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: Understanding the problem
We need to find a two-digit number that satisfies the given conditions.

Step 2: Expressing the number algebraically
Let the tens digit be 'x' and the units digit be 'y'. The two-digit number can be expressed as 10x + y.

Step 3: Formulating the equation
According to the given information, two times the two-digit number is equal to 9 times the number obtained by reversing the digits. Mathematically, we can express this as:
2(10x + y) = 9(10y + x)

Step 4: Simplifying the equation
Expanding the equation, we get:
20x + 2y = 90y + 9x

Simplifying further, we get:
20x - 9x = 90y - 2y
11x = 88y

Step 5: Analyzing the equation
From the equation, we can see that 11x must be divisible by 88, as there are no common factors other than 11 between them.

Step 6: Considering the possible values of x and y
Since the sum of the digits is 9, x + y = 9. Given that both x and y are non-negative integers, the possible values for x and y are as follows:

- x = 1, y = 8
- x = 2, y = 7
- x = 3, y = 6
- x = 4, y = 5
- x = 5, y = 4
- x = 6, y = 3
- x = 7, y = 2
- x = 8, y = 1
- x = 9, y = 0 (Not possible as y cannot be 0)

Step 7: Checking the values of x and y
Let's substitute the possible values of x and y back into the equation and check if they satisfy the condition.

For x = 1 and y = 8:
11(1) = 88(8)
11 = 704 (Not satisfied)

For x = 2 and y = 7:
11(2) = 88(7)
22 = 616 (Not satisfied)

For x = 3 and y = 6:
11(3) = 88(6)
33 = 528 (Not satisfied)

For x = 4 and y = 5:
11(4) = 88(5)
44 = 440 (Not satisfied)

For x = 5 and y = 4:
11(5) = 88(4)
55 = 352 (Not satisfied)

For x = 6 and y = 3:
11(6) = 88(3)
66 = 264 (Not satisfied)

For x = 7 and y = 2:
11(7) = 88(2)
77 = 176 (Not satisfied)

For x = 8 and y = 1:
11(8) = 88(1)
88 = 88 (Satisfied)

Therefore, the two-digit number is