Van Diagram II

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hakyology: Senior | Next Rank: 100 Posts; Posts: 66; Joined: Tue Jun 05, 2007 1:39 pm; Thanked: 1 times

by hakyology » Mon May 26, 2008 4:50 pm

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?
2
5
6
8
9

The answer is 6. Please explain. Thank you in advance.

59= 22+27+28-6+x
59=71+x (????)

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Source: Beat The GMAT — Problem Solving | Mon May 26, 2008 4:50 pm

DavidP: Site Admin; Posts: 387; Joined: Sun Feb 17, 2008 4:25 pmFollowed by: 29 members; Thanked: 17 times

by DavidP » Mon May 26, 2008 6:06 pm

Total # of signups that all students made is 77 (22 + 27 + 28 = 77)

Assume that:

x = # of students who signed up for 1 class

y = # of students who signed up for 2 classes

z = # of students who signed up for 3 classes

Based on the conditions of the question, we know the following 3 equations are true:

1. 77 = x + 2y + 3z

2. x + y + z = 59

3. y = 6 (since 6 students signed up for exactly 2 classes)

So then, we can deduce the value for x

x = 59 - y - z = 59 - 6 -z = 53 - z

Finally, we can calculate a value for z

77 = (53 - z) + 2*6 + 3z = 65 + 2z

2z = 77 - 65 = 12

z = 6

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Sunny143: Junior | Next Rank: 30 Posts; Posts: 10; Joined: Sat Mar 15, 2008 1:10 pm

Re:

by Sunny143 » Mon May 26, 2008 6:12 pm

I could not attach the Venn diagram, so you got to visualize here. Consider the following variables

p - no. of student JUST attending Poetry club
h - no of students attending JUST the history club
w - no. of students attending JUST the writing club
x - just attending poetry and history(part above all 3 area)
y - just attending history and writing
z - just attending writing and poetry
c - attending all the 3 (which we intend to find).

The equations are:
p+h+w+(x+y+z)+c = 59 =>p+h+w+c = 59-6=53 .......1
p+x+z+c = 22.........2
h+x+y+c = 27.........3
w+y+z+c = 28.........4

adding 2, 3 and 4
p+h+w+2(x+y+z)+3c = 77
=>p+w+h+3c = 65 ................5

5-1 gives
2c = 12 => c = 6.

The equations might seems daunting at first, but once iu have the venn diagram in front of you, these are effortless and moreover you dont have to remember any formulae. This system follows from the question stem but one has to be careful with the calculations as with any GMAT problem.

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