Given Information:
- Two times a two-digit number is 9 times the number obtained by reversing the digits.
- The sum of the digits is 9.

Let's solve the problem step by step:

Step 1: Understand the problem
We are given a two-digit number, and its double is equal to 9 times the number obtained by reversing its digits. Additionally, the sum of the digits is 9. We need to find the original number.

Step 2: Represent the number
Let's assume the two-digit number as AB, where A represents the tens digit and B represents the units digit.

Step 3: Form equations
From the given information, we can form the following equations:
1. 2(10A + B) = 9(10B + A) (equation 1) - This equation represents the double of the two-digit number being equal to 9 times the number obtained by reversing its digits.
2. A + B = 9 (equation 2) - This equation represents the sum of the digits being equal to 9.

Step 4: Solve the equations
Let's solve the equations to find the values of A and B.

Solving equation 2:
Since A + B = 9, we can rearrange the equation to get:
A = 9 - B

Substituting the value of A in equation 1:
2(10(9 - B) + B) = 9(10B + (9 - B))
Simplifying the equation:
2(90 - 9B + B) = 9(10B + 9 - B)
180 - 18B + 2B = 90B + 81 - 9B
180 - 16B = 81B + 81
180 - 81 = 97B
99 = 97B
B = 99/97 (which is not possible as B should be an integer)

Since B cannot be a fraction, it means our assumption that the number is a two-digit number is incorrect.

Step 5: Adjust the assumption
Since the assumption of a two-digit number did not yield an integer value for B, let's adjust our assumption.

New assumption:
Let's assume the two-digit number as BA, where B represents the tens digit and A represents the units digit.

Step 6: Form equations
Using the new assumption, we can form the following equations:
1. 2(10B + A) = 9(10A + B) (equation 1)
2. A + B = 9 (equation 2)

Step 7: Solve the equations
Let's solve the equations to find the values of A and B.

Solving equation 2:
Since A + B = 9, we can rearrange the equation to get:
B = 9 - A

Substituting the value of B in equation 1:
2(10(9 - A) + A) = 9(10A + (9 - A))
Simplifying the equation:
2(90 - 10