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Each of the hundred students in a class enrolled in at least one of

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Bunuel
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Each of the hundred students in a class enrolled in at least one of [#permalink] New post  01 Feb 2023, 12:45
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Each of the hundred students in a class enrolled in at least one of the three activities: Singing, Dancing, and Theatre. If 40 participated in Singing, 50 participated in Dancing and 40 participated in Theatre and if 10 participated in all the three activities, how many students participated in at least two activities?

A. 10
B. 20
C. 24
D. 30
E. 32
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yashikaaggarwal
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Re: Each of the hundred students in a class enrolled in at least one of [#permalink] New post  01 Feb 2023, 15:55
The answer is B, you can refer to the solution in the picture in the attachment.

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Re: Each of the hundred students in a class enrolled in at least one of [#permalink] New post  02 Feb 2023, 03:26
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Bunuel wrote:
Each of the hundred students in a class enrolled in at least one of the three activities: Singing, Dancing, and Theatre. If 40 participated in Singing, 50 participated in Dancing and 40 participated in Theatre and if 10 participated in all the three activities, how many students participated in at least two activities?

A. 10
B. 20
C. 24
D. 30
E. 32


Solution:

  • We can make the following venn diagram based on this diagram
    Attachment:
    singdance.png
    singdance.png [ 11.82 KiB | Viewed 448 times ]
  • According to the venn diagram, \(a+b+c+d+e+f+10=100\) or \(a+b+c+d+e+f=90........(i)\)
  • We also know
    • 40 participated in Singing i.e., \(a+d+e+10=40\) or \(a+d+e=30........(ii)\)
    • 50 participated in Dancing i.e., \(b+e+f+10=50\) or \(b+e+f=40........(iii)\)
    • 40 participated in Theatre i.e., \(c+d+f+10=40\) or \(c+d+f=30........(iv)\)
  • We are asked the value of \(d+e+f+10\)

  • Upon adding equations ii, iii and iv, we get: \((a+b+c+d+e+f)+d+e+f=30+40+30\)
    \(⇒90+d+e+f=100\)
    \(⇒d+e+f=10\)
    \(⇒d+e+f+10=20\)

Hence the right answer is Option B
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Re: Each of the hundred students in a class enrolled in at least one of [#permalink] New post  02 Feb 2023, 10:15
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Bunuel wrote:
Each of the hundred students in a class enrolled in at least one of the three activities: Singing, Dancing, and Theatre. If 40 participated in Singing, 50 participated in Dancing and 40 participated in Theatre and if 10 participated in all the three activities, how many students participated in at least two activities?

A. 10
B. 20
C. 24
D. 30
E. 32


We can create the following equation:

Total = Sing + Dance + Theater - Double - 2(Triple) + Neither

100 = 40 + 50 + 40 - D - 2 x 10 + 0

100 = 130 - D - 20

100 = 110 - D

10 = D

Thus, 10 + 10 = 20 students participate in at least two activities.

Answer: B
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Re: Each of the hundred students in a class enrolled in at least one of [#permalink]
02 Feb 2023, 10:15
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