Bunuel wrote:
At a software company with 21 employees, the average salary is $80,000 and the median salary is $70,000. Which of the following must be true?
I. At least one employee has a salary of more than $90,000.
II. At least one employee has a salary of less than $55,000.
III. No employee has a salary of more than $285,000.
A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III
Note that since the average salary is $80,000 and since there are 21 employees in the company, the sum of the salaries of all the employees is 21 x 80,000 = 1,680,000.
Let’s test each Roman numeral:
Roman numeral I: At least one employee has a salary of more than $90,000.
Let’s determine the minimum possible value of the highest paid employee. To keep the value of the highest paid employee at the minimum, we must maximize the salaries of the remaining employees. We also need to take into account that the median salary is $70,000; therefore, we take the salaries of the first 11 employees to be $70,000.
If the salaries of the first 11 employees are $70,000, their sum is 11 x 70,000 = 770,000. The sum of the salaries of the remaining 10 employees is 1,680,000 - 770,000 = 910,000. We actually know at this point that there is at least one employee with a salary of more than $90,000, because if the salaries of the remaining 10 employees were all less than $90,000, then the sum of their salaries would have been less than $900,000. Roman numeral I is true.
Roman numeral II: At least one employee has a salary of less than $55,000.
We actually illustrated that Roman numeral II is not necessarily true through the analysis of Roman numeral I. In our example above, the first 11 employees had salaries of $70,000 and the rest of the salaries added up to $910,000. We could say, for instance, that the remaining 10 employees have salaries of $91,000, and this shows that no employee necessarily has to have a salary that is less than $55,000.
Roman Numeral III: No employee has a salary of more than $285,000.
To test whether this Roman numeral is true, let’s calculate the maximum possible value of the highest paid employee. To achieve the maximum, we need to make the salaries of the remaining 20 employees as low as possible. For this purpose, let’s take the salaries of the 10 employees below the median to be 0 and the salaries of the next 10 employees, including the median, to be $70,000. Then, the sum of the salaries of the 20 employees besides the highest-paid employee is 0 + 10 x 70,000 = $700,000. Then, we find that the salary of the highest-paid employee can be as high as 1,680,000 - 700,000 = $980,000. This is greater than $285,000, so Roman numeral III is false.
Answer: A