Problem:
The sum of the digits of a two-digit number is 9. When we interchange these digits, it is found that the resulting new number is greater than the original number by 27. What is the two-digit number?

Solution:

The problem states that the sum of the digits of a two-digit number is 9. Let's assume the original number to be AB, where A represents the tens digit and B represents the units digit.

Step 1: Determine the equation for the sum of the digits
The sum of the digits is given as A + B = 9.

Step 2: Determine the equation for the difference between the original number and the new number
The difference between the original number AB and the new number BA is given as 10A + B - (10B + A) = 27.

Step 3: Simplify the equation for the difference
Expanding the equation, we have 10A + B - 10B - A = 27.
This simplifies to 9A - 9B = 27.

Step 4: Simplify the equation for the sum of the digits
Since A + B = 9, we can rewrite the equation as A = 9 - B.

Step 5: Substitute the value of A in the equation for the difference
Substituting the value of A in the equation 9A - 9B = 27, we get (9 - B) * 9 - 9B = 27.

Step 6: Solve the equation for B
Expanding and simplifying the equation, we have 81 - 9B - 9B = 27.
Combining like terms, we get 81 - 18B = 27.
Subtracting 81 from both sides, we have -18B = -54.
Dividing both sides by -18, we get B = 3.

Step 7: Substitute the value of B in the equation for A
Since A + B = 9, substituting B = 3, we get A + 3 = 9.
Subtracting 3 from both sides, we have A = 6.

Step 8: Determine the two-digit number
The two-digit number is AB, which is 63.

Conclusion:
The two-digit number is 63. The original number is 63, and when we interchange the digits, we get 36, which is greater than the original number by 27.

Assume the digit in ones place as x and the digit in tens place is y The original number is (10y + x) Number obtained by reversing the digits = (10x + y) Now check for the condition Given that sum of digits of the number is 9 That is x + y = 9   → (1) Second condition is, number obtained by interchanging the digits is greater than the original number by 27 That is (10x + y) = (10y + x) 27 ⇒ 10x + y – 10y – x = 27 ⇒ 9x – 9y  = 27 ∴ x – y = 3 → (2) Add (1) and (2), we get x + y = 9 x – y = 3 ----------- 2x = 12 ∴ x = 6 Put x = 6 in (1), we get 6 + y = 9 ⇒ y = 9 – 6 = 3 The original number = 10y + x = 10(3) + 6 = 36