I assume you mean 2.272727272727···

let 

  x = 2.272727272727···  write the repeating digits several times:

Multiply that equation by 100 put the one above under it, and
subtract:

100x = 227.2727272727··· 
   x =   2.272727272727···
------------------------
 99x = 225 the repeating decimals cancel out when you subtract them.

  x = 225/99 which reduces to 25/11

Answer 25/11:

Checking:
         2.2727···
     11)25.0000···
        22
         3 0
         2 2
           80
           77
            30
            22
             80     
             77
              3···

Edwin