Estimation — sanity-checking with rounded numbers
Estimation isn't laziness; it's a critical skill. AQA expects you to round each input to 1 significant figure, calculate with the rounded values, and use the result to spot calculator errors or to give a quick "good enough" answer.
Round each number to 1 s.f.
To 1 s.f.:
- 487 → 500.
- 21 → 20.
- 0.062 → 0.06.
- 0.00387 → 0.004.
- 4951 → 5000.
The first non-zero digit is the only one kept; the next digit decides whether to round up or down.
Then compute with the rounded numbers
Worked example: estimate 487 × 21.
- 500 × 20 = 10,000.
- (Actual answer: 10,227. Estimate is within 3%.)
Worked example: estimate (4.92 × 8.07) ÷ 0.493.
- ≈ (5 × 8) ÷ 0.5 = 40 ÷ 0.5 = 80.
- (Actual answer: 80.55…)
Estimating square roots
If the number under the root isn't a perfect square, find the two perfect squares it lies between.
Estimate √60: √49 = 7 and √64 = 8, and 60 is closer to 64 → ≈ 7.7. (Actual: 7.746.)
Estimate √0.4: think √(40/100) = √40 / 10 ≈ 6.3 / 10 = 0.63.
Estimating with awkward decimals
Make every decimal a "nice" decimal: 0.485 ≈ 0.5; 1.97 ≈ 2; 0.0309 ≈ 0.03.
Worked example: estimate (0.485 × 19.7) ÷ 4.96.
- ≈ (0.5 × 20) ÷ 5 = 10 ÷ 5 = 2.
Bounding the estimate
When asked whether the estimate is "an underestimate" or "an overestimate", look at how each rounding affected the value:
- Rounding numerators UP makes the result bigger.
- Rounding denominators UP makes the result smaller.
In the example above: 0.485 → 0.5 (up), 19.7 → 20 (up), 4.96 → 5 (up). Numerator both rounded up (so it grew); denominator rounded up (so the result shrank). Net effect is unclear without more care — that's why mark schemes accept any reasonable answer with sound reasoning.
When estimation is enough
- "Roughly how many beads in a jar?" — estimation is the intended method.
- "Is the calculator answer sensible?" — compute the estimate; if your calc result is wildly different, recheck.
⚠Common mistakes— Common mistakes (examiner traps)
- Rounding to 2 s.f. when 1 s.f. is asked for. Quicker, but loses the M1 if the question specifies 1 s.f.
- Rounding to the nearest integer for small decimals. 0.487 to 1 s.f. is 0.5, not 0 or 1.
- Mixing exact and rounded values. Round all inputs first, then compute.
- Forgetting to interpret the sign of error (over- vs under-estimate).
- Using estimation to write the final answer when the question asks for an exact value. Read the wording carefully.
➜Try this— Quick check
Estimate 39.7 × 0.0203 ÷ 0.51.
- 39.7 ≈ 40; 0.0203 ≈ 0.02; 0.51 ≈ 0.5.
- (40 × 0.02) ÷ 0.5 = 0.8 ÷ 0.5 = 1.6.
- (Actual: 1.581…)
AI-generated · claude-opus-4-7 · v3-deep-number