Estimation and Checking
Estimation and Checking
Estimation is the skill of producing a quick, approximate answer that is close enough to be useful. In mathematics, it serves two essential purposes: forming a rough answer before a precise calculation (to guide your approach), and checking a calculated result (to catch errors, particularly those arising from technology or mental arithmetic). A well-judged estimate can immediately reveal whether an answer is wildly wrong.
Rounding to 1 Significant Figure
The standard method for estimation is to round every value in the calculation to 1 significant figure (1 sf) before calculating. The first significant figure is the first non-zero digit. Subsequent digits are replaced by zeros.
| Original number | Rounded to 1 sf | Reason |
|---|---|---|
| 4 738 | 5 000 | First sf is 4; next digit is 7 ≥ 5, so round up |
| 0.0623 | 0.06 | First sf is 6; next digit is 2 < 5, so round down |
| 295 000 | 300 000 | First sf is 2; next digit is 9 ≥ 5, so round up |
| 1.84 | 2 | First sf is 1; next digit is 8 ≥ 5, so round up |
Using Estimation to Check
After obtaining a calculated result, round the inputs to 1 sf and perform the same operation mentally. If the estimate and the calculated answer are of similar magnitude, the answer is likely correct. If they differ by a factor of 10 or more, an error has been made (often a misplaced decimal point or a forgotten power of ten).
For square roots and other roots, use knowledge of nearby perfect squares or cubes rather than the 1 sf rounding rule.
Worked Examples
Round 0.00835, 47 200 and 9.51 to 1 significant figure.
0.00835: first non-zero digit is 8; next digit is 3 < 5, so round down: \( \approx 0.008 \)
47 200: first sf is 4; next digit is 7 ≥ 5, so round up: \( \approx 50,000 \)
9.51: first sf is 9; next digit is 5 ≥ 5, so round up: \( \approx 10 \)
Estimate: \( \frac{4.87 \times 312}{0.52} \) and \( \sqrt{97.8} \)
First expression: Round to 1 sf: \( \frac{5 \times 300}{0.5} = \frac{1500}{0.5} = 3000 \)
(Actual value: \( \approx 2923 \) — the estimate is close.)
Square root: \( \sqrt{97.8} \approx \sqrt{100} = 10 \)
No 1 sf rounding needed — use the nearest perfect square instead.
A student calculates \( 28.4 \times 6.3 = 178.92 \). Another calculates it as \( 17.892 \). Use estimation to identify the correct answer.
Estimate: \( 30 \times 6 = 180 \).
178.92 is close to 180 ✓. 17.892 is 10 times too small ✗.
The first answer, £178.92, is correct. The second has a misplaced decimal point.
Key Takeaways
- Estimation uses 1 significant figure rounding to produce a quick approximate answer.
- The first significant figure is the first non-zero digit. Subsequent digits become zero.
- For square roots, use the nearest perfect square rather than 1 sf rounding.
- If a calculated answer and its estimate differ by an order of magnitude (factor of 10), a decimal point or power-of-ten error is likely.
- State clearly whether an estimate is an overestimate or underestimate where possible — this requires checking whether each rounded value is above or below the original.