Let the two digits be $x$ and $y$.
Original number$=xy=10x+y$
$ x + y = 12 \quad \to(1)$
Given that new number formed by reversing the digits is greater than the original number by $54,$
$ y * 10 + x = 10 * x + y + 54 $
$ 10 y + x = 10 x + y +54 $
$ 9y-9x = 54 $
$ x - y = -6 \quad \to(2)$
From $(1)$ and $(2)$
$ 2 x = 6 $
$ x = 3 $ and $ y = 9$
Hence, Original number $=xy=10x+y= 39$
Correct Answer: $A$