Choosing efficient exact/decimal representations | Year 8

01 / LEARN

Understand the idea.

Exact does not mean decimal

An exact value loses no information. Fractions, roots and expressions containing pi can be exact even without a short decimal form. A terminating decimal can also be exact: $\frac{3}{8} = 0.375$ . Choose the form that makes the next calculation clear.

A finite approximation to a recurring or irrational decimal loses information. Write $\frac{1}{3} \approx 0.33$ , but $\frac{1}{3} = 0. \bar{3}$ . The symbols $=$ and $\approx$ distinguish an unchanged value from one that has been rounded or estimated.

Round at the end

Keep exact forms or extra calculator digits through intermediate steps, then round the final result. Rounding several inputs first can accumulate error, even when each individual approximation seems close. A final decimal cannot recover digits already discarded.

Match the precision and form to the task. Money is usually stated in cents; an exact share may be clearer as a fraction. Follow the requested decimal places, and avoid suggesting greater measurement precision than the original information supports.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Three rounded thirds

Compare adding three exact thirds with adding three copies of 0.33.

Worked solution

1
Add the exact fractions.
$\frac{1}{3} + \frac{1}{3} + \frac{1}{3} = \frac{3}{3} = 1$
The three equal thirds make one whole exactly.
2
Add the rounded inputs.
$0.33 + 0.33 + 0.33 = 0.99$
Each input has already been shortened to two decimal places.
3
Compare the totals.
$1 - 0.99 = 0.01$
The three small rounding losses accumulate to one hundredth.

Answer

The exact sum is 1; the rounded-input sum is 0.99.

EXAMPLE 02

A sensible final length

A circular track has circumference $12 π$ metres. Give it to one decimal place using $π \approx 3.14159$ .

Worked solution

1
Keep the exact circumference visible.
$C = 12 π m$
The expression containing pi retains the full given value.
2
Use the specified approximation.
$C \approx 12 \times 3.14159 = 37.69908 m$
Keeping these digits avoids rounding the product prematurely.
3
Round only the final result.
$C \approx 37.7 m$
For one decimal place, the hundredths digit nine rounds the tenths digit six up to seven.

Answer

Approximately 37.7 m.

03 / YOUR TURN

Try it yourself.

Check your understanding

Is $\frac{1}{8} = 0.125$ exact or approximate?

Show answer & reasoning

Exact.

Multiplying both numerator and denominator by $125$ gives $\frac{1}{8} = \frac{125}{1000} = 0.125$ . The conversion terminates without rounding, so it is exact.

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Write $\frac{3}{8}$ as an exact decimal.
2 marks
Round $\frac{1}{3}$ to two decimal places and use the correct equality or approximation symbol.
2 marks
Find the exact total of three one-third-metre pieces. Explain why adding three rounded lengths of 0.33 m gives a different result.
3 marks
The square root of 18 is approximately 4.24. Which expression gives its exact value: $\sqrt{18}$ or 4.24? Explain.
2 marks
Four students share AUD 9 equally. Find each share and decide whether cents require rounding.
2 marks
Use $π \approx 3.14159$ to round $5 π$ cm to one decimal place.
2 marks

Read the worked solutions

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

Write $\frac{3}{8}$ as an exact decimal.
2 marks
$\frac{3}{8} = \frac{375}{1000} = 0.375$ .
0.375
Round $\frac{1}{3}$ to two decimal places and use the correct equality or approximation symbol.
2 marks
The decimal 0.333… continues beyond two places.
$\frac{1}{3} \approx 0.33$ .
Find the exact total of three one-third-metre pieces. Explain why adding three rounded lengths of 0.33 m gives a different result.
3 marks
$3 \times \frac{1}{3} = 1$ .
Three rounded copies lose a total of 0.01 m.
1 m; early rounding gives 0.99 m.
The square root of 18 is approximately 4.24. Which expression gives its exact value: $\sqrt{18}$ or 4.24? Explain.
2 marks
Eighteen is not a perfect square, so the exact root is irrational.
$\sqrt{18}$ gives the exact root; 4.24 is a rounded approximation to it.
Four students share AUD 9 equally. Find each share and decide whether cents require rounding.
2 marks
$9 \div 4 = 2.25$ .
This terminates exactly at two decimal places.
AUD 2.25 each; no rounding is needed.
Use $π \approx 3.14159$ to round $5 π$ cm to one decimal place.
2 marks
$5 \times 3.14159 = 15.70795$ .
The hundredths digit is 0, so the tenths value remains 15.7.
15.7 cm

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Four operations with integers; signed fraction/decimal calculations; order of operations; terminating and recurring decimals; rational and irrational numbers; estimating square roots and recognising pi; choosing efficient exact/decimal representations.

Official curriculum source ↗

In this lesson

Understand the idea.

Exact does not mean decimal

Round at the end

See the method in action.

Three rounded thirds

Add the exact fractions.

Add the rounded inputs.

Compare the totals.

A sensible final length

Keep the exact circumference visible.

Use the specified approximation.

Round only the final result.

Try it yourself.

Check your understanding

Keep practising