The sum of the digits of a 2 digit number is 9. When 27 is added to the number the digits get interchanged. Find the number.
A. 45
B. 36
C. 18
D. 27

Options

A. C
B. D
C. B
D. A

Answer

To solve the problem, we need to find a two-digit number where the sum of its digits is 9, and when 27 is added to the number, the digits are interchanged. Let's denote the two-digit number as \(10Y + X\), where \(Y\) is the tens digit and \(X\) is the units digit.

### Step 1: Set up the equations
1. The sum of the digits is given as:
\[ X + Y = 9 \quad \text{(Equation 1)} \]

2. When 27 is added to the number, the digits get interchanged:
\[ 10Y + X + 27 = 10X + Y \quad \text{(Equation 2)} \]

### Step 2: Rearrange Equation 2
Rearranging Equation 2 gives:
\[ 10Y + X + 27 = 10X + Y \]
Subtract \(Y\) and \(X\) from both sides:
\[ 10Y - Y + X - X + 27 = 10X - X \]
This simplifies to:
\[ 9Y - 9X + 27 = 0 \]
Or:
\[ 9Y - 9X = -27 \]
Dividing the entire equation by 9:
\[ Y - X = -3 \quad \text{(Equation 3)} \]

### Step 3: Solve the system of equations
Now we have two equations:
1. \(X + Y = 9\) (Equation 1)
2. \(Y - X = -3\) (Equation 3)

We can solve these equations simultaneously. From Equation 3, we can express \(Y\) in terms of \(X\):
\[ Y = X - 3 \]

### Step 4: Substitute \(Y\) in Equation 1
Substituting \(Y\) in Equation 1:
\[ X + (X - 3) = 9 \]
This simplifies to:
\[ 2X - 3 = 9 \]
Adding 3 to both sides:
\[ 2X = 12 \]
Dividing by 2:
\[ X = 6 \]

### Step 5: Find \(Y\)
Now substituting \(X = 6\) back into Equation 1 to find \(Y\):
\[ 6 + Y = 9 \]
So:
\[ Y = 3 \]

### Step 6: Find the two-digit number
Now we can find the two-digit number:
\[ \text{Number} = 10Y + X = 10(3) + 6 = 30 + 6 = 36 \]

### Conclusion
Thus, the two-digit number is **36**.