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overlapping sets

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by coolbreeze2011 » Sat Jul 30, 2011 10:28 am
Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all three instruments, 20 play only one instrument. How many people play two instruments?
A. 3
B. 6
C. 9
D. 12
E. 15

Can someone please explain.
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Source: — Quantitative Reasoning |
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by coolbreeze2011 » Sat Jul 30, 2011 10:39 am
I don't know the correct answer, I just found this problem on some forum. But IMO its C.
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by Fractal » Sun Jul 31, 2011 9:23 am
First step:
10 + 11 + 14 = 35

Second step:
35 - 20 - 3 = 12

Hence I would go with D
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by Ozlemg » Mon Aug 01, 2011 6:56 am
there is a formula for this. Memorize this :

the Q asks exactly how many people play excatly 2 instruments?

s(AUBUC) :A+B+C -2(A and B)-2(A and C)-2(B and C)+ (A+B+C)
So, 10+11+14 = 20+2*x+2*x+2*x+3 = x is 2

2*x*3--> 12
Hence, D
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by coolbreeze2011 » Tue Aug 02, 2011 9:14 am
Thank you so much. I always get confused solving overlapping sets problems. This is very helpful :)
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by sumgb » Sun Aug 14, 2011 11:32 pm
I believe the answer should be 6. here's why -
piano, P = 10
guitar, G = 11
violin, V =14
so P+G+V = 10+11+14 = 35
since only 3 people play all three instrument, subtracting 3 from each group, we get
7+8+11 = 26
but 26 includes people playing only 1 instrument and people playing only 2 instrument
since it is given 20 people play 2 instrument, subtracting that value we get
26 - 20 = 6 people play only 1 instrument.
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