Rounding, Estimation and Limits of Accuracy — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers rounding numbers, estimating the answer to a calculation, and judging how accurate a result can sensibly be. By the end of this guide you should be able to round to decimal places and to significant figures, and know the difference.

You should also be able to estimate a calculation by rounding to 1 significant figure, decide whether an estimate is above or below the true value, round sensibly in context, and recognise when rounding too early has spoiled an answer.

The organising idea is that rounding trades precision for usefulness, and the question decides how much you can afford to lose. Writing a population as 68,000,000 rather than 68,349,721 loses detail but gains something more valuable: a number people can hold in mind and compare. The same trade governs estimation, where deliberately crude figures give a fast sanity check, and it governs accuracy, where an answer quoted to more digits than the input data supports is claiming a precision that was never there. Every part of this topic is that one judgement applied in a different setting.

Key terms and definitions

Decimal place (d.p.) — a digit position after the decimal point.

Significant figure (s.f.) — a digit that carries information about size, counted from the first non-zero digit.

Estimate — an approximate answer, usually found by rounding each number to 1 s.f.

Order of magnitude — the rough size of a number, in powers of ten.

Overestimate — an approximation larger than the true value. Underestimate — smaller.

Truncate — cut off digits rather than rounding them.

Limits of accuracy — the range a rounded measurement could really have come from.

Core concepts

Rounding to decimal places

Count the required number of digits after the decimal point, then look at the next digit.

If it is 5 or more, round up; if it is less than 5, leave the digit as it is.

So 3.847 to 2 d.p. is 3.85, because the next digit is 7. And 3.842 to 2 d.p. is 3.84, because the next digit is 2.

Only the very next digit matters. In 3.8449 to 2 d.p., the next digit is 4, so the answer is 3.84 — the 9 further along is irrelevant and must not be used to round the 4 up first.

Rounding in stages like that is a genuine error that changes answers, and it is worth being explicit about.

Rounding to significant figures

Significant figures are counted from the first non-zero digit, wherever it sits.

In 0.00427, the first significant figure is the 4, so to 2 s.f. the answer is 0.0043. The leading zeros locate the number but carry no information about precision.

In 47,382, the first significant figure is the 4, so to 2 s.f. the answer is 47,000. Here the zeros are needed, as placeholders keeping the 4 and the 7 in the right columns.

That is the key difference from decimal places: with significant figures you must preserve the size of the number, either with placeholder zeros or with standard form.

Which zeros are significant

Leading zeros are never significant: 0.0052 has 2 s.f.

Zeros between non-zero digits always are: 4007 has 4 s.f.

Trailing zeros in a whole number are ambiguous, which is one reason standard form is useful — 4.7 × 10⁴ unambiguously shows 2 s.f.

Estimating a calculation

To estimate, round every number to 1 significant figure, then calculate.

For 4.87 × 21.3, round to 5 × 20 = 100. The true value is 103.7, so the estimate is close and confirms the answer's size.

The purpose is a sanity check rather than accuracy. An estimate that differs from your calculator answer by a factor of ten usually means a decimal point or a key has gone astray, which is exactly what the estimate exists to catch.

Use the approximately equal symbol ≈ rather than =, since the statement is not an equality.

For division, rounding to numbers that divide neatly is acceptable and often better. For 391 ÷ 19, rounding to 400 ÷ 20 = 20 is far more useful than 400 ÷ 19.

Estimating with roots

Where a square root appears, round to a nearby perfect square rather than to 1 s.f.

To estimate √48, use √49 = 7 rather than √50, which needs a calculator itself.

The same applies to cube roots: estimate ∛30 using ∛27 = 3.

Overestimate or underestimate

Questions often ask whether an estimate is above or below the true value, and the reasoning is about what each rounding did.

Rounding a number up in a multiplication pushes the answer up; rounding it down pushes the answer down.

Division reverses for the denominator: rounding the bottom of a fraction up makes the answer smaller.

For 4.87 × 21.3 estimated as 5 × 20, one number went up and the other down, so the direction is not obvious and the honest answer is that it cannot be decided without more care.

But for 5.2 × 31 estimated as 5 × 30, both were rounded down, so the estimate is definitely an underestimate. Questions are usually built so both roundings point the same way.

Rounding in context

The situation decides how to round, and sometimes it overrides the usual rule.

Money is normally given to 2 d.p., and lengths to a sensible unit.

Where the answer counts objects, round to a whole number in the direction the situation requires. Working out how many 12-seat minibuses are needed for 50 people gives 4.17, which must be rounded up to 5 — four buses would leave people behind, even though the usual rule says round down.

