Overview
What you’ll learn
- Round numbers to a given number of decimal places.
- Round numbers to a given number of significant figures, including very large and very small numbers.
- State the range of values a rounded number could have taken, and identify rounding errors in context.
- Estimate the value of calculations by rounding to 1 significant figure first.
- Decide when a real-world answer must be rounded up or down (people, boxes, trips), not just to the nearest value.
In 30 seconds
- 1Decimal places count from the decimal point; significant figures count from the first non-zero digit.
- 2Look at the next digit: 4 or less → leave it; 5 or more → round up.
- 3Placeholding zeros are not significant: 0.0041 has two significant figures.
- 4A number given as 3.6 (1 d.p.) actually lies between 3.55 and 3.65 — that range is the approximation error.
- 5To estimate, round everything to 1 significant figure, then compute the friendly numbers.
- 6In context, some answers must round up (boxes, buses, people) — nearest-value rounding would break the situation.
Notes · 1 of 5
Rounding to decimal places
Decimal places count digits after the point — the next digit decides whether to round up.
- 1Find the cut-off digit
Count places after the decimal point: 5.2476 to 2 d.p. cuts after 4.
- 2Look next door
The digit after the cut-off is 7, which is 5 or more.
- 3Round and trim
Increase the cut-off digit and drop the rest: 5.2476 → 5.25.
| Instruction | Digit that decides | Answer |
|---|---|---|
| 5.2476 to 1 d.p. | 4 (hundredths) | 5.2 |
| 5.2476 to 2 d.p. | 7 (thousandths) | 5.25 |
| 5.2476 to 3 d.p. | 6 (ten-thousandths) | 5.248 |
Notes · 2 of 5
Rounding to significant figures
Significant figures start at the first non-zero digit — leading zeros are only placeholders.
For a number like 4 183, the first significant figure (1 s.f.) is the 4. But for 0.004 183, the zeros are placeholders: the first significant figure is the 4 in the thousandths place. Trailing zeros inside a number do count: in 20 507, the digits 2, 0, 5 are the first three significant figures.
| To | Next digit | Answer |
|---|---|---|
| 1 s.f. | 0 | 300 000 |
| 2 s.f. | 4 | 300 000 |
| 3 s.f. | 8 → round up, carry | 305 000 |
| 4 s.f. | 6 → round up, carry | 304 900 |
Watch the carry: 19 999 to 3 s.f. starts as 19.9… → the third 9 rounds up, which cascades to 20 000. Write the rounded value with its place value made clear — “20 000 to 3 s.f.”, not just “20”.
Notes · 3 of 5
Approximation errors
Every rounded value hides a range of possible originals — the error is the gap between them.
If a mass is stated as 12.6 kg to 1 d.p., the true value could be anything from 12.55 kg up to (but not including) 12.65 kg. That interval is the range of possible values. The largest possible absolute error is half of the last rounded unit: here, 0.05 kg.
| Given value | Precision | Possible range |
|---|---|---|
| 24 cm (nearest cm) | whole number | 23.5 ≤ length < 24.5 |
| 3.7 s (1 d.p.) | tenths | 3.65 ≤ time < 3.75 |
| 6 400 m (2 s.f.) | hundreds | 6 350 ≤ distance < 6 450 |
In real-world contexts the error matters: a delivery quoted “about 500 g” could be up to 5 g short. Questions often ask whether a specific value (say 12.64) could be the true value behind a rounded one — test it by rounding the candidate and comparing.
Notes · 4 of 5
Estimating calculations
Round to 1 significant figure first, then do easy arithmetic — the estimate catches wrong answers.
- 1Round everything
Round each number to 1 s.f.: 48.7 → 50, 21.4 → 20.
- 2Compute simply
50 × 20 = 1 000.
- 3Compare later
The exact value 1 042.18 is close to the estimate — the working looks right.
Estimates also expose order-of-magnitude mistakes. If you estimate 41.3 × 5.9 ≈ 240 but your exact working produces 24, you dropped a factor of ten — go back and check. Estimation is the calculator-free lie detector for long computations.
Notes · 5 of 5
Rounding in real-world contexts
Sometimes “nearest” is wrong: people, boxes and buses force an answer up or down.
| Situation | Round… | Why |
|---|---|---|
| Buses needed for 168 students, 40 per bus | Up → 5 | 4 buses only carry 160; 8 students are stranded |
| Boxes for 4 870 pens, 24 per box | Up → 203 | A remainder still needs a box |
| Books each person can buy with $50 at $12 each | Down → 4 | 5 books cost $60, which is over budget |
| Your weight, height or a bill | Nearest (as instructed) | No whole-object constraint |
The habit to build: after computing, ask what the number counts. Whole objects force one direction; measurements just follow the rounding instruction in the question.
Revise
Key terms
The words examiners expect you to use precisely.
- Decimal place (d.p.)
- A position after the decimal point; 3.47 is given to 2 decimal places.
- Significant figure (s.f.)
- A digit of a number starting from the first non-zero digit; leading zeros are placeholders.
