Rounding, estimation and sensible accuracy
Choose a rounding place deliberately and use an estimate to judge the size of a calculated answer.
Core and Extended.
Before you begin
- Identify each decimal place value.
- Calculate using the four operations.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which digit determines 3.746 rounded to two decimal places?
What you will learn
- Round to decimal places, significant figures and named place values.
- Estimate calculations using convenient rounded values.
- Choose an appropriate accuracy in context.
Choose the last digit to keep
Decimal places count positions to the right of the decimal point. To round 4.386 to two decimal places, keep the hundredths digit 8 and inspect the next digit 6: the result is 4.39. To the nearest hundred, round 5 764 by inspecting its tens digit: it becomes 5 800.
The usual school rounding convention increases the retained magnitude when the next digit is five or more. A carry can cross the decimal point: 9.997 to two decimal places is 10.00. Keep the zeros when they communicate the requested accuracy; 10.00 records hundredths, while 10 by itself does not.
Significant figures start with meaningful magnitude
Count significant figures from the first non-zero digit. In 0.004086, the first significant digit is 4. To three significant figures the retained digits are 4, 0 and 8; the next digit 6 rounds the result to 0.00409. Leading zeros locate the number but do not count as significant figures.
Zeros between non-zero digits count. A trailing zero after a decimal can also communicate precision, as in 2.50. Standard form makes large-number precision clear: 1.20 × 10⁵ has three significant figures. Rounding 99.6 to two significant figures gives 1.0 × 10², which makes the intended two figures explicit.
Estimate before calculating exactly
An estimate uses nearby convenient quantities to predict scale. For (19.7 × 4.12)/0.49, replace the values by 20, 4 and 0.5: the estimate is 80/0.5 = 160. Dividing by one half doubles the amount. An exact result around 160 is plausible; around 16 or 1600 suggests an input error.
Round the original values at the start of an estimate, then calculate with those approximations. Do not repeatedly round every intermediate result in an exact calculation. If the question specifies one significant figure for the estimate, follow that instruction even when another rounding choice looks convenient.
Accuracy also depends on what the answer means
A money total is normally reported to the nearest cent, but a count of buses needed may require rounding up to the next whole bus. For 97 passengers and 24 seats per bus, 97/24 is just over four, so five buses are needed. Rounding to the nearest integer would leave one passenger without a seat.
In the Cambridge calculator papers, non-exact final values normally use three significant figures, or one decimal place for angles, unless the question specifies otherwise. Exact fractions and surds should stay exact when requested. Keep unrounded results for later parts and round only the final reported quantity.
Pause and explain
What is 0.004086 to three significant figures?
Worked example
Estimate (19.7 × 4.12)/0.49 by rounding each input to one significant figure.
Show the worked solution
- The inputs become 20, 4 and 0.5.
- Multiply to get 80, then divide by 0.5.
- The estimate is 160; use it to check the scale of an exact calculation.
Answer 160
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Round 7.286 to two decimal places and two significant figures.
Give me a hint
Locate the final retained digit separately for the two requests.
Compare my reasoning
- For two decimal places, retain 7.28; the next digit six gives 7.29.
- For two significant figures, retain 7.2; the next digit eight gives 7.3.
- The two answers differ because the requested rounding positions differ.
7.29; 7.3
Look for these in your work
- Distinguished decimal places from significant figures.
- Used the first discarded digit.
Round 0.006487 to two significant figures and 99.96 to one decimal place.
Give me a hint
Ignore leading zeros for significant figures, and allow a carry when rounding.
Compare my reasoning
- In the first value, retain 6 and 4; the next eight gives 0.0065.
- In the second, the hundredths digit is six, so 99.9 rounds up to 100.0.
- The zero in 100.0 preserves the requested one-decimal-place accuracy.
0.0065; 100.0
Look for these in your work
- Counted significant figures from the first non-zero digit.
- Carried the rounding change across the integer boundary.
A trip of 298 km uses 21.4 litres of fuel. Estimate kilometres per litre using one significant figure, then explain why 140 km/L is implausible.
Give me a hint
Use 300 km and 20 litres for the estimate.
Compare my reasoning
- The estimate is 300/20 = 15 km/L.
- A claimed 140 km/L is nearly ten times the estimated size.
- At that efficiency, 21.4 litres would cover almost 3000 km, inconsistent with the stated distance.
About 15 km/L; 140 km/L has the wrong scale.
Look for these in your work
- Rounded inputs before estimating.
- Used the estimate and units to challenge an implausible result.
Common mistakes
- Counting leading zeros as significant figures.
- Truncating when the next digit requires rounding up.
- Rounding intermediate values in an otherwise exact calculation.
What can you explain now?
Round 12 749 to the nearest hundred.
Compare with the explanation
12 700
The tens digit is four, so the hundreds digit stays seven.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Round 0.009956 to three significant figures. Explain why a transport problem may require rounding up even when ordinary rounding would go down.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources