CAT Overlapping Sets-Club Requirements Problem

[curioustester]

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?

a)2
b)5
c)6
d)8
e)9

I tried using the formula:
Total=P+H+W-(PH+PW+HW)+PHW
59=22+27+28-(6)+PHW but this doesn't work out to an answer choice...

The solution uses a venn diagram and different equations, but I had seen this formula on another thread for another overlapping sets and tried to apply it to this problem.

[JK]

You need to start thinking like the test maker. Use the answer choices and work your way from the inside out on a venn diagram. Start with C - you can then eliminate two answer choices if it is wrong.

C says 6 for three clubs
Assume Poetry and History share the 6 for students in two clubs

P = 22 - 6 (for two clubs) - 6 (for three clubs) = 10
H = 27 - 6 (for two clubs) - 6 (for three clubs) = 15
W = 28 - 6 (for three clubs) = 22

P+H+W+twoclubs+threeclubs = 59
10 + 15 + 22 + 6 + 6 = 59

Boom

[JonathanSchneider]

The reason that your formula failed you here is because that middle part is not actually 6. PH as you've written it means the entire overlap between Poetry and History, but this INCLUDES those who are also taking Writing. Likewise, PW and HW each include those who are engaged in all three clubs (on the Venn Diagram, this can be seen as the middle section). In this problem, however, the 6 represents the number of people engaged in exactly two clubs (not two or more). As a result, you cannot simply insert the 6 there.

If you still want to think about it with a formula, you need to adjust. Since we're no longer canceling out the middle part three times when we subtract the 6, we need to subtract it twice still:

Total = P + H + W - 6 (this is just the people in two clubs) - 2PHW (we subtract this twice because it has been counted three times, once per single letter)

As the PHW is what we're after, call that x and insert the rest of the known figures:

59 = 22 + 27 + 28 - 6 - 2x
59 = 71 - 2x
-12 = -2x
x = 6

[malikrulzz]

I agree with both of u here but I am confised because ques says "If 6 students sign up for exactly two clubs". Now 6 is PW, PH and HW. How can u place 6 in each group. What I mean to say is

P = 22 - 6 (for two clubs) - 6 (for three clubs) = 10
H = 27 - 6 (for two clubs)- I didn't get this - 6 (for three clubs) = 15

[JonathanSchneider]

Perhaps you mistook my meaning? I don't mean that you should subtract the 6 twice. In fact, the 6 doesn't fit very nicely with the standard formula. This is because the formula uses the symbols PW, PH, and HW to represent the total overlap between each of those two groups, including the part of each overlap that ALSO overlaps with the third group. But the 6 in this problem represents the total number of people involved in only two groups. In fact, this does not fit evenly into our formula. Thus, we need to adjust conceptually, as I explained above.