What the exam asks
Approximation questions give expressions with ugly decimals such as 24.98% of 639.9 or sqrt(1443). You are expected to round smartly and choose the nearest option. Usually 3 to 5 questions in Prelims; the target is 15 to 20 seconds each. Options are normally far enough apart that clean rounding gives the right answer.
Rounding rules
- Round each number to the nearest convenient value: 24.98% becomes 25%, 639.9 becomes 640, 11.97 becomes 12.
- For a product, if you round one number up, try to round the other down to keep the error small.
- For a division, round both numbers in the same direction (up and up, or down and down).
- Square roots: go to the nearest perfect square. sqrt(1443) is about sqrt(1444) = 38; sqrt(626) is about 25.
- Cube roots: go to the nearest perfect cube. Cube root of 2,195 is about 13 (2,197).
- Percentages: change them to friendly fractions (24.9% to 1/4, 33.2% to 1/3, 12.6% to 1/8).
Friendly values to recognise
| Looks like | Use |
|---|---|
| sqrt(195), sqrt(197) | 14 (196) |
| sqrt(483), sqrt(485) | 22 (484) |
| sqrt(1023) | 32 (1,024) |
| sqrt(2499) | 50 (2,500) |
| cube root of 3,373 | 15 (3,375) |
| cube root of 9,259 | 21 (9,261) |
| 16.66% / 14.28% / 11.11% | 1/6, 1/7, 1/9 |
Step-by-step method
- Scan the options first. If they are 50 apart, you can round aggressively; if they are 5 apart, be careful.
- Round every number and write the clean expression.
- Apply BODMAS to the clean expression.
- Pick the option closest to your value. If two options are close, check the direction of your rounding error.
Worked example
39.97% of 649.8 + 17.02 x 14.99 = ?
40% of 650 = 260. 17 x 15 = 255. Total about 515. With options 480, 515, 545, 570, 600, the answer is 515.
Worked example 2
(45.02)² - (31.97)² = ?
Use a² - b² = (a + b)(a - b) with 45 and 32: 77 x 13 = 1,001. The answer is about 1,000.
Worked example 3
sqrt(1443) x 4.98 + 199.99 / 8.01 = ?
38 x 5 = 190; 200 / 8 = 25; total about 215.
Error control (when options are close)
- If you rounded 1443 to 1444, the true root is slightly less than 38 - so the real answer is slightly less than your estimate.
- When a big number is multiplied, even a small rounding (4.98 to 5) adds error. Estimate the direction: 38 x 0.02 = 0.76 too high.
- Never round something like 1.49 to 1 or 2 inside a product; keep 1.5.
Common traps
- Rounding 49.5% of 802 as 50% of 800 is fine, but rounding 4.5 x 19.8 as 5 x 20 = 100 overshoots the true 89.1 badly.
- Forgetting that division by a rounded-down number raises the result.
- Calculating exactly - it wastes time; the question rewards estimation.
Time target: 15 seconds each. Practise squares up to 50 and cubes up to 25 so that roots come instantly.