GCSE Maths Estimation & bounds revision

To round, find the digit in the place you are asked for, look at the next digit, and round up if it is 5 or more. Rounding 7.842: to 1 d.p. the next digit is 4, so 7.8; to 2 d.p. it is 2, so 7.84. Significant figures count from the first non-zero digit, so 0.004 589 starts counting at the 4: to 1 s.f. it is 0.005, to 2 s.f. it is 0.0046. For whole numbers the trailing digits become zeros: 3486 to 2 s.f. is 3500, and to 1 s.f. is 3000.

Keep the place value when you round: 3486 to 2 s.f. is 3500, not 35. Measures round the same way — a length of 14.68 cm to the nearest centimetre is 15 cm. Watch zeros: in 0.0305 the first significant figure is the 3, and the zero between 3 and 5 still counts, so to 2 s.f. the next digit 5 rounds it up to 0.031.

An estimate replaces every number with its value to 1 significant figure: (6.8 × 41.2) ÷ 19.5 becomes (7 × 40) ÷ 20 = 280 ÷ 20 = 14. The true value is 14.4 (to 3 s.f.), so the estimate says a calculator answer of 1.44 or 144 must be a slip. Square roots estimate by bracketing: √98 sits just under 10 because 9.9² = 98.01 — so √98 ≈ 10 to 1 s.f.

Estimation questions say “estimate” or “without a calculator” — they want the rounded working shown. Write the rounded calculation as a separate line: “≈ 7 × 40 ÷ 20”. That line carries the method mark even if the mental arithmetic slips. The word “approximate” in the answer (“≈ 14”) also signals you know the value is not exact.

If w = 4.7 to 1 d.p., the original could be anything from 4.65 up to (but not including) 4.75, because 4.75 itself would round to 4.8. In inequality notation: 4.65 ≤ w < 4.75 — closed at the bottom, open at the top. The half-unit rule works at any accuracy: 350 to 2 s.f. means 345 ≤ x < 355.

Truncation (chopping digits without rounding) gives a different interval: if 8.94 is a truncated value to 2 d.p., the original sat anywhere from 8.94 up to 8.95, so 8.94 ≤ x < 8.95 — closed at the bottom because truncation only ever removes digits. Exam questions state “by truncation” explicitly; assuming rounding when the word says truncation is a guaranteed mark loss.

Standard units at GCSE: length in mm, cm, m, km; mass in mg, g, kg, tonnes; time in seconds, minutes, hours; money in pounds and pence (£1 = 100p). Calculations often need decimals: 45 minutes is 0.75 hours, 350 g is 0.35 kg, 235p is £2.35. Keep units consistent before calculating — a speed mixing minutes and hours is the classic error. Compound units (speed, density, pressure) get their own treatment in Chapter 13; here the skill is converting single units and writing results with the right unit attached.

Higher. Each rounded value carries a hidden interval, and a calculation should be tested at its extremes. Take p = 4.5 and q = 3.2 (both to 1 d.p.), so p sits in [4.45, 4.55) and q in [3.15, 3.25). Then: p + q runs from 4.45 + 3.15 = 7.6 to 4.55 + 3.25 = 7.8; p × q from 4.45 × 3.15 = 14.0175 to 4.55 × 3.25 = 14.7875. For division the biggest answer uses the biggest over the smallest: p ÷ q has UB = 4.55 ÷ 3.15 ≈ 1.44 and LB = 4.45 ÷ 3.25 ≈ 1.37. Differences flip: p − q has UB = 4.55 − 3.15 = 1.4 and LB = 4.45 − 3.25 = 1.2.

Context problems follow the same logic. A car travels 40 m (to the nearest metre) in 5 s (to the nearest second): the fastest speed uses the largest distance and the smallest time, UB = 40.5 ÷ 4.5 = 9 m/s, and the slowest is 39.5 ÷ 5.5 ≈ 7.18 m/s. Phrases decide which bound matters: “definitely under the limit” compares with the UB, “at least” with the LB.