In practical mathematics, answers are rarely needed to many decimal places. Rounding and significant figures are tools for expressing answers with an appropriate level of precision for the context.
KEY TAKEAWAY: Always match your precision to the context. Money → 2 decimal places. Building measurements → nearest mm or cm. Populations → nearest thousand.
Rule: Look at the digit immediately to the right of your rounding position.
- If it is $\geq 5$, round up
- If it is $< 5$, round down (truncate)
Examples:
| Original | Rounded to 2 d.p. | Rounded to 1 d.p. | Rounded to nearest whole |
|---|---|---|---|
| $3.467$ | $3.47$ | $3.5$ | $3$ |
| $12.845$ | $12.85$ | $12.8$ | $13$ |
| $0.0354$ | $0.04$ | $0.0$ | $0$ |
Significant figures (s.f.) count meaningful digits, starting from the first non-zero digit.
| Number | 1 s.f. | 2 s.f. | 3 s.f. |
|---|---|---|---|
| $4567$ | $5000$ | $4600$ | $4570$ |
| $0.03842$ | $0.04$ | $0.038$ | $0.0384$ |
| $12.06$ | $10$ | $12$ | $12.1$ |
How to count significant figures:
- Non-zero digits always count: $342$ has 3 s.f.
- Zeros between non-zero digits count: $3042$ has 4 s.f.
- Leading zeros do not count: $0.0045$ has 2 s.f.
- Trailing zeros after the decimal point count: $3.50$ has 3 s.f.
EXAM TIP: When a question says “correct to 2 significant figures”, find the first two non-zero digits and round from there.
Always round to 2 decimal places (nearest cent):
$$\$12.567 \to \$12.57$$
Context determines the rounding direction:
| Situation | Rule | Why |
|---|---|---|
| Buying paint (litres needed) | Round up | Don’t run short |
| Number of buses needed | Round up | Everyone must fit |
| Money to give as change | Round to nearest | Convention |
| Manufacturing tolerance | Round to nearest | Accuracy required |
COMMON MISTAKE: Students round $2.1\text{ L}$ of paint to $2\text{ L}$ because “it’s closest”. In context, you need $3\text{ L}$ (a whole can) to have enough.
A builder needs $23.4\text{ m}$ of timber. Timber is sold in $3\text{ m}$ lengths. How many lengths are needed?
$\$23.4 \div 3 = 7.8 \text{ lengths}$$
Round up to $8$ lengths (can’t buy $0.8$ of a length).
Total timber purchased: \$8 \times 3 = 24\text{ m}$
Waste: \$24 - 23.4 = 0.6\text{ m}$
APPLICATION: Rounding decisions in real life often have cost implications. Knowing when to round up — even if the number is close to the lower value — is a critical practical skill.
