Interpreting precision and reasonable answers | Year 9

01 / LEARN

Understand the idea.

Round once, at the end

An exact expression such as $36 π$ retains information that a short decimal loses. Use the calculator value of $π$ and keep intermediate values unrounded. Round only the final result to the accuracy requested in the question.

To round to one decimal place, inspect the hundredths digit. For $113.097 \dots$ , the tenths digit is $0$ and the following digit is $9$ , so the rounded result is $113.1$ . Always include the unit.

36 π \approx 113.097 \dots

, so to one decimal place use

113.1

Ask whether the size makes sense

Estimate the expected size before accepting a calculator answer. Check whether the quantity is a length, area, volume or capacity, and confirm that every dimension uses compatible units. A correct numerical product can still carry the wrong unit.

Extra displayed digits do not improve the original measurements. Follow any specified rounding instruction; otherwise choose precision that suits the measurements and context. A tank several metres across should hold thousands of litres rather than just a few bottles.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Avoid rounding pi early

A closed cylinder has radius 2 cm and height 7 cm. Find its surface area to one decimal place.

Worked solution

1
Find the two end areas.
$A_{ends} = 2 π (2^{2}) = 8 π {cm}^{2}$
A closed cylinder includes both circular ends.
2
Find the wall area.
$A_{wall} = 2 π (2) (7) = 28 π {cm}^{2}$
The wall area is circumference multiplied by height.
3
Add before rounding.
$A = 8 π + 28 π = 36 π {cm}^{2}$
Retaining $π$ avoids accumulated approximation error.
4
Round the final value.
$36 π = 113.097 \dots \approx 113.1 {cm}^{2}$
The hundredths digit is $9$ , so the tenths digit rounds up.

Answer

113.1 cm²

EXAMPLE 02

Check a capacity claim

A tank has internal dimensions 3 m by 2 m by 1.25 m. A student says its capacity is 7.5 L. Explain and correct the answer.

Worked solution

1
Calculate volume with its unit.
$V = 3 \times 2 \times 1.25 = 7.5 m^{3}$
Three dimensions measured in metres give cubic metres.
2
Convert the volume to litres.
$7.5 m^{3} = 7.5 \times 1000 = 7500 L$
One cubic metre holds one thousand litres.
3
Explain the original error.
$7.5 L = 0.0075 m^{3}$
The proposed capacity is one thousand times too small: the product was incorrectly labelled in litres.

Answer

7500 L; the student confused cubic metres with litres.

03 / YOUR TURN

Try it yourself.

Check your understanding

Round $10 \sqrt{2}$ cm to one decimal place.

Show answer & reasoning

14.1 cm.

Calculate $10 \sqrt{2} = 14.142 \dots cm$ . The hundredths digit is $4$ , so rounding to one decimal place gives $14.1 cm$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

A cylinder has radius 4 cm and height 5 cm. Give its volume to one decimal place.
2 marks
A calculated length is $\sqrt{50}$ m. Round it to two decimal places.
2 marks
A square has a stated side length of 2.35 m. Calculate its perimeter.
2 marks
A rectangular plot is 6.8 m by 4.2 m. Give its area to one decimal place.
2 marks
A height is recorded as 1.20 m to the nearest 0.01 m. State the maximum rounding error in metres.
2 marks
A student uses $\sqrt{2} \approx 1.4$ to calculate $10 \sqrt{2}$ and reports 14.0 cm. Find the correct answer to one decimal place and explain the difference.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A cylinder has radius 4 cm and height 5 cm. Give its volume to one decimal place.
2 marks
$V = π (4^{2}) (5) = 80 π \approx 251.327$ cm³, giving 251.3 cm³.
251.3 cm³
A calculated length is $\sqrt{50}$ m. Round it to two decimal places.
2 marks
$\sqrt{50} = 7.071067 \dots$ , so the answer is 7.07 m.
7.07 m
A square has a stated side length of 2.35 m. Calculate its perimeter.
2 marks
$P = 4 (2.35) = 9.40$ m, written as 9.4 m.
9.4 m
A rectangular plot is 6.8 m by 4.2 m. Give its area to one decimal place.
2 marks
$6.8 \times 4.2 = 28.56$ m², which rounds to 28.6 m².
28.6 m²
A height is recorded as 1.20 m to the nearest 0.01 m. State the maximum rounding error in metres.
2 marks
The recording step is 0.01 m. Its half-step is 0.005 m.
0.005 m
A student uses $\sqrt{2} \approx 1.4$ to calculate $10 \sqrt{2}$ and reports 14.0 cm. Find the correct answer to one decimal place and explain the difference.
3 marks
$10 \sqrt{2} = 14.142 \dots$ cm. Rounding only at the end gives 14.1 cm.
14.1 cm; the square root was rounded too early.

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Curriculum reference & lesson scope

Australian Curriculum v9 / South Australian Curriculum

Year 9 surface area and volume of right prisms and cylinders, capacity, absolute/relative/percentage measurement error, precision and practical measurement problems.

Official curriculum source ↗

In this lesson

Understand the idea.

Round once, at the end

Ask whether the size makes sense

See the method in action.

Avoid rounding pi early

Find the two end areas.

Find the wall area.

Add before rounding.

Round the final value.

Check a capacity claim

Calculate volume with its unit.

Convert the volume to litres.

Explain the original error.

Try it yourself.

Check your understanding

Keep practising