The number obtained by interchanging the two digits of a two-digit number is lesser than the original number by 54. If the sum of the two-digit number is 10, then what is the original number?

Options

A. 28
B. 39
C. 82
D. Cannot be determined

Answer

To solve the problem, we need to find a two-digit number based on the given conditions. Let's break it down step by step.

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

### Step 2: Set Up the Equations
According to the problem:
1. The sum of the digits is 10:
\[ x + y = 10 \quad \text{(Equation 1)} \]
2. The number obtained by interchanging 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 by 9:
\[ x - y = 6 \quad \text{(Equation 2)} \]

### Step 3: Solve the System of Equations
Now we have a system of equations:
1. \(x + y = 10\) (Equation 1)
2. \(x - y = 6\) (Equation 2)

We can solve these equations simultaneously.

#### Add Equation 1 and Equation 2:
\[ (x + y) + (x - y) = 10 + 6 \]
This simplifies to:
\[ 2x = 16 \]
Dividing by 2:
\[ x = 8 \]

#### Substitute \(x\) back into Equation 1:
\[ 8 + y = 10 \]
Subtracting 8 from both sides:
\[ y = 2 \]

### Step 4: Find the Original Number
Now that we have \(x\) and \(y\):
- The original two-digit number is:
\[ 10x + y = 10(8) + 2 = 80 + 2 = 82 \]

### Conclusion
The original two-digit number is **82**.

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