Question

In an office with 8 employees, the average rating of all employees is 30. The average rating of the top five employees is 38, and the average rating of the bottom three employees is 25. Which of the following is not possible?

• A. One of the top five employees has a rating of 50.
• B. The lowest rating among the bottom three employees is 20.
• C. The highest rating among the top five employees is 40.
• D. One of the bottom three employees has a rating of 24.

Hint:

To check feasibility, verify that the sum of individual parts matches the total and that averages align. Always double-check when combining smaller groups.

Answer 1

The Correct Option is C

Total Ratings:

Total ratings of all employees:

\[ \text{Total} = 8 \times 30 = 240 \]

Top Five and Bottom Three Totals:

Total ratings of the top five employees:

\[ \text{Top Five Total} = 5 \times 38 = 190 \]

Total ratings of the bottom three employees:

\[ \text{Bottom Three Total} = 3 \times 25 = 75 \]

Verification of Totals:

The total ratings of all employees:

\[ \text{Sum of Top Five + Bottom Three} = 190 + 75 = 265 \]

This exceeds the total ratings of 240, so adjustments are required.

Analyze the Options:

(A) One of the top five employees has a rating of 50:

If one top employee has a rating of 50, the remaining total for the top four is:

\[ 190 - 50 = 140, \quad \text{Average for four} = \frac{140}{4} = 35 \]

This is possible, as the remaining ratings align with the data.

(B) The lowest rating among the bottom three employees is 20:

If one bottom employee has a rating of 20, the remaining total for the other two is:

\[ 75 - 20 = 55, \quad \text{Average for two} = \frac{55}{2} = 27.5 \]

This is possible, as the averages match.

(C) The highest rating among the top five employees is 40:

If the highest top employee rating is 40, then the remaining total for the other four is:

\[ 190 - 40 = 150, \quad \text{Average for four} = \frac{150}{4} = 37.5 \]

This contradicts the given average of 38, making this impossible.

(D) One of the bottom three employees has a rating of 24:

If one bottom employee has a rating of 24, the remaining total for the other two is:

\[ 75 - 24 = 51, \quad \text{Average for two} = \frac{51}{2} = 25.5 \]

This is possible, as it satisfies the conditions.

Thus, the correct answer is (C).