Arvind was asked to calculate the arithmetic mean of twenty positive integers each of which had two digits. By mistake, he interchanged the two digits, say x and y, in one of

these twenty integers. As a result, his answer for the arithmetic mean was 1.35 more than what it should have been. Then (y — x) equals

Let's analyze the problem step by step.

### Step 1: Setup
Let the correct number be \( 10x + y \), where \( x \) and \( y \) are the digits of a two-digit number (with \( x \) in the tens place and \( y \) in the units place). 

Due to a mistake, Arvind interchanged the digits and used the incorrect number \( 10y + x \) in his calculation instead of the correct number \( 10x + y \). 

The average of the twenty numbers was higher than it should have been by 1.35, which means the sum of the numbers used in his incorrect calculation was higher than the sum of the correct numbers by:
\[
20 \times 1.35 = 27
\]
This indicates that the difference between the incorrect number and the correct number is 27.

### Step 2: Find the difference
The difference between the incorrect number \( 10y + x \) and the correct number \( 10x + y \) is:
\[
(10y + x) - (10x + y) = 10y + x - 10x - y = 9y - 9x = 9(y - x)
\]
So, the difference between the incorrect and correct numbers is \( 9(y - x) \). We are told that this difference is 27:
\[
9(y - x) = 27
\]

### Step 3: Solve for \( y - x \)
Now, solving for \( y - x \):
\[
y - x = \frac{27}{9} = 3
\]

Thus, \( y - x = \boxed{3} \).

Would you like a more detailed breakdown of any part of this problem?

Here are some related questions to explore:
1. What would happen if Arvind interchanged the digits of two numbers instead of just one?
2. How would the arithmetic mean change if more digits were interchanged incorrectly?
3. What are some other errors that could affect the calculation of an arithmetic mean?
4. How does the number of numbers being averaged affect the impact of such mistakes?
5. How would the calculation change if the integers involved were not restricted to two digits?

**Tip:** Whenever a mistake involves transposing digits, focus on how much each digit contributes to the difference. Often, the solution will involve simple algebra.

Explore how to calculate the arithmetic mean of twenty two-digit positive integers with a digits interchange problem. Understand the impact of digit transposition on arithmetic mean calculations with a detailed step-by-step solution.

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