The difference between a two-digit number and the number obtained by interchanging the digits is 27. What is the differences between the digits of the number ?

Options

A. 3
B. 5
C. 6
D. Can’t be determined

Answer

To solve the problem, we need to find the difference between the digits of a two-digit number given that the difference between the number and the number obtained by interchanging its digits is 27.

### Step-by-Step Solution:

1. **Define the Digits**:
Let the two-digit number be represented as \(10x + y\), where \(x\) is the tens digit and \(y\) is the units digit.

2. **Interchange the Digits**:
When the digits are interchanged, the number becomes \(10y + x\).

3. **Set Up the Equation**:
According to the problem, the difference between the two numbers is 27. We can write this as:
\[ (10x + y) - (10y + x) = 27 \]

4. **Simplify the Equation**:
Simplifying the left-hand side:
\[ 10x + y - 10y - x = 27 \]
This simplifies to:
\[ 9x - 9y = 27 \]

5. **Factor Out the Common Term**:
Dividing the entire equation by 9 gives:
\[ x - y = 3 \]

6. **Consider the Reverse Case**:
If we consider the case where the larger digit is \(y\) and the smaller digit is \(x\), we set up the equation:
\[ (10y + x) - (10x + y) = 27 \]
Simplifying this gives:
\[ 9y - 9x = 27 \]
Dividing by 9 results in:
\[ y - x = 3 \]

7. **Interpret the Results**:
We have two equations from the two cases:
- From the first case: \(x - y = 3\)
- From the second case: \(y - x = 3\)

Both equations indicate that the difference between the digits \(x\) and \(y\) is 3.

8. **Conclusion**:
The difference between the digits of the number is \(3\).