I will basically repeat the response from the other tutor, displaying the calculations differently. Then I will use the computations from that process to demonstrate a shortcut for solving a problem like this.
The given number is D = 0.5727272... with the "72" repeating.
Multiply the number by 10 so that the entire decimal part is repeating:
10D = 5.727272....
Since 2 digits are repeating, multiply by 10^2=100 to get another number in which the decimal part of the number is the same:
1000D = 572.727272....
Subtract the two numbers; the decimal parts will cancel:
1000D = 572.727272....
10D = 5.727272....
-----------------------
990D = 567
Perform the division...
D = 567/990
...and simplify the fraction
D = 63/110
Now look at how the numerator and denominator of the fraction were obtained:
The numerator 567 is the difference between "572" and "5". The string "572" is the non-repeating part of the given number plus one cycle of the repeating part; the "5" is the non-repeating part.
The denominator 990 is the difference between 10 and 1000. The two 9's are because there are 2 repeating digits in the given number; the single 0 is because there is 1 non-repeating digit in the given number.
The process can then be performed using this shortcut:
numerator: 572-5=567
denominator: 990 because 2 repeating digits and 1 non-repeating digit
ANSWER: 567/990
Here are a couple of quick examples using the shortcut....
(1) 0.3455555...
numerator: 345-34=311
denominator: 900
fraction: 311/900
(2) 0.123456745674567....
numerator: 1234567-123=1234444
denominator: 9999000
fraction: 1234444/9999000
You can check those examples, and others of your own devising, with a calculator.