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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 $:\text{B}$