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??
least value venn diagram
- nithi_mystics
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have no idea about this oneagganitk 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??
- nithi_mystics
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I am not sure if my approach is right or if I have missed something in my calculation. But here it goes..
We need to find the minimum percentage of people who like all three. So, I take the minimum value for the percentage of people who like each fruit.
70% of people like apples,
75% like bananas and
80% like cherries
Our aim is to minimize the number of people who like all three. So whatever explanation is given below is aiming at that.
1) 80 like cherries and 20 do not.
2) 75 like bananas and 25 do not.
Now lets say the 25 who don't like bananas like cherries and out of the 75 who like banana, 20 do not like cherries and the remaining 55% like both.
====> ie 55 like both cherries and bananas and the remaining 45 like either cherries or bananas but not both.
3) 70 like apples and 30 do not.
Lets say, 30 who do not like apples, like cherries and bananas. So thats 55-30 = 25.
So out of 70 who like apples, 25 like all three.
There must be a much simpler way of solving this problem. But I am not able to think of any.
Agganitk, can you post the OA?
We need to find the minimum percentage of people who like all three. So, I take the minimum value for the percentage of people who like each fruit.
70% of people like apples,
75% like bananas and
80% like cherries
Our aim is to minimize the number of people who like all three. So whatever explanation is given below is aiming at that.
1) 80 like cherries and 20 do not.
2) 75 like bananas and 25 do not.
Now lets say the 25 who don't like bananas like cherries and out of the 75 who like banana, 20 do not like cherries and the remaining 55% like both.
====> ie 55 like both cherries and bananas and the remaining 45 like either cherries or bananas but not both.
3) 70 like apples and 30 do not.
Lets say, 30 who do not like apples, like cherries and bananas. So thats 55-30 = 25.
So out of 70 who like apples, 25 like all three.
There must be a much simpler way of solving this problem. But I am not able to think of any.
Agganitk, can you post the OA?
Thanks
Nithi
Nithi
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70% eat apples means: it includes those who eat
75% eat Bananas means: it includes those who eat
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
- 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
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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?"
Stuart Kovinsky | Kaplan GMAT Faculty | Toronto
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I noticed in this problem that there are no numbers given for any groups, and there is no total number. MY questions are:
1- Do I use the two equations above for this kind of overlapping sets only ?
2- I tried to use the solution for the following question to solve for the above question, but didn't work. What is the difference ?
3- Do approaches differ from one overlapping sets problem to another ?
In a group of 68 students, each student is registered for at least one of three classes - History, Math and English. Twenty-five students are registered for History, twenty-five students are registered for Math, and thirty-four students are registered for English. If only three students are registered for all three classes, how many students are registered for exactly two classes?
[spoiler]History students: a + b + c + 3 = 25
Math students: e + b + d + 3 = 25
English students: f + c + d + 3 = 34
TOTAL students: a + e + f + b + c + d + 3 = 68
The question asks for the total number of students taking exactly 2 classes. This can be represented as b + c + d.
If we sum the first 3 equations (History, Math and English) we get:
a + e + f + 2b +2c +2d + 9 = 84.
Taking this equation and subtracting the 4th equation (Total students) yields the following:
a + e + f + 2b + 2c +2d + 9 = 84
-[a + e + f + b + c + d + 3 = 68]
b + c + d = 10
[/spoiler]
1- Do I use the two equations above for this kind of overlapping sets only ?
2- I tried to use the solution for the following question to solve for the above question, but didn't work. What is the difference ?
3- Do approaches differ from one overlapping sets problem to another ?
In a group of 68 students, each student is registered for at least one of three classes - History, Math and English. Twenty-five students are registered for History, twenty-five students are registered for Math, and thirty-four students are registered for English. If only three students are registered for all three classes, how many students are registered for exactly two classes?
[spoiler]History students: a + b + c + 3 = 25
Math students: e + b + d + 3 = 25
English students: f + c + d + 3 = 34
TOTAL students: a + e + f + b + c + d + 3 = 68
The question asks for the total number of students taking exactly 2 classes. This can be represented as b + c + d.
If we sum the first 3 equations (History, Math and English) we get:
a + e + f + 2b +2c +2d + 9 = 84.
Taking this equation and subtracting the 4th equation (Total students) yields the following:
a + e + f + 2b + 2c +2d + 9 = 84
-[a + e + f + b + c + d + 3 = 68]
b + c + d = 10
[/spoiler]
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