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let the present age of Ajay is $2x$ and Vijay present age is $3x$.

It is given that Ajay is $6$ year younger than Vijay: $2x=3x-6\implies x=6$

such a present age of Ajay=$12$ year

present age of Vijay=$18$

So the ratio of Ajay to Vijay age after $6$ year $=\frac{12+6}{18+6}\implies\frac{18}{24}=\frac{3}{4}$

So Option $(D)$ is correct.

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