The average of 20 real numbers is zero. Out of these 20 numbers, what is the maximum possible number of terms that can be strictly greater than zero?

Introduction / Context:
This question checks your conceptual understanding of averages and how positive and negative numbers can combine to give an average of zero. You are asked to determine the maximum possible count of numbers that can be greater than zero while the overall average of 20 numbers remains zero. It is a logical reasoning question based on the idea of balancing positive and negative values rather than heavy calculations.

Given Data / Assumptions:

• Total number of numbers = 20.
• Average of the 20 numbers = 0.
• Numbers can be positive, negative, or zero.
• We want to maximize how many numbers are > 0.

Concept / Approach:
If the average of 20 numbers is zero, the sum of all 20 numbers must be zero. That is: sum of all 20 numbers = 0. To have many positive numbers, the remaining numbers must be sufficiently negative (or zero) to bring the total sum back to zero. We want to see how many numbers can be strictly positive while still allowing the sum to remain zero.

Step-by-Step Solution:
Step 1: Let k be the number of positive numbers. Step 2: Then the remaining 20 - k numbers are either negative or zero. Step 3: For the sum of all 20 numbers to be zero, the sum of the positive numbers must be balanced by the sum of the negatives. Step 4: To maximize k, we make exactly one number negative and all the others positive, with no zeros. So k can be 19, and the remaining 1 number is negative. Step 5: For example, suppose 19 numbers are +1 each, and the remaining number is -19. Then the sum is 19 * 1 + (-19) = 19 - 19 = 0 and the average is 0 / 20 = 0. Step 6: It is impossible to have all 20 numbers greater than zero because then the sum would be strictly positive and the average would also be positive, not zero.
Verification / Alternative check:
We have demonstrated one concrete example with 19 positive numbers and 1 negative number that yields an average of zero. This proves that 19 positive numbers are possible. We also argued that 20 positive numbers are impossible, so 19 is the maximum.
Why Other Options Are Wrong:
Option A (0): It is not necessary that all numbers be non-positive; we clearly constructed a case with many positive numbers. Option B (1): You can certainly have more than one positive number; our example has 19. Option C (19): This is the correct numeric count, but it is not listed as a separate option in the original set, so within the given options, the correct choice is "None of these". Option E (20): Having all 20 numbers positive would make the sum and thus the average strictly positive, contradicting the given average of zero.
Common Pitfalls:
Some students incorrectly assume that the numbers must be symmetrically distributed, such as 10 positive and 10 negative, which is not required. Others may overlook the possibility of using a single large negative number to balance many small positive numbers.
Final Answer:
The maximum possible number of positive numbers is 19, which is not explicitly listed, so the correct option is None of these.