Question

Express 0.234 234 .................. as a fraction in its simplest form

Answer 1

Let  n = .234....  

Now our job is to get the entire part that repeats on the left of the decimal point.
If we multiply the number by 1000 exactly one 2, one 3 and one 4 will move to the left.

So    1000n = 234.234...    now lets remove our original number
             -  n =      -.234...    Notice how all the repeaters minus all the repeaters
           999n = 234             leaves only the 234 that we move to the left of
                                                       the decimal point 
Now lets divide both side by 999
999n   = 234
999        999  this is the answer in unreduced form 9 can go into both = 26
                                                                                                                         111
  
       Try it on your calculator (divide 26 by 111)