According to a survey, at least 70% of people like apples, at least 75

Question

According to a survey, at least 70% of people like apples, at least 75% like bananas and at least 80% like cherries. What is the minimum percentage of people who like all three?

A. 15% 
B. 20% 
C. 25% 
D. 30% 
E. 35%

Bunuel · Accepted Answer

First of all, let's simplify the question: say there are 100 people. So, we have that at least 70 people like apples, at least 75 like bananas and at least 80 like cherries. Since we want to minimize the group which likes all three, then let's minimize the groups which like each fruit:

80 people like cherries;
75 people like bananas;
70 people like apples.

-----(-----------) ----  80 people like cherries and 20 don't (each red dash represents 5 people who like cherries);
 ----- (-----------)----   75 people like bananas and 25 don't (each blue dash represents 5 people who like bananas).

So, we can see that minimum 55 people like both cherries and bananas (11 dashes).

To have minimum overlap of 3, let 20 people who don't like cherries and 25 who don't like bananas to like apples. So, we distributed 20+25=45 people who like apples and 70-45=25 people still left to distribute. The only 25 people who can like apples are those who like both cherries and bananas. Consider the diagram below:

-----(-----)------ ----  80 people like cherries and 20 don't (each red dash represents 5 people who like cherries);
 ----- (-----)----------   75 people like bananas and 25 don't (each blue dash represents 5 people who like bananas);
  -----(-----) ------ ----   70 people like apples and 30 don't (each green dash represents 5 people who like apples).

Therefore the minimum number of people who like all three is 25.

Answer: C (OA is not correct).

Nizomiddin · Answer

I guess it can be solved a bit quicker. 30 people out of 100 don't like apples. As we must consider the worst case scenario, let's assume that 25 people who don't like bananas are among 70 who like apples. So, 70-25=45. Also, let's assume that 20 people who don't like cherry is among these 45. So 45-20=25. It is the worst case scenario and the minimum number of people who like all three fruits is 25.

EvaJager · Answer

Let's consider 100 people and drop using %.
Since we are looking for the minimum overlap, let's consider the case with exactly 70, 75 and 85. If the numbers are bigger, then the overlap between the three sets increases.
Since 70 + 75 = 145 >100, at least 45 people like both A(pple) and B(anana). Again, we take the minimum overlap.
It means that there are 70 - 45 = 25 who like A and not B, and 75 - 45 = 30 who like B and not A.
If all these people like C(herries), then 80 - (25 + 30) = 25, who are in neither of the above two categories, they must like all (A, B, and C).
Those who like C, necessarily must be among those who like either A or B, because 70 + 75 - 45 = 100, are all the people.

Answer 25 - should be C and not B.

Observer · Answer

To find the minimal intersection of the people who like apples and the people who like bananas, we need a number x such that 70+(75-x)=100. x=45. The group of people who like both apples and bananas and the group of people who like cherries must intersect because 45+80 > 100. So we need a number y such that 45+(80-y)=100. y=25. Answer is C.

ROFLZZZ · Answer

I got the right answer but not sure why it worked. I added up 80%+75%+70% = 225%. Thought of this in my head as a three level venn diagram, with 100% being the maximum. 225 is 125 above the maximum, so going to the first intersect level is 125, which is still above the maximum of 100, so going to the second and final intersect level is 25, which was my answer, and I have no idea if this was just luck.

avinashrao9 · Answer

Hi Bunuel, Can the explanation be given with a diagram.. I m still confused

Bunuel · Answer

It's a hard question (62% of the users got it wrong!), so I can understand that. We have a similar question on the site to which Karishma gave a venn diagram solution: in-a-village-of-100-households-75-have-at-least-one-dvd-98257.html#p825632 Hope it helps.

SVaidyaraman · Answer

It seems to me the answer is D.

Here's how:

1. Let us keep the number as 70, 75 and 80 to minimize the likes.
2. Number of people who like apples is formed by the people who like A only, who like both A and B only , who like both A and C only and who like A, B and C. This is what is equal to 70. i.e., A + AB+AC+ABC =70
3. Similarly B +AB+BC+ABC= 75 and C+AC+BC+ABC=80
4. Summing all the three we have A+B+C+2AB+2BC+2AC+3ABC=225
5. If A=0, B=5, C=10, AB=BC=CA=35, ABC can be 0.

That is the minimum number of people who like all the 3 can be 0.

Bunuel · Answer

The correct answer is C, not D.

In your case total = 3*35+10+5 = 120%.

SVaidyaraman · Answer

OOPS! sorry about the oversight. There is actually a straightforward solution based on the line of reasoning in my previous post.

