Question

Last year the average (arithmetic mean) salary of the 10 employees of Company X was \$42,800. What is the average salary of the same 10 employees this year?
(1) For 8 of the 10 employees, this year's salary is 15 percent greater than last year's salary.
(2) For 2 of the 10 employees, this year's salary is the same as last year's salary.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

In problems involving percentage changes on a total sum, remember that the change in the total depends on what portion of the sum the percentage is applied to. If the split is unknown, a unique answer cannot be found.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
To find the average salary for this year, we need to calculate the total sum of salaries for this year and divide by the number of employees (10).
From the prompt, we can calculate the total sum of salaries for last year:
Total Salary (Last Year) = Average Salary \(\times\) Number of Employees
Total Salary (Last Year) = \$42,800 \(\times\) 10 = \$428,000.
Step 2: Detailed Explanation:
Analyze Statement (1): For 8 of the 10 employees, this year's salary is 15 percent greater than last year's salary.
This statement tells us about the change for 8 employees but provides no information about the remaining 2 employees. Their salaries could have increased, decreased, or stayed the same, and we don't know their initial salaries. We cannot calculate this year's total salary. Statement (1) is not sufficient.
Analyze Statement (2): For 2 of the 10 employees, this year's salary is the same as last year's salary.
This statement tells us about the change for 2 employees but provides no information about the remaining 8 employees. We cannot calculate this year's total salary. Statement (2) is not sufficient.
Analyze Both Statements Together:
We know that 8 employees received a 15% raise, and 2 employees' salaries remained the same. Let \(S_A\) be the sum of last year's salaries for the 8 employees who got a raise, and \(S_B\) be the sum of last year's salaries for the 2 employees who did not.
We know \(S_A + S_B = \$428,000\).
This year's total salary is calculated as: Total Salary (This Year) = \(1.15 \times S_A + 1 \times S_B\)
The problem is that we don't know the values of \(S_A\) and \(S_B\). The distribution of the \$428,000 between the two groups affects the final total.

Scenario 1: Assume the 8 employees with raises had low salaries, e.g., \(S_A = \$80,000\). Then \(S_B = \$428,000 - \$80,000 = \$348,000\). This Year's Total = \(1.15(\$80,000) + \$348,000 = \$92,000 + \$348,000 = \$440,000\). Average = \$44,000.
Scenario 2: Assume the 8 employees with raises had high salaries, e.g., \(S_A = \$400,000\). Then \(S_B = \$428,000 - \$400,000 = \$28,000\). This Year's Total = \(1.15(\$400,000) + \$28,000 = \$460,000 + \$28,000 = \$488,000\). Average = \$48,800.
Since we can get different values for this year's average salary, the combined statements are not sufficient.
Step 3: Final Answer:
Even with both statements, the average salary for this year cannot be uniquely determined.