Let the present age of Barun’s be $x$ years,
Therefore, Arun’s Present age $=40\%$ of $x=\frac{40}{100} \times x = \frac{2x}{5}$ years.
Let, after $t$ years Arun’s age will be half of Barun’s age.
Now, $\frac{2x}{5} + t = \frac{1}{2}(x + t)$
$\Rightarrow \frac{2x+5t}{5} = \frac{x+t}{2}$
$\Rightarrow4x+10t = 5x+5t$
$\Rightarrow x = 5t $
$\therefore$ The Barun’s age increased by $= \left[\frac{(x+t)-x}{x}\right]\times100 \% = \frac{t}{x}\times 100 \% = \frac{t}{5t}\times 100 \% = 20\%.$
$\textbf{Short Method:}$ Let the present age of Barun’s be $100$ years,
Therefore, Arun’s Present age $ = 40$ years.
Let, after $t$ years Arun’s age will be half of Barun’s age.
Now, $40 + t = \frac{1}{2}(100 + t)$
$\Rightarrow 80 + 2t = 100 + t $
$\Rightarrow t = 20$
$\therefore$ The Barun’s age increased by $= \dfrac{20}{100} \times 100\% = 20\%.$
Correct Answer $:\text{B}$