50 people responded to a survey carried out to find the likes and disl

The problem can be visualized as a venn diagram. Each flavor is represented by a circle - if someone is in it then they like that flavor, otherwise they don't.

For part 2, let me show an example of why 0 people can like only Mango and Vanilla:



- The total is 10 + 4 + 4 + 8 + 24 + 0 + 0 = 50.
- Mango has 10 + 0 + 24 + 4 = 38 likes.
- Vanilla has 0 + 0 + 24 + 8 = 32 likes.
- Strawberry has 4 + 8 + 24 + 4 = 40 likes.

I hope that helped!

Thank you so much for the detailed explanation Bismuth83. Yeah, the whole explanation does make sense and now it's very clear to me. Thank you!

Cannot understand the “only” reasoning behind the data........

Liking only two flavors means that for all other flavors, the person must dislike them.

Let me know if there is anything else I can help clarify!

Just think of the most extreme situation.
Question 1: The most extreme situation is that 10 people who dislike strawberry all like mango AND vanilla.
Question 2: The most extreme situation is that 40 people who like strawberry also like mango AND vanilla. And for the rest 10 people who dislike strawberry, they either like mango or vanilla but not both.

Hi Bismuth83,

is there a way of finding out the minimum number of people that like Mango and Vanilla purely through the inequalities you derived? or should I always construct the Venn diagram to check for these type of questions?

i didnt understand the approach of the this question could some one explain with clarity

While I do think it is possible to construct answers to these 3-set Venn problems with inequalities, I think that is generally a dangerous game as the inequalities can change slightly depending on how the information is presented. Bismuth83's visual approach is far an away the one that I would choose as it is applicable in ALL 3-set Venn cases regardless of how the wording is tweaked!
HOWEVER, this problem doesn't have to be visualized as a 3-Set Venn problem as there is no information on the "cross-likes" of anything (X people liked exactly 2 flavors, or Y people liked strawberry only, etc). This question can actually collapse into a 2-Set Venn, in which case, I'd use the double set matrix. Here's what I mean.

The question text is collapsing everything into two groups: strawberry and Mango+Vanilla (the people who like BOTH of these have become one group).

Side Note on the language of BOTH (since another poster asked about it). Since the language says "only mango and vanilla" that combines mango+vanilla and implies that someone only likes them both as opposed to also liking strawberry. This is subtly different from saying the people who like "only mango or only vanilla." In that case, you'd be adding together the people who like only mango (and not strawberry or vanilla) with the people who like only vanilla (but not strawberry or mango). Okay - back to the solution!

We need to be a bit careful here, since you can't just add the Mango and Vanilla as there is definitely overlap there (we just don't know how much). For example, these groups can't be completely distinct, as there can't be 38 people who liked Mango but not vanilla and 32 people who liked vanilla and not mango because there aren't 38+32=70 people. There is an overlap of at least 20 people (so at least 20 people like both Mango+vanilla, while no more than 32 can like both (since only 32 people like vanilla).

So here is what we know, and we can map this on a double-set venn.
- 40 people like strawberry
- 10 people do not like strawberry
- between 20-32 people like both mango+vanilla

Because this is a Max/Min problem, testing the answer choices is a good move, so we are trying to figure out if the max of the answer choices (10) and the min of the answer choices (0) is possible.


In both cases, I chose the minimum in the 20-32 range to test first (20), and in both cases, the math worked (all rows and columns summed correctly).



If it hadn't I would have tested the higher end of 32 to see if it was possible in that scenario. In this case, everywhere in the range makes 0 and 10 possible for the total number of people who said they ONLY liked Mango and Vanilla (meaning they liked Mango and Vanilla but did not like Strawberry).

Hope this possible variation helps! For me, the double-set matrix makes it easier to think about which numbers I'm going to test as I only have to make up 1-2 numbers and then compute the rest (rather than coming up with numbers for all of the spaces in a 3-set venn)!


Whit

Liked only Mango and Vanilla means 'Liked Mango (38),' 'Liked Vanilla (32)' but disliked Strawberry (10).

At most how many people can be common to all three? Only 10 because of the 50 people only 10 disliked strawberry. So at most 10 can be common to all three. Hence select 10 for first blank.

At least how many people must be common to all three? We have 50 people. No one needs to be common between 10 of disliked strawberry and 32 of liked vanilla. They could easily be different people. So select 0 for second blank.

Maximum Only Mango and Vanilla
Question: What's the maximum number who can like Mango AND Vanilla but NOT Strawberry?

50 people total.
40 like Strawberry -> 10 do not like Strawberry
Only these 10 can be in the group that likes both Mango & Vanilla but not Strawberry
Since 38 like Mango and 32 like Vanilla, there's room for overlap, but the max is limited by those not in Strawberry group.
Answer: Maximum = 10


Minimum number who could have liked only Mango and Vanilla
To minimize the number who liked only Mango and Vanilla, we maximize overlap with Strawberry.
Assume all 32 people who like both Mango and Vanilla also like Strawberry.
So, M∩V but not S = 0.
Minimum = 0