06/18/2020 (2011: Countdown Round, Problem 4)

Q: Laura will have seven test scores at the end of the semester. Each of the scores is an integer less than or equal to 100. The first five scores are 80, 83, 87, 94, and 96. How many different values are possible for the median of Laura's seven scores after she takes the last two tests?

A: 15

So say that both of the scores are less than 80, that means that the median is 83. If both the scores are greater than 96, that means the median is 94. That means 80 ≤  median ≤  94, but every score is an integer, so the median is only integral numbers between 80 and 94 inclusive, meaning there are 1 + 94 - 80 = 15 possible medians.