Choosing efficient exact/decimal representations | Year 8

01 / LEARN

The idea, step by step

Exact does not mean decimal

An exact answer preserves the value completely. Fractions, roots and expressions containing pi can all be exact. Some decimals are also exact: $\frac{3}{8} = 0.375$ . By contrast, a finite decimal cannot exactly represent $\frac{1}{3}$ . Use $=$ for an exact equality and $\approx$ when a value has been rounded or estimated.

Round at the end

Use exact values or extra calculator digits during intermediate steps, and round the final result to the precision asked for. Money is normally reported in cents, while a fraction may be the clearest answer for a share. Precision should fit the context: a measurement reported to a centimetre does not justify many decimal places in the final length.

A common trap

A decimal answer is always more accurate than a fraction.

Instead: A rounded decimal can lose information. An exact fraction may be more accurate and easier to use.

02 / WORKED EXAMPLES

See the thinking

Example 1

Three rounded thirds

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

$\frac{1}{3} + \frac{1}{3} + \frac{1}{3} = 1$ .
But $0.33 + 0.33 + 0.33 = 0.99$ .
Each early approximation loses a small amount, so the losses add.

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

Example 2

A sensible final length

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

$12 \times 3.14159 = 37.69908$ .
The hundredths digit is 9, so 37.6 rounds up to 37.7.
Keeping pi in the calculation avoids using the rough value 3.1 too early.

Approximately 37.7 m.

Check your understanding

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

Show answer

Exact.

$\frac{1}{8} = \frac{125}{1000}$ , with no rounding.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF 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

Checking saved progress…

Ready for the topic test?

Review the chapter, then choose a practice test.

Topic review

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 ↗

Back to Integers and rational numbers