The average of five numbers is 27. If one number is excluded, the average of the remaining four numbers becomes 25. What is the value of the excluded number?

Introduction:
This question checks your understanding of how removing one element from a set changes the average. You are given the average of five numbers and the average of four numbers obtained after excluding one number. You must find the excluded number.

Given Data / Assumptions:
Average of five numbers = 27. Average of four numbers (after excluding one number) = 25. Let the excluded number be x.

Concept / Approach:
We use the relation between average and total: Total of five numbers = 5 * 27. Total of remaining four numbers = 4 * 25. The excluded number is the difference between these two totals.

Step-by-Step Solution:
Step 1: Total sum of the original five numbers. Total (5 numbers) = 5 * 27 = 135. Step 2: Total sum of the remaining four numbers. Total (4 numbers) = 4 * 25 = 100. Step 3: Find the excluded number. Excluded number x = 135 - 100 = 35.

Verification / Alternative Check:
Imagine the original five numbers sum to 135. Removing 35 leaves 100, and 100 / 4 = 25, exactly matching the given new average. Thus, the calculation is correct.

Why Other Options Are Wrong:
25: This would lead to a total of 110 for the remaining four numbers, which is inconsistent with the given average of 25. 45 or 55: These would produce too large a difference between the two totals and would not result in the correct averages. 30: This does not give the correct new total of 100 when subtracted from 135.

Common Pitfalls:
Some students mistakenly subtract the two averages (27 - 25) and think the result is the excluded number or part of it. The correct approach is to convert both averages to total sums first, then subtract to get the exact excluded value.

Final Answer:
The excluded number is 35.