Bounds

#### Tier: #Foundation #Higher

Description

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.

Links

Linear inequalities Substitution Rounding Error intervals