Bounds & Limits of Accuracy

Error intervals, upper and lower bounds for rounded measurements, and calculations with bounds.

When a number is rounded, the true value lies in an interval. You must find upper and lower bounds and use them in calculations (especially for maximum and minimum values).

Error interval for a rounded value

If a number is rounded to the nearest unit (e.g. nearest 10, nearest 0.1), the error is half of that unit either side. Example: 7.3 rounded to 1 d.p. Lower bound = 7.25 Upper bound = 7.35 Error interval: 7.25 ≤ x < 7.35 (Usually the upper bound is written with ≤ or < depending on the specification – WJEC typically uses ≤ x < for continuous measures.)

Worked example

A length is given as 12 cm to the nearest centimetre. Write the error interval.

Solution

Half unit = 0.5 cm Lower bound = 11.5 Upper bound = 12.5 **11.5 ≤ length < 12.5**

Bounds in calculations

To find the maximum or minimum of a calculated value: **Maximum of a + b:** upper A + upper B **Minimum of a + b:** lower A + lower B **Maximum of a − b:** upper A − lower B **Minimum of a − b:** lower A − upper B **Maximum of a × b:** upper A × upper B (for positive values) **Minimum of a × b:** lower A × lower B **Maximum of a ÷ b:** upper A ÷ lower B **Minimum of a ÷ b:** lower A ÷ upper B

Worked example

a = 8.4 (1 d.p.), b = 2.1 (1 d.p.). Both positive. Find the maximum value of a ÷ b.

Solution

Upper a = 8.45, lower b = 2.05 Max a ÷ b = 8.45 ÷ 2.05 ≈ **4.122** (to 3 d.p.)

Practice questions

Try these without looking at the answers first.

Q1.Number given as 40 to nearest 10. Error interval?

Show answer

35 ≤ x < 45

Q2.Mass 3.6 kg to 1 d.p. Lower and upper bounds?

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3.55 kg and 3.65 kg

Q3.p = 5.2 (1 d.p.), q = 1.4 (1 d.p.). Minimum of p − q?

Hint: lower p − upper q = 5.15 − 1.45 = 3.7

Show answer

3.7

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