When measured values are rounded to a given accuracy, each measurement has an upper and lower bound. A calculation combining such values therefore produces a result that also has a range. The challenge is knowing which combination of input bounds to use in order to find the maximum or minimum possible result of that calculation.
How do you find the upper and lower bounds of a single value?
If a measurement is given to some degree of accuracy, the upper bound is half a unit above the stated value, and the lower bound is half a unit below — where "a unit" is the last decimal place given.
- Value given to nearest 10: bounds are ±5
- Value given to nearest whole number: bounds are ±0.5
- Value given to 1 d.p.: bounds are ±0.05
- Value given to 2 d.p.: bounds are ±0.005
- Value given to 2 s.f. (e.g. 3,700 to 2 s.f.): bounds are ±50
Example: A length l = 6.4 cm (to 1 d.p.)
- Lower bound: 6.4 − 0.05 = 6.35 cm
- Upper bound: 6.4 + 0.05 = 6.45 cm
Error interval: 6.35 ≤ l < 6.45 (upper bound is not included when rounded — the value 6.45 itself would round to 6.5, not 6.4).
What are the rules for bounds in calculations?
The operation determines which bounds you combine to find the maximum or minimum result.
| Operation | Maximum result | Minimum result |
|---|---|---|
| a + b | UB(a) + UB(b) | LB(a) + LB(b) |
| a − b | UB(a) − LB(b) | LB(a) − UB(b) |
| a × b | UB(a) × UB(b) | LB(a) × LB(b) |
| a ÷ b | UB(a) ÷ LB(b) | LB(a) ÷ UB(b) |
The subtraction and division rows are the tricky ones. For a − b: to maximise a difference, make the value being subtracted (b) as small as possible. For a ÷ b: to maximise a quotient, make the divisor (b) as small as possible.
Worked examples
Addition: A rectangle has length a = 8.3 cm and width b = 4.7 cm, both measured to 1 d.p. Find the maximum possible perimeter.
- UB(a) = 8.35, UB(b) = 4.75
- Perimeter = 2(a + b), so maximum perimeter = 2 × (8.35 + 4.75) = 2 × 13.1 = 26.2 cm
Subtraction: Two lengths: p = 15 m (nearest metre) and q = 9 m (nearest metre). Find the minimum possible value of p − q.
- LB(p) = 14.5, UB(q) = 9.5
- Minimum (p − q) = 14.5 − 9.5 = 5 m
Multiplication: A field is a = 120 m (nearest 10 m) by b = 85 m (nearest 5 m). Find the maximum possible area.
- UB(a) = 125, UB(b) = 87.5
- Maximum area = 125 × 87.5 = 10,937.5 m²
- Minimum area = LB(a) × LB(b) = 115 × 82.5 = 9,487.5 m²
Division: Speed = distance ÷ time. Distance d = 240 km (nearest 10 km), time t = 3 hours (nearest hour). Find the maximum possible speed.
- UB(d) = 245, LB(t) = 2.5
- Maximum speed = 245 ÷ 2.5 = 98 km/h
- Minimum speed = LB(d) ÷ UB(t) = 235 ÷ 3.5 = 67.14... km/h (67.1 to 1 d.p.)
How do you handle truncation instead of rounding?
Truncation cuts off digits without rounding, so the truncated value is always less than or equal to the true value. If a value is truncated to, say, 1 decimal place:
- Lower bound = the truncated value exactly
- Upper bound = the truncated value + 0.1 (i.e. one unit of the last place)
Example: Length l is truncated to 5.3 cm (to 1 d.p.)
- Lower bound: 5.3 cm
- Upper bound: 5.4 cm (not included: 5.3 ≤ l < 5.4)
Frequently asked questions
Why does the maximum of a − b use LB(b) and not UB(b)?
Think about it in plain language: to make a − b as large as possible, you want a to be as large as possible (use UB(a)) and b to be as small as possible (use LB(b)), because you're subtracting b. A smaller subtracted value leaves more behind.
Do bounds calculations appear on Foundation or Higher tier?
Bounds and error intervals are on Higher GCSE in most exam board specifications (AQA, OCR, Edexcel/Pearson). Finding bounds of a single value appears at Foundation; using bounds in multi-step calculations is Higher only.
How precise should I give my answer?
Do not round intermediate bounds — use them exactly in the calculation. Only round the final answer if the question specifies a degree of accuracy. Premature rounding can push your answer outside the correct range.
What if one of the inputs to a division can be zero?
If LB(b) = 0 (meaning b could be zero), the division a ÷ b is undefined at that bound. In practice, GCSE questions are designed to avoid this — they give measurements that cannot realistically be zero. If you encounter it, note the issue and use the smallest positive value that the bounds allow.
For Socratic GCSE bounds and calculations practice with Professor Pi, see aitutors.me.