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Among 100 students in a class, 57 like football and 68 like basketball

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Bunuel
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Among 100 students in a class, 57 like football and 68 like basketball [#permalink] New post   06 Feb 2024, 02:30
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E

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35% (medium)

Question Stats:

79% (02:14) correct 21% (02:13) wrong based on 90 sessions
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Among 100 students in a class, 57 like football and 68 like basketball. The maximum possible number of students who like both football and basketball is represented by 'a', and the minimum possible number of students who like both football and basketball is represented by 'b'. What is the value of 'a' minus 'b'?

A. 12
B. 19
C. 23
D. 28
E. 32




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Among 100 students in a class, 57 like football and 68 like basketball. The maximum possible number of students who like both football and basketball is a and the minimum possible number of students who like both football and basketball is b. What is the value of a-b?
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E
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KarishmaB
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Among 100 students in a class, 57 like football and 68 like basketball [#permalink] New post   06 Feb 2024, 03:32
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The overlap is minimum when Neither = 0 and it is maximum when one set lies inside the other set.

Here is a video discussing this with the help of diagrams: Max Min in Overlapping Sets

When Neither = 0,
Total = F + B - Both
100 = 57 + 68 - Both
Both = 25 (minimum value)

Maximum value of Both = 57

Difference = 57 - 25 = 32

Answer (E)
Bunuel wrote:
Among 100 students in a class, 57 like football and 68 like basketball. The maximum possible number of students who like both football and basketball is represented by 'a', and the minimum possible number of students who like both football and basketball is represented by 'b'. What is the value of 'a' minus 'b'?

A. 12
B. 19
C. 23
D. 28
E. 32




Show Spoiler
Among 100 students in a class, 57 like football and 68 like basketball. The maximum possible number of students who like both football and basketball is a and the minimum possible number of students who like both football and basketball is b. What is the value of a-b?

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Among 100 students in a class, 57 like football and 68 like basketball [#permalink] New post   06 Feb 2024, 08:25
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Maximum possible number of students who like both = 57

total number of students who like both ,57 + 68 = 125

57+68-b=100 , b=57+68-100=25

A-B
=57-25
=32

IMO-E­
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Re: Among 100 students in a class, 57 like football and 68 like basketball [#permalink] New post   06 Feb 2024, 15:17
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­Correct option is E


­We have our equation set as:
F+B-a+y = 100 (y = number of students who like neither of the games)
=> a-y = 68+57-100 = 25
We can have y=0 in which case we shall have a=25 which is the minimum

Or We can have a scenario in which there is a possibility that all the students who like football also like basketball, in which case we shall have the intersection value = a = 57 which is the maximum

So difference = 57 - 25 = 32

Correct option is E

We can approach it in another beautiful way:

From fig 1 we can say the maximum value of a=57
From fig 2 we can say minimum value of a = 25
(shaded region is the intersection = a)

So difference = 57-25 = 32
­­­­­
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Re: Among 100 students in a class, 57 like football and 68 like basketball [#permalink]
06 Feb 2024, 15:17
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