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?"
least value venn diagram
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life is a test wrote:My attempt at an explanation in the attached diag.
hope that helps.
where do we get the AB =20 part? for the minimum value. you must have maximized AB, but how?
First, start with only 2 sets: If we want to minimize the intersection of A=70 and B=75 from 100, we set the "empty space" in the venn diagram to 0 (there are no people who like none of the fruits), so that
100 = A + B - AB + 0
100 = 70 + 75 - AB
AB = 45
This is the minimum number of people who like both apples and bananas.
Now we introduce the cherries. Again, we want to minimize the intersection of AB=45 and C=80 from 100:
100 = AB + C - ABC
100 = 45 + 80 - ABC
ABC = 25
100 = A + B - AB + 0
100 = 70 + 75 - AB
AB = 45
This is the minimum number of people who like both apples and bananas.
Now we introduce the cherries. Again, we want to minimize the intersection of AB=45 and C=80 from 100:
100 = AB + C - ABC
100 = 45 + 80 - ABC
ABC = 25
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I understand this approach, but where are we maximizing something to find the minimum? Is it when we give a value for 30%, 25% and 20%?wontheGMAT wrote:70% eat apples means: it includes those who eat
- Only Apple -OA
Apples, and Bananas - AB
Apples and Cherry - AC
Apples, Bananas and Cherry =ABC
75% eat Bananas means: it includes those who eat
- Only Bananas -OB
Bananas and Apple - AB
Bananas and Cherry - BC
Apples, Bananas and Cherry =ABC
Total in the set=100% = OA + OB + OC + AB+AC+BC+ABC
70 = Only OA + AB + AC + ABC
75 = Only OB + AB + BC + ABC
80 = Only OC + BC + AC + ABC
70+75+80= OA + OB + OC + 2AB+2AC+2BC+3ABC
225= 100 + AB + AC + BC + 2ABC .........eq1
Now remaining
30% eat = BC
25% Eat = AC
20% Eat = AB
30+25+20 = AB + AC + BC = 75...........eq2
Substitute this in eq1
225= 100 + 75 + 2ABC
225-175 = 2 ABC
ABC = 25 ... People who eat all three fruits
This question is simpler than many of the explanations make it seem.
Use the following formula twice.
X + Y - XY = 100 (where X is one group, Y is a second group, and XY is people who are in both groups)
In reality, X + Y - XY could be less than 100, but since we want to find the smallest XY, we need to set it equal to 100.
Do the formula for two groups A and B.
A + B - AB = 100
70 + 75 - AB = 100
so 45 = AB
Do the formula a second time for groups AB and C.
AB + C - ABC = 100
45 + 80 - ABC = 100
25 = ABC
Use the following formula twice.
X + Y - XY = 100 (where X is one group, Y is a second group, and XY is people who are in both groups)
In reality, X + Y - XY could be less than 100, but since we want to find the smallest XY, we need to set it equal to 100.
Do the formula for two groups A and B.
A + B - AB = 100
70 + 75 - AB = 100
so 45 = AB
Do the formula a second time for groups AB and C.
AB + C - ABC = 100
45 + 80 - ABC = 100
25 = ABC
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It is minimum when there are no members belonging to "ONLY" category. Only A+ Only B + Only c = 0. Rest is easy.
AB+BC+AC+ABC = 100
2AB+2BC+2AC+2ABC = 200
(AB+AC)+(AB+BC)+(AC+BC) +2ABC = 200
(70-ABC)+(75-ABC)+(80-ABC) + 2ABC = 200
225-3ABC+2ABC = 200
225-ABC=200
ABC = 25
AB+BC+AC+ABC = 100
2AB+2BC+2AC+2ABC = 200
(AB+AC)+(AB+BC)+(AC+BC) +2ABC = 200
(70-ABC)+(75-ABC)+(80-ABC) + 2ABC = 200
225-3ABC+2ABC = 200
225-ABC=200
ABC = 25
My approach is below:
At least 70% like Apples...hence at most 30% dislike apples
At most 25% dislike Bananas and at most 20% dislike cherries
Hence at most: 30+20+25=75% dislike all three fruits
Which follows at least 100-75= 25% like all three fruits (Complementary probability function)
At least 70% like Apples...hence at most 30% dislike apples
At most 25% dislike Bananas and at most 20% dislike cherries
Hence at most: 30+20+25=75% dislike all three fruits
Which follows at least 100-75= 25% like all three fruits (Complementary probability function)
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I think there's a fairly intuitive explanation to this one. If the minimum number of people who like fruit of each type is 70%, 75% and 80% respectively then the maximum number of people who don't like fruit of each type is 30%, 25% and 20% respectively. Together the maximum number of people who don't like fruit of any type is 75%. Then what is the minimum number of people who like either 1 fruit, or 2 fruits or all 3 fruits? The answer is 100%-75%=25%. Obviously the lowest possible value of this will be people who like all 3 fruits. Other groups (like 1 fruit or like 2 fruits) can only be bigger than this. So the answer is 25%.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??
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