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Construct a fraction from a recurring decimal

Start a set of 8
Step 2 of 9

Cancel the repeated tail

Multiplication by a suitable power of ten shifts a whole block. Subtracting two equations with identical tails gives an exact equation

  1. 1Name the whole recurring value Let x represent the positive magnitude of the entire decimal. Do not cut off its repeating tail \(\displaystyle x=0.4\overline{27}\)
  2. 2Align 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 \(\displaystyle \begin{aligned}1{,}000x&=427.\overline{27}\\10x&=4.\overline{27}\end{aligned}\)
  3. 3Subtract, then divide Subtract both sides. The infinite tails cancel exactly, leaving an ordinary linear equation \(\displaystyle \begin{aligned}990x&=423\\x&=\frac{423}{990}=\frac{47}{110}\end{aligned}\)
  4. 4Restore 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 \(\displaystyle -0.4\overline{27}=-\frac{47}{110}\)
Year 11, VCE Units 1 & 2, Chapter 1, 4 difficulty levels, No calculatorBuild a worksheet

Exact recurring conversion

A finite prefix does not prevent rationality. The bar determines the repeating block. Use algebra rather than a rounded calculator display

  1. Name the whole recurring value Let x represent the positive magnitude of the entire decimal. Do not cut off its repeating tail \(\displaystyle x=0.4\overline{27}\)
  2. 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 \(\displaystyle \begin{aligned}1{,}000x&=427.\overline{27}\\10x&=4.\overline{27}\end{aligned}\)
  3. Subtract, then divide Subtract both sides. The infinite tails cancel exactly, leaving an ordinary linear equation \(\displaystyle \begin{aligned}990x&=423\\x&=\frac{423}{990}=\frac{47}{110}\end{aligned}\)
  4. 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

\[2.1\overline{54}\]
  1. \(\displaystyle x=2.1\overline{54}\)Name the complete positive value
  2. \(\displaystyle 1{,}000x=2{,}154.\overline{54}\)Move the prefix and the two-digit block before the decimal point
  3. \(\displaystyle 10x=21.\overline{54}\)Move only the one-digit prefix
  4. \(\displaystyle 990x=2{,}133\)Subtract the equations so that the matching tails cancel
  5. \(\displaystyle x=\frac{2{,}133}{990}=\frac{237}{110}\)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 block
Using 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 not
Losing the sign
✕\(\displaystyle -0.\overline6=\frac23\)✓\(\displaystyle -0.\overline6=-\frac23\)
A sign before the decimal applies to the whole number

Difficulty 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