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M70-28

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M70-28  [#permalink]

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New post 03 Sep 2018, 06:10
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

56% (01:08) correct 44% (02:00) wrong based on 9 sessions

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Re M70-28  [#permalink]

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New post 03 Sep 2018, 06:10
1
Official Solution:


Of the 27 participants in a talent show, 13 are singers, 14 are musicians, and 9 are dancers. If 4 people are both singers and dancers, 3 are both singers and musicians and 1 person is all three, how many people are both dancers and musicians?


A. 0
B. 1
C. 2
D. 3
E. 4


We’ll go for ALTERNATIVE because the numbers in the answers are easy to use.

Since we have three overlapping sets, we’ll draw a Venn diagram, and instead of going into redundant calculations, we’ll try the median answer (C) 2 and see whether it works out:

The third layer, consisting of all three groups, is made up of 1 person. This makes the number of people who are singers and musicians (but not dancers) 3 – 1 = 2, and the number of singer-dancers (but not musicians) 4 – 1 = 3. The number of people who only sing is 13 – 1 – 2 – 3 = 7. Now, if answer (C) is correct, the number of those who only dance is 9 – 3 – 1 – 1 = 4, and the number of those who are only musicians is 14 – 2 – 1 – 1 = 10. So the overall number of participants is: 7 + 3 + 1 + 2 + 10 + 1 + 4 = 28, instead of 27. (C) is eliminated.

We can just check a random answer, but note that the answer we need to check is actually a larger one in order to make both ‘just musicians’ and ‘just dancers’ groups smaller (and thus making the total number of participants smaller). Let’s see answer choice (D) 3:

As we’ve seen before, we have 1 person in all three groups; 2 who are singers and musicians (but not dancers); 3 singer-dancers (but not musicians); and 7 only sing. Now, if answer (D) is correct, the number of those who only dance is 9 – 3 – 1 – 2 = 3, and the number of those who are only musicians is 14 – 2 – 1 – 2 = 9. So the total number of participants is: 7 + 3 + 1 + 2 + 9 + 2 + 3 = 27. That’s our answer!


Answer: D
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Re: M70-28  [#permalink]

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New post 07 Sep 2018, 08:02
1
shouldn't it be 4?
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Re: M70-28  [#permalink]

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New post 09 Oct 2018, 07:10
can you please explain with the help of Venn diagram
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Re: M70-28  [#permalink]

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New post 02 Nov 2018, 02:04
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since one person is all three, it makes 3 people (both singers and dancers but not musician) and 2 people (both singers and musician but not dancer). So, the number of people who can only sing is 7. If the number of people who is both musician and dancer but not singer is 2 (plus one person who does all three), musician alone is 9 and dancer alone is 3. Singer (7+2+1+3=13), Musician (2+1+2+9=14), Dancer (1+3+2+3=9).


Total = 7+2+1+9+2+3+3=27
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Re: M70-28  [#permalink]

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New post 03 Dec 2018, 04:52
I'm getting a 4 too. (13-3-2-1 +14-2-1 +9-3-1) = 7+11+5= 23, (27-23)=4 . Where am I going wrong ?
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Re: M70-28  [#permalink]

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New post 18 Feb 2019, 01:21
1
Bunuel wrote:
Of the 27 participants in a talent show, 13 are singers, 14 are musicians, and 9 are dancers. If 4 people are both singers and dancers, 3 are both singers and musicians and 1 person is all three, how many people are both dancers and musicians?


A. 0
B. 1
C. 2
D. 3
E. 4


If S=singers, M=musicians and D=dancers, we have:
D∩S=4, S∩M=3 and S∩M∩D=1
We are looking for D∩M=1+x (because S∩M∩D=1)
Dx is the part of circle D without the intersections with the other circles
Mx is the part of circle M without the intersections with the other circles
Thus, we have 3 equations with 3 unknowns:
Mx+Dx+13+x=27
Mx+3+x=14 (Because m=14)
Dx+4+x=9 (Because d=9)

Which gives x=2 and D∩M=1+x=3

Answer (D)

PS: I couldn't upload a file because I have less than 5 posts, but if you make a Venn diagram, you'll see what I am talking about.
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Re M70-28  [#permalink]

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New post 01 Jun 2019, 03:11
I think this is a high-quality question and I don't agree with the explanation. If we are considering D) to be correct while substituting options why are we plugging in 2 instead of 3?
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Re M70-28   [#permalink] 01 Jun 2019, 03:11
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