Recurring decimals to fractions

Objective. Convert simple recurring decimals into fractions algebraically.

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Learn: the key idea

A recurring decimal has a digit or group of digits that repeats forever, such as 0.333... or 0.272727..., and every recurring decimal can be written exactly as a fraction. The algebraic method sets x equal to the decimal, multiplies by a power of 10 that shifts the repeating block to align, and subtracts to eliminate the recurring part. For example, to convert 0.555... to a fraction, let x = 0.555..., so 10x = 5.555..., and subtracting gives 9x = 5, so x = 5/9. The power of 10 used depends on how many digits repeat: one repeating digit needs ×10, two repeating digits need ×100.

Worked example

Write the recurring decimal 0.272727… as a fraction in its simplest form.

1 $Let x = 0.272727\dots; the block 27 repeats every 2 digits.$
2 $Multiply by 100: 100x = 27.272727\dots$
3 $Subtract: 100x - x = 27, so 99x = 27 and x = 27/99.$
4 $Simplify by 9: x = 3/11.$

Answer: 3/11

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Common mistake. Pupils sometimes multiply by the wrong power of 10, such as using ×10 for a two-digit repeating block, which fails to cancel the recurring part. Match the power of 10 to the number of repeating digits.

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