A student finds the average of 10, 2 - digit numbers. If the digits of one of the numbers is interchanged, the average Increases by 3.6. The difference between the digits of the 2-digit numbers is:

Options

A. 4
B. 3
C. 2
D. 5

Answer

To solve the problem step by step, we need to analyze the information given and formulate equations based on that.

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

**Hint:** Remember that in a two-digit number, the first digit represents the tens place and the second digit represents the units place.

### Step 2: Interchange the digits
When the digits are interchanged, the new number becomes \( 10y + x \).

**Hint:** Think about how the place value changes when digits are swapped.

### Step 3: Calculate the change in average
The average of the original 10 numbers is given by:
\[ \text{Average} = \frac{S}{10} \]
where \( S \) is the sum of the 10 numbers.

When one number is changed from \( 10x + y \) to \( 10y + x \), the new sum becomes:
\[ S' = S - (10x + y) + (10y + x) = S - 10x - y + 10y + x = S - 9x + 9y \]

The new average becomes:
\[ \text{New Average} = \frac{S'}{10} = \frac{S - 9x + 9y}{10} \]

### Step 4: Set up the equation for the change in average
According to the problem, the average increases by 3.6:
\[ \frac{S - 9x + 9y}{10} = \frac{S}{10} + 3.6 \]

Multiplying through by 10 to eliminate the denominator gives:
\[ S - 9x + 9y = S + 36 \]

### Step 5: Simplify the equation
By simplifying the equation, we get:
\[ -9x + 9y = 36 \]
Dividing through by 9:
\[ -y + x = 4 \quad \text{or} \quad x - y = 4 \]

**Hint:** This equation tells us the difference between the tens digit and the units digit.

### Step 6: Conclusion
The difference between the digits \( x \) and \( y \) is:
\[ x - y = 4 \]

Thus, the difference between the digits of the two-digit number is **4**.

**Final Answer:** The difference between the digits of the two-digit number is **4**.