = 28. Is this an over-estimate or an under-estimate? Explain.
- 6.7 was rounded up but 4.3 was rounded down, so the two roundings pull in opposite directions.
- Work out the exact value to decide: 6.7 × 4.3 = 28.81.
Answer: An under-estimate, because 28 is less than 28.81. When one number is rounded up and another down, check rather than assume.
Example 5. Estimate the total cost of 29 tickets at £4.85 each.
- Round: 30 × £5.
- Both numbers were rounded up.
Answer: ≈ £150, which is an over-estimate.
Common mistakes
- Rounding to the nearest whole number instead of 1 s.f. 82 rounds to 80, not 82.
- Using exact values in an estimate. An estimate must show the rounded numbers.
- Rounding a root badly. Round inside the root to a square number you know.
- Saying an estimate is too high without a reason. Explain which numbers were rounded up or down.
- Giving an exact answer with ≈. The ≈ answer should come from the rounded calculation.
How estimation is examined
Estimation is a non-calculator skill tested at both tiers of the Edexcel GCSE (1MA1) and International GCSE (4MA1) papers, often worth 2 or 3 marks. Questions ask you to:
- estimate a calculation by rounding to 1 significant figure
- estimate an expression containing a square or root
- say whether an estimate is an over-estimate or an under-estimate, with a reason
- use an estimate to check or criticise an answer
The rounding and estimation worksheet has questions with answers, and the GCSE number revision engine creates more.
Estimation uses significant figures and the square numbers from powers and roots.
Frequently asked questions
How do you estimate a calculation?
Round each number to 1 significant figure, then work out the easier calculation.
How do you estimate a square root?
Round to the nearest square number you know. √38 ≈ √36 = 6.
How do you know if an estimate is an over-estimate?
If you rounded up the numbers being multiplied, or rounded down a number you divide by, the estimate is too large.