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The ratio between Sumit's and Prakash's age at present is $2:3$. Sumit is $6$ years younger than Prakash. The ratio of Sumit's age to Prakash's age after six years will be :

1. $1:2$
2. $2:3$
3. $3:4$
4. $3:8$

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 $(C).$
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