The Non-Terminator

All fractions other than the ones of the form $\frac{1}{2^n5^m}$ will have non-terminating decimals.

Therefore the numbers which have terminating decimal representation are $1,2,4,5,8,10,16,20,25,32,40,50$

Hence there would be $38$ fractions which are non-terminating.

1) Find the number of terminating decimals of the form 1/k (where "k" ranges from 1 to 50)

   In any base B, 1/k has terminating decimals only if "k" has primes which are included in B, 
  So in our case, Base(B)=10, the numbers of the form, 1/2, 1/5, 1/10, 1/4, 1/25, 1/50, etc are the numbers with terminating decimals while the remaining are with non terminating decimals.

Base 10 = 2*5, so "k" must be of the form (2^a) * (5^b). Following are the possible values of "k" ("k" ranges from 1 to 50)

k = 2^(0 to 5)                            below 51      :         => 6
k = 5^( 1 or 2)                           below 51      :         => 2
k = 2*5, (2*5)^2                          below 51      :         => 2                
k = (2^2)*5,  (2^3)*5                     below 51      :         => 2

Total  count of "k"= 6+2+2+2 = 12

2) Subtract the number of terminating decimals from the total "n" numbers (n=50) to get the number of non-terminating decimals.

 No. of non-terminating decimals = (n - No. of terminating decimals )
                                                                          = 50 - 12
                                                                          = 38