The sum of a two - digt number and the number obtained by reversing its digits in 121 . Find the number, if the its units place digit is greater than the tens place digit by 7 .

Answer

To solve the problem step by step, we will define the two-digit number and set up equations based on the information provided.

### Step 1: Define the digits
Let the two-digit number be represented as \(10x + y\), where:
- \(x\) is the digit in the tens place.
- \(y\) is the digit in the units place.

### Step 2: Set up the first equation
According to the problem, the sum of the two-digit number and the number obtained by reversing its digits is 121. The number obtained by reversing the digits is \(10y + x\). Therefore, we can write the equation:
\[ (10x + y) + (10y + x) = 121 \]
This simplifies to:
\[ 11x + 11y = 121 \]

### Step 3: Simplify the first equation
Dividing the entire equation by 11 gives us:
\[ x + y = 11 \quad \text{(Equation 1)} \]

### Step 4: Set up the second equation
The problem states that the unit place digit \(y\) is greater than the tens place digit \(x\) by 7. This can be expressed as:
\[ y = x + 7 \quad \text{(Equation 2)} \]

### Step 5: Substitute Equation 2 into Equation 1
Now, we will substitute \(y\) from Equation 2 into Equation 1:
\[ x + (x + 7) = 11 \]
This simplifies to:
\[ 2x + 7 = 11 \]

### Step 6: Solve for \(x\)
Subtract 7 from both sides:
\[ 2x = 11 - 7 \]
\[ 2x = 4 \]
Now, divide by 2:
\[ x = 2 \]

### Step 7: Find \(y\)
Now that we have \(x\), we can find \(y\) using Equation 2:
\[ y = x + 7 = 2 + 7 = 9 \]

### Step 8: Form the original number
Now we can form the original two-digit number:
\[ 10x + y = 10(2) + 9 = 20 + 9 = 29 \]

### Conclusion
The original two-digit number is **29**.