Question

The sum of the digits of a two-digit number is 12. The number formed by interchanging the digits is 54 more than the original number. What is the original number?

Answer 1

Let X and Y be the digits.
If a two-digit number is written XY, then its value is 10x+y.
When the digits of the number are interchanged, it's YX, and its value is 10y+x.

The sum of the digits is 12.
The number formed by interchaning the digits is 54 more than the original number.


`x+y=12 \\ 10x+y+54=10y+x \\ \\ x+y=12 \\ 10x-x+y-10y=-54 \\ \\ x+y=12 \\ 9x-9y=-54 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |/ 9 \\ \\ x+y=12 \\ \underline{x-y=-6} \\ x+x=12-6 \\ 2x=6 \\ x=(6)/(2) \\ x=3 \\ \\ x+y=12 \\ 3+y=12 \\ y=12-3 \\ y=9`

The digits X and Y are 3 and 9, so the original number is 39.