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2015-09-18T19:02:59+05:30

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5.4178  bar is  the rational number with  the last four digits  repeating infinitely.
         5.4178 4178 4178 4178 ....

Simple rule to write in the rational number form is :  4178 is four digits. so take 0.0001.  here 1 is in the 4 th decimal place.  subtract it from 1 and get q.    the numerator p  = 0.4178, the given repeating number without bar on top.

             p / q =  5.0 +  [ 0.4178 / (1 - 0.0001) ] =  5.4173 / 0.9999
                     = 54173 / 9999

This can be obtained by using  geometric progression formula too.

     5.0 +  0.4178 + 0.00004178 + 0.0000 0000 4178 + ...
   = 5.0 +  0.4178 [ 1 + 0.0001 + 0.0000 0001 + .. ]
   = 5.0 + 0.4178 * 1 / [ 1 - 0.0001 ]
     = 5.0 + 0.4178 / 0.9999  = 5.4173 / 0.9999  = 54173 / 9999


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