When a length is given as "12 cm to the nearest cm", it is not exactly 12 — it is anything that rounds to 12, i.e. from 11.5 up to (but not including) 12.5. Those two edges are the lower bound and upper bound. Real measurements are always rounded, so knowing how far off they might be is genuine, examinable maths.
Why this matters: engineers, builders and scientists must know the worst case. A shelf cut to "the nearest cm" could be up to half a centimetre too long — enough to jam a bracket. Cambridge tests bounds every series, and marks are lost when students combine bounds the wrong way round in a division.
1. Finding the bounds — half the last unit
- Halve the degree of rounding and add/subtract it. Rounded to the nearest 1 → ±0.5; nearest 10 → ±5; nearest 0.1 → ±0.05.
- Lower bound = value − half unit; upper bound = value + half unit.
- The upper bound is the value the measurement reaches but never equals — we still write it as the neat number.
| Measurement | Lower bound | Upper bound |
|---|---|---|
| 12 cm (nearest cm) | 11.5 cm | 12.5 cm |
| 250 g (nearest 10 g) | 245 g | 255 g |
| 4.6 s (nearest 0.1 s) | 4.55 s | 4.65 s |
2. Calculating with bounds
To get the extremes of a calculation, pick the bound of each measurement that pushes the answer highest or lowest. Addition and multiplication are intuitive; division and subtraction flip.
| Operation | Maximum uses | Minimum uses |
|---|---|---|
| a + b | a_upper + b_upper | a_lower + b_lower |
| a − b | a_upper − b_lower | a_lower − b_upper |
| a × b | a_upper × b_upper | a_lower × b_lower |
| a ÷ b | a_upper ÷ b_lower | a_lower ÷ b_upper |
Worked (area): A rectangle is 8 cm by 5 cm, each to the nearest cm. Upper bound of area = 8.5 × 5.5 = 46.75 cm². Lower bound = 7.5 × 4.5 = 33.75 cm².
Worked (speed): A car travels 100 m (nearest m) in 8 s (nearest s). Maximum speed = distance_upper ÷ time_lower = 100.5 ÷ 7.5 = 13.4 m/s. Minimum speed = 99.5 ÷ 8.5 = 11.7 m/s (3 s.f.).
Common exam mistake: for a maximum division, dividing upper by upper. It must be big ÷ small. Also, the upper bound of "12 to the nearest cm" is 12.5, not 12.4 or 12.49 — always use the exact half-unit.
Exam technique: the command word Calculate the upper bound wants the bound stated and then substituted — write "UB length = 8.5" before multiplying. Method marks reward the correct bound choice even if the arithmetic slips.