Quant Cat exam 5 #23 overlapping sets

[sanyu4]

Per question below using the intuitive method, how do you get to the final answer of 6 from 12 duplicates? I understand all the steps until 12 duplicates.

Thank you

Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?

Alternatively, we can use a more intuitive approach to solve this problem. If we add up the total number of club sign-ups, or registrations, we get 22 + 27 + 28 = 77. We must remember that this number includes overlapping registrations (some students sign up for two clubs, others for three). So, there are 77 registrations and 59 total students. Therefore, there must be 77 - 59 = 18 duplicate registrations.

We know that 6 of these duplicates come from those 6 students who sign up for exactly two clubs. Each of these 6, then, adds one extra registration, for a total of 6 duplicates. We are then left with 18 - 6 = 12 duplicate registrations. These 12 duplicates must come from those students who sign up for all three clubs.

For each student who signs up for three clubs, there are two extra sign-ups. Therefore, there must be 6 students who sign up for three clubs:

12 duplicates / (2 duplicates/student) = 6 students

Between the 6 students who sign up for two clubs and the 6 students who sign up for all three, we have accounted for all 18 duplicate registrations.

The correct answer is C.

[arekgalat]

Nice approach sanyu4.

I used other one.
We have 79 registrations vs 59 students
we know that we have 6 students in two groups. Therfore, we have: 77 - (6*2) = 65 reg. and 59 - 6 = 53 stud.
We have 65 reg. vs. 53 stud.
We need to get eventually the equal numbers so we need to eliminate the three group students. To do so we put the equation: 65 - 3x = 53 - x
12=2x
x=6

[sanyu4]

thanks

[tim]

:)

[rmm135]

I recently saw this question on a CAT and am looking for another approach. Is it possible to do this with the inside-out Venn Diagrams? I like sanyu4's approach but I would like to stay consistent with my technique on these problems and I like the Venn Diagrams but I am having a hard time using it here. Any thoughts?

[RonPurewal]

What are "inside-out" venn diagrams?