Exact recurring conversion
A finite prefix does not prevent rationality. The bar determines the repeating block. Use algebra rather than a rounded calculator display
- Name the whole recurring value Let x represent the positive magnitude of the entire decimal. Do not cut off its repeating tail \(\displaystyle x=0.4\overline{27}\)
- Align identical tails If there are m prefix digits and k repeating digits, multiply by ten to the power of their combined lengths and by ten to the power of the prefix length. The two fractional tails then match \(\displaystyle \begin{aligned}1{,}000x&=427.\overline{27}\\10x&=4.\overline{27}\end{aligned}\)
- Subtract, then divide Subtract both sides. The infinite tails cancel exactly, leaving an ordinary linear equation \(\displaystyle \begin{aligned}990x&=423\\x&=\frac{423}{990}=\frac{47}{110}\end{aligned}\)
- Restore a negative sign If the original decimal is negative, apply that sign to the resulting fraction. A minus sign before the decimal applies to the entire value \(\displaystyle -0.4\overline{27}=-\frac{47}{110}\)
Worked example
Convert to an exact fraction
\[2.1\overline{54}\]
- \(\displaystyle x=2.1\overline{54}\)Name the complete positive value
- \(\displaystyle 1{,}000x=2{,}154.\overline{54}\)Move the prefix and the two-digit block before the decimal point
- \(\displaystyle 10x=21.\overline{54}\)Move only the one-digit prefix
- \(\displaystyle 990x=2{,}133\)Subtract the equations so that the matching tails cancel
- \(\displaystyle x=\frac{2{,}133}{990}=\frac{237}{110}\)Divide by 990 and simplify by nine. No rounding was used
Common mistakes
Ignoring a prefix
✕\(\displaystyle 0.2\overline7=\frac{27}{99}\)✓\(\displaystyle 0.2\overline7=\frac{27-2}{100-10}=\frac5{18}\)
The non-repeating 2 requires different powers of ten from a purely repeating two-digit blockUsing a finite approximation
✕\(\displaystyle 0.\overline3=\frac{333}{1000}\)✓\(\displaystyle 0.\overline3=\frac13\)
The finite fraction stops after three decimal digits; the complete recurring value does notLosing the sign
✕\(\displaystyle -0.\overline6=\frac23\)✓\(\displaystyle -0.\overline6=-\frac23\)
A sign before the decimal applies to the whole numberDifficulty levels
introPure recurring blocks with one or two digits and magnitude below one
standardWhole parts, signs and a one-digit non-repeating prefix
challengeLonger prefixes, two-digit cycles and signed exact values
examMixed exact conversions without a supplied setup; not a claim of a Unit 1 external exam
Questions about this topic
Must the fraction be simplified?+
In this practice topic an equivalent fraction is accepted. The worked solutions simplify it to make the equivalence clear
Does a correct fraction prove I can write the general argument?+
No. The fraction shows the conversion result. Practise the ungraded written argument separately and compare each logical step with the model
What if the repeating block starts with zero?+
Keep its full digit length. For a two-digit block such as 03, shift two places, not one
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Specialist Mathematics, chapter 1: Number, sets and proof
In this chapter
Classify real numbers and recognise undefined formsExplain number-system structureConvert between fractions and decimals, and between mixed and improper formsConstruct a fraction from a recurring decimalWork with sets and interval notationCount subsets and power-set elementsExpress a number in prime-factor index formInterpret propositions and quantifiersReason with direct and indirect proof methodsBuild and check mathematical induction arguments