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         A               C                         B
we assume that Ramesh and Dinesh travel with their constant speeds.

they meet at C  after time T.
Speed of Ramesh = AC/T                                speed of Dinesh = BC/T

Ramesh travels CB in 4 hrs                             Dinesh travels CA in 1 hr
speed = CB/4                                                 speed = CA/1

total distance AB
total time duration for Ramesh = T+4 hrs            for Dinesh = T+1 hr
avg Speed of Ramesh = AB/(T+4)                      for  Dinesh :  AB / (T+1)

Ratio of their avg speeds = (T+1) / (T+4)

average speeds of Ramesh and Dinesh for the two legs of travel:

AC/T = CB/4 = (AB-AC)/4                                      BC/T= CA
=>  T = 4 AC/(AB - AC)                               T = BC / CA= (AB-AC) / CA

=> T * T = 4    => T = 2 hrs          by multiplying the above two expressions

substituting the value of T, we get 
=>  BC = 2 CA        and     AB = 3 CA

Ramesh:  Avg speed = AC/2 = AB/6 = BC/4
Dinesh:  Avg speed = CA/1 = AB/3 = BC/2

Ratio of average speeds = (T+1)/(T+4) = 3/6 = 1/2

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