Given the following survey results, how many people like C only?

  1. 21 people like Product A
  2. 14 people like Products A and B
  3. 12 people like Products A and C
  4. 8 people like Products A, B and C
  5. 26 people like Product B
  6. 14 people like Products B and C
  7. 39 people like Product C

1 Answer

21

Explanation:

For each product, there will be people who like it only (ex. A only), people who like it and one other (ex. A and B only) and those who like all three. Let's work through this.

I'll note that this question appears to be a more fleshed-out version of this one posted earlier in the week: https://socratic.org/questions/in-a-survey-it-was-found-that-21-people-liked-product-a-26-liked-product-b-and-3 where the total number of likes for C was 39. I'll use that figure in here.

For product A, 21 people in total like it. 14 people like A and B (and might like C), 12 like C and A (and might like B), and 8 like A, B and C. This means that we need to subtract the 8 (those who like all three) from the numbers listed for people liking two products. This will give us:

People who like A and B only #=14-8=6#

People who like A and C only #=12-8=4#

#overbrace("Likes A")^(21)-overbrace("Likes A, B only")^(6)-overbrace("Likes A, C only")^(4)-overbrace("Likes A, B, C ")^8=overbrace("Likes A only")^(color(red)3)#

For Product B, 26 people like it. 14 like A and B (and might like C), 14 people like B and C (and might like A), and 8 like all three. We can do the same sort of math for this that we did for A:

People who like A and B only #=14-8=6#

People who like B and C only #=14-8=6#

#overbrace("Likes B")^(26)-overbrace("Likes A, B only")^(6)-overbrace("Likes B, C only")^(6)-overbrace("Likes A, B, C ")^8=overbrace("Likes B only")^(color(red)6)#

We can now do C. 39 people like C. 4 people like C and A only. 6 people like B and C only. And 8 like A, B, and C. Therefore:

#overbrace("Likes C")^(39)-overbrace("Likes A, C only")^(4)-overbrace("Likes B, C only")^(6)-overbrace("Likes A, B, C ")^8=overbrace("Likes C only")^(color(red)21)#

~~~~~

I want to note that when I first started working with A that if we take the numbers given as "only", we'd end up with something that made no sense:

A total likes = 21

A, B likes = 14
A, C likes = 12

which puts us at 26 likes, which is more than the 21 given. And so I realized we needed to subtract out from the A, B likes, for instance, those people who also liked C.