Conversely, working out how many whole 30 cm shelves can be cut from a 2 m plank gives 6.67, which must be rounded down to 6, since a partial shelf is no use.

Read what the answer is for before rounding it.

Rounding too early

Keep full accuracy throughout a calculation and round only the final answer.

Rounding partway through introduces an error that the later steps magnify, and in a multi-stage calculation it frequently changes the digit the question asked for.

On a calculator, use the memory or the answer key rather than writing down and retyping a rounded intermediate value.

Limits of accuracy

A rounded value does not identify a single number; it identifies a range.

A length given as 24 cm to the nearest cm could really be anything from 23.5 cm up to but not including 24.5 cm.

That range is the error interval, written 23.5 ≤ x < 24.5, and it is the starting point for calculating upper and lower bounds.

The practical consequence is that any answer computed from rounded data is itself uncertain, which is why quoting a result to many decimal places when the inputs were measured to the nearest centimetre claims a precision that does not exist.

Worked examples

Example 1: Significant figures with placeholder zeros

Round 68,349 to 2 significant figures, and 0.004062 to 3 significant figures.

For 68,349, the first two significant figures are 6 and 8. The next digit is 3, which is less than 5, so the 8 stays.

The answer is 68,000, with the zeros as placeholders keeping the 6 and 8 in the right columns.

For 0.004062, the first significant figure is the 4, so the first three are 4, 0 and 6. The next digit is 2, so the 6 stays.

The answer is 0.00406. The leading zeros are not significant but must be written to locate the number.

Example 2: Estimating and judging direction

Estimate 6.2 × 48.7, and say whether your estimate is an over- or underestimate.

Round each to 1 s.f.: 6 × 50 = 300.

The first number was rounded down from 6.2 to 6, which pulls the product down. The second was rounded up from 48.7 to 50, which pushes it up.

The two effects work against each other, so the direction cannot be determined without further work. The true value is 301.94, so in fact the estimate is slightly low — but an exam answer should say the roundings conflict rather than guess.

Had the question been 6.2 × 51.3, estimated as 6 × 50, both roundings would have been downwards and the estimate would be a definite underestimate.

Example 3: Rounding in context

A school needs to transport 137 pupils in coaches seating 45 each. How many coaches are needed?

Divide: 137 ÷ 45 = 3.04 to 2 d.p.

The usual rule would round 3.04 down to 3, but three coaches seat only 135 pupils, leaving two behind.

The answer must be rounded up: 4 coaches.

Stating the reason — that a partial coach is not possible and everybody must travel — is what earns the mark, since the arithmetic alone points the other way.

Common mistakes and how to avoid them

Rounding in stages. Look only at the next digit, not the one beyond it.

Counting leading zeros as significant. Significant figures start at the first non-zero digit.

Dropping placeholder zeros. 47,382 to 2 s.f. is 47,000, not 47.

Using = instead of ≈ in an estimate. The statement is an approximation.

Rounding to 1 s.f. inside a square root. Round to a nearby perfect square instead.

Rounding partway through a calculation. Keep full accuracy and round only at the end.

Applying the usual rule in context. Coaches round up; shelves cut from a plank round down.

Exam technique for "Rounding and Estimation"

Underline the digit you are rounding and circle the one after it. That single habit prevents both the stage-rounding error and the wrong-digit error.

Write the rounded values on their own line in an estimation question, using ≈, before calculating. The rounding itself carries a method mark.

For direction questions, say what each rounding did and how it affects the answer, rather than asserting over or under.

Keep the calculator's full value between stages, and say so if the question asks about accuracy.

In context questions, state why you rounded the way you did.

Match the accuracy of your answer to the data you were given: results from measurements rounded to the nearest centimetre do not justify four decimal places.

Quick revision summary

Rounding trades precision for usefulness, and the question decides how much can be lost.

To round to decimal places, look only at the next digit: 5 or more rounds up. Never round in stages.

Significant figures are counted from the first non-zero digit. Leading zeros never count; zeros between digits always do; placeholder zeros must be kept, so 47,382 to 2 s.f. is 47,000.

To estimate, round every number to 1 s.f. and use ≈. For roots, round to a nearby perfect square instead.

An estimate is an underestimate if every number was rounded down, and an overestimate if every one was rounded up; rounding the denominator up makes the answer smaller.

In context, round the way the situation demands — coaches up, shelves down — and give the reason.

Never round partway through; keep full accuracy and round the final answer only.

A rounded measurement is a range, so an answer cannot be more precise than the data behind it.