Rukasha Maths

GCSE maths Beta

Recurring decimals and exact fractions · Revision guide

Higher · Number · 1 page in the PDF

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Learning objectives

  • Recognise the repeating block in a decimal.
  • Convert recurring decimals to simplified exact fractions.

Key ideas

An overbar identifies the digits that repeat forever.

0.36‾=0.363636…0.\overline{36}=0.363636\ldots

Multiply by powers of ten so the recurring tails match.

Subtract the aligned equations to remove the infinite recurring tails.

Divide to find the fraction, then simplify using a common factor.

Worked examples

Example 1

Convert the decimal to an exact fraction; only the digit six repeats.

x=0.16‾x=0.1\overline6
10x=1.6‾,100x=16.6‾10x=1.\overline6,\qquad100x=16.\overline6
100x−10x=16−1⇒90x=15100x-10x=16-1\quad\Rightarrow\quad90x=15
x=1590=16x=\frac{15}{90}=\frac16

Example 2

Convert the repeating decimal to a fraction.

x=0.45‾⇒100x−x=45x=0.\overline{45}\quad\Rightarrow\quad100x-x=45
x=4599=511x=\frac{45}{99}=\frac5{11}

Watch out

Match the recurring tails before subtracting. Do not treat a non-repeating digit as part of the repeating block.