When measurements are rounded, the actual value lies within an interval defined by the lower and upper bounds.
For a value $x$ rounded to degree of accuracy $d$: $$x - \frac{d}{2} \leq \text{actual value} < x + \frac{d}{2}$$
Bound calculations:
- Adding bounds: lower + lower = lower bound; upper + upper = upper bound
- Subtracting: lower − upper = lower bound; upper − lower = upper bound
- Multiplying: lower × lower = lower bound; upper × upper = upper bound
- Dividing: lower ÷ upper = lower bound; upper ÷ lower = upper bound
Example: $a = 6.4$ (to 1 d.p.) and $b = 3.2$ (to 1 d.p.). Find the upper bound of $\frac{a}{b}$. $$\frac{6.45}{3.15} \approx 2.048...$$
Remember: to maximise a quotient, maximise the numerator and minimise the denominator.
Common error: using the same bound type for both values in a division, or forgetting to use strict inequality at the upper bound.