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Number · Foundation & Higher

Bounds & accuracy

When a measurement is given to a certain degree of accuracy, the true value lies within an upper and lower bound.

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Key facts to remember

  • 1If a value is rounded to the nearest unit, the bounds are ± half a unit.
  • 2Lower bound = stated value − half the unit of rounding.
  • 3Upper bound = stated value + half the unit of rounding (but the upper bound itself is never reached).
  • 4For maximum of a sum: add upper bounds. For minimum of a sum: add lower bounds.
  • 5For maximum of a quotient A÷B: use max A and min B.
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Formulas

Lower bound
Rounded value − (rounding unit ÷ 2)
Upper bound
Rounded value + (rounding unit ÷ 2)
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Worked examples

Example 1

A length is measured as 8.4 cm to 1 d.p. Write down the upper and lower bounds.

Working

  1. Rounding unit = 0.1 cm, so half = 0.05 cm
  2. Lower bound = 8.4 − 0.05 = 8.35 cm
  3. Upper bound = 8.4 + 0.05 = 8.45 cm
AnswerLB = 8.35 cm, UB = 8.45 cm
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Common mistakes

✗Halving the last significant figure incorrectly.
✗Mixing up which operation needs upper/lower bounds (e.g. for division, max = max ÷ min).
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Exam tips

✓Write "LB =" and "UB =" clearly — examiners must see both.
✓For multi-step calculations with bounds, think about what combination gives the max or min result.

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