When a measurement is rounded, there is a range of values it could actually be. Bounds and error intervals let you state that range precisely. GCSE exams ask you to find the upper and lower bound of a rounded value and use them to work out the maximum or minimum result of a calculation.
What is a bound and what is an error interval?
A bound is the limiting value that a rounded measurement could just reach. There are two:
- The lower bound (LB) is the smallest value the true measurement could be.
- The upper bound (UB) is the largest value the true measurement could be.
An error interval states both bounds together using inequality notation:
$$\text{lower bound} \leq x < \text{upper bound}$$
Notice that the inequality sign on the left is ≤ (the value can equal the lower bound) but on the right is < (the value cannot quite equal the upper bound — if it did, it would round up to the next value).
How do you find bounds for a value rounded to the nearest unit?
The rule for rounding is:
Half the rounding unit is added to and subtracted from the stated value.
Worked example: A length is given as 7 m, rounded to the nearest metre.
- Rounding unit = 1 m, so half = 0.5 m
- Lower bound = 7 − 0.5 = 6.5 m
- Upper bound = 7 + 0.5 = 7.5 m
- Error interval: 6.5 ≤ length < 7.5
| Stated value | Rounded to | Lower bound | Upper bound | Error interval |
|---|---|---|---|---|
| 40 kg | nearest 10 kg | 35 kg | 45 kg | 35 ≤ x < 45 |
| 3.6 cm | 1 d.p. | 3.55 cm | 3.65 cm | 3.55 ≤ x < 3.65 |
| 2400 | nearest 100 | 2350 | 2450 | 2350 ≤ x < 2450 |
| 0.08 s | 2 d.p. | 0.075 s | 0.085 s | 0.075 ≤ x < 0.085 |
What is the difference between rounding and truncation?
Truncation simply cuts off digits without rounding. The lower bound is therefore the value itself; the upper bound is the value plus the truncation unit.
Example: A number is truncated to 2 decimal places to give 4.73.
- Lower bound = 4.73 (the number could be exactly 4.73)
- Upper bound = 4.74 (it could be just below 4.74 but could not reach 4.74)
- Error interval: 4.73 ≤ x < 4.74
The crucial difference is that truncation always rounds down — there is no "round up" for large decimal parts. That shifts the error interval upwards compared to rounding.
How do you apply bounds to compound measure calculations?
Bounds questions often involve calculating the maximum or minimum of a quantity that depends on two measured values.
The key rules for finding a maximum or minimum:
| Calculation | To MAXIMISE | To MINIMISE |
|---|---|---|
| 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) |
Worked example: A car travels 120 km (to the nearest 10 km) in 2 hours (to the nearest hour). Find the maximum possible average speed.
- Distance: LB = 115 km, UB = 125 km
- Time: LB = 1.5 h, UB = 2.5 h
- Maximum speed = maximum distance ÷ minimum time = 125 ÷ 1.5 = 83.3 km/h (1 d.p.)
To get maximum speed, divide the biggest distance by the smallest time — a larger numerator and smaller denominator both push the result higher.
How do you write bounds for numbers given to significant figures?
The method is identical — just identify the rounding unit correctly.
- 3400 (to 2 s.f.): rounding unit is 100. LB = 3350, UB = 3450.
- 0.0056 (to 2 s.f.): rounding unit is 0.0001. LB = 0.00555, UB = 0.00565.
A common pitfall: students use the full stated value rather than the rounding unit. Always ask "what is the smallest change that would produce a different rounded answer?" — that is your rounding unit.
What notation does the mark scheme expect?
GCSE mark schemes accept these forms for an error interval:
- Inequality notation: 6.5 ≤ x < 7.5
- Interval notation: [6.5, 7.5)
Use ≤ at the lower end and < at the upper end. Using < at both ends, or ≤ at the upper end, will cost you a mark. When the question says "write down the error interval", they expect the double inequality format.
Frequently asked questions
Why is the upper bound never actually reached?
Because if a measurement reached the upper bound exactly, it would round up to the next value, not round down to the stated value. For example, 7.5 rounds to 8, not 7. So the true length must be less than 7.5, not equal to it — hence the strict inequality (<) at the upper end.
Do I always subtract/add half the rounding unit?
For rounding, yes — always half the rounding unit. For truncation, the lower bound is the stated value itself and the upper bound adds the full rounding unit (not half). Make sure you read the question to see whether the value was rounded or truncated.
What if the question asks for the lower bound of a subtraction like a − b?
To minimise a − b, you want the smallest a and the largest b, so use LB(a) − UB(b). Students commonly make the error of using LB(a) − LB(b), which does not give the minimum result. Always think: to make the answer as small as possible, make the top as small as possible and the bottom as large as possible.
How accurate should I give my answer to bounds calculations?
Unless the question specifies, give at least 3 significant figures and do not round intermediate steps. If the question says "find the maximum speed to 1 decimal place," round only your final answer, not intermediate values. Premature rounding is the most common source of method-mark loss in these questions.
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