What you'll learn
This topic covers upper and lower bounds and error intervals — the range a rounded or truncated value could actually take. In this guide you will learn how to find the bounds of a rounded measurement, how to write error intervals, how to use bounds in calculations, and how to give an answer to a suitable degree of accuracy. These are key skills for accuracy questions.
Key terms and definitions
Upper bound — the largest value a rounded measurement could be.
Lower bound — the smallest value a rounded measurement could be.
Error interval — the range of possible values, written a ≤ x < b.
Degree of accuracy — how precisely a value is given (nearest cm, 1 d.p., etc.).
Truncation — cutting off digits rather than rounding.
Core concepts
Finding bounds of a rounded value
A measurement rounded to a unit could be up to half a unit either side. A length of 24 cm to the nearest cm has a lower bound of 23.5 cm and an upper bound of 24.5 cm. The halfway value rounds up, so the lower bound is included and the upper bound is not.
Writing error intervals
The error interval uses inequalities: for 24 cm to the nearest cm, 23.5 ≤ x < 24.5. The ≤ shows the lower bound is possible and the < shows the upper bound is the limit but not reached. Match the half-unit to the rounding (0.05 for 1 d.p., etc.).
Bounds in addition and multiplication
For a sum or product, the maximum uses the upper bounds of both values and the minimum uses both lower bounds. For example, the greatest area of a rectangle uses the upper bound of each side.
Bounds in subtraction and division
For a difference or quotient, the maximum uses the upper bound of the first value and the lower bound of the second, and the minimum reverses this. This catches many students out, so think about what makes the result biggest.
Suitable degree of accuracy
When upper and lower bounds of a calculation agree when rounded to a certain accuracy, you can give the answer to that accuracy. This is how you justify a sensible final answer in bounds problems.
Worked examples
Example 1: Error interval
A mass is 5 kg to the nearest kg. Write the error interval.
4.5 ≤ m < 5.5.
Example 2: Maximum sum
Two lengths are each 8 cm to the nearest cm. Find the maximum total.
8.5 + 8.5 = 17 cm.
Example 3: Maximum quotient
Greatest value of a ÷ b where a = 20 and b = 5, both to the nearest whole?
Upper a, lower b: 20.5 ÷ 4.5 = 4.56 (3 s.f.).
Common mistakes and how to avoid them
Using the wrong half-unit. Match it to the rounding (0.5, 0.05, 0.005…).
Wrong inequality. Lower bound uses ≤, upper bound uses <.
Same bounds for division. Max quotient uses upper ÷ lower, not upper ÷ upper.
Rounding too early. Use exact bounds throughout.
Forgetting to justify accuracy. Compare bounds to state a suitable final answer.
Exam technique for Bounds and Error Intervals
Add or subtract half a unit for the bounds.
Write error intervals as a ≤ x < b.
Use both upper bounds for a maximum sum or product.
Use upper ÷ lower for a maximum quotient (and the reverse for a minimum).
Compare bounds to give a suitable degree of accuracy.
Quick revision summary
A value rounded to a unit could be up to half a unit either side: 24 cm to the nearest cm has bounds 23.5 ≤ x < 24.5, with the lower bound included (≤) and the upper bound not reached (<). Match the half-unit to the rounding (0.05 for 1 d.p., 0.005 for 2 d.p.). In calculations, a sum or product has its maximum from both upper bounds and minimum from both lower bounds. A difference or quotient has its maximum from the upper bound of the first and the lower bound of the second (and the reverse for the minimum) — a frequent trap. When the upper and lower bounds of a calculation agree to a certain accuracy, you can give the answer to that accuracy. The common errors are using the wrong half-unit, the wrong inequality, the wrong bounds for division, and rounding too early. Add/subtract half a unit, write intervals carefully, choose bounds by what makes the result biggest or smallest, and justify the final accuracy.