least value venn diagram

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by anujjj » Sun Sep 23, 2012 1:58 am
An easy way to solve this would be using a easy formula :-

The least value of a venn diagram is

total- People who like a= 100%-70%=30%

Total - People who like b= 100%- 75%=25%

Total- People who like C = 100%- 80%= 20%

Sum all the remaining and subtract from the total:- 100%- ( 30% + 25% + 20%)= 25%

This formula works for all minimum area of venn diagram question try it follow it.

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by mparakala » Sat Nov 03, 2012 10:03 am
Stuart Kovinsky wrote:
agganitk 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%

Ans??
Let's say we have 100 people to make things simple. We want to minimize the triple group, so let's minimize how many people like each kind of fruit, giving us:

70 apple lovers, 75 banana lovers and 80 cherry lovers.

Now, 70 + 75 + 80 = 225, so we have 225 "fruit loves" spread out among 100 people.

Therefore, there are 125 more "fruit loves" than there are people.

Our job is to minimize the number of people who love all 3; to do so, we want to maximize the number of people who love exactly 2 of the 3 and minimize the number of people who love exactly 1 of the 3.

So, we can come up with two equations:

AB + AC + BC + 2ABC = 125

and

AB + AC + BC + ABC = 100

AB = number who like just apple/banana
AC = number who like just apple/cherry
BC = number who like just banana/cherry
ABC = number who like all 3

The first equation is derived from the triple-overlapping set equation:

True # of objects = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in exactly 2 groups) - 2(# in all 3 groups)

100 = 70 + 75 + 80 - AB - AC - BC - 2(ABC)

and when we rearrange to get all variables on one side:

AB + AC + BC + 2ABC = 125

The second equation is derived from another version of the triple-overlapping set equation:

True # of objects = (total in exactly 1 group) + (total in exactly 2 groups) + (total in exactly 3 groups)

We set the "total in exactly 1 group" to 0, so we get:

100 = 0 + AB + AC + BC + ABC

So, back to our equations:

AB + AC + BC + 2ABC = 125
AB + AC + BC + ABC = 100

If we subtract the second from the first, we get:

ABC = 25... done!

Now, at this point you may be saying, "umm.. ok.. but I asked for a simple way to solve, that seemed super complicated and time consuming!"

However, if you understand the concepts behind triple-overlap (or double-overlap) questions, it's fairly intuitive; the complicated part is getting to the stage at which you have that deeper understanding.

Of course, this exact question won't appear on the GMAT. So, as always, after you do a question you ask yourself: "what did I learn from this question that's going to help me on future questions?"

Here's our takeaways:

1) there are multiple ways to solve overlapping sets questions. The more you familiarize yourself with the 3 major approaches (equations/venn diagrams/matrices(the last only works when there are 2 overlapping sets, unless you're really good at drawing a 3-dimensional matrix)), the more likely it is that the quickest approach will jump out at you on test day.

2) if you're shooting for a 600+, learn the two equations noted above.

3) whenever you're asked to minimize something, think "what do I need to maximize to achieve that result?"
Thank you so much for your explanation !

In summary, where everything is expressed as a percentage.. i.e., out of 100... the difference between the sums 125-100 = 25. Right?

1st eq is vey clear.. i do not understand why the 2nd eq assumes a particular quantity to be zero! please explain.

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by rajeshsinghgmat » Sat Jan 26, 2013 5:52 pm
C) 25 the answer

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by hchandana23 » Mon Jun 17, 2013 2:40 am
Hello,

Could you please let me know if the following approach is incorrect:

Atleast 70 like apple implies maximum 30 dont like apples.
Atleast 75 like bananas implies maximum 25 do not like bananas.
Atleast 80 like cherries implies maximum 20 do no like cherries.

We need to get the minimum number of people who like all 3... so if we assume that all the above set of people (30, 25 and 20) are all different then their sum comes to 75.
Hence total number of people remaining will be 100-75=25.

So, 25% minimum people will like all 3...

Thanks and Regards,
Harsha

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by Mathsbuddy » Tue Nov 26, 2013 12:30 am
To start:

Minimum %:
A only = 40
B only = 50
C only = 75
A+B only = 25
B+C only = 0
A+C only = 0
A+B+C = 5

add them up to check:
A = 40 + 25 + 5 = 70
B = 50 + 25 = 75
C = 75 + 5 = 80

Note that the "+" symbol here represents an intersection.

However the total becomes 195%

By combining percentages between A,B and C we can reduce it to 100% which leaves 25% in the middle.

If you draw it in a Venn diagram it is clearer.