1. we have A+B+C+AB+BC+CA+ABC=100 - (1)
2. We also have A+B+C+2AB+2BC+2CA+3ABC= 225 - (2)

Just plug in the values:

ABC = 0
we have from (1) and (2) the impossibility AB+BC+CA=125 i.e., >100

for ABC=20
we have the impossibility AB+BC+CA +ABC=105 i.e., >100

similarly for ABC=15

Only for ABC=25 we have AB+BC+CA+ABC=100.
It is easy to figure out the values and verify.

Though for ABC=35, there is nothing contradictory, 25 is smaller and hence the solution.

SVaidyaraman · Answer

Here's another quick way to arrive at the solution:

1. A+B+C+AB+BC+CA+ABC=100 --- (1)
2. A+B+C+2AB+2BC+2CA+3ABC = 225 --- (2)

(2)- (1) -> AB+BC+CA+2ABC = 125 --- (3)

(3) - (1) -> ABC-A-B-C = 25 ---(4)

As A, B and C are >=0, the minimum value of ABC is 25.

Therefore the number of people who like all the three is 25%.

gracie · Answer

30% don't like apples +
25% don't like bananas +
20% don't like cherries =
75%=maximum % of people who don't like at least one fruit
100%-75%=25%=minimum % of people who like all three fruits

Alex75PAris · Answer

apples 70% like 30% dislike
bananas 75% like 25% dislike
cherries 80% like 20% dislike

We want the minimum of people who like for all this fruits.

Imagine that all the people who dislike bananas like apple : 70-25 = 45
idem for cherries, 45 - 20 = 25, answer choice C

EMPOWERgmatRichC · Answer

Hi All,

We're told that according to a survey, AT LEAST 70% of people like apples, AT LEAST 75% like bananas and AT LEAST 80% like cherries. We're asked for the MINIMUM percentage of people who like ALL THREE. For this type of question, you can draw pictures and/or do math. Your goal is to find a way to make the MINIMUM number occur... Here's how you can TEST VALUES.

If there are 100 people, then we have 
70 who like apples 
75 who like bananas 
80 who like cherries

Clearly, some people have been counted in more than 1 category. Here's how to figure out the minimum number who like all 3....

There are 80 who like cherries; that means there are 20 who DON'T. 
Of the 75 who like bananas, we can say that 20 DON'T like cherries (this helps to minimize the number who like cherries AND bananas). 
Now we have: 
55 who like cherries AND bananas 
25 who like just cherries 
20 who like just bananas

Next, we factor in the apples. Of those 70 people, we can spread them around and maximize the number who like apples AND cherries OR apples and bananas, BUT NOT ALL 3. 
So, let's split the 70 into: 
25 who like apples and cherries 
20 who like apples and bananas

That leave's 25 who MUST like cherries, bananas AND apples. 25/100 = 25%

Final Answer:  C

GMAT assassins aren't born, they're made, 
Rich

mikecal · Answer

graph-paper-graph-paper-template-graph-paper-a4-graph-paper-printable-Favim.com-4736346.jpeg I was able to solve this pretty much instantly by just picturing this is my head. The green representing apples, yellow for bananas, and red for cherries. The minimum overlap of the 3 is solid.

nick1816 · Answer

percentage of person who dislike apple= 100-70=30
percentage of person who dislike banana= 100-75=25
percentage of person who dislike cherry= 100-80 = 20
Minimum percentage of person who like all three= 100-(30+25+20)=25

GMATinsight · Answer

Answer: Option C

Step-by-Step Video solution by  GMATinsight

https://www.youtube.com/watch?v=YhEaxZ3GMU4

snehal2306 · Answer

Here’s how I solved. 
Everything is in percentage, so we can take direct percentage values. 
1) the least people who like apples: 70% , hence people who dislike apples are at the most = 30%
2) the least people who like bananas: 75% , hence people who dislike bananas are at the most = 25%
3) the least people who like cherries: 80% , hence people who dislike cherries are at the most =20%

The minimum people who dislikes everything = total- (maximum number of people who dislikes all three) 
= 100-(20%+25%+30%) = 25%

Posted from my mobile device

Peracha1 · Answer

So basically, what you did was NOT luck — it was actually correct thinking.
You added:
 70 + 75 + 80 = 225
Now think of it like this:
 There are only 100% people in total, but the total preferences add up to 225%.
So the “extra” is:
 225 − 100 = 125
That extra 125% has to come from overlap (people liking more than one fruit).
Now to find the  minimum who like all three , we try to push as much overlap as possible into people liking only 2 fruits (not all 3).
Why? Because we are minimizing the 3-way overlap.
So:
 
 People who like 2 fruits account for 1 extra count each  
 People who like all 3 account for 2 extra counts each   
We first use 2-fruit overlaps to absorb as much of the 125 as possible.
Max we can use = 100 (since total people are 100)
So:
 125 − 100 = 25 left
This leftover 25 can only come from people who are in all 3 groups.
So minimum people who like all three = 25%

According to a survey, at least 70% of people like apples, at least 75

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According to a survey, at least 70% of people like apples, at least 75

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