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M08-07

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New post 16 Sep 2014, 00:36
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Among 200 people, 56% like strawberry jam, 44% like apple jam, and 40% like raspberry jam. If 30% of the people like both strawberry and apple jam, what is the largest possible number of people who like raspberry jam but do not like either strawberry or apple jam?

A. 20
B. 60
C. 80
D. 86
E. 92

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New post 16 Sep 2014, 00:36
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Official Solution:

Among 200 people, 56% like strawberry jam, 44% like apple jam, and 40% like raspberry jam. If 30% of the people like both strawberry and apple jam, what is the largest possible number of people who like raspberry jam but do not like either strawberry or apple jam?

A. 20
B. 60
C. 80
D. 86
E. 92


Look at the diagram below:

Image

Notice that "30% of the people like both strawberry and apple jam" doesn't mean that among these 30% (60) cannot be some people who like raspberry as well. Both strawberry and apple jam is the intersection of these two groups, if we refer to the diagram it's the yellow segment in it.

Next, no formula is needed to solve this question: 112 like strawberry jam, 88 like apple jam, 60 people like both strawberry and apple jam. So the # of people who like either strawberry or apple (or both) is \(112+88-60=140\) (on the diagram it the area covered by Strawberry and Apple). So there are TOTAL of \(200-140=60\) people left who "do not like either strawberry or apple jam". Can ALL these 60 people like raspberry? As \(\text{Raspberry}=80 \ge 60\), then why not! So, maximum # of people who like raspberry and don't like either strawberry or apple jam is 60 (grey segment on the diagram). Notice here that in this case the # of people who like none of the 3 jams (area outside three circles) will be zero.

Side note: minimum # of people who like raspberry and don't like either strawberry or apple jam would be zero (consider Raspberry circle inside Strawberry and/or Apples circles). In this case those 60 people (who "do not like either strawberry or apple jam") will be those who like none of the 3 jams.


Answer: B
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Re: M08-07  [#permalink]

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New post 03 May 2015, 03:00
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I like your way of doing it bunuel but its much easier to keep with the formulas :P
In case everyone in this group likes at least one jam (the case of max)
Image
112+88+80-(60-x)-z-y-2*x = 200
x+y+z = 20
So the resulting amount of people is 80-20 = 60 who could like raspberry jam but not any other.
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New post 11 Mar 2015, 22:08
Hi Bunuel

I marked B

Can you pls advise if my method mentioned below is correct?

Total = R + A + S - (Sum of Exactly 2 group overlap) - 2 (All 3 )
200 = 80 + 88 + 112 - 60 - 2 (All 3 )

Solving we get: (All 3) = 10

Since R = 80 and (All 3 ) = 10, then Max Raspberry can have = 70
But since 70 is not mentioned in the options, the next best answer would be 60

Is this method correct?

Thanks
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M08-07  [#permalink]

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New post 21 Apr 2015, 01:55
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Hi buddyisraelgmat,
If you don't mind, let me try to answer why your solution is wrong.

Your equation:
Total = R + A + S - (Sum of Exactly 2 group overlap) - 2 (All 3 )

As Bunuel stated, "Notice that "30% of the people like both strawberry and apple jam" doesn't mean that among these 30% (60) cannot be some people who like raspberry as well."

Key Error: It was wrong for you to put the value 60 as "exactly those who like apple and strawberry only". The sixty could include those who like Strawberry, Apple, AND Raspberry. This error led you to multiply the number of those who like all (All 3) by 2.

Your equation should have been,
Total =R + A + S - (Sum of 2 groups) - (All 3)
200 = 112 + 88 + 80 - 60 - x
x = 20. This means that of the 60 who like both apple and strawberry, 20 like raspberry too.

Therefore number of people who like raspberry only is 80-20 =60

I'm no GMAT expert but I hope my explanation makes sense to you and is right. lol
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Re: M08-07  [#permalink]

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New post 21 Oct 2015, 23:19
Dear bunuel,

Thanks for the detailed reply. I have one doubt however, is it assumed that at least one person likes all three types of jams ? In case no , then can we not have a situation where 80 people like only Raspberry jam and 60 like both apple and strawberry jams ? The answer in such case would be 80 and not 60 ?

Looking forward to a reply. Thanks in advance
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New post 21 Oct 2015, 23:30
spetznaz wrote:
Dear bunuel,

Thanks for the detailed reply. I have one doubt however, is it assumed that at least one person likes all three types of jams ? In case no , then can we not have a situation where 80 people like only Raspberry jam and 60 like both apple and strawberry jams ? The answer in such case would be 80 and not 60 ?

Looking forward to a reply. Thanks in advance


It's not possible 80 people to like raspberry jam but do not like either strawberry or apple jam because in this case the total number of people would be 80 + 140 = 220 > 200.

Hope it's clear.
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Re: M08-07  [#permalink]

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New post 18 Dec 2015, 10:20
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Zhenek wrote:
I like your way of doing it bunuel but its much easier to keep with the formulas :P
In case everyone in this group likes at least one jam (the case of max)
Image
112+88+80-(60-x)-z-y-2*x = 200
x+y+z = 20
So the resulting amount of people is 80-20 = 60 who could like raspberry jam but not any other.


We wouldn't need y and z. Since we are trying to find teh largest possible value, assume x and y to be 0.