- Round up
- Increase the final kept digit by 1, done when the next digit is 5 or more.
- Round down
- Leave the final kept digit unchanged, done when the next digit is 4 or less.
- Approximation error
- The difference between a rounded value and the true value it came from.
- Range of possible values
- All values that would round to a given number, e.g. 3.65 ≤ x < 3.75 for “3.7 to 1 d.p.”.
Show 3 more key termsShow fewer key terms
- Estimate
- A quick approximate value found by rounding inputs first, used to check calculations.
- Order of magnitude
- The power of ten a number sits at; estimates catch errors of this size.
- Exact value
- The value before any rounding is applied.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Counting leading zeros as significant figures, e.g. calling 0.042 “3 s.f.”.
The zeros only mark the decimal place; they carry no precision of their own.
Instead: Start counting at the 4: 0.042 has two significant figures.
Mistake: Rounding 2.3618 to 2 d.p. as 2.37 “because the last digit is 8”.
The decision digit is the one just after the cut-off (the 1 in the thousandths place), not the final digit of the original.
Instead: 2.3618 → the thousandths digit 1 is less than 5, so 2.36.
Show 8 more mistakesShow fewer mistakes
Mistake: Writing 19 999 to 3 s.f. as 19 900.
The next digit is 9, so the kept 9 rounds up and the carry runs along: 19.9 → 20.0.
Instead: 19 999 = 20 000 to 3 s.f. Carries can change earlier digits.
Mistake: Claiming a value “to 1 d.p. of 3.6” must be exactly 3.6.
Rounding destroys information: every value from 3.55 up to 3.65 produces the same rounded number.
Instead: The true value satisfies 3.55 ≤ x < 3.65.
Mistake: Answering “202 boxes” for a packing question that computes 202.9.
You cannot order 0.9 of a box, and 202 boxes leave pens unpacked.
Instead: Whole-object contexts round up when there is any remainder: 203 boxes.
Mistake: Using exact arithmetic when a question says “estimate”.
The marks reward the rounding step; exact answers skip the skill being tested.
Instead: Write the 1-s.f. rounded numbers first (e.g. 80 ÷ 2), then compute the estimate.
Mistake: Letting the final digit decide, e.g. rounding 4.049 to 1 d.p. as 4.1 “because of the 9”.
Only the digit immediately after the cut-off votes — the hundredths digit 0, not the 9 further along.
Instead: 4.049 to 1 d.p. is 4.0; cover everything after the deciding digit first.
Mistake: Saying 2.30 has two significant figures because “the last zero is empty”.
A trailing zero after the decimal point records the precision of the measurement, so it counts.
Instead: 2.30 (to 2 d.p.) has three significant figures: 2, 3 and the final 0.
Mistake: Writing the range behind 470 (2 s.f.) as 469.5 to 470.5.
That treats 470 as correct to the nearest unit. Two significant figures mean the nearest ten, since the last kept digit is in the tens place.
Instead: 470 to 2 s.f. hides 465 ≤ N < 475 — half a ten, not half a unit.
Mistake: Reporting a rounded shortcut as the exact answer, e.g. writing 8 × 3 = 24 for 8.47 × 3.2.
The 1-s.f. version is only an estimate; the exact product is 27.104, and estimates do not replace it unless the question asks.
Instead: Keep exact values through the working and round only the final answer as instructed — use estimates to sanity-check, not to substitute.
Check yourself
Quick quiz
18 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Round to decimal places
Round 3.047 to 1 decimal place.
Ready for more practice on Chapter 3: Approximation and Estimation?
Generate a fresh 5-question drill built from this chapter’s key points. Free with your study account.
Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 6Foundation
- 5Standard
- 4Challenging
15 questions worth 54 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 3 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Chapter 3.
Mark scheme
PROChapter 3 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
- Term 1Coming soonWeighted Assessment 1 (WA1)
- Term 4Coming soonEnd-of-Year Examination (EOY)
Questions
Frequently asked questions
What is the difference between decimal places and significant figures?
Decimal places count digits after the decimal point. Significant figures start counting at the first non-zero digit of the number, wherever it is — so 0.0041 to 2 s.f. keeps the 4 and the 1, not two digits after the point.
How do I round a number like 19 999 to 3 significant figures?
The first three significant figures are 1, 9, 9, and the next digit is 9, so the third 9 rounds up. The carry runs along and the answer is 20 000. Rounding can change digits you did not expect.
What range hides behind a rounded value like 12.6 kg (1 d.p.)?
Any value from 12.55 up to but not including 12.65 rounds to 12.6. The gap between those bounds — half the last unit on each side — is the approximation error.
How do examiners want me to estimate?
Round every number to 1 significant figure first, then compute with the friendly numbers, showing the rounded values. For example, estimate 198 ÷ 3.9 as 200 ÷ 4 = 50.
When must I round up instead of to the nearest value?
When the answer counts whole objects that must all fit — boxes, buses, trips, people. If the division leaves a remainder, the remainder still needs one more box or bus, so round up.
Up next
Chapter 4: Algebraic Expressions and Formulae