Question

Amol was asked to calculate the arithmetic mean of 10 positive integers, each of which had 2 digits. By mistake, he interchanged two digits, say \( a \) and \( b \), in one of these 10 integers. As a result, his answer for the arithmetic mean was 1.8 more than what it should have been. Then \( b - a \) equals:

• A. 1
• B. 2
• C. 3
• D. None of these

Hint:

When working with averages, check how changes in individual numbers affect the overall mean.

Answer 1

The Correct Option is B

Let the original sum of the numbers be \( S \), and the sum after interchanging \( a \) and \( b \) be \( S' \). The difference in the mean is: \[ \frac{S' - S}{10} = 1.8. \] Thus, \( S' - S = 18 \). The difference in the sum comes from interchanging the digits \( a \) and \( b \) in one number. The difference in the value is \( 10b + a - (10a + b) = 9b - 9a = 9(b - a) \). Thus: \[ 9(b - a) = 18 \Rightarrow b - a = 2. \] Hence, the Correct Answer is \( b - a = 2 \).