Accuracy in A Level Math: significant figures, exact answers and rounding

If you could win back only one kind of mark across all of Cambridge A Level Math, accuracy would be a strong candidate. Almost every examiner report, for every paper, mentions it. The advice is consistent, simple and easy to forget under exam pressure. This guide sets it out in one place.

This guide is based on a reading of the examiner reports for recent sessions. The advice is summarised in our own words.

The rule on the front of the paper

Unless a question says otherwise, answers that are not exact must be given correct to three significant figures. Angles in degrees must be given correct to one decimal place. Examiners cannot award an accuracy mark if the final answer is not given to the required accuracy, even when the method is perfect.

Work to more figures than you give

This is the most repeated piece of advice in the reports, in Pure Math, Mechanics and Statistics alike.

• To give a final answer correct to 3 significant figures, work with at least 4 significant figures in the steps before it.
• Rounding in the middle of a calculation can push the final digit over. Reports describe students who did everything correctly and still lost the accuracy mark because they rounded an early value to 3 significant figures.
• Where possible, use the calculator's stored value for an earlier answer instead of retyping a rounded one.
• If a question gives you an answer to an earlier part, check whether the next part needs the unrounded value.

Statistics has a particular trap. In normal approximation questions, rounding a value too early before reading from the tables can give a noticeably different probability. Keep extra figures until the final step.

Significant figures are not decimal places

Reports mention students who gave three decimal places when three significant figures were asked. The two behave differently:

• 0.04567 to 3 significant figures is 0.0457, not 0.046.
• For a value smaller than 1, the leading zeros do not count as significant figures. Reports for Statistics mention this separately because it affects many probability answers.
• Angles in radians are given to 3 significant figures, not to 1 decimal place. Angles in degrees are the ones that use 1 decimal place.

Do not round what is exact

Some answers should not be rounded at all.

• Exact answers: if a question says "exact", or the answer is naturally a surd, a fraction, a multiple of π or a logarithm, give it in that form. A decimal approximation will not earn the mark.
• Exact decimals: if a calculation gives a decimal that ends, such as exactly 0.12345, give it in full. Reports remind students that only non-exact answers should be rounded to 3 significant figures.
• Fractions: reports for Statistics note that there is no need to convert fractions to decimals, and that doing so is both slower and a source of lost accuracy. Probabilities can be left as fractions. An improper fraction that is equivalent to a whole number should be given as the whole number.
• Answers to be shown: when the printed result is exact, your working should be exact too.

Values that have been rounded are not exact solutions

Reports include a version of this mistake: a student finds an approximate value and then treats it as if it were the exact solution to an equation, or substitutes a rounded decimal into the next step. If a later question asks for an exact value, the earlier approximate value cannot be reused.

Angles: degrees and radians

• Check which unit the question uses and set the calculator mode before starting.
• Do not convert a radian answer into degrees, and then back again, in the middle of a question. Conversion loses accuracy and gives more chances for error. In circular measure, work in radians from the start.
• In arc length and sector area formulae, the angle must be in radians.

A quick checklist for the last line of every numerical answer

1. Did the question ask for an exact answer?
2. If not, is it correct to 3 significant figures (or 1 decimal place for an angle in degrees)?
3. Did my working keep at least 4 significant figures?
4. Are the units and the variable named in the question present?
5. Does the size of the answer make sense?

That last question catches more errors than it should. A probability above 1, a negative length or a mass of 4000 kg for a parcel are all signs of an earlier mistake.

Questions students ask

How many significant figures should I give in A Level Math?

Unless the question says otherwise, answers that are not exact are given correct to three significant figures, and angles in degrees to one decimal place. Examiners cannot award the accuracy mark if the final answer is not to the required accuracy, even when the method is perfect.

Why do I lose accuracy marks when my method is right?

Usually because of early rounding. Work with at least 4 significant figures in the steps before a 3 significant figure answer, and use the calculator's stored value instead of retyping a rounded one.

What is the difference between significant figures and decimal places?

Significant figures count from the first non-zero digit. 0.04567 to 3 significant figures is 0.0457, not 0.046. Leading zeros do not count. Angles in radians are given to 3 significant figures, and only angles in degrees use 1 decimal place.

When should I not round at all?

When the question asks for an exact answer, or the answer is naturally a surd, fraction, multiple of π or logarithm. Do not switch to decimals. A decimal that ends exactly should be given in full, and probabilities can stay as fractions.

Should I convert between degrees and radians in the middle of a question?

Avoid it. Conversion loses accuracy and gives more chances for error. Check the unit before you start, and in circular measure work in radians from the beginning, because arc length and sector area formulae need radians.

How can I check my final answer quickly?

Ask whether it should be exact, whether it is correct to 3 significant figures, whether your working kept 4, whether units and variable names are present, and whether the size makes sense. A probability above 1 or a negative length signals an earlier mistake.