Step 1: Count the number of recurring digits

For example, if we want to change 0.567567(567) ̅ to fraction, there are three recurring digits: 5,6, and 7.

Step 2: Convert to a fraction

We write the first set of the recurring digits as the numerator and the denominator with a series of 9s depending on the count from Step 1. Since we counted three and we have 567 as the numerator, denominator would have three 9’s.

Step 3: Simplify the fraction

Cancel common factors to simplify the fraction.

Examples of how to work out recurring decimals

Q1) Express the following recurring decimals to fractions.


a. To change 0.123123(123) ̅ to a fraction, we first count the number of digits that are recurring.


We would have three 9s on the denominator and 123 as the numerator.

0.(123) ̅=123/999

Look for common factors shared by the numerator and denominator to simplify the fraction.


b. Now to change 0.5123123123(123) ̅, we first separate the recurring and nonrecurring digits.


Express 0.5 as a fraction: .5= 5/10. To express 0.0(123) ̅ as a fraction, we can use the value of (123) ̅ from a. So, we have (123) ̅= 123/99=41/33.

Since there’s only 1 zero before the recurring digits, we multiply 41/33 by 1/10.


Now, to find the value of 0.5(123) ̅, we combine the values of the decimals.