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According to a survey, at least 70% of people like apples, at least 75 [#permalink]

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19 Sep 2009, 22:28

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75% (hard)

Question Stats:

52% (02:14) correct
48% (02:00) wrong based on 71 sessions

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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?

Re: According to a survey, at least 70% of people like apples, at least 75 [#permalink]

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20 Sep 2009, 01:14

thanks but i dont understand that solution. it is a very intuitive answer and i'm not at that level. i need some formulas. build from the ground up first then i'll try to understand it at a highlevel.

i guess A+B has to have some minimal overlap since it has to be max 100. But why not 40 or 35 etc.. because it only has to be less than 100 it doesn't have to add to 100.

Re: According to a survey, at least 70% of people like apples, at least 75 [#permalink]

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15 Oct 2009, 02:46

70% like A 75% like B 80% like C

If 70% like A and 75% like B then the ones which like them both are 70+75-100 = 45% 80% like C so all the C lovers plus the ones which like A an B both are: 45+80 = 125 so the ones which like A + B + C are 125- 100= 25%

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?

Overlapping elements (ie everything that is in more than one set) = A+B+C - 100= 125

A&B+B&C+C&A+A&B&C= 125 => 45-x+55-x+50-x + x = 125. Solved it. Wrong answer.

Or Method 2:

A&B= A+B-100= 45 B&C= B+C-100=55 C&A= C+A-100= 50

Let X equal overlap of A&B&C

Overlapping elements (ie everything that is in more than one set) = A+B+C - 100= 125

A&B+B&C+C&A + (-3x+x) = 125

Since X is counted 3 times in the calculation above you need to take it out but you need 1 X that represent A&B&C thus:

A&B+B&C+C&A -2x = 125

Solving em. Wrong.

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.