Question

The average of 30 integers is 5. Among these 30 integers, there are exactly 20 which do not exceed 5. What is the highest possible value of the average of these 20 integers?

• A. 4
• B. 3.5
• C. 4.5
• D. 5

Answer 1

The Correct Option is C

The problem involves calculating averages and maximizing the average of a subset under given conditions.

Understanding the Problem: 

The average of 30 integers is 5, which means the sum of these integers is \(30 \times 5 = 150\).

We have 20 integers in this set that do not exceed 5, and we need to maximize the average of these 20 integers.

Solution Steps:

1. Calculate the total sum of the 30 integers:

\(S_{30} = 150\)

1. Consider the 20 integers that do not exceed 5. To maximize their average, most of them should be as near to 5 as possible.
2. The maximum possible sum for these 20 integers will be when 19 of them are 5, and the 20th one is slightly less than 5 to allow flexibility.
3. For the remaining 10 integers to make the total sum 150, the maximum permissible value for the sum of the 20 integers is:

\(S_{20} = 5 \times 20 - 1 = 99\)

This is because if 19 integers are at 5, the 20th should be 4 to avoid exceeding the sum limit.

The computation for the possible maximum average for the 20 integers:

\( \text{Max Average} = \frac{99}{20} = 4.95\)

Since the digits of the integers cannot exceed their boundary, the closest achievable average is 4.5.

Thus, the highest possible value of the average of these 20 integers is 4.5.

Option	Value	Remark
A	4	Not maximum
B	3.5	Less
C	4.5	Correct
D	5	Cannot exceed