The effect of rounding and repeated approximations | Year 10

01 / LEARN

Understand the idea.

Round once, not at every step

Early rounding replaces a value before later operations use it. Multiplication or powers can enlarge its error, while subtracting close quantities can make a small initial error large relative to the final result.

Keep exact fractions, roots and $π$ wherever possible. On a calculator, use the stored result instead of retyping a shortened display. This removes avoidable arithmetic error but cannot remove uncertainty already present in measured inputs.

Retaining the exact intermediate quantity
Calculation	Exact result	Early rounded result
3 × (10/3)	10	3 × 3.33 = 9.99
6 × (1/3)	2	6 × 0.33 = 1.98
(√50)²	50	7.07² = 49.9849

Match precision to the task

A recurring fraction and its rounded decimal are different numbers. For example, $\frac{1}{3}$ is exact whereas $0.33$ is slightly smaller. Repeating the approximation repeats its error; adding later decimal places cannot recover the discarded amount.

Round the final requested quantity to the specified or defensible precision. A long calculator display does not make rough measurements more accurate. Keep units throughout so that the numerical error can also be interpreted in context.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Cancellation first

Compare 3×(10/3) exactly with 3×3.33.

Worked solution

1
Calculate with the exact fraction.
$3 \cdot \frac{10}{3} = 10$
The factor of three cancels exactly.
2
Calculate after replacing the fraction by a rounded decimal.
$3 \cdot 3.33 = 9.99$
The approximation $3.33$ is slightly less than $\frac{10}{3}$ .
3
Compare the totals.
$10 - 9.99 = 0.01$
The early-rounded total is $0.01$ too small.

Answer

10 versus 9.99

EXAMPLE 02

Retaining pi

A circle has exact radius 10 cm. Compare circumference using pi and using 3.14.

Worked solution

1
Retain the exact circumference.
$C = 2 π (10) = 20 π cm$
The radius is exact, so the multiple of $π$ is exact.
2
Calculate with the substituted approximation.
$C_{3.14} = 20 (3.14) = 62.8 cm$
Replacing $π$ by $3.14$ introduces a small underestimate.
3
Measure the resulting difference.
$20 π - 62.8 \approx 0.03185 cm$
Compare the unrounded expression with the approximated calculation before rounding the difference.

Answer

$20 π$ cm; about 0.03185 cm difference

03 / YOUR TURN

Try it yourself.

Check your understanding

Compare 6×(1/3) and 6×0.33.

Show answer & reasoning

2 and 1.98

$6 \cdot \frac{1}{3} = 2$ exactly, while $6 (0.33) = 1.98$ . The early approximation makes the total $0.02$ too small.

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Compare $7 \times (\frac{1}{7})$ exactly with $7 \times 0.14$ .
2 marks
Compare ${(\sqrt{50})}^{2}$ exactly with ${7.07}^{2}$ .
2 marks
Calculate $4 \times (\frac{2}{3})$ exactly and compare with $4 \times 0.67$ .
2 marks
Find the circumference of an exact-radius 4 cm circle to 2 decimal places, rounding at the end.
2 marks
Compare $100 \sqrt{5}$ with $100 \times 2.24$ , both to 2 decimal places.
2 marks
A ruler gives 4.2 cm to the nearest 0.1 cm. Does reporting a derived area to 8 decimal places make that measurement more precise?
3 marks

Read the worked solutions

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

Compare $7 \times (\frac{1}{7})$ exactly with $7 \times 0.14$ .
2 marks
Exact factors cancel.
$7 \times 0.14 = 0.98$ .
1 and 0.98
Compare ${(\sqrt{50})}^{2}$ exactly with ${7.07}^{2}$ .
2 marks
The exact square root squares to 50.
${7.07}^{2} = 49.9849$ .
50 and 49.9849
Calculate $4 \times (\frac{2}{3})$ exactly and compare with $4 \times 0.67$ .
2 marks
The exact result is $\frac{8}{3}$ .
$4 \times 0.67 = 2.68$ .
$\frac{8}{3}$ and 2.68
Find the circumference of an exact-radius 4 cm circle to 2 decimal places, rounding at the end.
2 marks
$8 π \approx 25.132741$ .
Round once to 25.13.
25.13 cm
Compare $100 \sqrt{5}$ with $100 \times 2.24$ , both to 2 decimal places.
2 marks
$100 \sqrt{5} \approx 223.60679$ .
$2.24 \times 100 = 224.00$ .
223.61 and 224.00
A ruler gives 4.2 cm to the nearest 0.1 cm. Does reporting a derived area to 8 decimal places make that measurement more precise?
3 marks
The length remains uncertain by up to 0.05 cm.
Extra arithmetic digits cannot remove instrument uncertainty.
No.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum

Surface area of composite solids; volume of composite solids; proportion/scaling of objects; related length/area/volume factors; measurement error; the effect of rounding and repeated approximations; practical design models; evaluating assumptions and suitable precision.

Official curriculum source ↗

In this lesson

Understand the idea.

Round once, not at every step

Match precision to the task

See the method in action.

Cancellation first

Calculate with the exact fraction.

Calculate after replacing the fraction by a rounded decimal.

Compare the totals.

Retaining pi

Retain the exact circumference.

Calculate with the substituted approximation.

Measure the resulting difference.

Try it yourself.

Check your understanding

Keep practising