The company at which Mark is employed has 80 employees, each of whom has a different salary. Mark’s salary of $43,700 is the second-highest salary in the first quartile of the 80 salaries. If the company were to hire 8 new employees at salaries that are less than the lowest of the 80 salaries, what would Mark’s salary be with respect to the quartiles of the 88 salaries at the company, assuming no other changes in the salaries?
The answer is
The fifth-lowest salary in the second quartile.
This doesn't make sense.
Imagine the employees ranked in order from highest salary to lowest salary. If we add 8 new employees who have salaries lower than the pre-existing 80, then we append 8 salaries to the bottom of the list. How does Mark's position in the queue change? Shouldn't his position stay the same?
The quartiles range from lowest to highest salaries (e.g., see Wikipedia's Quartile). Thus, the first quartile is the lowest quarter range of the salaries. As there's $\frac{80}{4} = 20$ salaries per quartile, this means Mark's salary occurs at position $62$, i.e., $19$ from the bottom. After adding $8$ more salaries at the bottom of the ranking, each quartile would now comprise $\frac{88}{4} = 22$ salaries. Thus, Mark's salary is now $27$ from the bottom, so it's in the second quartile, being the fifth-lowest salary (i.e., since $27 - 22 = 5$) in that group.