A number consists of two digits. The sum of the digits is 11, reversing the digits, the number decreases by 45, the number is :

Options

A. 38
B. 65
C. 74
D. 83

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 digit in the tens place
- \( y \): the digit in the units place

### Step 2: Set up the equations
According to the problem:
1. The sum of the digits is 11:
\[ x + y = 11 \quad \text{(Equation 1)} \]

2. Reversing the digits decreases the number by 45:
The original number can be expressed as \( 10x + y \).
The number after reversing the digits is \( 10y + x \).
Therefore, we can set up the equation:
\[ 10x + y - 45 = 10y + x \]

### Step 3: Simplify the second equation
Rearranging the second equation:
\[ 10x + y - x - 10y = 45 \]
This simplifies to:
\[ 9x - 9y = 45 \]
Dividing the entire equation by 9 gives us:
\[ x - y = 5 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations
Now we have two equations:
1. \( x + y = 11 \)
2. \( x - y = 5 \)

We can add these two equations to eliminate \( y \):
\[ (x + y) + (x - y) = 11 + 5 \]
This simplifies to:
\[ 2x = 16 \]
Dividing by 2 gives:
\[ x = 8 \]

### Step 5: Find \( y \)
Now substitute \( x = 8 \) back into Equation 1:
\[ 8 + y = 11 \]
Solving for \( y \):
\[ y = 11 - 8 = 3 \]

### Step 6: Form the original number
The original two-digit number is:
\[ 10x + y = 10(8) + 3 = 80 + 3 = 83 \]

### Final Answer
The original number is **83**.

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