The sum of two digits of a two-digit number is 11. If 63 is added to the number, the digits get reversed. Find the number.

Answer

To solve the problem step by step, we will define the two digits of the two-digit number. Let's denote:

- The first digit (tens place) as \( x \)
- The second digit (units place) as \( y \)

### Step 1: Set up the equations based on the problem statement.

From the problem, we know:

1. The sum of the two digits is 11:
\[ x + y = 11 \quad \text{(Equation 1)} \]

2. If 63 is added to the number, the digits get reversed. The original number can be expressed as \( 10x + y \). When 63 is added, the new number becomes \( 10x + y + 63 \), and this is equal to the number with reversed digits, which is \( 10y + x \):
\[ 10x + y + 63 = 10y + x \quad \text{(Equation 2)} \]

### Step 2: Simplify Equation 2.

Rearranging Equation 2:
\[ 10x + y + 63 = 10y + x \]
Subtract \( x \) and \( y \) from both sides:
\[ 10x - x + y - y + 63 = 10y - y \]
This simplifies to:
\[ 9x + 63 = 9y \]
Now, subtract 63 from both sides:
\[ 9x = 9y - 63 \]
Dividing the entire equation by 9 gives:
\[ x = y - 7 \quad \text{(Equation 3)} \]

### Step 3: Substitute Equation 3 into Equation 1.

Now we will substitute Equation 3 into Equation 1:
\[ x + y = 11 \]
Substituting \( x \) from Equation 3:
\[ (y - 7) + y = 11 \]
Combining like terms:
\[ 2y - 7 = 11 \]
Adding 7 to both sides:
\[ 2y = 18 \]
Dividing by 2:
\[ y = 9 \]

### Step 4: Find \( x \).

Now that we have \( y \), we can find \( x \) using Equation 1:
\[ x + 9 = 11 \]
Subtracting 9 from both sides:
\[ x = 2 \]

### Step 5: Find the original number.

The original two-digit number can be found using:
\[ \text{Number} = 10x + y = 10(2) + 9 = 20 + 9 = 29 \]

### Final Answer:
The two-digit number is **29**.

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