Let the fraction be x.  Then

x = 0.6898989898

Since it repeats the 2 digits ("89") over and over,
we multiply both sides by 102 or 100.

100x = 68.989898989

Now put the original equation underneath that,
lining up the decimal points

100x = 68.98989898
   x =  0.6898989898
--------------------

Continue the upper equation so that the decimals go
as far as they do on the lower equation, and subtract.
Then the string of decimals will cancel:

100x = 68.9898989898
   x =  0.6898989898
--------------------
 99x = 68.3000000000

 99x = 68.3

Now remove the decimal by multiplying both sides by 10

990x = 683

   x = 

That doesn't reduce.  Check by dividing out the
fraction, changing it back to a decimal

         .6898----
  990)683.0000----
      594 0
       89 00
       79 20
        9 800
        8 910
          8900
          ....

We don't need to continue because we reached
the same remainder 8900 a second time, so
the decimals will keep repeating "89"

Answer:  

Edwin