A measurement rounded to a given accuracy lies in a range. The lower bound is the smallest value that rounds to it; the upper bound is the limit it cannot reach.
Rounded to the nearest 10 cm, 240 cm lies in 235 \leq x < 245. To 1 d.p., 6.3 lies in 6.25 \leq x < 6.35.
Take half the degree of accuracy either side: if accurate to the nearest 5, add and subtract 2.5.
Numbers rounding to 5
Bounds in calculations
Addition: lower + lower, upper + upper.
Subtraction: a - b has upper bound a\text{upper} - b\text{lower} and lower bound a\text{lower} - b\text{upper}.
For positive bounds only: multiplication uses lower × lower and upper × upper. Division uses upper ÷ lower for the upper bound, and lower ÷ upper for the lower bound; the denominator must remain positive.
For signed intervals, compare endpoint products rather than assuming lower × lower works. If a divisor interval contains zero, these simple finite quotient rules do not apply.
Using bounds
Area of a rectangle 8.5 cm by 4.2 cm (1 d.p.): upper = 8.55 \times 4.25, lower = 8.45 \times 4.15.
To give an answer to a stated accuracy, work out both bounds and check they round to the same value; that value is the answer to that accuracy.
Error intervals
Written as a \leq x < b. Use \leq for the lower bound and < for the upper bound.
The ≤ lower / < upper form here describes positive measurements rounded using the standard GCSE convention. Bounds of a derived calculation can have different endpoint inclusion; a limiting bound need not be attainable.
スライド
Sign up free to view the lesson slides
Step through every slide for this topic — plus flashcards and revision notes — with a free account.
