The sum of digits of a two digit number is 11. If the digit of ten's place is increased by 5 and the digit at unit's place is decreased by 5 the digit of the number are found to be reversed. Find the origional number.

Answer

To solve the problem step by step, we will define the two-digit number using its digits and then set up equations based on the conditions provided.

### Step 1: Define the digits
Let the two-digit number be represented as \( 10x + y \), where:
- \( x \) is the digit in the tens place
- \( y \) is the digit in the units place

### Step 2: Set up the first equation
According to the problem, the sum of the digits is 11. Therefore, we can write:
\[ x + y = 11 \quad \text{(1)} \]

### Step 3: Set up the second equation
The problem states that if the digit in the tens place is increased by 5 and the digit in the units place is decreased by 5, the digits are reversed. This gives us the equation:
\[ 10(x + 5) + (y - 5) = 10y + x \]

### Step 4: Simplify the second equation
Expanding the left side:
\[ 10x + 50 + y - 5 = 10y + x \]
This simplifies to:
\[ 10x + y + 45 = 10y + x \]

### Step 5: Rearranging the equation
Now, rearranging the equation gives us:
\[ 10x - x + y - 10y + 45 = 0 \]
Which simplifies to:
\[ 9x - 9y + 45 = 0 \]
Dividing through by 9:
\[ x - y + 5 = 0 \quad \text{(2)} \]

### Step 6: Solve the system of equations
Now we have a system of equations:
1. \( x + y = 11 \) (from step 2)
2. \( x - y + 5 = 0 \) (from step 5)

From equation (2), we can express \( x \) in terms of \( y \):
\[ x = y - 5 \quad \text{(3)} \]

### Step 7: Substitute equation (3) into equation (1)
Substituting equation (3) into equation (1):
\[ (y - 5) + y = 11 \]
This simplifies to:
\[ 2y - 5 = 11 \]
Adding 5 to both sides:
\[ 2y = 16 \]
Dividing by 2:
\[ y = 8 \]

### Step 8: Find \( x \)
Now, substituting \( y = 8 \) back into equation (1):
\[ x + 8 = 11 \]
Thus,
\[ x = 3 \]

### Step 9: Write the original number
The original two-digit number is:
\[ 10x + y = 10(3) + 8 = 30 + 8 = 38 \]

### Final Answer
The original number is **38**.