Let the Sumit's and Prakash's age at present is $2x$ years and $3x$ years respectively.
Sumit is $6$ years younger than Prakash, so $2x = 3x-6$
$\implies x = 6$
$\therefore$ The present age of Sumit $ = 2x = 12$ years and the present age of Prakash $ = 3x = 18$ years.
Now, the ratio of Sumit's age to Prakash's age after six years will be $ = \dfrac{12+6}{18+6} = \dfrac{18}{24} = \dfrac{3}{4}$
Hence, The ratio of Sumit's age to Prakash's age after six years will be $3:4.$
So, the correct answer is $(C).$