Highmont private school polled the graduating class of 125 for the musical genres they listen to. 88% indicated at least one of the following genres: indie rock, classical, country, and electronica. Of these students, 40% responded they listen to only one of the four genres. The average number of students within each distinct group of those who listen to exactly two genres is greater than the total number of those who listen to exactly three genres. If the only lack of overlap in musical tastes was in country and electronica, what is the greatest number of students who could listen to classical, country, and indie rock?
Answer and Explanation
For this week’s question, Now That’s Quite a Playlist, I found the easiest way to organize all the information—and there is a lot of it—is to use a modified Venn diagram. We know that there is an overlap in all of the genres except country and electronica. Therefore, we can draw three circles, one each for classical, indie rock, and country, in the standard Venn diagram format. Next, we can draw a circle for electronica, overlapping it with classical and indie rock, but make sure that it doesn’t overlap with country. Now, we can start plugging in the relevant numbers.
First off, only 110 students answered the survey to begin with. 44 of those listened to one distinct genre. We don’t have to worry where those 44 go, since our focus is on the overlapping regions. What’s important is that leaves us with 66 for the overlapping regions. If we look at our diagram (and you should draw one out, if you haven’t already), there are five regions that encompass exactly two genres. The average of these has to be greater than the total. Play around with a few numbers. For instance, if we assume the average that encompasses exactly two genres is 10, then that would leave us with 16 (too many). Upping the average to 11 would give us 11 left over, which is almost what we want.
At this point, there is a little twist. We shouldn’t assume that since 11 didn’t work that we should plug in 12 (which would give us 6 left over). In other words, the average in each overlapping region of two genres doesn’t have to be an integer. Therefore, we could have 56 total in those regions that represent exactly two genres, leaving us with a total of 10 to divvy up. Remember, the only non-overlap is in country and electronica (surprise!), so each region of exactly three genres must have at least one. This leaves us with a maximum of ‘9’ that can represent classical, country, and electronica. Answer: (C).