POP

E1.10 Limits of Accuracy

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01 — Explanation

How do I find the upper and lower bound of a rounded measurement?

Any measurement rounded to the nearest unit $u$ could actually be anywhere within half a unit either side of the stated value:

\text{LB} = \text{value} - \tfrac{u}{2} \qquad \text{UB} = \text{value} + \tfrac{u}{2}

For example, a length of $200\text{ m}$ measured to the nearest $10\text{ m}$ has $\text{LB}=195\text{ m}$ and $\text{UB}=205\text{ m}$ .

02 — Explanation

How do I find the bound of a sum or difference of two measurements?

Operation	To maximise the result	To minimise the result
$A+B$	$\text{UB}_A + \text{UB}_B$	$\text{LB}_A + \text{LB}_B$
$A-B$	$\text{UB}_A - \text{LB}_B$	$\text{LB}_A - \text{UB}_B$

Subtraction is the one that catches people out: to make $A - B$ as large as possible, you want $A$ big and $B$ small — so it's $\text{UB}_A - \text{LB}_B$ , not $\text{UB}_A - \text{UB}_B$ .

03 — Explanation

How do I find the bound of a product or quotient of two measurements?

Operation	To maximise the result	To minimise the result
$A\times B$	$\text{UB}_A \times \text{UB}_B$	$\text{LB}_A \times \text{LB}_B$
$A \div B$	$\text{UB}_A \div \text{LB}_B$	$\text{LB}_A \div \text{UB}_B$

The underlying pattern: to make a fraction as large as possible, maximise the top and minimise the bottom, and vice versa.

04 — Explanation

Which bounds do I use to maximise or minimise a calculated quantity like speed?

Identify the formula, then apply the product/quotient rule above term by term.

Worked Example

A distance of $d = 200\text{ m}$ is measured correct to the nearest $10\text{ m}$ , and a time of $t = 25\text{ s}$ is measured correct to the nearest second. Find the upper bound of the average speed, in m/s.

Bounds for $d$ : $\text{LB}_d = 195$ , $\text{UB}_d = 205$ .
Bounds for $t$ (half-unit $0.5$ ): $\text{LB}_t = 24.5$ , $\text{UB}_t = 25.5$ .
Speed $=\dfrac{d}{t}$ ; to maximise it, divide the largest possible distance by the smallest possible time:

\text{UB}_{\text{speed}} = \frac{205}{24.5} = 8.3673\ldots \approx 8.37 \text{ m/s (3 s.f.)}

Examiner Tip: It's tempting to pair "upper bound of speed" with "upper bound of time" — but a smaller denominator makes a fraction bigger, so maximising a quotient always means minimising the bottom. Keep exact bound values through the whole calculation and round only the final answer.

05 — Explanation

Why do we sometimes write bounds with a strict inequality on one side?

Strictly speaking, a measurement rounded to the nearest unit can equal its lower bound exactly, but can never reach its upper bound — a value of exactly $205\text{ m}$ would itself round to $210\text{ m}$ , not $200\text{ m}$ . This is why a fully rigorous bound is written as $195 \leqslant x < 205$ . At this level you are not expected to justify this asymmetry in every answer, but you should recognise the notation if it appears in a question, and never write $205 \leqslant x$ .

06 — Explanation

How do I find the bound of a quantity involving more than two measurements, like a formula with three variables?

Apply the same maximise/minimise logic to each variable in turn, based on whether it appears as a numerator-type or denominator-type quantity in the formula.

Worked Example

The volume of a cuboid is $V = lwh$ , where $l = 12\text{ cm}$ , $w=8\text{ cm}$ , and $h=5\text{ cm}$ , each measured correct to the nearest centimetre. Find the lower bound of the volume.

Every variable is multiplied (numerator-type), so minimising $V$ means minimising all three at once.
Bounds: $\text{LB}_l = 11.5$ , $\text{LB}_w = 7.5$ , $\text{LB}_h = 4.5$ .
Multiply: $V_{\min} = 11.5 \times 7.5 \times 4.5 = 388.125\text{ cm}^3$ .