M08-07

Question

Out of 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 maximum 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

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

Bunuel · Accepted Answer

Official Solution:

Out of 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 maximum 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

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

Consider the diagram below:
 
 https://gmatclub.com/gmat-practice-tests/resources/import/img/m08-07.png 
 
Notice that "30% of the people like both strawberry and apple jam" doesn't imply that none of these 30% (60 people) can also like raspberry jam. The intersection of the strawberry and apple jam groups is represented by the yellow segment in the diagram.
 
No specific formula is needed to solve this question: 112 people like strawberry jam, 88 people like apple jam, and 60 people like both strawberry and apple jam. Thus, the number of people who like either strawberry or apple jam (or both) is  112 + 88 - 60 = 140  (the area covered by Strawberry and Apple in the diagram). Therefore, there are a total of  200 - 140 = 60  people who "do not like either strawberry or apple jam." Can all of these 60 people like raspberry jam? Since there are 80 people who like raspberry jam ( 	ext{Raspberry} = 80 \ge 60 ), it is possible! The maximum number of people who like raspberry jam and don't like either strawberry or apple jam is 60 (the gray segment in the diagram). In this case, the number of people who don't like any of the three jams (the area outside the three circles) would be zero.
 
Side note: The minimum number of people who like raspberry jam and don't like either strawberry or apple jam would be zero (if the Raspberry circle is entirely inside the Strawberry and/or Apple circles). In this case, the 60 people who "do not like either strawberry or apple jam" would be those who don't like any of the three jams.

Answer: B

Zhenek · Answer

I like your way of doing it bunuel but its much easier to keep with the formulas

In case everyone in this group likes at least one jam (the case of max) https://i.imgur.com/EPnw0Ie.png 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.

GMATNinja · Answer

And here's a video explanation. Enjoy!

https://www.youtube.com/watch?v=uPrhAld0VNQ

buddyisraelgmat · Answer

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

sanchito · Answer

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

spetznaz · Answer

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

Bunuel · Answer

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.

swanidhi · Answer

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

Abhigyanashakuntalam · Answer

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.

maitysourav · Answer

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.

Bunuel · Answer

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.

Kinshook · Answer

Please see the solution in image.

IMO B

AndrewN · Answer

I would like to point out that any time a percentage question is based on a unit of 100, calculating the number of items for any given category becomes extremely easy.

56% of people like strawberry jam = 56 (per 100) * 2, or 112 people
44% like apple jam = 44 * 2, or 88 people
40% like raspberry jam = 40 * 2, or 80 people
30% like both strawberry and apple jam = 30 * 2, or 60 people

Considering just the strawberry and apple subgroup, the 60 people who like both must be deducted from  each  total to avoid double counting:

112 - 60 = 52 people who like strawberry but not apple jam
88 - 60 = 28 people who like apple but not strawberry jam

So far, we have 52 + 28 + 60 people, or 140 people, who like strawberry jam, apple jam, or both strawberry and apple jam. Thus, there are only 60 people remaining of the 200 who are unaccounted for.  Since the question asks about this very group of people , and since there are at least 60 people who like raspberry jam, it is reasonable to say that  there can be no more than 60 people who like raspberry jam but neither strawberry nor apple jam . Choice (B) it is.

I will be honest: I wrote nothing down for this question, and mental math, tracing the above line of thought, allowed me to figure out the question in about 90 seconds. If I can do it, then so can you.  Sometimes the question gives you an easy out: make sure you know how to spot it when it does appear.

Good luck with your studies.

- Andrew

Vegita · Answer

Hi  Bunuel

I understand why the maximum number of people who like only raspberry jam cannot be 80 but I do not understand why the number of people who don't like any of the 3 jams is euqal to 0.

Could you please clarify?

Thank you.

AndrewN · Answer

Hello,  Vegita . The answer to your question lies in the way the question itself is framed:  what is the  largest possible  number of people who like raspberry jam but do not like either strawberry or apple jam?  Thus, we need to consider an extreme possibility in which we maximize this target group. If we consider an overlap in which certain people enjoy all three jams, we will only  decrease  the number of people who prefer raspberry jam exclusively.

I hope that makes sense. You will find that quite often, the difference between a Hard question and an Easy one is that the former simply requires a more careful  reading —the math might not be any tougher at all.

If you have further questions, please let me know. Good luck with your studies.

- Andrew

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

KarishmaB · Answer

We can consider the 200 people later on and just deal with percentages for now. Assume 100 people. 56 like strawberry, 44 like apple and 30 like both. This means that 56 + 44 - 30 = 70 people like at least one of strawberry and apple jams. Hence out of 100, max 30 people can like ONLY raspberry. We know that 40 people like raspberry so 30 people can like ONLY raspberry. But we have 200 people, not 100 so 60 people can like ONLY raspberry. Answer (B) Check this blog post: https://anaprep.com/sets-statistics-thr ... ping-sets/

einstein801 · Answer

Confusing, there are 60 people who do not like either strawberry or apple jam. but if there are 80  peolpe who like R, where does the 20 shortfall go?

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