Rounding and Estimation
Most numbers in the real world are approximate. Rounding is how we decide how much to keep, estimation is how we check an answer makes sense, and error intervals are how we say exactly how uncertain a rounded number is.
National curriculum KS3: round to decimal places and significant figures; estimate calculations; use inequality notation for error intervals. GCSE adds upper and lower bounds in calculations.1 · Rounding is choosing the nearest
To round 3.7 to the nearest whole number, ask: is it closer to 3 or to 4? Look at the number line. The decision is made by the digit just after the place you are rounding to. 5 or more and you round up; 4 or less and you round down. That digit is called the deciding digit on this page.
What about exactly halfway? 3.5 is the same distance from 3 and 4. The school convention is to round halves up: 3.5 → 4, 2.65 → 2.7. (Scientists and some programming languages sometimes round halves to the even neighbour to avoid bias, so Python's round(2.5) gives 2. For exams, halves go up.)
2 · Decimal places versus significant figures
These are two different ways of saying where to round. Decimal places (d.p.) count from the decimal point. Significant figures (s.f.) count from the first non-zero digit, wherever it is. Significant figures work for any size of number, which is why science uses them.
Type a number and watch which digits are kept (teal), which one decides (orange), and what happens to the rest.
Which zeros count?
| Number | Significant figures | Why |
|---|---|---|
| 0.0048 | 2 | Leading zeros only show the size. They never count. |
| 4.08 | 3 | A zero between non-zero digits always counts. |
| 4.80 | 3 | A trailing zero after the decimal point counts: it says "measured to the hundredth". |
| 4800 | 2, 3 or 4 | Ambiguous. Are the zeros measured or just place-holders? This is exactly why standard form exists: 4.8 × 10³ vs 4.80 × 10³ settles it. |
Two traps. (1) Keep the place-holders. 36 482 to 2 s.f. is 36 000, not 36. The zeros keep the 3 in the ten-thousands column. (2) Carrying. 199.96 to 1 d.p. is 200.0, not 199.10: the 6 rounds the 9 up, which carries all the way along. Try both in the tool above.
3 · Estimation: round first, then calculate
Before you touch a calculator, or when there isn't one, round every number to 1 significant figure and do the easy sum. It takes seconds and it catches the error where you typed 0.3 instead of 0.03. The symbol ≈ means "approximately equal to".
Rules of thumb for a good estimate:
- 1 s.f. is the default. Occasionally 2 s.f. is better for a number like 15 (rounding to 20 is a big change) or when it makes a division neat (e.g. round 7.8 ÷ 3.9 to 8 ÷ 4, not 8 ÷ 4 … that one's already neat).
- Say whether the estimate is an over- or under-estimate. Rounding the top of a fraction up or the bottom down makes the answer too big.
- Estimation questions in exams give marks for the rounded numbers written down, not just the final answer. Show them.
4 · Error intervals: how wrong could it be?
A sign says a bridge is 12 m long, measured to the nearest metre. The true length isn't exactly 12. It's anything that rounds to 12: as small as 11.5, up to (but not including) 12.5. That range is the error interval. The ends are the lower bound and upper bound.
The recipe never changes: half the unit of rounding each side. Rounded to the nearest 10? ± 5. To 1 d.p.? ± 0.05. The notation matters too: ≤ on the lower bound (11.5 does round to 12) but < on the upper (12.5 rounds up to 13, so it isn't included). Writing 11.5 ≤ x < 12.5 with the right symbols is worth a mark on its own.
Truncation is different. If a number was truncated (chopped, not rounded) to 12, the true value is anything from 12 up to but not including 13: 12 ≤ x < 13. Ages work like this: "I'm 15" means 15 ≤ age < 16. Read the question for which one it says.
5 · Bounds in calculations (GCSE)
If measurements are uncertain, anything calculated from them is uncertain too. A rectangle 8 cm by 5 cm, each to the nearest cm, doesn't have area exactly 40 cm². Its biggest possible area uses both upper bounds; its smallest uses both lower bounds. For a division it's trickier: the biggest answer comes from the biggest top and the smallest bottom.
| To get the… | a + b | a − b | a × b | a ÷ b |
|---|---|---|---|---|
| upper bound | UB + UB | UB − LB | UB × UB | UB ÷ LB |
| lower bound | LB + LB | LB − LB | LB × LB | LB ÷ UB |
Subtraction and division are the ones people get wrong: to make a difference as big as possible you take the most from the first and the least from the second.
6 · Where it shows up
Science practicals
A ruler reads to the nearest mm, so every length is ± 0.5 mm. Results are given to the number of significant figures the equipment justifies, usually 2 or 3. Quoting 9 digits from a calculator is a mark lost.
Money
Prices are rounded to the nearest penny. A supermarket's "3 for £2" works out at 66.666…p each, so one item is charged at 67p. Interest calculations round at each step.
Engineering tolerances
A bolt specified as 10 mm ± 0.1 mm is an error interval. A part that falls outside it is rejected. Whole factories run on upper and lower bounds.
Computing
Floating-point numbers store about 7 (float) or 16 (double) significant figures, so 0.1 + 0.2 comes out as 0.30000000000000004. Rounding for display, and never comparing floats with ==, follows from this page.
Sanity-checking
A mortgage calculator says a £200 000 loan costs £85 a month. Estimate: 200 000 ÷ 25 years ÷ 12 ≈ £700 before any interest. The £85 is wrong by a factor of ten.
News and statistics
"Population 103 000" is a 3 s.f. figure; the true count is anywhere from 102 500 to 103 499. Headlines that compare 103 000 with 102 800 as if the difference were real are comparing noise.
Worked examples
Round 0.030849 to 3 significant figures
Estimate (58.3 × 0.193) ÷ 2.87
Error interval: a mass is 4.6 kg to 1 decimal place
Exam-style: a car travels 120 m (nearest 10 m) in 8.2 s (nearest 0.1 s). Find the upper bound of its speed.
Quick check
What you've learned
Rounding picks the nearest value at a chosen place, with the next digit deciding and halves going up. Decimal places count from the point; significant figures count from the first non-zero digit and work at any scale, with care over which zeros count. Estimation rounds everything to 1 s.f. first and is your check against calculator slips. A rounded number hides an error interval of half a unit either side, written with ≤ below and < above, and when rounded numbers go into a calculation, the bounds of the answer come from choosing the right bounds of the inputs.