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

Options

A. `78`
B. `87`
C. `88`
D. `77`

Answer

To solve the problem step by step, let's denote the two-digit number as \(10x + y\), where \(x\) is the digit in the tens place and \(y\) is the digit in the units place.

### Step 1: Set up the equations based on the problem statement.
1. The sum of the digits is 15:
\[ x + y = 15 \quad \text{(Equation 1)} \]
2. The number obtained by interchanging the digits exceeds the original number by 9:
\[ 10y + x = 10x + y + 9 \quad \text{(Equation 2)} \]

### Step 2: Simplify Equation 2.
Rearranging Equation 2:
\[ 10y + x - y - 10x = 9 \]
This simplifies to:
\[ 9y - 9x = 9 \]
Dividing through by 9 gives:
\[ y - x = 1 \quad \text{(Equation 3)} \]

### Step 3: Solve the system of equations.
Now we have two equations:
1. \(x + y = 15\) (Equation 1)
2. \(y - x = 1\) (Equation 3)

From Equation 3, we can express \(y\) in terms of \(x\):
\[ y = x + 1 \]

### Step 4: Substitute \(y\) in Equation 1.
Substituting \(y\) in Equation 1:
\[ x + (x + 1) = 15 \]
This simplifies to:
\[ 2x + 1 = 15 \]
Subtracting 1 from both sides:
\[ 2x = 14 \]
Dividing by 2:
\[ x = 7 \]

### Step 5: Find \(y\).
Now substitute \(x = 7\) back into Equation 3 to find \(y\):
\[ y = 7 + 1 = 8 \]

### Step 6: Write the original number.
The original two-digit number is:
\[ 10x + y = 10(7) + 8 = 70 + 8 = 78 \]

### Final Answer:
The original number is **78**.

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