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

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According to a survey, at least 70% of people like apples [#permalink] New post 15 Oct 2012, 01:00
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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. 0%
E. 35%
[Reveal] Spoiler: OA

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Re: According to a survey, at least 70% of people like apples [#permalink] New post 15 Oct 2012, 02:58
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Skientist wrote:
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. 0%
E. 35%

I'm really interested in a detailed explaination of how to approach this problem.

Thanks

Skientist


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).
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Re: According to a survey, at least 70% of people like apples [#permalink] New post 15 Oct 2012, 04:47
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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.
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Re: According to a survey, at least 70% of people like apples [#permalink] New post 15 Oct 2012, 07:28
Skientist wrote:
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. 0%
E. 35%

I'm really interested in a detailed explaination of how to approach this problem.

Thanks

Skientist



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.
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Re: According to a survey, at least 70% of people like apples [#permalink] New post 15 Jul 2013, 23:39
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Re: According to a survey, at least 70% of people like apples [#permalink] New post 16 Jul 2013, 11:37
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Skientist wrote:
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. 0%
E. 35%

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.
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Re: According to a survey, at least 70% of people like apples [#permalink] New post 17 Jul 2013, 14:07
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.
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Re: According to a survey, at least 70% of people like apples [#permalink] New post 17 Jul 2013, 19:34
Bunuel wrote:
Skientist wrote:
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. 0%
E. 35%

I'm really interested in a detailed explaination of how to approach this problem.

Thanks

Skientist


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).


Hi Bunuel,

Can the explanation be given with a diagram.. I m still confused :(
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Re: According to a survey, at least 70% of people like apples [#permalink] New post 17 Jul 2013, 21:33
Expert's post
avinashrao9 wrote:
Bunuel wrote:
Skientist wrote:
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. 0%
E. 35%

I'm really interested in a detailed explaination of how to approach this problem.

Thanks

Skientist


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).


Hi Bunuel,

Can the explanation be given with a diagram.. I m still confused :(


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.
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Re: According to a survey, at least 70% of people like apples [#permalink] New post 18 Jul 2013, 00:40
EvaJager wrote:
Skientist wrote:
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. 0%
E. 35%

I'm really interested in a detailed explaination of how to approach this problem.

Thanks

Skientist


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.
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Re: According to a survey, at least 70% of people like apples [#permalink] New post 18 Jul 2013, 00:48
Expert's post
SravnaTestPrep wrote:
EvaJager wrote:
Skientist wrote:
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. 0%
E. 35%

I'm really interested in a detailed explaination of how to approach this problem.

Thanks

Skientist


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.


The correct answer is C, not D.

In your case total = 3*35+10+5 = 120%.
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COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: According to a survey, at least 70% of people like apples [#permalink] New post 18 Jul 2013, 03:17
Quote:
The correct answer is C, not D.

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


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.
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Re: According to a survey, at least 70% of people like apples [#permalink] New post 18 Jul 2013, 06:05
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%.
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Re: According to a survey, at least 70% of people like apples [#permalink] New post 22 Oct 2013, 15:32
For the same problem , the maximum value would be 70.

Is that correct? can some one please verify if that is correct?

Thanks
Re: According to a survey, at least 70% of people like apples   [#permalink] 22 Oct 2013, 15:32
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