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Re: In a class of 345 students, equal number of students enrolled in Engli [#permalink]
Bunuel wrote:
In a class of 345 students, equal number of students enrolled in English club, Math club and Science club. 30 students enrolled both in English club and Math club, 26 students enrolled both in Math club and Science club, 28 students enrolled both in Science club and English club and 14 students enrolled in all three clubs. If there are 43 students who didn’t enroll in any of the three clubs, how many students enrolled in both English and Math clubs but not in Science club?

A. 108
B. 124
C. 178
D. 246
E. 286

Are You Up For the Challenge: 700 Level Questions


Given:
1. In a class of 345 students, equal number of students enrolled in English club, Math club and Science club.
2. 30 students enrolled both in English club and Math club, 26 students enrolled both in Math club and Science club, 28 students enrolled both in Science club and English club and 14 students enrolled in all three clubs.

Asked: If there are 43 students who didn’t enroll in any of the three clubs, how many students enrolled in both English and Math clubs but not in Science club?

Please see the solution in attached image.

Number of students enrolled in both English and Math clubs but not in Science club = 80 + 82 + 16 = 178

IMO C
Attachments

Screenshot 2020-04-25 at 11.55.21 PM.png
Screenshot 2020-04-25 at 11.55.21 PM.png [ 47 KiB | Viewed 6465 times ]

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Re: In a class of 345 students, equal number of students enrolled in Engli [#permalink]
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Expert Reply
Bunuel wrote:
In a class of 345 students, equal number of students enrolled in English club, Math club and Science club. 30 students enrolled both in English club and Math club, 26 students enrolled both in Math club and Science club, 28 students enrolled both in Science club and English club and 14 students enrolled in all three clubs. If there are 43 students who didn’t enroll in any of the three clubs, how many students enrolled in both English and Math clubs but not in Science club?

A. 108
B. 124
C. 178
D. 246
E. 286

Are You Up For the Challenge: 700 Level Questions


We can create an equation using the following formula:

Total = N(E) + N(M) + N(S) - N(E and M) - N(M and S) - N(E and S) + N(all three) + N(none)

345 = n + n + n - 30 - 26 - 28 + 14 + 43

345 = 3n - 84 + 57

345 = 3n - 27

372 = 3n

124 = n

The number of students enrolled in either the English club or the Math club but not in the Science club is:

Total - N(S) - N(none)

345 - 124 - 43 = 178

Answer: C

In a class of 345 students, equal number of students enrolled in Engli [#permalink]
CrackVerbalGMAT wrote:
In a question on Venn diagrams, we try to evaluate the word problem in small chunks without trying to consume the whole question in one go.
When the Venn diagram is being drawn, we try to populate the common regions first followed by the exclusive regions i.e. we proceed outwards from the inner regions.
We also ensure that we don’t ignore the numbers that can be outside the circles but inside the rectangle.

Considering all the above, we can draw a Venn diagram for the given question which looks like the one below:

Attachment:
25th April 2020 - Reply 4 - 1.jpg


The question says that equal number of students enrolled for the English, Math and Science clubs. This essentially means that the total number of people in each of these circles should be the same.
Therefore,

a + 44 = b + 42 which gives us a = b – 2

c + 40 = b + 42 which gives us c = b + 2.

Total number of people inside the circles = 345 – People who opted for none of the clubs = 345 – 43 = 302.

Therefore, a + b + c + 14 + 16 + 14 + 12 = 302 OR a + b + c = 246.

Substituting the values of a and c in the above equation and simplifying, we get b = 82, a = 80 and c = 84.
We do not want the students enrolled in the Science Club, therefore, we consider the regions outside the Science circle (marked in red) i.e. a+16 + b = 80 + 16 + 82 = 178.

Attachment:
25th April 2020 - Reply 4 - 2.jpg


The correct answer option is C
.
Hope that helps!


Thanks for the solution, Sir. I understood what you have done. However, I have a doubt about how the question is phrased - "BOTH English and Maths but not Science. Should the answer then not be 16? That is the only area where students are enrolled in both but not in the third subject.

CrackVerbal
ArvindCrackVerbal

Originally posted by iammanas on 23 Aug 2020, 04:57.
Last edited by iammanas on 05 Sep 2020, 01:28, edited 1 time in total.
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Re: In a class of 345 students, equal number of students enrolled in Engli [#permalink]
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iammanas wrote:
CrackVerbalGMAT wrote:
In a question on Venn diagrams, we try to evaluate the word problem in small chunks without trying to consume the whole question in one go.
When the Venn diagram is being drawn, we try to populate the common regions first followed by the exclusive regions i.e. we proceed outwards from the inner regions.
We also ensure that we don’t ignore the numbers that can be outside the circles but inside the rectangle.

Considering all the above, we can draw a Venn diagram for the given question which looks like the one below:

Attachment:
25th April 2020 - Reply 4 - 1.jpg


The question says that equal number of students enrolled for the English, Math and Science clubs. This essentially means that the total number of people in each of these circles should be the same.
Therefore,

a + 44 = b + 42 which gives us a = b – 2

c + 40 = b + 42 which gives us c = b + 2.

Total number of people inside the circles = 345 – People who opted for none of the clubs = 345 – 43 = 302.

Therefore, a + b + c + 14 + 16 + 14 + 12 = 302 OR a + b + c = 246.

Substituting the values of a and c in the above equation and simplifying, we get b = 82, a = 80 and c = 84.
We do not want the students enrolled in the Science Club, therefore, we consider the regions outside the Science circle (marked in red) i.e. a+16 + b = 80 + 16 + 82 = 178.

Attachment:
25th April 2020 - Reply 4 - 2.jpg


The correct answer option is C
.
Hope that helps!


Thanks for the solution, Sir. I understood what you have done. However, I have a doubt with how the question is phrased - "BOTH E and M but not S. Should the reason not be 16? That is the only area where students are enrolled in both but not in the third subject.




Absolutely!!....The question is not asking for the number of students enrolled in Math or English but not Science, it is saying Math AND English but not Science. So the answer should be 16.
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Re: In a class of 345 students, equal number of students enrolled in Engli [#permalink]
I understood the solution but my doubt is : Isn't the final statement confusing? ; "how many students enrolled in both English and Math clubs but not in Science club?". Doesn't this mean E Int M-E int M int S.?
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Re: In a class of 345 students, equal number of students enrolled in Engli [#permalink]
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