Pure repeating decimals
Put the repeating block over as many 9s as it has digits, then simplify. One digit goes over 9, two digits over 99, three digits over 999. Example: 0.123123... = 123/999 = 41/333.
Decimals with a non-repeating start
Take all the digits through the first repeat, subtract the non-repeating part, and divide by 9s followed by 0s. Use one 9 per repeating digit and one 0 per non-repeating digit. Example: 0.2333... = (23 - 2) / 90 = 21/90 = 7/30.
Spot the common ones
Many answers are familiar fractions. 0.1666... is 1/6, 0.8333... is 5/6, and 0.0909... is 1/11. Recognizing these saves the simplifying step.
Practice problems
- 0.333... as a fraction
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One repeating digit, so 3/9 = 1/3.
- 0.272727... as a fraction
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Two repeating digits, so 27/99 = 3/11.
- 0.1666... as a fraction
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(16 - 1) / 90 = 15/90 = 1/6.
References
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