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Here's one way to convert the number to the ratio of integers.
- Count the number of repeating digits. Call this number <em>n</em>. Here, only 5 has an overline, so there is only one repeating digit. That is, <em>n </em>= 1.
- Multiply the number by 10^(-n) and subtract that from the original number. Call the result <em>p</em>. Here, doing this gives
- Compute the value <em>q</em> = 1 - 10^(-n). The result of step 2 is <em>q</em> times the original number. Here, <em>q</em> = 1 - 10^-1 = 0.9.
- Create the fraction <em>p/q</em>. Note that both <em>p</em> and <em>q</em> have a finite number of decimal digits, so this is easily converted to the ratio of integers, by moving the decimal point an equal number of places in the numerator and denominator. Reduce the resulting fraction as may be required. This is the fraction you're looking for. Here, p/q = 3.23/0.9 = 323/90. No reduction of this fraction is possible.
The fraction you're looking for is ...