N14, N15NumberFoundation & HigherGrades 2–5

Rounding & Estimation

Rounding replaces a number with a simpler one that is close to it. Estimating uses rounding to predict an answer before you calculate — the fastest way to catch a slipped decimal point. Nearly every calculator question ends with "give your answer to 3 significant figures", so this lesson protects marks across the whole paper.

35 min Video by Maths Genie AQA GCSE Maths

The big picture

Decimal places count digits after the decimal point. Significant figures count digits from the first non-zero digit, wherever the point is. So 0.004 72 to 2 significant figures is 0.0047, while 4720 to 2 significant figures is 4700 — the zeros hold place value.

AQA estimation questions expect you to round every number to 1 significant figure, calculate, and then often explain whether your estimate is too big or too small. That reasoning mark is about which way each number was rounded.

Rounding to Decimal Places and Significant FiguresCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

What you'll learn

Round to a given number of decimal places
Round to a given number of significant figures
Estimate calculations by rounding to 1 significant figure
Decide whether an estimate is an overestimate or underestimate

Decimal places

Count the required number of digits after the decimal point, then look at the $\textbf{next}$ digit. If it is 5 or more, round up; otherwise leave the last digit as it is. Remove the digits after it.

Keep any trailing zero that the accuracy requires: 3.497 to 2 d.p. is $3.50$ , not $3.5$ .

Example 1Round

12.8462

to 2 decimal places.

1Keep two decimal places:

12.84|62

2The next digit is 6 (5 or more), so round up.

Answer

12.85

Your turnSame method, new numbers — try it before moving on.

Round $7.3951$ to 2 decimal places.

Your answer

Show hint

Rounding 7.39 up changes two digits — and keep the zero.

Show worked solution

Solution

1 $7.39|51$ : the next digit is 5, so round up.
2 $7.39 \to 7.40$ (the zero shows 2 d.p.).

Answer: $7.40$

Significant figures

The first significant figure is the first non-zero digit. Count from there, round in the same way, and fill any gap before the decimal point with zeros to keep the place value.

Zeros between non-zero digits are significant: $4.07$ has 3 significant figures. Leading zeros ( $0.00$ …) are not.

Example 1Round (a)

58{,}372

to 2 s.f. (b)

0.006\,183

to 3 s.f.

1(a) First two figures:

58|372

. Next digit 3, so round down. Fill with zeros:

58{,}000

2(b) First significant figure is 6:

0.00618|3

. Next digit 3:

0.00618

Answer

(a)

58{,}000

(b)

0.00618

Your turnSame method, new numbers — try it before moving on.

Round $0.040\,76$ to 2 significant figures.

Your answer

Show worked solution

Solution

1First significant figure is 4: $0.040|76$ .
2The next digit is 7, so round up: $0.041$ .

Answer: $0.041$

Example 2Round

396{,}500

to 2 significant figures.

39|6500

: the next digit is 6, so round up.

39 \to 40

, then fill with zeros.

Answer

400{,}000

Your turnSame method, new numbers — try it before moving on.

Round $7{,}849$ to 1 significant figure.

Your answer

Show worked solution

Solution

1 $7|849$ : the next digit is 8, so round up.
2 $8{,}000$

Answer: $8{,}000$

Tip — A rounded number should be about the same size as the original: $58{,}372 \to 58$ is wrong because it has lost place value.

Estimating calculations

Round every number to 1 significant figure, then calculate. Choose numbers that are easy to work with — especially for division, where a "friendly" pair can make a calculation mental.

State your rounded calculation clearly, not just the final estimate.

Example 1Estimate the value of

\dfrac{612 \times 4.87}{0.198}

1Round to 1 s.f.:

\dfrac{600 \times 5}{0.2}

2Numerator:

3000

. Dividing by

0.2

is multiplying by 5.

Answer

15{,}000

Your turnSame method, new numbers — try it before moving on.

Estimate the value of $\dfrac{38.7 \times 9.14}{0.52}$ by rounding each number to 1 significant figure.

Your answer

Show worked solution

Solution

1 $\dfrac{40 \times 9}{0.5}$ .
2 $\dfrac{360}{0.5} = 720$ .

Answer: $720$

How good is an estimate?

The exact value is $15{,}052.7\ldots$ , so the estimate of 15,000 is remarkably close. That is why estimation works as a check: a calculator answer of 1505 or 150,527 would immediately look wrong.

Over- or underestimate?

Rounding a number $\textbf{up}$ in a multiplication (or the numerator of a fraction) makes the estimate bigger. Rounding the $\textbf{denominator}$ up makes the estimate smaller, because you are dividing by more.

Example 1Estimate

\dfrac{21.4 \times 3.2}{4.6}

and state whether your estimate is an overestimate or an underestimate.

1Round:

\dfrac{20 \times 3}{5} = 12

2The numerator was rounded down (both numbers), making it smaller.

3The denominator was rounded up, making the result smaller still.

Answer

12

, an underestimate

Think like an examiner

For estimates, show the rounded numbers. A correct final number with no rounding shown may score only 1 mark.
Round to 1 s.f. unless told otherwise — "600 × 5" rather than "612 × 4.87 ≈ 610 × 4.9".
Explanations of over/underestimates must refer to how specific numbers were rounded.

