Rounding and Estimation
Rounding keeps the useful detail in a number and replaces the rest with a simpler nearby value. Estimation uses those simpler numbers to make a quick, sensible calculation.
Foundation & Higher: start with place value, then learn whole-number rounding, decimal places, significant figures and practical estimation.
Start with place value
A digit’s position tells you its value. In 4,376, the 4 means four thousands, not four units.
| Place | Digit and value in 4,376 |
|---|---|
| Thousands | 4 → 4000 |
| Hundreds | 3 → 300 |
| Tens | 7 → 70 |
| Units (ones) | 6 → 6 |
After the decimal point come tenths, hundredths and thousandths. In 12.483:
| Place | Digit and value |
|---|---|
| Tens | 1 → 10 |
| Units | 2 → 2 |
| Tenths | 4 → 0.4 |
| Hundredths | 8 → 0.08 |
| Thousandths | 3 → 0.003 |
12.483 = 10 + 2 + 0.4 + 0.08 + 0.003
Rounding replaces a number with a nearby, simpler value. The requested accuracy tells you which place matters.
Example Question:
Round 47 to the nearest 10.
The neighbouring multiples of ten are 40 and 50. Which is closer? Example 1 shows this on a number line.
Exam Tip
Read the requested accuracy before you change any digits: nearest 10, decimal places or significant figures.
Always round directly from the original number. Keep full calculator values until the final answer unless the question asks for an estimate.
Method — Choose the nearest value
1. Find the rounding position. For the nearest 10, it is the tens digit. For the nearest 100, it is the hundreds digit.
2. Look at the next digit to the right. This deciding digit tells you whether the number is below halfway, or at least halfway, to the next rounded value.
3. If the deciding digit is 0–4, keep the rounding digit. If it is 5–9, increase the rounding digit by 1. Carry into the next place if necessary.
For the positive numbers here, an exact halfway value rounds to the larger of the two choices. That is why a deciding digit of 5 rounds up.
4. Finish at the required accuracy. Use placeholder zeros in whole numbers. Remove unwanted decimal digits, but keep any zero needed to show the requested decimal places.
The symbol ≈ means “approximately equal to”. Rounding usually changes the value, so use ≈ rather than =.
Worked Examples
Example 1 — Round to the nearest 10
Round 47 to the nearest 10.
47 is 7 away from 40 and only 3 away from 50. It is closer to 50.
47 ≈ 50 to the nearest 10.
The midpoint is 45. Numbers below 45 are closer to 40; numbers above 45 are closer to 50. Exactly 45 rounds to 50 using our halfway rule.
The ones digit 7 tells us that 47 has passed the halfway point. This explains the digit shortcut.
Example 2 — Choose hundreds or thousands
Round 4,376 to the nearest 100.
The hundreds digit is 3. The deciding tens digit is 7, so increase 3 to 4 and replace the tens and units with zeros.
4,376 ≈ 4,400 to the nearest 100.
Now round the original 4,376 to the nearest 1000.
The thousands digit is 4. The deciding hundreds digit is 3, so keep 4.
4,376 ≈ 4,000 to the nearest 1000.
These answers use different accuracies. Neither is meant to keep every digit.
Decimal places — count after the point
Decimal places, often shortened to d.p., count digits after the decimal point. One decimal place keeps tenths. Two keep hundredths.
Example 3 — Round to one decimal place
Round 3.746 to 1 decimal place.
The tenths digit 7 is the last digit to keep. The next digit, 4, decides:
3.746
The deciding digit is below 5, so keep 7 and remove the remaining decimal digits.
3.746 ≈ 3.7 to 1 d.p.
Example 4 — Two decimal places and useful zeros
Round 3.746 to 2 decimal places.
Keep through the hundredths digit 4. The deciding thousandths digit is 6:
3.746
Increase the hundredths digit from 4 to 5.
3.746 ≈ 3.75 to 2 d.p.
Zero counts as a decimal digit too. In 5.04, the 0 is the tenths digit and the 4 is the hundredths digit.
5.04 ≈ 5.0 to 1 d.p. Keep the final zero to show one decimal place.
Sometimes increasing a digit causes a carry. For 9.96 to 1 d.p., the deciding digit 6 increases the tenths digit 9:
9.96 ≈ 10.0 to 1 d.p.
Write 10.0, not just 10, when one decimal place is requested.
Significant figures — start at the first non-zero digit
Significant figures, shortened to s.f., count from the first non-zero digit. Count every digit after that, including zeros.
Zeros before the first non-zero digit only locate the number in place value; they do not start the significant-figure count.
Keep the requested number of significant figures, then use the next digit to decide whether to round up.
Example 5 — Round a large number to significant figures
Round 4,783 to 2 significant figures.
The first two significant digits are 4 and 7. The next digit is 8, so the 7 becomes 8.
4,783 ≈ 4,800 to 2 s.f.
The zeros keep this a number in the thousands. Writing 48 would change its size.
Now round 305,000 to 2 significant figures.
