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Of a group of people, 10 play piano, 11 play guitar, 14 play

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Re: Overlapping Sets Problem [#permalink] New post 27 Aug 2011, 11:22
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those who play single instrument = 20
those who don't play a single instrument (play 2 or 3) = 35-20 = 15

those who play 2 instruments = 15-3 = 12 (3 play all three instruments)
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Re: Overlapping Sets Problem [#permalink] New post 06 Sep 2011, 11:43
I worked on this using Venn Diagram. Got the answer 6 in one shot. The answer is how many play 2 instruments so we need to count the people who also play 3 instruments ( given already 3).
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Re: Overlapping Sets Problem [#permalink] New post 06 Sep 2011, 20:57
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MBAhereIcome wrote:
those who play single instrument = 20
those who don't play a single instrument (play 2 or 3) = 35-20 = 15

those who play 2 instruments = 15-3 = 12 (3 play all three instruments)


The 35 that you counted: 10 + 11 + 14 double counts the number of people who play only 2 instruments and triple counts the number of people who play all 3 instruments. The circle of 10 of piano includes the red and the green region. The circle of 11 of Guitar includes the red and green region too. Similarly, the circle of 14 of violin includes the red and the green region too. So each red region is counted twice and the green region is counted 3 times.

Attachment:
Ques2.jpg
Ques2.jpg [ 14.89 KiB | Viewed 4098 times ]


So 35 - 20 = 15
15 - 3 = 12 (Subtract once the triple counted green region. Now everything is double counted)
12/2 = 6 = Total number of people who play 2 or 3 instruments.
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink] New post 05 Feb 2012, 01:18
slingfox wrote:
Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument. How many play 2 instruments?

A. 3
B. 6
C. 9
D. 12
E. 15


Total folks = 10 + 11 + 14 = 35
20 Play only one instrument. So no of people playing two or more instruments = 35 - 20 = 15.
From the venn diagrams we know that 15 represents the intersection part between Piano, Guitar and Violin. In the venn diagram, people having all the three capabilities are counted thrice. Hence people playing exactly two instruments = 15 - 3 (3) = 6

Hence B.
Direct formulas may not be applicable all the time. Focus on which area is being counted and how many times.
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink] New post 05 Feb 2012, 01:47
The way I did it was add up all the instruments together, so 10 + 11 +14 = 35. I knew 3 people played all 3, so I subtract 3 from each of the instruments, so 7+8+11 = 26. I know this is the amount of instruments played by a person who plays 1 or 2 instruments. Since the amount of people that play 1 instrument is 20, you subtract all the single instrument from the total. 26-20=6, so 6 instrument. We know that people in this group play 2 instruments, so there is 3 people playing 6 instruments. 3 people who plays exactly 2 instruments + 3 people who plays exactly 3 = 6 people who play 2 or more instruments.
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Re: Overlapping Sets Problem [#permalink] New post 05 Feb 2012, 02:19
VeritasPrepKarishma wrote:
MBAhereIcome wrote:
those who play single instrument = 20
those who don't play a single instrument (play 2 or 3) = 35-20 = 15

those who play 2 instruments = 15-3 = 12 (3 play all three instruments)


The 35 that you counted: 10 + 11 + 14 double counts the number of people who play only 2 instruments and triple counts the number of people who play all 3 instruments. The circle of 10 of piano includes the red and the green region. The circle of 11 of Guitar includes the red and green region too. Similarly, the circle of 14 of violin includes the red and the green region too. So each red region is counted twice and the green region is counted 3 times.

Attachment:
Ques2.jpg


So 35 - 20 = 15
15 - 3 = 12 (Subtract once the triple counted green region. Now everything is double counted)
12/2 = 6 = Total number of people who play 2 or 3 instruments.

Hi, Dont you think the answer should be A i.e. 3 since the question asks for how many people who play 2 instruments and not atleast 2 instruments.
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink] New post 05 Feb 2012, 02:47
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slingfox wrote:
Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument. How many play 2 instruments?

A. 3
B. 6
C. 9
D. 12
E. 15

To elaborate more.

Look at the diagram below:
Attachment:
Union_3sets.gif
Union_3sets.gif [ 11.63 KiB | Viewed 3938 times ]

To solve this question one should fundamentally understand two things:
1. What does the question ask: "How many play 2 instruments?" So, we should find the sum of the sectors 1, 2, 3, and 4. Notice that those who play two instruments include also those who play all three instruments, (sector 4);

2. What happens when we sum all three groups, 10 piano players, 11 guitar players and 14 violin players? When we add these three groups, we'll get 10+11+14=35 but some sections are counting more than once in this number: sections 1, 2, and 3 are counted twice and section 4 thrice. Now, if we subtract those who play only one instrument (inner white sections on the diagram), we'll get 35-20=15, so twice sections 1, 2, and 3 plus thrice section 4 equals to 15.

