A student finds the average of ten 2-digit numbers. While copying numbers, by mistake, he writes one number with its digits interchanged. As a result his answer is 1.8 less than the correct answer. The difference of the digits of the number, in which he made mistake, is :

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the problem, we need to find the difference of the digits of the two-digit number that was mistakenly written with its digits interchanged. Let's denote the two-digit number as \( 10a + b \), where \( a \) is the tens digit and \( b \) is the units digit.

### Step 1: Understand the average calculation
The average of ten numbers is calculated by dividing the sum of the numbers by the total count of numbers. In this case, the average of the correct numbers is:
\[ \text{Correct Average} = \frac{S}{10} \]
where \( S \) is the sum of the ten numbers.

### Step 2: Identify the incorrect number
When the student interchanges the digits of one number, the number becomes \( 10b + a \). The sum of the numbers with the mistake becomes:
\[ S' = S - (10a + b) + (10b + a) = S - 9a + 9b \]

### Step 3: Calculate the incorrect average
The average with the mistake is:
\[ \text{Incorrect Average} = \frac{S'}{10} = \frac{S - 9a + 9b}{10} \]

### Step 4: Set up the equation based on the difference in averages
According to the problem, the incorrect average is 1.8 less than the correct average:
\[ \frac{S - 9a + 9b}{10} = \frac{S}{10} - 1.8 \]

### Step 5: Clear the fractions
Multiply both sides by 10 to eliminate the denominators:
\[ S - 9a + 9b = S - 18 \]

### Step 6: Simplify the equation
Subtract \( S \) from both sides:
\[ -9a + 9b = -18 \]
Now, divide the entire equation by -9:
\[ a - b = 2 \]

### Step 7: Find the difference of the digits
The difference of the digits \( a \) and \( b \) is:
\[ |a - b| = 2 \]

### Final Answer
The difference of the digits of the number in which he made the mistake is **2**.

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