Assume total to be 100
100 = 56 + 44 + 40 - (30 - x) - 2x
100 = 110 - x
x = 10.

Double the total and value of x.
Total = 2*100 = 200
x = 10*2 = 20.
Final answer - 40*2-20 = 60 :)
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Re: M08-07  [#permalink]

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New post 13 Jan 2016, 07:46
First, assume that Rasberry is isolated circle not having any combined portion. If so, answer would be 40% of 200=80. But is it?
Lets check. If so, (112+88-60)+80=220, but you can't exceed 200,that is total number. So, rasberry must have combined portion by 20. Since we need only such people who like rasberry but don't like others, we must deduct 20 from 80. 80-20=60.
Alternatively,
112 +88-60=140
But we have 200 people in total. So, 200-140=60 gives us our desired portion.
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Re: M08-07  [#permalink]

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New post 20 Feb 2016, 05:06
This is a good question and like always, a good question has an easier solution. (without Venn diagrams)
First let's consider that there are 100 people in sample space as we are processing percentage data.
So draw a line, write 0 at left end and 100 at the right extreme end (baseline)
56 people like strawberry, so draw a line above the baseline starting from the left end (0) to a point more than half of the original line and write 56 at the right end of this line.
As per the question we need to maximize the number of raspberry liking fellows who do not like strawberry and apple.
So we will try to maximize the people who like both strawberry and apple.
Now apple is liked by 44 people and 30 percent like both apple and strawberry.
So draw another line on top of the raspberry line starting from left end and draw till value 30.
Now the remaining 14 apple lovers will be drawn after 56 (previous extent of raspberry lovers)
so from 56, draw a line til 70. (70 - 56 = 14)
So now we have fitted apple and raspberry as per the question.
Now from 70 to 100 we have an open space of 30, where 30 of the 40 raspberry lovers can find peace.
So in 100 people, 30 is the maximum number of required raspberry lovers and hence in 200 they will double to 60.
Thats the answer.
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Re: M08-07  [#permalink]

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New post 19 May 2016, 00:47
Zhenek wrote:
I like your way of doing it bunuel but its much easier to keep with the formulas :P
In case everyone in this group likes at least one jam (the case of max)
Image
112+88+80-(60-x)-z-y-2*x = 200
x+y+z = 20
So the resulting amount of people is 80-20 = 60 who could like raspberry jam but not any other.


Hi chetan2u,

I am wondering, why can't we take X, Y and z as 0
the question asks- LARGEST possible number of people who like raspberry jam but do not like either strawberry or apple jam?

If we take X, Y and z as 0 then the largest no can be 80 as well.. :roll:

What I am missing..?
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New post 19 May 2016, 01:30
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PrakharGMAT wrote:
Zhenek wrote:
I like your way of doing it bunuel but its much easier to keep with the formulas :P
In case everyone in this group likes at least one jam (the case of max)
Image
112+88+80-(60-x)-z-y-2*x = 200
x+y+z = 20
So the resulting amount of people is 80-20 = 60 who could like raspberry jam but not any other.


Hi chetan2u,

I am wondering, why can't we take X, Y and z as 0
the question asks- LARGEST possible number of people who like raspberry jam but do not like either strawberry or apple jam?

If we take X, Y and z as 0 then the largest no can be 80 as well.. :roll:

What I am missing..?


Yes, you can do that, BUT there is a restriction on TOTAL, which is 200...
If you make all these x, y and z as 0...
this means 60 are COMMON only to S and A..
so total number of S and A = 112 + 88 - 60 = 140..
Now this leaves 200-140 = 60 for R, but that number is 80..


so the numbers do not match..
Therefore you will have to find the OVERLAP, as it has to be there to fit in all our quantities
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Re: M08-07  [#permalink]

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New post 12 Dec 2016, 09:42
Why cant we have Raspberry completely as a separate circle (max value 80)
It is not told in the question that all 3 circles need to intersect.
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New post 04 Aug 2019, 00:38
Bunuel wrote:
Among 200 people, 56% like strawberry jam, 44% like apple jam, and 40% like raspberry jam. If 30% of the people like both strawberry and apple jam, what is the largest possible number of people who like raspberry jam but do not like either strawberry or apple jam?

A. 20
B. 60
C. 80
D. 86
E. 92



Please see the solution in image.

IMO B
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New post 19 Nov 2019, 01:00
maitysourav

We cannot have Raspberry=80.

because if we assume that Strawberry&Apple&Raspberry = 0 ;
then, Strawberry&Apple=60 ; Strawberry=52 ; Apple = 28 ; Strawberry&Raspberry = 0 ; Apple&Raspberry = 0;
This is not possible since now if we add only raspberry = 80
We will get 60 + 52 + 28 + 0 + 0 + 80 = 220 (>200 [total])
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New post 19 Nov 2019, 02:44
sidney123 wrote:
I think this is a poor-quality question and I don't agree with the explanation. Largest possible number of people who like Raspberry only, would contain 0 people who like Raspberry & Strawberry; Raspberry & Apple and Raspberry, Stawberry and Apple. So maximum number is 40% of 200=80. I do not agree with the explanation.


Here is an alternate explanation, if you wanna consider 0% as the overlap between all three JAMS.
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Re: M08-07   [#permalink] 19 Nov 2019, 02:44
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