Since, 15 counts section 4, those who play all the instruments, thrice then of 15-3=12 counts these section twice. So, now 12 counts all sections 1, 2, 3 and 4 twice. We need to count them once thus divide this number by 2 --> 12/2=6 play 2 instruments.

Answer: B.

Detailed analysis of this concept is here: formulae-for-3-overlapping-sets-69014.html#p729340

subhajeet wrote:
Hi, Dont you think the answer should be A i.e. 3 since the question asks for how many people who play 2 instruments and not atleast 2 instruments.


No, if it were the case question would ask: "how many play EXACTLY 2 instruments?" How many play 2 instruments, means how many play at least 2 instruments, hence this group includes also those who play all 3 instruments.

Refer to the link above for more on this issue.

Hope it helps.
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink] New post 06 Feb 2012, 01:20
Bunuel wrote:
slingfox wrote:
Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument. How many play 2 instruments?

A. 3
B. 6
C. 9
D. 12
E. 15

To elaborate more.

Look at the diagram below:
Attachment:
Union_3sets.gif

To solve this question one should fundamentally understand two things:
1. What does the question ask: "How many play 2 instruments?" So, we should find the sum of the sectors 1, 2, 3, and 4. Notice that those who play two instruments include also those who play all three instruments, (sector 4);

2. What happens when we sum all three groups, 10 piano players, 11 guitar players and 14 violin players? When we add these three groups, we'll get 10+11+14=35 but some sections are counting more than once in this number: sections 1, 2, and 3 are counted twice and section 4 thrice. Now, if we subtract those who play only one instrument (inner white sections on the diagram), we'll get 35-20=15, so twice sections 1, 2, and 3 plus thrice section 4 equals to 15.

Since, 15 counts section 4, those who play all the instruments, thrice then of 15-3=12 counts these section twice. So, now 12 counts all sections 1, 2, 3 and 4 twice. We need to count them once thus divide this number by 2 --> 12/2=6 play 2 instruments.

Answer: B.

Detailed analysis of this concept is here: formulae-for-3-overlapping-sets-69014.html#p729340

subhajeet wrote:
Hi, Dont you think the answer should be A i.e. 3 since the question asks for how many people who play 2 instruments and not atleast 2 instruments.


No, if it were the case question would ask: "how many play EXACTLY 2 instruments?" How many play 2 instruments, means how many play at least 2 instruments, hence this group includes also those who play all 3 instruments.

Refer to the link above for more on this issue.

Hope it helps.


Bunnel: Thanks for the explanation. But it becomes a bit confusing here. Looks like i have to extra vigilent for these type of statements.
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Of a group of people, 10 play piano, 11 play guitar [#permalink] New post 22 Aug 2012, 08:49
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Question wordings are very poor or at least very ambiguous.
"How many play 2 instruments?" can be inferred in 2 ways (both of which are right) which creates unnecessary confusion
1) Play Exactly 2 instruments
2) Play at least 2 instruments
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink] New post 15 Aug 2013, 13:58
Yes its language is not correct someone should edit it "how many play at least two instrument" or Change the OA to - A.

I also got 3 :( :roll: :shock:

when i saw 26% correct answers i thought its a tough problem but later i discovered this issue, which everyone else has also reported.
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink] New post 15 Aug 2013, 14:18
slingfox wrote:
Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument. How many play 2 instruments?

A. 3
B. 6
C. 9
D. 12
E. 15


guitar = 11
piano = 10
violin = 14

1 or 2 instruments player = 10-3 +11-3 +14-3 = 35-9 = 26
1 instruments player = 20
so 2 instruments player = 26-20 = 6 =B
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Re: Overlapping Sets Problem [#permalink] New post 15 Aug 2013, 19:05
VeritasPrepKarishma wrote:
MBAhereIcome wrote:
those who play single instrument = 20
those who don't play a single instrument (play 2 or 3) = 35-20 = 15

those who play 2 instruments = 15-3 = 12 (3 play all three instruments)


The 35 that you counted: 10 + 11 + 14 double counts the number of people who play only 2 instruments and triple counts the number of people who play all 3 instruments. The circle of 10 of piano includes the red and the green region. The circle of 11 of Guitar includes the red and green region too. Similarly, the circle of 14 of violin includes the red and the green region too. So each red region is counted twice and the green region is counted 3 times.

Attachment:
Ques2.jpg


So 35 - 20 = 15
15 - 3 = 12 (Subtract once the triple counted green region. Now everything is double counted)
12/2 = 6 = Total number of people who play 2 or 3 instruments.


