assume, the total number of students be $x$
avg marks increased by $\dfrac{1}{2}$ due to an increase of $83 - 63$ = $20$ marks.
But the total increase in the marks = $\dfrac{1}{2}×x$ = $\dfrac{x}{2}$
Hence, we can write as
$\dfrac{x}{2} = 20 $
$\Rightarrow x = 20 \times 2 = 40$