least value venn diagram
- Deependra1
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Stuart, I like your approach but, I found difficult to understand your formula saying that you maximized the equation by making zero the quantities A,B &C.
A different approach would be that regardless the distribution of AB+BC+AC it will have to be equal to 75 as it should be maximized the amount of 2 shared qty's to minimize the 3 shared qty's.
This should help out understanding The concept without having to memorize an extra formula that may not be applicable for other problems.
A different approach would be that regardless the distribution of AB+BC+AC it will have to be equal to 75 as it should be maximized the amount of 2 shared qty's to minimize the 3 shared qty's.
This should help out understanding The concept without having to memorize an extra formula that may not be applicable for other problems.
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Super Short cut method to solve these type of problems:
subtract each of these % by 100 and then the final sum again by 100.
(100-70) + (100-75) + (100-80) = 75%
Subtracting again by 100 = 100-75 = 25 %
Off course for the actual solution you can refer to methods given by experts .
subtract each of these % by 100 and then the final sum again by 100.
(100-70) + (100-75) + (100-80) = 75%
Subtracting again by 100 = 100-75 = 25 %
Off course for the actual solution you can refer to methods given by experts .
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Thought of the following way
A - 70 Like - 30 Non Like
B - 80L - 20NL
C - 75L - 25NL
To find min number of likes , try to maximize the non-likes
starting with A , try to fit B, 80-30 = 50 now try to fit C , 50-25 = 25
so ans = 25 = 80-(30+25) = 70 -(20+25) = 75 -(30+20)
A - 70 Like - 30 Non Like
B - 80L - 20NL
C - 75L - 25NL
To find min number of likes , try to maximize the non-likes
starting with A , try to fit B, 80-30 = 50 now try to fit C , 50-25 = 25
so ans = 25 = 80-(30+25) = 70 -(20+25) = 75 -(30+20)
NSK
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Thanks Stuart for your explanation.Can you please explain why we need to substitute zero for total in exactly 1 group in the second equation.I am not able to figure it out.Please help
True # of objects = (total in exactly 1 group) + (total in exactly 2 groups) + (total in exactly 3 groups)
True # of objects = (total in exactly 1 group) + (total in exactly 2 groups) + (total in exactly 3 groups)
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Well, I learnt this rule today only.....Rule applicable only for least number
Here is my take:
A+B+C-2t= Ans
70+75+80-2*100 =25%
Here is my take:
A+B+C-2t= Ans
70+75+80-2*100 =25%
I solved this in 1 min. All i did is: suppose there are 100 people for survey
80 like cherries
75 like bananas
70 like apples
From 80 and 75 what will the least common number from 100?
When 20 people dont like cherries and 25 dont like bananas and you have to find the common of both. I would subtract 20 from 75.
Similarly for three number i would subtract the 20 + 25 ( i.e. 45) from 70.
so my outcome is 25
i.e. 25 %
80 like cherries
75 like bananas
70 like apples
From 80 and 75 what will the least common number from 100?
When 20 people dont like cherries and 25 dont like bananas and you have to find the common of both. I would subtract 20 from 75.
Similarly for three number i would subtract the 20 + 25 ( i.e. 45) from 70.
so my outcome is 25
i.e. 25 %
- kul512
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My approach-
Cherries- 80%
So those who don't like cherries- 100-80= 20% (these include only apple lover, only banana lover and apple+ banana lover)
Same way banana-75%
So those who don't like banana-25%
Apple-70%
Those who don't like apple-30%
When we add up these figures, we will get following information-
2* people who like only one type of fruit+ those who like any two type of fruit (this can be shown by venn diagram.)
so
2(Apple+ banana+ cherries)+ (apple or banana+ cherries or apple+ cherries or banana)=75%
Rewrite as-
Apple+ banana+ cherries +(apple+ cherries+ banana+ (apple or banana+ cherries or apple+ cherries or banana))=75%---------------------------------------(1)
"(apple+ cherries+ banana+(apple or banana+ cherries or apple+ cherries or banana)" is nothing but the count of people who don't like all three fruits.
