The sum of the digits of a two - digit number is `15`. The number obtained by interchanging the digits exceeds the given number by `9`. The number is

Options

A. `96`
B. `69`
C. `87`
D. `78`

Answer

To solve the problem step by step, we will define the two-digit number and set up equations based on the conditions given in the question.

### Step 1: Define the digits
Let:
- \( x \) = the digit in the tens place
- \( y \) = the digit in the units place

The two-digit number can be expressed as:
\[ 10x + y \]

### Step 2: Set up the first equation
According to the problem, the sum of the digits is 15. Therefore, we can write the first equation as:
\[ x + y = 15 \quad \text{(Equation 1)} \]

### Step 3: Set up the second equation
The problem states that the number obtained by interchanging the digits exceeds the original number by 9. The number after interchanging the digits is:
\[ 10y + x \]
According to the condition, we can set up the second equation:
\[ 10y + x = (10x + y) + 9 \]
Simplifying this gives:
\[ 10y + x = 10x + y + 9 \]
Rearranging the equation results in:
\[ 10y - y + x - 10x = 9 \]
This simplifies to:
\[ 9y - 9x = 9 \]
Dividing the entire equation by 9 gives us:
\[ y - x = 1 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \( x + y = 15 \) (Equation 1)
2. \( y - x = 1 \) (Equation 2)

We can solve these equations simultaneously. From Equation 2, we can express \( y \) in terms of \( x \):
\[ y = x + 1 \]

### Step 5: Substitute \( y \) in Equation 1
Now substitute \( y \) in Equation 1:
\[ x + (x + 1) = 15 \]
This simplifies to:
\[ 2x + 1 = 15 \]
Subtracting 1 from both sides gives:
\[ 2x = 14 \]
Dividing by 2 results in:
\[ x = 7 \]

### Step 6: Find \( y \)
Now substitute \( x = 7 \) back into the equation for \( y \):
\[ y = x + 1 = 7 + 1 = 8 \]

### Step 7: Form the original number
Now that we have both digits, \( x = 7 \) and \( y = 8 \), we can form the original two-digit number:
\[ 10x + y = 10(7) + 8 = 70 + 8 = 78 \]

### Conclusion
The two-digit number is:
\[ \boxed{78} \]