Limits of accuracy and bounds

What are bounds?

When a measurement is given to a stated degree of accuracy (e.g. "to the nearest cm" or "to 2 d.p."), the true value could lie anywhere in a range — the error interval.

For a value x rounded to accuracy δ:

• Lower bound = x − δ/2
• Upper bound = x + δ/2
• Error interval: lower bound ≤ true value < upper bound (note: upper is strict <)

Example: length = 8.4 cm (to 1 d.p.). δ = 0.1. Error interval: 8.35 ≤ length < 8.45.

Truncation vs rounding

Rounding to nearest unit: 7.4 → lower = 7.35, upper < 7.45. Truncating to 1 d.p.: 7.4 means any value from 7.4 to < 7.5. Lower = 7.4, upper < 7.5. Edexcel may specify truncation — check the question wording.

Bounds in calculations

To maximise a product (a × b): use upper bound of a × upper bound of b. To minimise a product: lower × lower. To maximise a quotient (a ÷ b): upper bound of a ÷ lower bound of b. To minimise a quotient: lower ÷ upper.

To maximise a difference (a − b): upper of a − lower of b. To minimise a sum (a + b): lower of a + lower of b.

Upper bound of calculated quantities

Example: speed = distance / time. distance = 240 m (nearest 10 m), time = 12 s (nearest second). Maximum speed = upper bound of distance ÷ lower bound of time = 245 ÷ 11.5 = 21.3 m/s.

Edexcel exam style

Edexcel Higher Papers often present a multi-step bounds question: "A formula is used to calculate V = ab. Given a = ... and b = ..., each measured to 1 d.p., find the upper bound of V and show whether a given value for V is guaranteed to be correct."