Problem Analysis:
Let's assume the two-digit number as '10x + y', where x and y are the digits of the number. According to the given conditions:
1. The number is 5 times the sum of its digits: 10x + y = 5(x + y)
2. When 9 is added to the number, the digits interchange their positions: 10x + y + 9 = 10y + x

Solution:
We can solve these two equations simultaneously to find the values of x and y, and then calculate the sum of the digits.

Solving the Equations:
Equation 1: 10x + y = 5x + 5y
Simplifying, we get: 5x - 4y = 0

Equation 2: 10x + y + 9 = 10y + x
Simplifying and rearranging, we get: 9x - 9y = -9

We can rewrite Equation 1 as: x = (4/5)y

Substituting this value of x in Equation 2, we get: 9(4/5)y - 9y = -9
Multiplying through by 5 to eliminate the fraction: 36y - 45y = -45
-9y = -45
y = (-45)/(-9)
y = 5

Substituting the value of y in Equation 1, we get: x = (4/5) * 5
x = 4

Calculating the Sum of the Digits:
The sum of the digits is x + y = 4 + 5 = 9

Final Answer:
The sum of the digits of the two-digit number is 9.

Explanation:
We started by assuming the two-digit number as '10x + y', where x and y are the digits. Then we formed two equations based on the given conditions: the number is 5 times the sum of its digits, and when 9 is added, the digits interchange their positions. By solving these equations simultaneously, we found the values of x and y to be 4 and 5 respectively. Finally, we calculated the sum of the digits as x + y = 4 + 5 = 9.