Each employee of a certain task force is either a manager or a director. What percent of the employees on...

Understanding the Question

We need to find: What percent of the employees are directors?

Let's clarify what we're looking for:

• The task force has only managers and directors (no other positions)
• We want: \(\frac{\mathrm{Number\ of\ Directors}}{\mathrm{Total\ Employees}} \times 100\%\)
• To be sufficient, we need to determine this exact percentage value

From the approach analysis, we discovered the Distance Principle: In weighted average problems, group sizes are inversely proportional to their distances from the average. This powerful insight will guide our analysis.

Analyzing Statement 1

Statement 1 tells us: The average manager salary is \(\$5,000\) less than the average salary of all employees.

This means managers "pull down" the overall average by \(\$5,000\). Think of it like a seesaw—managers are on one side, pulling down. But here's the crucial missing piece: we don't know how much directors "pull up" the average on the other side.

Let's visualize with concrete scenarios:

• Scenario A: If directors earn \(\$1,000\) more than average → We'd need many managers (pulling down by \(\$5,000\) each) to balance just a few directors (pulling up by only \(\$1,000\) each)
• Scenario B: If directors earn \(\$50,000\) more than average → We'd need fewer managers to balance these "heavier" directors

Since different director salaries lead to different percentages of directors, Statement 1 alone is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: The average director salary is \(\$15,000\) greater than the average salary of all employees.

This time, directors "pull up" the overall average by \(\$15,000\). But again, we're missing the other side of the seesaw—we don't know how much managers pull down.

Testing different scenarios:

• Scenario A: If managers earn \(\$1,000\) less than average → We'd need very few directors (each pulling up by \(\$15,000\)) to balance many managers (each pulling down by only \(\$1,000\))
• Scenario B: If managers earn \(\$30,000\) less than average → We'd need many more directors to overcome the strong downward pull from managers

Different manager salaries yield different percentages of directors, so Statement 2 alone is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Statements

Now we have both pieces of the puzzle:

• Managers: \(\$5,000\) below the overall average (pulling down)
• Directors: \(\$15,000\) above the overall average (pulling up)

Here's where the Distance Principle becomes powerful. For the weighted average to balance:

• The total "pull down" must equal the total "pull up"
• \(\mathrm{Managers} \times \$5,000 = \mathrm{Directors} \times \$15,000\)

Let's solve this balance equation:

• \(\mathrm{Managers} \times \$5,000 = \mathrm{Directors} \times \$15,000\)
• \(\mathrm{Managers} = 3 \times \mathrm{Directors}\)

This tells us that for every 1 director, there must be exactly 3 managers.

Therefore:

• In any group of 4 employees: 3 are managers, 1 is a director
• Directors = 1 out of 4 = \(25\%\)

The statements together are sufficient to determine that exactly \(25\%\) of employees are directors.

[STOP - Sufficient!] This eliminates choice E.

The Answer: C

Both statements together give us the exact distances from the average for each group. Using the Distance Principle, we can determine that directors make up exactly \(25\%\) of the task force.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."