The sum of the digits of a two-digit number is 9. Also 9 times this number is twice the number obtained by reversing the order of the digits. Find the number.

Let's denote the two-digit number as \( 10a + b \), where \( a \) is the tens digit and \( b \) is the units digit. We are given two key pieces of information:

1. The sum of the digits is 9:
   \[
   a + b = 9
   \]

2. Nine times the number is equal to twice the number obtained by reversing the digits:
   \[
   9 \times (10a + b) = 2 \times (10b + a)
   \]

### Step 1: Set up the equations
From the first condition, we have:
\[
a + b = 9
\]

From the second condition:
\[
9(10a + b) = 2(10b + a)
\]
Expanding both sides:
\[
90a + 9b = 20b + 2a
\]

### Step 2: Simplify the equation
Rearrange the terms to group \(a\) and \(b\) on opposite sides:
\[
90a - 2a = 20b - 9b
\]
\[
88a = 11b
\]

### Step 3: Solve for \( a \) and \( b \)
Simplify the equation by dividing both sides by 11:
\[
8a = b
\]

### Step 4: Substitute into the sum equation
Substitute \( b = 8a \) into the first equation \( a + b = 9 \):
\[
a + 8a = 9
\]
\[
9a = 9
\]
\[
a = 1
\]

Now, substitute \( a = 1 \) back into \( b = 8a \):
\[
b = 8 \times 1 = 8
\]

### Step 5: Find the number
The number is \( 10a + b = 10(1) + 8 = 18 \).

### Step 6: Verify the solution
- The sum of the digits of 18 is \( 1 + 8 = 9 \), which satisfies the first condition.
- Reversing the digits gives 81, and checking the second condition:
  \[
  9 \times 18 = 162 \quad \text{and} \quad 2 \times 81 = 162
  \]
  Both sides are equal, confirming the solution.

The number is \( \boxed{18} \).

Would you like more details or have any questions? 

Here are 5 related questions:

1. What happens if the digits sum up to a different number, like 8 or 10?
2. Can you derive a formula for similar problems involving sums of digits?
3. How would the solution change if the multiplier was a different number, like 8 instead of 9?
4. What if the number was three digits long? How would the approach change?
5. Could this type of problem have more than one solution?

**Tip:** Always verify your solution by substituting the found values back into the original problem conditions.

Discover how to solve a mathematical problem involving the sum of digits in a two-digit number. This problem requires understanding equations and digit manipulation. Suitable for students in Grades 6-8.

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