Concept: Any two-digit number contains is of the form "ab" where the "b" is in the "ones" place and "A" is in the "tens" place.

The value of "a" is 10a and the value of "b" is 1.

According to the question, the sum of the digits of the two-digit number is 9.
Therefore, we assume that the digit units place be "9-x" and at the ten’s place be "x".

Then, the sum of the digits = 9 - x + x = 9 (As given in the question)

Then, the original number = 10x + (9-x) = 9x+9

This is because the value of x will be "10 times x as it is on ten's place" 

When we interchange the digits, the digits at one’s place and tens place will be x and 9-x.

Thus, the new number will be 10(9 - x)+x = 90-9x

Now, New Number = Original Number + 27

90 -9x = 9x + 9+ 27

90 – 9x = 9x + 36

90- 36 = 18x

X = 3 and 9-x = 6

Thus, the two digit number is 9x + 9 = 9 (3)+9 =36