The sum of the digits of a two digit number is 5. If the digits are reversed, the number is reduced by 27. Find the number.

Answer

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

### Step 1: Define the digits
Let the two-digit number be represented as:
- \( x \): the tens digit
- \( y \): the units digit

### Step 2: Set up the first equation
According to the problem, the sum of the digits is 5. Therefore, we can write:
\[ x + y = 5 \quad \text{(Equation 1)} \]

### Step 3: Set up the second equation
The problem states that if the digits are reversed, the number is reduced by 27. The original number can be expressed as \( 10x + y \) and the reversed number as \( 10y + x \). Thus, we can write:
\[ (10x + y) - (10y + x) = 27 \]
Simplifying this gives:
\[ 10x + y - 10y - x = 27 \]
\[ 9x - 9y = 27 \]
Dividing the entire equation by 9, we get:
\[ x - y = 3 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations
Now we have a system of equations:
1. \( x + y = 5 \)
2. \( x - y = 3 \)

We can add these two equations to eliminate \( y \):
\[ (x + y) + (x - y) = 5 + 3 \]
This simplifies to:
\[ 2x = 8 \]
Dividing both sides by 2 gives:
\[ x = 4 \]

### Step 5: Substitute \( x \) back to find \( y \)
Now that we have \( x \), we can substitute it back into Equation 1 to find \( y \):
\[ 4 + y = 5 \]
Subtracting 4 from both sides gives:
\[ y = 1 \]

### Step 6: Form the two-digit number
Now that we have both digits, \( x = 4 \) and \( y = 1 \), the two-digit number is:
\[ 41 \]

### Final Answer
Thus, the two-digit number is **41**.

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