How to Convert an Infinite Decimal to a Fraction

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Infinite decimals can be tricky to convert to fractions because you cannot simply put the decimal over the appropriate multiple of 10. Converting an infinite decimal to a fraction can better help you to represent the number. For example, 0.3636... may be harder to grasp than 36/99. You can only convert repeating infinite decimals to fractions. For example, pi does not terminate or repeat so while it is commonly approximated as 22/7, it is not exact.

  • Set the repeating fraction equal to x. For example, if your infinite decimal is 0.18232323... you would write x=0.182323...

  • Determine the repeating length of the decimal. The repeating length is the number of digits in the repeating pattern. For example, 0.182323... has a repeating length of 2 because the pattern is "23." If your decimal was 0.485485485.... the repeating length would be 3.

  • Multiply each side of the equation from step 1 by 10^R, where R is the repeating length. For example, since 0.182323... has a repeating length of 2, and 10^2 is 100, you would get 100x=18.2323...

  • Subtract the equation in Step 1 from the equation in Step 3. For example, you would subtract x=0.182323... from 100x=18.2323... and you would get 99x=18.05.

  • Solve the equation in Step 4 for x. For example, with 99x=18.05 you would divide by 99 on both sides so you would have x=18.05/99, or 1805/9900.

  • Simplify the fraction found in Step 4. For example, 1805/9900 simplifies to 361/1980.

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