The sum of the digits of a 2-digit number is 12. If we add 54 to the number, the new number obtained is a number formed by interchange of the digits. Find the number.

Let the tens digit be x and the units digit be y.
The number is 10x+y.
Sum of digits: x+y=12 --- (Equation 1)

When 54 is added, digits are interchanged:

10x+y+54=10y+x
9y−9x=54
y−x=6
--- (Equation 2)

Adding Equation 1 and Equation 2:

(x+y)+(y−x)=12+6
2y=18⟹y=9
Substitute y=9 into Equation 1:

x+9=12⟹x=3
Hence, the number is 10(3)+9=39.