The sum of the digits of a two-digits number is 12. If the number formed by reversing the digits is less than the original number by 54, find the original number.

Options

A. 29
B. 39
C. 92
D. 93

Answer

To solve the problem step-by-step, we can follow these steps:

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

### Step 2: Set Up the Equations
From the problem, we have two pieces of information:
1. The sum of the digits is 12:
\[ x + y = 12 \quad \text{(Equation 1)} \]
2. The number formed by reversing the digits is less than the original number by 54:
\[ (10x + y) - (10y + x) = 54 \]
Simplifying this gives:
\[ 10x + y - 10y - x = 54 \]
\[ 9x - 9y = 54 \]
Dividing the entire equation by 9:
\[ x - y = 6 \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations
Now we have a system of equations:
1. \( x + y = 12 \)
2. \( x - y = 6 \)

We can solve these equations simultaneously.

### Step 4: Add the Equations
Adding Equation 1 and Equation 2:
\[ (x + y) + (x - y) = 12 + 6 \]
This simplifies to:
\[ 2x = 18 \]
Dividing by 2:
\[ x = 9 \]

### Step 5: Substitute to Find \( y \)
Now substitute \( x = 9 \) back into Equation 1:
\[ 9 + y = 12 \]
Subtracting 9 from both sides:
\[ y = 3 \]

### Step 6: Form the Original Number
The original two-digit number is:
\[ 10x + y = 10(9) + 3 = 90 + 3 = 93 \]

### Step 7: Verify the Solution
1. The sum of the digits \( 9 + 3 = 12 \) (correct).
2. The reversed number is \( 39 \), and \( 93 - 39 = 54 \) (correct).

Thus, the original number is **93**.

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