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

Introduction / Context:
This question uses basic average and total relationships to identify a single missing or excluded number. It is a very common pattern in aptitude tests, where the overall average and a new average after removing an item are given, and you need to recover the removed item from these two pieces of information.

Given Data / Assumptions:

• There are initially 5 numbers.

• Average of all 5 numbers = 27.

• After removing one number, the average of the remaining 4 numbers = 25.

• All numbers are real values and the arithmetic mean is used.

Concept / Approach:
The average of a set is the total sum divided by the number of elements. From this, we can find the total sum for the 5 numbers and the total sum for the remaining 4 numbers. The excluded number is exactly the difference between these two sums. So the plan is: compute total for 5, compute total for 4, and subtract to find the excluded number.

Step-by-Step Solution:
Total of 5 numbers = 27 * 5 = 135. Total of remaining 4 numbers = 25 * 4 = 100. Excluded number = total of 5 numbers − total of 4 numbers. Excluded number = 135 − 100 = 35.

Verification / Alternative check:
Assume the numbers are 4 unknowns plus 35. Their total is 100 + 35 = 135. The average with all 5 numbers is 135 / 5 = 27, which matches the original average. When you remove 35, you are left with the 4 numbers totalling 100, giving an average of 100 / 4 = 25, which matches the given new average. This confirms that 35 is indeed the excluded number.

Why Other Options Are Wrong:
If the excluded number were 25, the remaining total would be 135 − 25 = 110 and the new average for 4 numbers would be 110 / 4 = 27.5, not 25. Similarly, excluding 45 would leave 90 and give an average of 22.5, and excluding 55 would leave 80 and yield an average of 20. None of these match the required average of 25, so those options are inconsistent with the given data.

Common Pitfalls:
Some students try to average the averages directly or attempt to distribute the change in average without computing totals first, which can be confusing. Others accidentally divide the total by the wrong number of elements. To avoid these errors, always convert each average into a total, perform subtraction or addition as needed, and then recheck by recomputing the average.

Final Answer:
The excluded number is 35.