And which can be re-written as "100%- (those who like all three fruits)"
So equation (1) simplifies to
Apple+ banana+ cherries+(100%- those who like all three fruits)=75
So
Those who like all three fruits=(100-75)%+ those who like only one fruit
Those who like all three fruits=25%+ those who like only one fruit
To get the minimum number of people who like all three fruit, right hand side must be minimum. So we will take the number of people who like only one fruit as Zero.
Hence minimum number of people who like all three fruits is 25%.
One point to note in this question is that information provided is less than required to get an exact answer that's why word minimum is used. To minimize the number we are free to consider something within limit. But we cannot take the "number of people who like only fruit" as negative, though that will minimize the required answer further.
Cherries- 80%
So those who don't like cherries- 100-80= 20% (these include only apple lover, only banana lover and apple+ banana lover)
Same way banana-75%
So those who don't like banana-25%
Apple-70%
Those who don't like apple-30%
When we add up these figures, we will get following information-
2* people who like only one type of fruit+ those who like any two type of fruit (this can be shown by venn diagram.)
so
2(Apple+ banana+ cherries)+ (apple or banana+ cherries or apple+ cherries or banana)=75%
Rewrite as-
Apple+ banana+ cherries +(apple+ cherries+ banana+ (apple or banana+ cherries or apple+ cherries or banana))=75%---------------------------------------(1)
"(apple+ cherries+ banana+(apple or banana+ cherries or apple+ cherries or banana)" is nothing but the count of people who don't like all three fruits.
And which can be re-written as "100%- (those who like all three fruits)"
So equation (1) simplifies to
Apple+ banana+ cherries+(100%- those who like all three fruits)=75
So
Those who like all three fruits=(100-75)%+ those who like only one fruit
Those who like all three fruits=25%+ those who like only one fruit
To get the minimum number of people who like all three fruit, right hand side must be minimum. So we will take the number of people who like only one fruit as Zero.
Hence minimum number of people who like all three fruits is 25%.
One point to note in this question is that information provided is less than required to get an exact answer that's why word minimum is used. To minimize the number we are free to consider something within limit. But we cannot take the "number of people who like only fruit" as negative, though that will minimize the required answer further.
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Lets consider this way:
AB - Overlapping of A&B (oncluding portion of A,B & C)
BC - -same-
CA - -same-
AuBuC=A+B+C-AB-BC-CA+ABC
100=70+75+80-AB-BC-CA+ABC
AB+BC+CA=125+ABC - (1)
AB+BC+CA-2ABC=100 - (2) - CONSIDERING SINGLE LIKES AS 0
Solving (1) and (2)
125+ABC=100+2ABC
ABC=25
Difference with my approach - Somehow i couldnt consider 70,75 & 80 % as a single like (not considering the overlapping between 2 or three).
AB - Overlapping of A&B (oncluding portion of A,B & C)
BC - -same-
CA - -same-
AuBuC=A+B+C-AB-BC-CA+ABC
100=70+75+80-AB-BC-CA+ABC
AB+BC+CA=125+ABC - (1)
AB+BC+CA-2ABC=100 - (2) - CONSIDERING SINGLE LIKES AS 0
Solving (1) and (2)
125+ABC=100+2ABC
ABC=25
Difference with my approach - Somehow i couldnt consider 70,75 & 80 % as a single like (not considering the overlapping between 2 or three).
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- Rastis
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Stuart, can you find a way to better explain the answer? Everything that I've seen posted is way too confusing. I attempted to solve it via Venn Diagram but couldn't get it and the Venn Diagram example that was posted doesn't help at all. Can someone help please!
- GMATGuruNY
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To MINIMIZE the percentage who like all 3 fruits, we must MAXIMIZE the percentage who like 2 of the fruits.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??
Since A=70, the maximum value of BC= 30.
Since B=75, the maximum value of AC = 25.
Since C=80, the maximum value of AB= 20.
Thus:
The MAXIMUM percentage who like 2 of the fruits = 30+25+20 = 75.
Thus:
The MINIMUM percentage who like all 3 fruits = 100-75 = 25.
The correct answer is C.
Last edited by GMATGuruNY on Fri Sep 07, 2012 6:40 am, edited 1 time in total.
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Why 2ABC?Stuart Kovinsky wrote: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: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??
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?"