Question

Eight employees of an organization have been rated on a scale of 1 to 50 for their performance. All ratings are integers. The overall average rating of the eight employees is 30. While the five employees with the highest ratings average 38, the five employees with the lowest ratings average 25. Which of the following, about the ratings obtained by the eight employees, is DEFINITELY FALSE?

• A. The second highest rating obtained is 38.
• B. The lowest rating obtained is 1.
• C. The third lowest rating obtained is 37.
• D. The median of the eight ratings is 37.5.
• E. The highest rating obtained is 40.

Hint:

When analyzing sets of numbers with specific conditions, break down the information into sums and averages to spot contradictions.

Answer 1

The Correct Option is C

To solve this problem, we need to analyze the given data and constraints about the ratings of the eight employees:

• Overall average rating of the eight employees is 30.
• The sum of all ratings is 30 \times 8 = 240.
• Average of the five highest ratings is 38.
• The sum of the five highest ratings is 38 \times 5 = 190.
• Average of the five lowest ratings is 25.
• The sum of the five lowest ratings is 25 \times 5 = 125.

Combining these, notice that:

The sum of all ratings calculated from the five highest and lowest ratings is 190 + 125 = 315.

However, we know the sum from the average calculation should be 240. This indicates an overlap of exactly two ratings between the highest and lowest groups (i.e., 315 - 240 = 75).

Now, let's analyze the options:

1. The second highest rating obtained is 38.
   • This is possible since one or more of the highest ratings can indeed be 38.
2. The lowest rating obtained is 1.
   • This is a broad possibility as theoretically, one of the lowest ratings can be 1.
3. The third lowest rating obtained is 37.
   • This is not possible because if the third lowest is 37, then the sum of the four lowest (including 37) would move beyond the average range stated as 25.
   • Average of lowest five cannot include high ratings such as 37, suggesting it is clearly not fitting the criteria of the lowest group.
4. The median of the eight ratings is 37.5.
   • If we consider that the median is 37.5, it can still fit within certain possible ratings allocations.
5. The highest rating obtained is 40.
   • This is possible because 40 is a reasonable high-end rating out of 1-50 range.

Therefore, the statement that is definitely false is: The third lowest rating obtained is 37.

Answer 2

Step 1: Understand the ratings distribution.
The average of the eight ratings is 30, so the total sum of all ratings is \( 30 \times 8 = 240 \).
Step 2: Apply the conditions for the highest and lowest ratings.
The highest five ratings average 38, so their total sum is \( 38 \times 5 = 190 \). The lowest five ratings average 25, so their total sum is \( 25 \times 5 = 125 \).
Step 3: Determine the false statement.
The sum of the ratings for the highest and lowest groups is \( 190 + 125 = 315 \), but the total sum is 240, so some conditions cannot hold true. Option (C) is the false statement because the third lowest rating cannot be 37.
Final Answer: \[ \boxed{\text{(C) The third lowest rating obtained is 37.}} \]