A 2-digit number is 3 times the sum of its digits. If 45 is added to the number, its digits are Interchanged. The sum of digits of the number is

Options

A. 11
B. 9
C. 7
D. 5

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 first condition states that the two-digit number is three times the sum of its digits:
\[ 10x + y = 3(x + y) \]
Simplifying this gives:
\[ 10x + y = 3x + 3y \]
Rearranging the equation:
\[ 10x - 3x = 3y - y \]
This simplifies to:
\[ 7x = 2y \quad \text{(Equation 1)} \]

### Step 2: Use the second condition.

2. The second condition states that if 45 is added to the number, the digits are interchanged:
\[ 10x + y + 45 = 10y + x \]
Rearranging this gives:
\[ 10x + y + 45 - x = 10y \]
Simplifying:
\[ 9x + 45 = 9y \]
Rearranging gives:
\[ 9x - 9y = -45 \]
Dividing through by 9:
\[ x - y = -5 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations.

Now we have two equations:
1. \( 7x = 2y \)
2. \( x - y = -5 \)

From Equation 2, we can express \( x \) in terms of \( y \):
\[ x = y - 5 \]

### Step 4: Substitute into the first equation.

Substituting \( x = y - 5 \) into Equation 1:
\[ 7(y - 5) = 2y \]
Expanding this gives:
\[ 7y - 35 = 2y \]
Rearranging gives:
\[ 7y - 2y = 35 \]
This simplifies to:
\[ 5y = 35 \]
Thus:
\[ y = 7 \]

### Step 5: Find \( x \).

Substituting \( y = 7 \) back into \( x = y - 5 \):
\[ x = 7 - 5 = 2 \]

### Step 6: Calculate the sum of the digits.

Now we have \( x = 2 \) and \( y = 7 \). The sum of the digits is:
\[ x + y = 2 + 7 = 9 \]

### Final Answer:
The sum of the digits of the number is **9**.

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