9514 1404 393
Answer:
115/198
Explanation:
The repeat is 2 digits, so you can multiply the number by 10 to that power.
x = 0.58080...(2-digit repeat)
100x = 58.08080...(2-digit repeat)
Then subtract the original number.
100x -x = 99x = 58.08080... -0.58080... = 57.5
Now, divide by the coefficient of x.
x = 57.5/99 = 115/198
This fraction cannot be reduced, so is the fraction you're looking for.
0.580...(2-digit repeat) = 115/198
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Additional comment
The number of 9s in the denominator is the number of repeating digits. If the repeat started at the decimal point, the fraction would be those two repeating digits divided by two 9s (99).
Your number can be considered to be ...
0.58080...(2-digit repeat) = 0.5 + 0.0808...(2-digit repeat)
This will be ...
0.58080... = 1/2 + 8/99 = 99/198 +16/198 = 115/198