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In a certain class, all the students like at least one of the followin

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Bunuel
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In a certain class, all the students like at least one of the followin [#permalink] New post   09 Oct 2022, 04:45
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In a certain class, all the students like at least one of the following TV shows: Game of Thrones, Friends, and Big Bang Theory. It was found that the percentage of students who like the shows are 55, 86, and 69 respectively. If x% of the students in the class like all three TV shows, what is the ratio of the maximum to the minimum value of x?

A. 11:1
B. 6:1
C. 11:2
D. 4:1
E. 3:1
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C
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ankitgoswami
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Re: In a certain class, all the students like at least one of the followin [#permalink] New post   09 Oct 2022, 09:01
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Bunuel wrote:
In a certain class, all the students like at least one of the following TV shows: Game of Thrones, Friends, and Big Bang Theory. It was found that the percentage of students who like the shows are 55, 86, and 69 respectively. If x% of the students in the class like all three TV shows, what is the ratio of the maximum to the minimum value of x?

A. 11:1
B. 6:1
C. 11:2
D. 4:1
E. 3:1


Maximum Value

All = GOT + F + BBT - only GOT&F - only F&BBT - only GOT&BBT - 2 (GOT&F&BBT) + Neither

100 = 55 + 86 + 69 - only 2 - 2 (all 3) + 0

Only 2 + 2 (all 3) = 210 - 100

To maximise all 3, only 2 be 0

2 (all 3) = 110

all 3 = 55 , which is equal to the min value of all 3 so it is possible

Minimum Value

Minimum Value of x means minimum overlap of all 3
That means max separation of all the sub-sets

So % of students not liking GOT = 45
% of students not liking F = 14
% of students not liking BBT = 31

% of students liking all 3 = 100 - 45 -14 -31 = 10

Alternatively, make a Venn diagram and try to fit the lowest value i.e. GOT (55) in the portion of only F and only BBT, the balance which cannot be placed goes into the 'all 3' portion. Same result, all 3 = 10

Ratio = \(\frac{55}{10}\)
= \(\frac{11}{2}\)

Ans is C
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Re: In a certain class, all the students like at least one of the followin [#permalink] New post   10 Oct 2022, 15:42
ankitgoswami wrote:
Bunuel wrote:
In a certain class, all the students like at least one of the following TV shows: Game of Thrones, Friends, and Big Bang Theory. It was found that the percentage of students who like the shows are 55, 86, and 69 respectively. If x% of the students in the class like all three TV shows, what is the ratio of the maximum to the minimum value of x?

A. 11:1
B. 6:1
C. 11:2
D. 4:1
E. 3:1


Maximum Value

All = GOT + F + BBT - only GOT&F - only F&BBT - only GOT&BBT - 2 (GOT&F&BBT) + Neither

100 = 55 + 86 + 69 - only 2 - 2 (all 3) + 0

Only 2 + 2 (all 3) = 210 - 100

To maximise all 3, only 2 be 0

2 (all 3) = 110

all 3 = 55 , which is equal to the min value of all 3 so it is possible

Minimum Value

Minimum Value of x means minimum overlap of all 3
That means max separation of all the sub-sets

So % of students not liking GOT = 45
% of students not liking F = 14
% of students not liking BBT = 31

% of students liking all 3 = 100 - 45 -14 -31 = 10

Alternatively, make a Venn diagram and try to fit the lowest value i.e. GOT (55) in the portion of only F and only BBT, the balance which cannot be placed goes into the 'all 3' portion. Same result, all 3 = 10

Ratio = \(\frac{55}{10}\)
= \(\frac{11}{2}\)

Ans is C



Can you explain how you got to the min value? I'm not understanding this portion -
So % of students not liking GOT = 45
% of students not liking F = 14
% of students not liking BBT = 31
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Re: In a certain class, all the students like at least one of the followin [#permalink] New post   11 Oct 2022, 01:27
ankitgoswami wrote:
Bunuel wrote:
In a certain class, all the students like at least one of the following TV shows: Game of Thrones, Friends, and Big Bang Theory. It was found that the percentage of students who like the shows are 55, 86, and 69 respectively. If x% of the students in the class like all three TV shows, what is the ratio of the maximum to the minimum value of x?

A. 11:1
B. 6:1
C. 11:2
D. 4:1
E. 3:1


Maximum Value

All = GOT + F + BBT - only GOT&F - only F&BBT - only GOT&BBT - 2 (GOT&F&BBT) + Neither

100 = 55 + 86 + 69 - only 2 - 2 (all 3) + 0

Only 2 + 2 (all 3) = 210 - 100

To maximise all 3, only 2 be 0

2 (all 3) = 110

all 3 = 55 , which is equal to the min value of all 3 so it is possible

Minimum Value

Minimum Value of x means minimum overlap of all 3
That means max separation of all the sub-sets

So % of students not liking GOT = 45
% of students not liking F = 14
% of students not liking BBT = 31

% of students liking all 3 = 100 - 45 -14 -31 = 10

Alternatively, make a Venn diagram and try to fit the lowest value i.e. GOT (55) in the portion of only F and only BBT, the balance which cannot be placed goes into the 'all 3' portion. Same result, all 3 = 10

Ratio = \(\frac{55}{10}\)
= \(\frac{11}{2}\)

Ans is C


I get the min part but could you please explain why does it have to be 2 times as bolded red above?
Thank you very much.
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Re: In a certain class, all the students like at least one of the followin [#permalink] New post   11 Oct 2022, 04:13
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GOT :55
Friends :86
TBBT :69

the maximum possible value for x is the smallest value of all the three which is 55.

percentage who doesn't like GOT is 100-55 = 45; TBBT is 100 -69 = 31; Friends is 100-86 = 14;
So the minimum value of x is 100-45-31-14 = 10

Hence the ratio is 55/10 = 11:2
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Re: In a certain class, all the students like at least one of the followin [#permalink] New post   23 Oct 2022, 03:47
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bellbell wrote:
I get the min part but could you please explain why does it have to be 2 times as bolded red above?
Thank you very much.


When you add GOT, F, and BBT, students who like all 3 get added thrice and hence we have to remove them twice to ensure they are counted once.
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Re: In a certain class, all the students like at least one of the followin [#permalink] New post   23 Oct 2022, 14:33
Further explain on the minimum Value of x (like all three):

first, to continue on the equation (provided by ankitgoswami) for the max value of X:
All = G + F + B - onlyG&F - onlyF&B - onlyG&B - 3x + Neither
100 = 55 + 86 + 69 - onlyG&F - onlyF&B - onlyG&B - 2x + 0

to get min value for x, we need to get the max of only (G&F, F&B and G&B )
the max of onlyG&F = G&F -x (because G&F contains a x that like all three TV shows)
onlyG&F = 55+86-100-x= 41-x
then
the max of onlyF&B= 86+69-100-x=55-x
The max of onlyG&B= 55+69-100-x=24-x

Return to the original equation and introduce above value for max of only (G&F, F&B and G&B)
100 = 55 + 86 + 69 - (41-x) - (55-x) - (24-x) - 2x + 0
100= 210-41-55-24+3x-2x = 90+x
x= 100-90=10
so the ratio of max to the min value of x is
55/10=11/2
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Re: In a certain class, all the students like at least one of the followin [#permalink]
23 Oct 2022, 14:33
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