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Limits of Accuracy

FoundationHigherAQAEdexcelOCR

This free Foundation and Higher GCSE Maths worksheet on limits of accuracy helps you revise upper and lower bounds. Questions build from straightforward to exam standard, with full worked answers in the mark scheme — ideal for revision or homework. Suitable for AQA, Edexcel and OCR. The bounds lie half a unit either side of a rounded value.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

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Higher

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Topic overview

When a measurement is rounded, the true value lies within a range rather than being exactly the figure given. The limits of accuracy describe that range.

A length given as \(8\)cm to the nearest centimetre could actually be anything from \(7.5\)cm up to but not including \(8.5\)cm. The bounds are found by adding and subtracting half the rounding unit.

The upper bound is the value at which the measurement would round up to the next figure. Although \(8.5\) itself would round to \(9\), it is conventionally written as the upper bound, so the range is \(7.5 \le x < 8.5\).

Revision notes

Finding the bounds

Halve the rounding unit, then add and subtract it from the given value.

For \(8\)cm to the nearest cm, half of \(1\) is \(0.5\), giving bounds of \(7.5\) and \(8.5\)cm.

Different rounding units

The rounding unit is whatever the measurement was rounded to.

For \(3.4\)kg to \(1\) decimal place, the unit is \(0.1\), so half is \(0.05\) and the bounds are \(3.35\) and \(3.45\)kg.

Writing the answer

Use the inequality form \(\text{lower} \le x < \text{upper}\).

The lower bound is included and the upper is not, which is why the signs differ.

Key points

  • A rounded value lies within a range.
  • Halve the rounding unit to find the bounds.
  • Add for the upper bound, subtract for the lower.
  • The lower bound is included.
  • The upper bound is not included.
  • Write as \(\text{lower} \le x < \text{upper}\).

Worked examples

Example 1

A length is \(12\)cm to the nearest cm. Find the bounds.

Working

\[1 \div 2 = 0.5\]halve the rounding unit
\[12 - 0.5 = 11.5 \text{ and } 12 + 0.5 = 12.5\]subtract and add
\[11.5 \le x < 12.5\]write the range

Example 2

A mass is \(4.6\)kg to 1 decimal place. Find the bounds.

Working

\[0.1 \div 2 = 0.05\]halve the rounding unit
\[4.55 \text{ and } 4.65\]subtract and add
\[4.55 \le x < 4.65\]write the range

Example 3

A distance is \(250\)m to the nearest \(10\)m. Find the bounds.

Working

\[10 \div 2 = 5\]halve the rounding unit
\[245 \text{ and } 255\]subtract and add
\[245 \le x < 255\]write the range

Common mistakes

  • Using the whole rounding unit instead of half.

    The bounds are half a unit either side, so 8cm gives 7.5 and 8.5, not 7 and 9.

  • Getting the rounding unit wrong.

    For 1 decimal place the unit is 0.1, so half is 0.05.

  • Using ≤ for the upper bound.

    The upper bound is not included, so the sign is <.

  • Rounding the bounds.

    The bounds are exact values and should not themselves be rounded.

Exam tips

  • Identify the rounding unit before doing anything else.
  • Halve it, then add and subtract.
  • Use ≤ for the lower bound and < for the upper.
  • Do not round the bounds themselves.

Key terms

Upper bound
The largest value a measurement could be.
Lower bound
The smallest value a measurement could be.
Rounding unit
The precision to which a value was rounded.
Limits of accuracy
The range within which the true value lies.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.