The sum of the digits of a two digit number is 17. When 9 is added to the number, the digits interchange their places. What is the original two digit number?

Introduction / Context:
This question is a classic number puzzle involving a two digit number, the sum of its digits, and a transformation in which adding a fixed value produces a reversed digit arrangement. It tests your ability to convert verbal conditions about digits into algebraic equations.

Given Data / Assumptions:
- The number is a two digit positive integer.
- The sum of its tens digit and units digit is 17.
- Adding 9 to the original number produces another two digit number whose digits are reversed.
- We need to find the original two digit number.

Concept / Approach:
A two digit number can be represented as 10x + y, where x is the tens digit and y is the units digit. The reversed number is 10y + x. The given conditions translate into two equations: x + y = 17 for the sum of digits, and 10x + y + 9 = 10y + x for the reversal after adding 9. Solving this system delivers the values of x and y.

Step-by-Step Solution:
Step 1: Let the tens digit be x and the units digit be y, so the original number is 10x + y.Step 2: From the digit sum condition, we have x + y = 17.Step 3: Adding 9 to the number gives 10x + y + 9. This equals the reversed number 10y + x.Step 4: Set up the equation: 10x + y + 9 = 10y + x.Step 5: Rearrange: 10x - x + y - 10y + 9 = 0, which simplifies to 9x - 9y + 9 = 0, or 9(x - y + 1) = 0.Step 6: So x - y + 1 = 0 which gives x - y = -1, so x = y - 1.Step 7: Substitute into x + y = 17 to get (y - 1) + y = 17, so 2y - 1 = 17, then 2y = 18, so y = 9 and x = 8.Step 8: The original number is 10x + y = 10 * 8 + 9 = 89.

Verification / Alternative check:
Check the conditions directly with 89. Its digits are 8 and 9 whose sum is 17, satisfying the first condition. Adding 9 gives 89 + 9 = 98. The number 98 is indeed formed by reversing digits of 89, so both conditions are met. No other choice matches both constraints simultaneously, confirming that 89 is correct.

Why Other Options Are Wrong:
Option 98 has digit sum 17 but adding 9 gives 107 which is not a two digit reversal. Option 78 has digit sum 15, so it fails the first condition. Option 87 has digit sum 15 and adding 9 gives 96, which is not a reversal of 87. Option 79 has digit sum 16 and adding 9 gives 88, which again does not reverse the digits.

Common Pitfalls:
Students sometimes reverse the condition and assume subtracting 9 reverses digits, or they misrepresent the number as x + y instead of 10x + y. Sign errors while manipulating the equation 10x + y + 9 = 10y + x are also common. It is important to correctly interpret digit places and consistently use algebraic notation.

Final Answer:
The original two digit number is 89.