Rounding, Estimation and Accuracy

Knowing standard units and how to convert between them is essential for measurement problems.

Length: $1\text{ km}=1000\text{ m};1\text{ m}=100\text{ cm};1\text{ cm}=10\text{ mm}$

Mass: $1\text{ kg}=1000\text{ g};1\text{ tonne}=1000\text{ kg}$

Capacity: $1\text{ litre}=1000\text{ ml};1{\text{ cm}}^3=1\text{ ml}$

Time: $1\text{ hour}=60\text{ min};1\text{ min}=60\text{ s}$

Compound measures combine two units:

Measure	Formula	Units
Speed	$\text{speed}=\frac{\text{distance}}{\text{time}}$	m/s, km/h, mph
Density	$\text{density}=\frac{\text{mass}}{\text{volume}}$	g/cm³, kg/m³
Pressure	$\text{pressure}=\frac{\text{force}}{\text{area}}$	N/m², Pa

Worked example — convert 90 km/h to m/s.

$90\text{ km/h}=90\times \frac{1000}{3600}\text{ m/s}=90\times \frac{5}{18}=25\text{ m/s}$ ✓

Estimation uses rounding to 1 significant figure to quickly approximate the result of a calculation.

Worked example — estimate the value of $\frac{4.87\times 29.3}{0.531}$

Round each number to 1 significant figure: $\frac{5\times 30}{0.5}=\frac{150}{0.5}=300$

(Exact answer: $\approx 268.7$; the estimate is in the right order of magnitude.) ✓

Uses of estimation:

• Checking whether a calculator answer is reasonable
• Answering "estimate" questions in exams (show the rounded values you use — this is how method marks are awarded)
• Quick mental calculations

When a question says "use approximations to estimate", you must show the rounded values — a bare final answer scores no method marks.

Rounding to $n$ decimal places (d.p.): look at the $(n+1)$th digit. Round up if $\ge 5$; round down (truncate) if $<5$.

Worked example — round 3.2473 to 2 d.p.: the 3rd decimal digit is 7 ≥ 5, so round up: $3.25$

Rounding to $n$ significant figures (s.f.): count from the first non-zero digit. Same up/down rule.

Number	1 s.f.	2 s.f.	3 s.f.
34,682	30,000	35,000	34,700
0.004306	0.004	0.0043	0.00431
3.995	4	4.0	4.00

Trailing zeros after a decimal point are significant: 4.0 has 2 significant figures; 4.00 has 3.

Truncation means cutting off digits without rounding up. Truncating 3.849 to 2 d.p. gives 3.84 (not 3.85).

Any rounded or truncated measurement has a range of values that could have produced it — the error interval.

For a value $x$ rounded to a given degree of accuracy, the error interval is written using inequality notation:

$\text{lower bound}\le x<\text{upper bound}$

Rounded to the nearest unit:

• 37 rounded to the nearest whole number: error interval $36.5\le x<37.5$

Rounded to 1 decimal place:

• 5.3 rounded to 1 d.p.: error interval $5.25\le x<5.35$

Truncated to 1 decimal place:

• 5.3 by truncation: error interval $5.3\le x<5.4$ (truncation never rounds up, so the lower bound equals the stated value)

Worked example — a length $L$ is given as 8.4 cm, rounded to 1 d.p. Write the error interval.

$8.35\le L<8.45$ ✓

Note: the upper bound uses strict inequality ($<$) because a value of exactly 8.45 would round up to 8.5, not 8.4.

Upper bound is the largest value a measurement could take. Lower bound is the smallest.

When performing calculations with measurements, the bounds of the result depend on which combination produces the largest/smallest answer:

Calculation	Maximum result	Minimum result
$a+b$	UB of $a$ + UB of $b$	LB of $a$ + LB of $b$
$a-b$	UB of $a$ − LB of $b$	LB of $a$ − UB of $b$
$a\times b$	UB of $a$ × UB of $b$	LB of $a$ × LB of $b$
$a÷b$	UB of $a$ ÷ LB of $b$	LB of $a$ ÷ UB of $b$

Worked example — a rectangle has length $l=7.3$ cm and width $w=4.8$ cm, both rounded to 1 d.p. Find the upper bound of the area.

Bounds: $7.25\le l<7.35$ and $4.75\le w<4.85$

Upper bound of area $=7.35\times 4.85=35.6475$ cm² ✓

1. Confusing significant figures with decimal places

"3 significant figures" and "3 decimal places" are different. 0.004306 to 3 s.f. is 0.00431 (leading zeros are not significant). To 3 d.p. it would be 0.004.

2. Error interval upper bound — strict or non-strict inequality

The upper bound always uses strict inequality ($<$) for rounded values, because the upper bound itself would round to the next value up. For truncated values, the lower bound is non-strict ($\le$) and the upper bound is strict ($<$).

3. Subtraction bounds — not using the reverse combination

For $a-b$, the maximum result uses UB of $a$ and LB of $b$ (subtracting less gives more). A common error is using UB of $a$ − UB of $b$.

4. Estimation — not rounding first

Substituting unrounded values into a calculation and then rounding the answer is not estimation. All values must be rounded first (usually to 1 s.f.), then the simplified calculation is performed.

Mistake	Correction
"0.004306 to 3 s.f. is 0.004"	0.00431 (3 s.f. counts from the first non-zero digit: 4, 3, 0 — but 0 rounds up to 1)
"Error interval for 6.0 (1 d.p.) is $5.95\le x\le 6.05$"	Upper bound must be strict: $5.95\le x<6.05$
"Max of $a÷b$ uses $\text{LB}(a)÷\text{UB}(b)$"	Use $\text{UB}(a)÷\text{LB}(b)$ — large divided by small gives the maximum