Open Math›Year 10›Measurement precision›Significant figures

Significant figures

Round numbers to a specified number of significant figures and apply the correct precision to the results of calculations.

Worked examples

Rounding to 2 significant figures

Straightforward

Problem

Round

62\,850

to 2 significant figures.

Identify the first two significant figures.

Starting from the leftmost non-zero digit: the first significant figure is

6

(ten-thousands place) and the second is

2

(thousands place).

Look at the digit immediately after the second significant figure.

The digit in the hundreds place is

8

Apply the rounding rule.

Since

8 \geq 5

, round up the second significant figure:

2

becomes

3

Replace all remaining digits with zeros.

The hundreds, tens and units digits all become zeros:

62\,850 \approx 63\,000

Answer

63\,000

Counting significant figures

Moderate

Problem

How many significant figures does

0.05070

have?

Identify and ignore leading zeros.

The zeros before

5

(that is,

0.0

) are leading zeros. They are placeholders that locate the decimal point, not significant figures.

Count from the first non-zero digit.

The digit

5

is the first significant figure.

Assess the zero between

5

and

7

This zero sits between two non-zero digits (

5

and

7

), so it is significant.

Assess the digit

7

The digit

7

is non-zero, so it is significant.

Assess the trailing zero.

The trailing

0

appears after the decimal point. A trailing zero after the decimal point is always significant — it was deliberately recorded to indicate precision.

Answer

0.05070

has 4 significant figures:

5

0

7

and

0

Rounding a calculation result to the correct significant figures

Challenging

Problem

Calculate

\dfrac{4.56 \times 1.8}{3.0}

and express the answer to the correct number of significant figures. Explain why writing more digits would be misleading.

Count the significant figures in each value.

4.56

has 3 significant figures;

1.8

has 2 significant figures;

3.0

has 2 significant figures.

Determine the required precision for the answer.

For multiplication and division, the answer is rounded to the same number of significant figures as the least precise value. The least precise values have 2 sig figs, so the answer must be rounded to 2 significant figures.

Perform the calculation.

4.56 \times 1.8 = 8.208

8.208 \div 3.0 = 2.736

Round to 2 significant figures.

The first two significant figures of

2.736

are

2

and

7

. The next digit is

3

(less than

5

), so round down:

2.7

Explain why extra digits are misleading.

Writing

2.736

implies the result is known to four significant figures. However,

1.8

and

3.0

are each only known to 2 significant figures (to the nearest

0.1

). The extra digits suggest a precision that the original measurements cannot support.

Answer

2.7

Practise

Q1·Straightforward

Round

38\,470

to 2 significant figures.

Explanation

The first significant figure is

3

(ten-thousands place) and the second is

8

(thousands place). The next digit is

4

, which is less than

5

, so round down. All remaining digits become zeros:

38\,000

Q2·Straightforward

Round

0.003847

to 2 significant figures.

Explanation

Leading zeros are placeholders, not significant figures. The first significant figure is

3

and the second is

8

. The next digit is

4

, which is less than

5

, so round down:

0.0038

Q3·Straightforward

Round

52\,630

to 3 significant figures.

Explanation

The first three significant figures are

5

2

and

6

. The fourth digit is

3

, which is less than

5

, so round down. The remaining digit becomes zero:

52\,600

Q4·Straightforward

Round

0.07254

to 2 significant figures.

Explanation

The leading zeros are not significant. The first significant figure is

7

and the second is

2

. The next digit is

5

, so round up:

0.073

Q5·Moderate

How many significant figures does

0.00360

have?

2345

Explanation

Leading zeros (before the

3

) are not significant — they are just placeholders. The digits

3

6

and the trailing

0

are all significant. The trailing zero appears after the decimal point, so it was deliberately recorded and indicates precision to that place. Total: 3 significant figures.

Q6·Moderate

How many significant figures does

40\,500

have?

2345

Explanation

The digit

4

is significant, the zero between

4

and

5

is significant (it lies between two non-zero digits), and

5

is significant. The two trailing zeros in this whole number (no decimal point shown) are ambiguous placeholders and are not counted as significant. Total: 3 significant figures.

Q7·Moderate

A scientist records a mass as

4762.3

g. Round this measurement to 3 significant figures.

Explanation

The first three significant figures are

4

7

and

6

. The fourth digit is

2

, which is less than

5

, so round down. The result is

4760

g (to 3 significant figures).

Q8·Moderate

How many significant figures does

0.02050

have?

2345

Explanation

Breaking down

0.02050

: the leading zeros before

2

are not significant. The digit

2

is significant. The zero between

2

and

5

is significant (between two non-zero digits). The digit

5

is significant. The trailing zero after

5

is significant (it appears after the decimal point). Total: 4 significant figures (

2

0

5

0

Q9·Challenging

Calculate

3.42 \times 8.7

and express the answer to the correct number of significant figures.

Explanation

3.42

has 3 significant figures and

8.7

has 2 significant figures. The least precise factor has 2 sig figs, so the answer must be rounded to 2 significant figures.

3.42 \times 8.7 = 29.754

. Rounded to 2 significant figures:

30

Q10·Challenging

Calculate

0.0456 \div 0.12

and express the answer to the correct number of significant figures.

Explanation

0.0456

has 3 significant figures and

0.12

has 2 significant figures. The answer must be rounded to 2 significant figures.

0.0456 \div 0.12 = 0.38

. This already has 2 significant figures, so no further rounding is needed.

Q11·Challenging

Calculate

\dfrac{6.83 \times 4.2}{0.75}

and express the answer to the correct number of significant figures.

Explanation

6.83

has 3 sig figs,

4.2

has 2 sig figs, and

0.75

has 2 sig figs. The least precise value has 2 sig figs, so round the answer to 2 sig figs.

6.83 \times 4.2 = 28.686

28.686 \div 0.75 = 38.248

. Rounded to 2 significant figures:

38

Q12·Challenging

A student calculates the area of a rectangle with sides

3.2

cm and

4.15

cm and records the answer as

13.28

^2

. Which of the following is the most appropriately rounded answer?

$13.28$ cm$^2$$13.3$ cm$^2$$13$ cm$^2$$10$ cm$^2$

Explanation

3.2

cm has 2 significant figures and

4.15

cm has 3 significant figures. For multiplication, the answer must be rounded to 2 significant figures.

3.2 \times 4.15 = 13.28

. Rounded to 2 significant figures:

13

^2

. Writing

13.28

^2

implies a false precision — the side

3.2

cm is only known to the nearest

0.1

cm, so the area cannot be reliably stated to four significant figures.