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by pranjal25 » Sat Feb 08, 2014 10:24 pm
agganitk 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%

Ans??c

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by pedrodona » Wed Mar 26, 2014 4:55 am
Could anyone help me out please?

I understand the solution that has been given when I see it, but I was trying to solve it multiplying probabilities and I don't understand why that approach is incorrect. Provided they are independent events, we get the probability of two things happening together by multiplying their individual probabilities. In this case:

Pr(A)=0.7, Pr(B)=0.75 and Pr(C)=0.8, so to have the Prob(AUBUC)=0.7 x 0.75 x 0.8=42%

Why is this wrong? Thanks in advance,

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by evs.teja » Fri Sep 12, 2014 3:55 am
Dear Kovinsky,

Will you please explain the below mentioned point again

Our job is to minimize the number of people who love all 3; to do so, we want to maximize the number of people who love exactly 2 of the 3 and minimize the number of people who love exactly 1 of the 3.

Why are we only maximizing AB BC and CA and and not only A ,B and C....won't that bring down the value of ABC as well ?

Thanks
Teja

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by nikhilgmat31 » Tue Jun 09, 2015 4:15 am
Mr GMATGuruNY,

Please answer this question

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by GMATGuruNY » Tue Jun 09, 2015 5:02 am
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 MINIMIZE the percentage who like all 3 fruits, we must MAXIMIZE the percentage who like exactly 2 of the fruits.

Since 70% like apples, the maximum percentage who could like only bananas and cherries = 30%.
Since 75% like bananas, the maximum percentage who could like only apples and cherries = 25%.
Since 80% like cherries, the maximum percentage who could like only apples and bananas = 20%.

Thus:
The MAXIMUM percentage who could like exactly 2 of the fruits = 30+25+20 = 75%.
Thus:
The MINIMUM percentage who could like all 3 fruits = 100-75 = 25%.

The correct answer is C.
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by akash.abhinav » Thu Jun 11, 2015 11:47 am
Hi Stuart,

Apologies if this sounds real stupid, but can you explain why are we minimizing the people who like exactly one fruit? I mean in order to minimize the people who like all the 3 fruits ABC, shouldn't we maximize the number of people who like either exactly one fruit or they exactly like two fruits?

Thanks in advance
Abhinav

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by Shri420 » Mon Jun 15, 2015 6:32 pm
Thank you GMATGuruNY.
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by MartyMurray » Mon Jun 15, 2015 8:05 pm
akash.abhinav wrote:Hi Stuart,

Apologies if this sounds real stupid, but can you explain why are we minimizing the people who like exactly one fruit? I mean in order to minimize the people who like all the 3 fruits ABC, shouldn't we maximize the number of people who like either exactly one fruit or they exactly like two fruits?

Thanks in advance
Abhinav
The point is we are seeking to minimize the number who like all three and we are told that a certain percentage of people like each fruit.

We have to somehow account for the people who like each type and the total cannot go over 100 percent.

So what we are seeking to do is use up as many likes as possible without putting them into the all three category.

We minimize the number who like only one, because saying that people like only one uses up percentage points of the total without accounting for a maximum number of likes.

For instance if we say that 20% like only apples, we have just used up 20 percentage points of the total and we still have to account for the 80% who like cherries, 75% who like oranges and the remaining 50% who like apples. Having done this we are stuck saying that at least 50% like all three, because between the 20% who like only apples and the 80% who like cherries we have used up all of our percentage points of the total. So the apple and orange likes have to all overlap the cherry likes because there is no room left for separate categories.

So instead we say that nobody likes just one type and we use up as many likes as possible by maximizing people who like exactly two types, and in so doing we are stuck with the minimum number of likes that need to go into the likes all three category.
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by dineshbp » Tue Jun 16, 2015 9:08 pm
Our objective is to find minimum overlap. So, while allocating ensure that minimum overlap happens.

Apple: 70 %
So by using balance 30% we will have at least 45 % overlap with Banana. With that we have three group.
25% only apple (A)
30% only banana (B)
45% both (A & B)

Now if we introduce cherry with the objective to have minimum overlap. Then first use single items (A or B) and then combination. Here we go...
1) Apple & Cherry: 25%
2) Banana & Cherry: 30%
3) Balance will be Apple + Banana + Cherry: 25%

Hence the answer 25%

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by kulsim » Mon Jun 22, 2015 6:32 am
Hi! Apologies if this sounds trivial but just trying to make sure I am getting this fully. I now understood why we got 25% and just trying to think about this problem in reverse. If we were asked to maximize the ABC, would the answer be simply 75%?