Limits of accuracy

Maths · Number. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

Test this topic free All Maths topics

The idea

A measurement that has been rounded is not exact. A length given as 7 cm to the nearest centimetre could be anything from 6.5 cm up to 7.5 cm. The smallest possible value is the lower bound and the largest is the upper bound.

To find the bounds, take half of the unit the value was rounded to. Subtract it for the lower bound and add it for the upper bound.

When rounded values are used in a calculation, the answer also has bounds. To make an answer as large as possible, choose the bounds that help: for a subtraction or a division this means the largest first number and the smallest second number.

Rules to remember

lower bound = value − half the rounding unit; upper bound = value + half the rounding unit
Nearest 10: ± 5. Nearest whole number: ± 0.5. 1 decimal place: ± 0.05. Nearest 5: ± 2.5.
Error interval: lower bound ≤ x < upper bound
Upper bound of a + b or a × b: use both upper bounds.
Upper bound of a − b: upper a − lower b. Lower bound of a − b: lower a − upper b.
Upper bound of a ÷ b: upper a ÷ lower b. Lower bound of a ÷ b: lower a ÷ upper b.

Common mistake

Using two upper bounds for a subtraction or a division. To make a − b or a ÷ b as large as possible, b must be as small as possible.

Worked example

a = 9 and b = 2, each correct to the nearest whole number. Find the upper bound of a ÷ b, to 3 significant figures.

Half the rounding unit is 0.5, so 8.5 ≤ a < 9.5 and 1.5 ≤ b < 2.5.
For the largest answer use upper a and lower b: 9.5 ÷ 1.5.
9.5 ÷ 1.5 = 19 ÷ 3 = 6.333…

Answer: 6.33

Ask about this

Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.

Read the full notes

Watch: Limits of Accuracy Part 1

Keep going

← Estimation Ratio and proportion →Take a Maths test