Count 3 as the first significant digit and the internal 0 as the second. This means rounding at the ten-thousands place.
The next digit is 5, so round up.
305,000 ≈ 310,000 to 2 s.f.
Example 6 — Round a small decimal
Round 0.006482 to 2 significant figures.
The leading zeros do not count. Start at 6, then count 4. The next digit is 8.
Increase 4 to 5 and keep the leading zeros in their original places.
0.006482 ≈ 0.0065 to 2 s.f.
The answer stays around six thousandths. Removing a leading zero would make it ten times too large.
Example 7 — Decimal places are not significant figures
Round 12.483 in two different ways.
To 2 d.p.: keep two digits after the point, 4 and 8. The next digit is 3, so keep 8.
12.483 ≈ 12.48 to 2 d.p.
To 2 s.f.: start at 1 and keep 1 and 2. The next digit is 4, so keep the units digit 2.
12.483 ≈ 12 to 2 s.f.
Decimal places count from the decimal point. Significant figures count from the first non-zero digit.
Estimation — choose easy nearby numbers
An estimate is a deliberately approximate answer. It helps you plan quickly or spot an unreasonable calculator result.
Round the numbers first, then calculate with the easier values. Rounding each number to 1 significant figure is a useful method, especially when a question asks for it.
Other sensible nearby numbers may be easier. If a question states a rounding method, follow it.
Example 8 — Estimate a multiplication
Estimate 19.8 × 3.1 by rounding each number to 1 significant figure.
19.8 ≈ 20
3.1 ≈ 3
20 × 3 = 60
19.8 × 3.1 ≈ 60.
The exact product is 61.38, so 60 is a useful quick check. A calculator result of 598 would be far too large.
An estimate being close does not prove a calculator answer is correct. Check the numbers and operations you entered.
Example 9 — Estimate a division
Estimate 202 ÷ 4.9 by rounding each number to 1 significant figure.
202 ≈ 200
4.9 ≈ 5
200 ÷ 5 = 40
202 ÷ 4.9 ≈ 40.
We are dividing about 200 into groups of about 5. An answer around 40 makes sense.
Example 10 — Check a money calculation
Eight tickets cost £7.95 each. A calculator result says the total is £636. Is that sensible?
Round the price to £8 and keep the exact count of 8 tickets:
8 × £8 = £64
The total should be around £64. £636 is much too large.
Now calculate using the original price:
8 × £7.95 = £63.60
The total is £63.60. This is close to the estimate.
Keep money answers to the nearest penny when needed. For measurements, use the accuracy stated in the question.
Choose the right kind of approximation
| Question asks for… | Where to count |
|---|---|
| Nearest 10, 100 or 1000 | Find the named place in the whole number. |
| Decimal places | Count digits after the decimal point. |
| Significant figures | Count from the first non-zero digit. |
| An estimate | Choose easy nearby values, following any stated rounding instructions. |
In a rounded whole number such as 4800, the written zeros alone may not tell you its stated accuracy. Keep the instruction “to 2 s.f.” with the answer when explaining your working.
Check that your answer is sensible
Read your answer as a number. Rounding a number near 5000 should not suddenly give a number near 50.
Check the deciding digit in the original number. Do not round in stages.
For example, 2.449 rounds directly to 2.4 at 1 d.p. Rounding first to 2.45 and then to 2.5 gives the wrong result.
Keep the right number of decimal places when requested. For instance, 5.0 shows 1 d.p.
An estimate should be easy to calculate and of a sensible size. Use ≈ for the approximate result, and keep the original units.
Choose an answer that fits the situation
Sometimes the context decides whether you need the next whole number up or down. This is different from rounding to the nearest whole number.
Worked example — enough buses
93 people need buses with 40 seats each. 93 ÷ 40 = 2.325. Two buses have only 80 seats, so 3 buses are needed. Rounding to the nearest whole number would leave people without seats.
For complete pieces, you may need to round down instead. From 8 m of ribbon, 8 ÷ 1.5 = 5.333… strips of length 1.5 m fit mathematically. You can cut only 5 complete strips, assuming no cutting loss.
Quick practice — (a) How many 30-seat buses are needed for 76 people? (b) How many complete 1.2 m pieces can be cut from 7.6 m of rope?
(a) 76 ÷ 30 = 2.533… . Two buses hold only 60 people. Round up: 3 buses.
(b) 7.6 ÷ 1.2 = 6.333… . Round down: 6 pieces. Check: 6 × 1.2 = 7.2 m; seven pieces would need 8.4 m.
Think Like a Mathematician
How can the same number give three different rounded answers?
For 4,462:
| Accuracy | Rounded value |
|---|---|
| Nearest 10 | 4,460 |
| Nearest 100 | 4,500 |
| Nearest 1000 | 4,000 |
Each answer chooses from a different set of neighbouring values.
For tens, compare 4460 and 4470. For hundreds, compare 4400 and 4500. For thousands, compare 4000 and 5000.
The coarser the accuracy, the less detail the answer keeps. Always say which accuracy you are using.