Accuracy

\text{next digit} \ge 5 \Rightarrow \text{round up}

The same rule for d.p. and s.f.

0.00\underline{6}18 \to 3 \text{ s.f.} = 0.00618

Count s.f. from the first non-zero digit.

\frac{a}{b}: \ b \uparrow \Rightarrow \text{estimate} \downarrow

Rounding a divisor up gives an underestimate.

Watch out for these

0.0527

to 1 s.f. is

0.1

The first significant figure is 5:

0.05

3.497

to 2 d.p. is

3.5

Show two decimal places:

3.50

4{,}762

to 2 s.f. is

48

Keep the place value:

4{,}800

Rounding in stages:

3.46 \to 3.5 \to 4

(to the nearest whole number).

Round from the original:

3.46 \to 3

Stretch yourself

A number rounded to 2 significant figures is 3,500. What is the smallest the number could be, and what is the largest whole number it could be?

Hint — Which numbers round to 3,500 when you keep 2 significant figures?

Reveal solution

SolutionRounding to 2 s.f. means rounding to the nearest 100. The smallest value is 3,450 (which rounds up to 3,500). The largest whole number is 3,549, since 3,550 would round to 3,600.

\blacksquare

Questions students ask

What is the difference between decimal places and significant figures?

Decimal places count digits after the point; significant figures count digits from the first non-zero digit.

0.0452

has 4 decimal places but 3 significant figures.

What accuracy should I give if the question does not say?

Give 3 significant figures for most measurements, 2 decimal places for money, and write down the full calculator display first.

Is 0 a significant figure?

Zeros between other digits (4.07) and trailing zeros after a decimal point that show accuracy (3.50) are significant. Leading zeros (0.004) are not.

Key takeaways

Look at the next digit: 5 or more rounds up.
Decimal places count after the point; significant figures count from the first non-zero digit.
Fill with zeros to keep place value when rounding large numbers.
Estimate by rounding every number to 1 s.f.
Rounding a divisor up makes the estimate smaller.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 22 marks

(a)Round $6.0847$ to 2 decimal places.[1]
Show mark scheme (a)
- B1 $6.08$
Answer: $6.08$
(b)Round $0.003\,965$ to 2 significant figures.[1]
Show mark scheme (b)
- B1 $0.0040$
Answer: $0.0040$

Question 2Grade 43 marks

(a)Work out an estimate for $\dfrac{8.3 \times 312}{0.38}$ .[2]
Show mark scheme (a)
- M1Rounds at least two numbers to 1 s.f., e.g. $\frac{8 \times 300}{0.4}$
- A1 $6000$
Answer: $6000$
(b)Is your answer to (a) an overestimate or an underestimate? Give a reason.[1]
Show mark scheme (b)
- B1Underestimate, because 8.3 and 312 were rounded down (smaller numerator) and 0.38 was rounded up (dividing by more)

Examiner note

"Underestimate" with no reason scores nothing. Refer to the direction each number was rounded.

Question 3Grade 53 marks

A lorry can carry 4,700 kg, to 2 significant figures. Each crate weighs 58 kg, to the nearest kilogram.

Using the least possible load limit and the heaviest possible crate, work out the greatest number of crates that can definitely be carried.[3]
Show mark scheme
- M1Least load limit $4650$ kg and greatest crate $58.5$ kg
- M1 $4650 \div 58.5 = 79.48\ldots$
- A1 $79$ crates
Answer: $79$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

Round $4.638$ to 1 decimal place.
- $4.6$
- $4.7$
- $4.64$
- $5$
Show answer and explanation
Answer: $4.6$
The next digit is 3.
Round $83{,}600$ to 1 significant figure.
- $80{,}000$
- $8$
- $84{,}000$
- $90{,}000$
Show answer and explanation
Answer: $80{,}000$
Next digit 3: round down, keep place value.
Round $0.0762$ to 1 significant figure.
- $0.08$
- $0.07$
- $0.1$
- $0.076$
Show answer and explanation
Answer: $0.08$
First s.f. is 7; next digit 6 rounds up.
How many significant figures does $40.05$ have?
- $4$
- $2$
- $3$
- $5$
Show answer and explanation
Answer: $4$
Zeros between non-zero digits count.
Estimate $\dfrac{49 \times 21}{9.8}$ .
- $100$
- $1000$
- $10$
- $105$
Show answer and explanation
Answer: $100$
$\frac{50 \times 20}{10}$ .
Round $2.996$ to 2 decimal places.
- $3.00$
- $2.99$
- $3.0$
- $2.10$
Show answer and explanation
Answer: $3.00$
$2.99|6$ rounds up to $3.00$ .
Estimating $\frac{58}{2.1}$ as $\frac{60}{2}$ gives…
- an overestimate
- an underestimate
- the exact answer
- cannot tell
Show answer and explanation
Answer: an overestimate
Numerator up and denominator down both increase the result.
Estimate $\sqrt{80.3} \times 3.9$ .
- $36$
- $320$
- $27$
- $40$
Show answer and explanation
Answer: $36$
$\sqrt{81} \times 4 = 9 \times 4$ .

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