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Among 100 students in a college, 50 students play soccer, 40 students 14 Apr 2019, 07:43
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Among 100 students in a college, 50 students play soccer, 40 students play hockey and 60 students play tennis. If each student plays at least 1 sport and the number of students who play exactly 2 sports is not less than 3 times the number of students who play all the 3 sports, what is the maximum number of students who play all the three sports?


A. 8
B. 10
C. 15
D. 25
E. 50
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B
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Among 100 students in a college, 50 students play soccer, 40 students 14 Apr 2019, 19:20
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mangamma
Among 100 students in a college, 50 students play soccer, 40 students play hockey and 60 students play tennis. If each student plays at least 1 sport and the number of students who play exactly 2 sports is not less than 3 times the number of students who play all the 3 sports, what is the maximum number of students who play all the three sports?


A. 8
B. 10
C. 15
D. 25
E. 50
Show more

Let the number of students playing exactly 2 and 3 games be x and y...and \(x\geq{3y}\)..
The formula would give us the equation..
100=50+60+40-x-2y....x+2y=50.......We have added two times twice and three times thrice when we add all three 50+60+40, so we subtract X once and y twice
We are trying to maximize y, so take least value of x, that is 3y..
So 3y+2y=50....y=10

B
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Among 100 students in a college, 50 students play soccer, 40 students 30 Apr 2020, 19:41
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chetan2u
mangamma
Among 100 students in a college, 50 students play soccer, 40 students play hockey and 60 students play tennis. If each student plays at least 1 sport and the number of students who play exactly 2 sports is not less than 3 times the number of students who play all the 3 sports, what is the maximum number of students who play all the three sports?


A. 8
B. 10
C. 15
D. 25
E. 50

Let the number of students playing exactly 2 and 3 games be x and y...and \(x\geq{3y}\)..
The formula would give us the equation..
100=50+60+40-x-2y....x+2y=50.......We have added two times twice and three times thrice when we add all three 50+60+40, so we subtract X once and y twice
We are trying to maximize y, so take least value of x, that is 3y..
So 3y+2y=50....y=10

B
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chetan2u

In your solution I couldn't understand how x becomes 3y. Could you explain how did you conclude x=3y ?
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Among 100 students in a college, 50 students play soccer, 40 students 07 May 2020, 20:05
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using venn diagram,
number of students who play only soccer and hockey = a
number of students who play only tennis and hockey = b
number of students who play only tennis and soccer = c
number of students who play all three = d

Given that, Every student plays atleast one sport. and a+b+c >= 3d

Hence, 100 = 50-a-b-d+40-a-c-d+60-b-d-c+a+b+c+d

=> 100 = 150-a-b-2d-c
=> 50-2d = a+b+c
=> 50-2d >= 3d
=> 50 >= 5d
=> d <= 10

Hence maximum value of d = 10
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Among 100 students in a college, 50 students play soccer, 40 students 07 May 2020, 20:09
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ammuseeru
chetan2u
mangamma
Among 100 students in a college, 50 students play soccer, 40 students play hockey and 60 students play tennis. If each student plays at least 1 sport and the number of students who play exactly 2 sports is not less than 3 times the number of students who play all the 3 sports, what is the maximum number of students who play all the three sports?


A. 8
B. 10
C. 15
D. 25
E. 50

Let the number of students playing exactly 2 and 3 games be x and y...and \(x\geq{3y}\)..
The formula would give us the equation..
100=50+60+40-x-2y....x+2y=50.......We have added two times twice and three times thrice when we add all three 50+60+40, so we subtract X once and y twice
We are trying to maximize y, so take least value of x, that is 3y..
So 3y+2y=50....y=10

B

chetan2u

In your solution I couldn't understand how x becomes 3y. Could you explain how did you conclude x=3y ?
Show more


It's given that number of students who play exactly 2 sports is not less than 3 times the number of students who play all the 3 sports.

That means, number of students who play exactly 2 sports is equal to or greater than 3 times the number of students who play all the 3 sports.

number of students who play exactly 2 sports = x
number of students who play all the 3 sports = y

=> x >= 3y
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Among 100 students in a college, 50 students play soccer, 40 students 27 Aug 2020, 22:52
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Given: Among 100 students in a college, 50 students play soccer, 40 students play hockey and 60 students play tennis.
Asked: If each student plays at least 1 sport and the number of students who play exactly 2 sports is not less than 3 times the number of students who play all the 3 sports, what is the maximum number of students who play all the three sports?

Let the number of students playing all three sports be x
Exactly Two >= 3x
Total = A + B + C - Exactly Two + -2* Three + None
100 <= 50 + 40 + 60 - 3x - 2x + 0
5x <= 50
x <=10

IMO B
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Among 100 students in a college, 50 students play soccer, 40 students 24 May 2025, 14:18
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