Most of us have an eclectic taste in music, but does it rival some of those at the hypothetical high school below? Find out in this week’s challenge question.

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?

(A) 5

(B) 6

(C) 9

(D) 10

(E) 13

As always, check in on Thursday for the explanation!

125*0.88=110

110*0.40=44 only one genre.

110-44=66 two , three and four genre

66=a(2 genre)+b(3 genre)+c(4 genre)

a/5>b

There is no group of those who listen to exactly 4 genres, because there is no overlap in musical tastes was in country and electronica. this means c=0

a=66-b 66-b>5b b<11

There are just 2 groups of students listening 3 genres

X. classical, country, and indie rock

Y. classical, electronica, and indie rock

X+Y<11 When Y=1 then we get greatest number of X<10 X=9

Answer is C) 9

Chris, Can we expect this kind of question on GRE?

Whoknows,

Apparently, you knows :).

Out of the 20 answers or so, we received you are the only one who got it right!

So, I’d say this question has more moving parts than a GRE math question. But the subtle twists at the end, which stymied a lot of people, will definitely be waiting on the GRE. I’m guessing the GRE would have maybe made a similar question but with three non-overlapping regions. But who really knows, maybe they would have

Interestingly, I solved this with a modified Venn Diagram that made it more straightforward–at least visually–than an algebraic approach. For instance, it was a lot easier to see that there were ‘5’ regions that overlapped two genres. Still, there were a couple of twists there, of course. But maybe with this approach the question wasn’t quite as tricky.

Why wouldn’t the greatest value of X be when Y is 0?

x – group liking 2 genre and y – group liking 3 genre. There will not be a group for liking all four.

2x+3y=66 and x>5y . Solving this, we get y<66/13 which implies y can be a maximum of 5. Hence answer A.

Hey Sriram,

You definitely got close, but it looks like you missed something at the very end :).

While it is true that y can have a maximum of 5, there are two ‘y’s, i.e., regions that overlap with three. So you are really looking for 2y, which would give you a maximum of 10. Then, you’ll need to divvy up that between the two three overlapping groups, making sure to give at least one to both groups (the only non-overlapping region involves country and electronica).

That means you have a maximum of 9 who listen to classical, country, and indie rock. (C).

It’s 13.

For whoknows’ post, his/her method is right but he/she should not have included c, 4 genre, which would not exist because electronica and country would not overlap. Also, he/she should be dividing 4 instead of 5, so it should be 66=a(2 genre)+b(3 genre), and a/4>b, thus 66-b>4b, and b<13.2, making the largest possible one 13.

William,

The first part I agree with you–there are no overlapping regions of ‘4’. However, there are five possible overlapping regions of 2, hence the a/5.

It’s 13.

125*0.88-44=66 those are students with two or three genres, since no one can be in four genres for electronica and country do not overlap.

a=two genres

b=three genres

66=a+b

a/4>b

66-b>4b

5b<66

b<13.2

Love the elegant algebra–I think yours was the most precise approach so far :).

However, you want to make sure that you divide a by 5, since there are 5 overlapping groups of 2. Since you are comparing the average of these 5 groups (a/5) to the total of the the two three-overlapping groups, you want 2b. So it should look like this:

a/5 > 2b

66 = a + b

Solving for ‘b’, you’ll end up getting b < 6

Since you have 2 'b's, that gives us a maximum of 10 for the three non-overlapping groups. Each of these two groups must have at least one, leaving a maximum of 9 for classical, country, and indie rock. Hence, answer (C).

Ans E.

Our aim is to maximize overlap between classical,indie,country.

This can be done by making overlap between indie,electronica and classical zero

and overlap between indie and classical completely overlapped by country

Payal,

According to the question the only non-overlapping regions are between electronic and country :).

Hi Chis,

still not convinced 🙁

my venn diagram satisfies the condition of no overlap between electronic and country.

I don’t know how it can be interpreted as nonzero 3 way overlap between indie,electronica and classical when we do have respective overlaps between exactly two genres

Hmm…I see what you mean. That is a little vague. Had the question said, “Nobody who listens to electronica, also listens to country” then that would be more consistent with my interpretation. It seems that saying “country and electronic don’t overlap” can be interpreted as meaning “there is no overlap between those who listen to only country and electronica”. Is this what you meant?

That’s what I meant.No overlap between only country and electronica.Rest all kinds of overlaps are game.We can configure them any way we want to maximize the overlap between indie,country,classical.I wish I could upload my Venn diagram.

But now that I look at the question from another angle I see what you mean.Only overlap between country and electronica is zero.Rest all are non zero.

This one surely is a twister.

Yeah, I think that is misleading, and I’m happy you caught that :). I should have worded that more explicitly, so the meaning is unambiguous. With future teasers, I’ll be even more aware of any possible ambiguities.

Thanks for keeping me on my toes!

Answer is 10

Nobody has been able to crack this nut so far. There are definitely a few twists. Try using a modified Venn diagram.

125*0.88=110

110*0.40=44 only one genre.

110-44=66 two , three and four genre

66=a(2 genre)+b(3 genre)+c(4 genre)

a/5>b

a=66-b-c 66-b-c>5b b<11-c/6 when c=1 b<10.83 b=10

IMO correct answer is D) 10

You’re close, in fact the closest anyone has come to getting this right :). There is just one little thing that you overlooked.

125*0.88=110

110*0.40=44 only one genre.

110-44=66 two , three and four genre

66=a(2 genre)+b(3 genre)+c(4 genre)

a/5>b

There is no group of those who listen to exactly 4 genres, because there is no overlap in musical tastes was in country and electronica. this means c=0

a=66-b 66-b>5b b<11

I have still answer D) 10 :-))

–

D 10

Close, but it looks like you missed one little thing 🙂

That’s a whopper, but is the answer (B) 6 ?

It’s definitely a whopper :). So far, nobody has answered it correctly. Maybe you can give it another try :).

The answer is B) 6.

No. of students who listen to at least one genre is 125 * 0.88 = 110.

No. of students who listen to only one genre is 40% of 110 which is 49 (although it is 48.4 which should be rounded off to 48, answer choices didn’t match with this number. Which is an ethical sin for mathematicians! 😛 Pardon me if I am wrong ;))

No. of students who listen to at least two genre is 110 – 49 = 61.

No. of combinations possible to choose two genre from a pool of four is 4!/(2*2!) = 6.

But of the possible combinations to choose two genre one genre is left behind, so it is 5.

The least number of students for which the condition, average of 5 is greater than the total number of students who listen to at least three genre is 55 (average being 11).

If the average is 10, then total number of students who listen to at least three genre is 11, which fails the condition.

Therefore 61-55 is 6. Which is the answer choice B).

Hmm…I think you mixed something up at the 40% of 110 part. Try it again, because it looks like your reasoning may work out 🙂

After running the following:

(125) x (80%) x (44%) = 44 people like at least one musical genre

That leaves (125-44) = 81 people to like at least 2 or 3 genres.

Because we know (average ppl with 2 genre) > (average ppl with 3 genre).

That’s as far as I got…

so I guessed 13 since it’s the highest number (E)

Hey Henry,

I think you misread the question: it says 88% :). Give it another shot. So far nobody has gotten the question right.

It is 10

Close, but there is a little twist at the end 🙂

6

That’s not quite it :). Take another stab at it. So far nobody has gotten right.