Option (D) is correct.
It is given that, total number of balls $=12$
Let the number of white balls $=x$
$\therefore$ Probability of getting a white ball = $\frac{x}{12}$
Now, $6$ white balls are added. $\therefore$ Total number of balls $ =12 + 6 = 18$
Number of white balls $= x + 6$
$\therefore$ Probability of getting a white ball $= \frac{x + 6}{18}$
According to the question,
$\begin{align} \frac{x + 6}{18} &= 2 \times \frac{x}{12} \\ \implies x+6 &= 3x \\ \implies \quad ~ 2x &= 6 \\ \implies \quad ~~~ x&=3 \end{align}$