Common mistakes to avoid
Confusing d.p. with s.f. Decide where counting starts before you round.
Counting leading zeros as significant. In 0.006482, the first significant digit is 6.
Changing earlier digits unnecessarily. Usually only the rounding digit changes. A carry, as in 9.96 to 10.0, can also change earlier places.
Dropping placeholder zeros. 4,783 to 2 s.f. is 4,800, not 48.
Rounding more than once. Use the original number and the requested accuracy directly.
Writing an estimate as exact. Use ≈, and do not treat an estimate as a guaranteed total.
Choosing awkward approximations. The point of estimating is to make the calculation easier.
Try it yourself
Begin with the rounding questions, then try estimates and money problems. Show the rounding accuracy and use ≈ where the value is approximate.
1. Round 63 and 265 to the nearest 10.
For 63, the deciding ones digit is 3, so keep the tens digit.
63 ≈ 60 to the nearest 10.
265 is halfway between 260 and 270. Our halfway rule chooses 270.
265 ≈ 270 to the nearest 10.
2. Round 4,376 to the nearest 100 and nearest 1000.
Nearest 100: the deciding tens digit is 7, so increase the hundreds digit.
4,376 ≈ 4,400.
Nearest 1000: the deciding hundreds digit is 3, so keep the thousands digit.
4,376 ≈ 4,000.
3. Round 6.284 to 1 decimal place.
Keep the tenths digit 2. The deciding hundredths digit is 8, so increase 2 to 3.
6.284 ≈ 6.3 to 1 d.p.
4. Round 7.056 to 2 decimal places.
The tenths digit is 0 and the hundredths digit is 5. Keep both places.
The deciding thousandths digit is 6, so increase the hundredths digit to 6.
7.056 ≈ 7.06 to 2 d.p.
5. Round 9.96 to 1 decimal place.
Keep tenths. The deciding hundredths digit 6 means round up.
Increasing nine tenths carries into the units, and then into the tens.
9.96 ≈ 10.0 to 1 d.p. Keep the final decimal zero.
6. Round 6,824 and 305,000 to 2 significant figures.
In 6824, keep 6 and 8. The next digit is 2, so do not increase 8.
6,824 ≈ 6,800 to 2 s.f.
In 305000, keep 3 and the internal 0. The next digit is 5, so round at the ten-thousands place.
305,000 ≈ 310,000 to 2 s.f.
7. Round 0.000738 to 2 significant figures.
Ignore the leading zeros when counting. Keep 7 and 3; the deciding digit is 8.
Increase 3 to 4, and preserve the leading zeros.
0.000738 ≈ 0.00074 to 2 s.f.
8. Round 0.08476 to 2 decimal places and to 2 significant figures.
For 2 d.p., keep 0.08. The next digit is 4, so the hundredths digit stays 8.
0.08476 ≈ 0.08 to 2 d.p.
For 2 s.f., start at 8 and keep 8 and 4. The next digit is 7, so increase 4 to 5.
0.08476 ≈ 0.085 to 2 s.f.
9. Estimate 29.7 × 4.2 by rounding each number to 1 significant figure.
29.7 ≈ 30
4.2 ≈ 4
30 × 4 = 120
29.7 × 4.2 ≈ 120. This is an estimate, not the exact product.
10. Estimate 398 ÷ 8.1 by rounding each number to 1 significant figure.
398 ≈ 400
8.1 ≈ 8
400 ÷ 8 = 50
398 ÷ 8.1 ≈ 50.
11. Six tickets cost £9.95 each. Estimate the total, then calculate exactly to decide whether £60 is enough.
Round £9.95 to £10 and keep the exact count of six tickets.
6 × £10 = £60
The estimated total is £60. Since this is close to the budget, use the original price to decide.
6 × £9.95 = £59.70
£60 is enough. The exact change is £0.30.
12. A calculator result for 4.98 × 19.8 is 986.04. Use an estimate to explain why this is unreasonable.
4.98 ≈ 5
19.8 ≈ 20
5 × 20 = 100
The answer should be around 100. 986.04 is much too large.
Re-entering the original calculation gives 98.604, which agrees in size with the estimate.
Frequently Asked Questions
Why do we round numbers?
A simpler nearby number can be easier to communicate or calculate with. Keep enough detail for the question you are answering.
What happens exactly halfway?
For the positive numbers in this lesson, choose the larger neighbouring rounded value. For example, 45 to the nearest 10 is 50.
Does zero count as a significant figure?
A zero before the first non-zero digit does not count. Once counting has started, zeros do count, including internal zeros such as the 0 in 305.
Why write 5.0 instead of 5?
They have the same numerical value, but 5.0 shows an answer given to one decimal place.
Must I always estimate using one significant figure?
No. It is a useful method, but other easy nearby values can work. Follow the method stated in an exam question.
Can an estimate be larger or smaller than the exact answer?
Yes. Rounding may move values up or down. An estimate is a check on size, not a promise of the exact answer.
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