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Number · Subtopic 5 of 5

Estimation and Limits of Accuracy

Round to significant figures or decimal places, and estimate answers by rounding every number to $1$ s.f. A measurement rounded to the nearest unit could really be up to half a unit either side: those are its bounds. For calculations, choose the bounds that make the answer biggest or smallest.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Round to decimal places and significant figures.
Estimate a calculation by rounding to $1$ significant figure.
Write down upper and lower bounds of a measurement.
Find bounds of sums, differences, products and quotients.

Pause and think

A sign says a bridge is " $3$ m high" (to the nearest metre). Could a $3.4$ m lorry pass under it safely?

Section 2 of 14

Core idea

Start hereHow long is "8 cm"?

A ruler reading of 8 cm, to the nearest cm, could be 7.6 or 8.4. Engineers designing a part must know the worst case. Bounds tell you the range a rounded number could really be.

The big idea

Rounded to a unit $u$ : true value within $\pm\frac u2$ . For calculations, choose the bound combination that makes the answer biggest (or smallest).

1. Rounding and bounds

8 cm to the nearest cm: the true length lies in this range

Half a unit either side of the rounded value:

7.5\le l<8.5

x-\tfrac u2\le\text{true}<x+\tfrac u2

$x$: rounded value.
$u$: rounding unit (1 cm, 0.1 s, 10 m...).

$0.04726$ : $0.047$ (2 s.f.), $0.05$ (2 d.p.).

2. Calculations with bounds

Which bounds to use

Drawing diagram…

For the minimum, swap every max and min.

Worked exampleRectangle 8 cm by 5 cm

Both to the nearest cm. Bounds of the area.

Goal: Upper, lower.
Strategy: Multiply matching bounds.

$8.5\times5.5=46.75$
Upper.
$7.5\times4.5=33.75$
Lower.

Answer

33.75\le A<46.75

cm².

✓ CheckThe stated value

40

is inside ✓.

Worked exampleSpeed

100 m (nearest 10 m) in 12.0 s (nearest 0.1 s).

Goal: Max and min speed.
Strategy: max ÷ min; min ÷ max.

$95\le d<105,\quad11.95\le t<12.05$
Bounds.
$\frac{105}{11.95}=8.79$
Maximum.
$\frac{95}{12.05}=7.88$
Minimum.

AnswerBetween

7.88

and

8.79

m/s.

✓ Check

100/12=8.33

lies between ✓.

Common misconceptionUpper bound with subtraction Max of

a-b

uses the lower bound of

b

Pause and think

Upper bound of $a-b$ with $a=10$ , $b=4$ (nearest whole)?

Summary

Bounds ± half a unit.
Choose bounds to push the answer the right way.

Upper bound: Largest value that rounds to the given value.

HistoryArchimedes bounds π. Around

250

BC Archimedes couldn't compute

\pi

exactly, so he trapped it between bounds:

3\tfrac{10}{71}<\pi<3\tfrac17

. Giving a lower and upper bound when the exact value is unknown is still how scientists report measurements.

Fun factRounding errors that mattered. In 1982 the Vancouver Stock Exchange started a new index at

1000

. It was recalculated thousands of times a day, each time truncated to

3

decimal places. Within two years it read about

520

instead of the correct value of about

1100

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