The average of a set of whole numbers is 27.2. When 20% of the elements are removed from this set, the average of the remaining numbers becomes 34. Which of the following can be the number of elements in the new set?

Introduction / Context:
This question is about how the average of a set of numbers changes when some elements are removed. It checks if you can relate total sum, average and number of elements and understand how removing low valued elements can increase the average.

Given Data / Assumptions:

• Original average of a set of whole numbers is 27.2.• Let the original number of elements be N.• Twenty percent of the elements are removed from the set.• The new average after removal is 34.• We must choose the possible number of elements in the new set from the options.

Concept / Approach:
Total sum of numbers equals average multiplied by the number of elements. Removing elements changes both the count and often the total sum. If removing some elements increases the average, the removed elements must have average less than the original average. Here, the new number of elements is 80% of N, that is 0.8N. We use the fact that the new total sum equals the new average times the new count. Given options represent the new count values.

Step-by-Step Solution:
Step 1: Original total sum S = 27.2 * N.Step 2: New number of elements = 80% of N = 0.8N.Step 3: New total sum S new = 34 * 0.8N = 27.2N.Step 4: Therefore S new = S, meaning the removed elements together contribute zero to the sum.Step 5: This implies that the removed elements must all be zero, since the numbers are whole numbers.Step 6: Thus any N where 20% of N is an integer and S is consistent is possible.Step 7: New count = 0.8N must match one of the options: 27, 35, 52 or 63.Step 8: For option 52 as new count, original N = 52 / 0.8 = 65, which is an integer.Step 9: For the other options, the corresponding original N is not an integer.

Verification / Alternative check:
If original N is 65, original sum S = 27.2 * 65 = 1768. New count = 0.8 * 65 = 52. New average with the same sum is 1768 / 52 = 34. This matches the given data, so 52 is a valid new count.

Why Other Options Are Wrong:
With new counts 27, 35 or 63, the implied original counts are 27/0.8, 35/0.8 and 63/0.8, which are not integers. That contradicts the idea that we started from a whole number of elements.

Common Pitfalls:
Candidates sometimes try to guess specific values instead of using average and total sum relationships. Others ignore the requirement that the number of elements must be a whole number both before and after removal.

Final Answer:
The number of elements in the new set can be 52.