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Manager  Status: D-Day is on February 10th. and I am not stressed
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Each of the 59 members in a high school class is required  [#permalink]

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5
30 00:00

Difficulty:   15% (low)

Question Stats: 83% (01:52) correct 17% (02:24) wrong based on 583 sessions

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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
Math Expert V
Joined: 02 Sep 2009
Posts: 62290

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3
8
manalq8 wrote:
QUESTION

why did u multiply 2*(two clubs)?

Attachment: Union_3sets.gif [ 11.63 KiB | Viewed 30238 times ]
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

Translating:
"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"
Total=59;
Neither=0 (as members are required to sign up for a minimum of one);
"27 students for the history club": H=27;
"28 students for the writing club": W=28;
"6 students sign up for exactly two clubs": {Exactly 2 groups members}=6, so sum of sections 1, 2, and 3 is given to be 6, (among these 6 students there are no one who is the member of ALL 3 clubs)

"How many students sign up for all three clubs": question is $$PnHnW=x$$. Or section 4 =?

Apply formula: $$Total=P+H+W -$${Sum of Exactly 2 groups members}$$-2*PnHnW + Neither$$ --> $$59=22+27+28-6-2*x+0$$ --> $$x=6$$.

For more check ADVANCED OVERLAPPING SETS PROBLEMS
Similar problem at: ps-question-94457.html#p728852

Hope it helps.
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Re: Each of the 59 members in a high school class is required  [#permalink]

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11
3
The following may be helpful to those struggling with the logic of the formula Total=P+H+W -{Sum of Exactly 2 groups members}-2*PnHnW + Neither .

First, forget this formula for a moment and rather recall the first "easier" formula: Total = A + B + C - (all cases of 2 group overlap, which equals AnB + AnC + BnC) + (the single 3 group overlap, which is AnBnC, or g) + (neither)

Recall that when we subtract all cases of group overlap, AnB + AnC + BnC, then we also subtract the 3 group overlap, g, 3 times. Since we add g 3 times when we add A +B + C, we are back at 0 regarding g, hence why we add it back following this subtraction, as you can see up there in the first formula.

In the gmat question we have above, the "simpler" first formula rendition would therefore go: 59 = 22 + 27 + 28 - (AnB + AnC + BnC) + g + 0

We know that the total number of members in exactly two groups (so NOT the ones in all three) is 6.

Now, can we make AnB + AnC + BnC just equal 6? No, since AnB, and the rest of them, also include those members of all three groups, and thus may be larger than 6. The work around is that we simply need to account for g in each case, by including it explicitly in the formula and then set the part that is NOT g equal to 6.

Hence, AnB = (some part of the total 6) + g, and the same for the others. Just to give 'the part of the total 6' on A and B's side a name, let's substitute 'anb'. Don't get confused here: anb is just AnB with g subtracted.

Hence AnB + AnC + BnC = (anb + g) + (anc + g) + (bnc + g)

But what does anb equal? We don't know, but what we DO know that anb + anc + bnc, all the sets that have ONLY members of two and no more or less, is equal to 6.

Hence anb + anc + bnc = 6

With this understood we just replace anb, anc, and bnc with 6.

Hence, (AnB + AnC + BnC) = (6 + g + g + g)

Back to the first formula rendition, we now have 59 = 22 + 27 + 28 - (6 + g + g +g) + g + 0
Simplified (a little): 59 = 22 + 27 + 28 - 6 - g - g - g + g + 0
Simplified (a little more): 59 = 22 + 27 + 28 - 6 - 2g + 0

But wait, this just is the second formula!

Second formula: Total = P+H+W -{Sum of Exactly 2 groups members}-2*PnHnW + Neither

Where we left off: 59 = 22 + 27 + 28 - 6 - 2g + 0

You can now hopefully see WHY 6, the sum of those in exactly 2 groups, is subtracted, as well as WHY 2g is subtracted.
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10
2
59 - poetry - history - writing + 2*(two clubs) + three clubs = 0
59 - 22 - 27 - 28 + 12 + three clubs = 0
three clubs = 6

C
Manager  Status: D-Day is on February 10th. and I am not stressed
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QUESTION

why did u multiply 2*(two clubs)?
Manager  Joined: 12 Feb 2012
Posts: 114
Re: Each of the 59 members in a high school class is required  [#permalink]

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Hey Bunuel,

Dude your awesome! Just a quick question, as I like to attack problems from different methods. The forumula you used is perfect but I also learned another cool formula from another problem which you can see here:

overlapping-sets-84100.html
ps-venn-diagrams-77473.html

The formula involves finding the minimum value for the intersection of all the sets (A and B and C) in a three overlapping set problem.
Here it is:
According to a survey, at least 70% of people like apples, at least 75% like bananas and at least 80% like cherries. What is the minimum percentage of people who like all three?

The way it is attacked is by finding the # of people that arent in the set

So the solution to the above problem I posted would be :
A and B and C = Total - [(Total-A) + (Total-B) + (Total-C)]

My question is if I were to use this method to find the solution I get weird numbers.

Total=59
A=Poetry=22
B=History=27
C=Writing=39

A and B and C =59-[(59-22)+(59-32)+(59-28)]=49. Actual answer is 6.

Why does this formula work on the problem I posted but not on this question?

Your my hero dude. Thank you!
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Re: Each of the 59 members in a high school class is required  [#permalink]

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Hi,
the question states that -A total of 22 students sign up for the poetry club which means 22 is the whole of poetry club including the intersections with other clubs....Can you please explain how do we know if poetry is exclusive of the intersections?
Math Expert V
Joined: 02 Aug 2009
Posts: 8298
Re: Each of the 59 members in a high school class is required  [#permalink]

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irishiraj87 wrote:
Hi,
the question states that -A total of 22 students sign up for the poetry club which means 22 is the whole of poetry club including the intersections with other clubs....Can you please explain how do we know if poetry is exclusive of the intersections?

it is inclusive....
P+N+W-(sum of two clubs)..... Means that P and N and W are inclusive and you are subtracting repeated elements from the SUM
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Re: Each of the 59 members in a high school class is required  [#permalink]

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2
I wasn't all that familiar with the overlapping concept, but I solved it through drawing a diagram -- posting in case anyone is looking for a different logic.

Attachment: IMG_0282.JPG [ 1.45 MiB | Viewed 5854 times ]

**The shaded areas cover the 6 students who joined exactly 2 clubs.

Total # of students = (3club) + (2club) + (1club).
3-club: X
2-club: 6
1-club: P+H+W-(2clubs)-(3clubs)
Note that when we add the given P H W values together, we have added each of the shaded 2-club areas twice, and the center 3-club area three times, so be sure to account for that in your equation.

59=X+6+(22+27+28-(2*6) - 3X)
2X=6+77-12-59
2X=12
X=6.
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Re: Each of the 59 members in a high school class is required  [#permalink]

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Hi house, i just went through the advanced overlapping set topic and it is very well helpful. God bless the hands responsible for putting it all together. However i have a question pls and it goes thus; How do i know when to use the first formular i.e TOTAL= A+B+C-(SUM OF 2-GROUP OVERLAPS)+ALL THREE+NEITHER and when to use second formular i.e TOTAL=A+B+C-(SUM OF EXACTLY 2-GROUP OVERLAPS)+2(ALL THREE)+NEITHER.
i need a quick response its just a few days to my test. Re: Each of the 59 members in a high school class is required   [#permalink] 03 Nov 2019, 01:16
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