The question only states that how many play 2 instruments.
Nowhere it is mentioned that it asks for only 2 instruments or atleast 2 instruments???
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Re: Overlapping Sets Problem [#permalink] New post 15 Aug 2013, 21:26
Expert's post
jaituteja wrote:
VeritasPrepKarishma wrote:
MBAhereIcome wrote:
those who play single instrument = 20
those who don't play a single instrument (play 2 or 3) = 35-20 = 15

those who play 2 instruments = 15-3 = 12 (3 play all three instruments)


The 35 that you counted: 10 + 11 + 14 double counts the number of people who play only 2 instruments and triple counts the number of people who play all 3 instruments. The circle of 10 of piano includes the red and the green region. The circle of 11 of Guitar includes the red and green region too. Similarly, the circle of 14 of violin includes the red and the green region too. So each red region is counted twice and the green region is counted 3 times.

Attachment:
Ques2.jpg


So 35 - 20 = 15
15 - 3 = 12 (Subtract once the triple counted green region. Now everything is double counted)
12/2 = 6 = Total number of people who play 2 or 3 instruments.


The question only states that how many play 2 instruments.
Nowhere it is mentioned that it asks for only 2 instruments or atleast 2 instruments???


I understand the confusion. But the language of sets is very mathematical and literal. Say, if I ask you, whether you play an instrument and you have to answer in yes or no, you will say yes even if you play 2/3/4 instruments. Similarly, if I ask you whether you play two instruments, you will answer yes even if you play 3/4 instruments.

No of people who play two instruments includes number of people who play more than 2 too since these people certainly do play 2 instruments (and they play some more).

If 'at least' or 'only' is mentioned, we will consider it accordingly.
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink] New post 16 Aug 2013, 00:57
Expert's post
PiyushK wrote:
Yes its language is not correct someone should edit it "how many play at least two instrument" or Change the OA to - A.

I also got 3 :( :roll: :shock:

when i saw 26% correct answers i thought its a tough problem but later i discovered this issue, which everyone else has also reported.


The correct answer is B, not A.
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink] New post 18 Aug 2013, 05:09
PiyushK wrote:
Yes its language is not correct someone should edit it "how many play at least two instrument" or Change the OA to - A.

I also got 3 :( :roll: :shock:

when i saw 26% correct answers i thought its a tough problem but later i discovered this issue, which everyone else has also reported.


The question never said exactly 2 instrument; had it done so your answer would have been correct, but the question said people who play two instrument which might include 3 people. hence the solution is exactly 2 once and all three once; 3+3=6
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink] New post 18 Aug 2013, 10:36
slingfox wrote:
Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument. How many play 2 instruments?

A. 3
B. 6
C. 9
D. 12
E. 15

play only piano 10-3=7
Play only guitar 11-3=8
Play only violin 14-3=11
so, 20=7+8+11-2overlapping set
2overlapping set=26-20=6
so, the best answer is (B)
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink] New post 14 Sep 2013, 02:09
saxenaashi wrote:
slingfox wrote:
Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument. How many play 2 instruments?

A. 3
B. 6
C. 9
D. 12
E. 15


Total folks = 10 + 11 + 14 = 35
20 Play only one instrument. So no of people playing two or more instruments = 35 - 20 = 15.
From the venn diagrams we know that 15 represents the intersection part between Piano, Guitar and Violin. In the venn diagram, people having all the three capabilities are counted thrice. Hence people playing exactly two instruments = 15 - 3 (3) = 6

Hence B.

Direct formulas may not be applicable all the time. Focus on which area is being counted and how many times.


Here "6" that you have got is counted twice i.e. 2(Exactly two) = 6 Therefore 3 will be exactly two..

Hence if question asks 2 instruments then it will be Exactly 2 + All = 3 + 3 = 6 people..
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Re: Tough Overlapping Set Problem- Veritas [#permalink] New post 20 Dec 2013, 08:15
35 TOTPeople - 20 individual instr - 3 ABC = 12 remain for 2 instr / 2 instr = ANS:6 people play 2
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink] New post 05 Feb 2014, 10:24
Hi,

I just used a very simple approach hope that might help some people who are trying to avoid cramming up the formula

People playing all three instruments = 3

Now subtract this from each group to remove the common part for all 3

Piano(Two or one) = 7
Guitar(Two or one) = 8
Violin(Two or one) = 11

Now if we sum all of the above we will get the number of people who play either two or one instrument
Sum(Two or one) = 26

Question tells us that there are 20 people who play exactly 1 instrument hence

26-20 = 6 = People playing two instruments

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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink] New post 24 Sep 2014, 07:36
Poorly written question,

if the question was how many people play at least 2 instruments than the answer would be 6

If the question was, as is implied by the majority of the answers, how many people played solely 2 instruments the answer would be 3.
Re: Of a group of people, 10 play piano, 11 play guitar, 14 play   [#permalink] 24 Sep 2014, 07:36
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