There are 40 students at Boston College. 35 students play Basketball

Question

There are 40 students at Boston College. Students are free to be a part of any number of sports teams they like (or none at all). 35 students play Basketball, 33 play Football and 22 play Baseball. If x and y are, respectively, the highest and smallest possible number of students that participate in all three teams, what is the value of x – y?

A 10
B 12
C 17
D 20
E 22

A 10
B 12
C 17
D 20
E 22

GMATinsight · Answer

Highest number of students who participate in all the games = Max.(35, 33, 22) = 22

Lowest number of students = y

Maximum Value of Football and Baseball combined = 40-35 = 5

Maximum Value of Football and Basketball combined = 40-22 = 18

Maximum Value of Basketball and Baseball combined = 40-33 = 7

i.e. 5+(18)+(7)+y = 40

i.e. y = 10

i.e. x-y = 22-10 = 12

Answer: Option B

Archit3110 · Answer

for all games the highest students who can participate in all games ; 35,33,22 is 22 = x
both games ; 40-35 + 40-33 + 40-22 ; 30
and least game common be y 
so 30+y=40
y=10
x-y = 22-10 ; 12
IMO B

ScottTargetTestPrep · Answer

Solution:

To find the highest number of students that participate in all three teams, assume all of the 22 baseball players also play the other two sports. (To see that this is possible, assume that 7 students play basketball only, 5 students play football only and 6 students play basketball and football, but not baseball. Then, 7 + 6 + 22 = 35 students play basketball, 5 + 6 + 22 = 33 students play football and 22 students play baseball, which agrees with the information given to us. Further, the total number of students is basketball only + football only + basketball & football only + all three + neither = 7 + 5 + 6 + 22 + 0 = 40, which also agrees with the given information.)

To find the smallest number of students that participate in all three sports, notice that we have:

#Basketball = #Basketball only + #Basketball & Football only + #Basketball & Baseball only + #All 3

#Football = #Football only + #Basketball & Football only + #Football & Baseball only + #All 3

#Baseball = #Baseball only + #Basketball & Baseball only + #Football & Baseball only + #All 3

Adding all three equalities, we obtain:

#Basketball + #Football + #Baseball = #Exactly one sport + 2*(#Exactly two sports) + 3 * #All 3

35 + 33 + 22 = #Exactly one sport + 2*(#Exactly two sports) + 3 * #All 3

#Exactly one sport + 2*(#Exactly two sports) + 3 * #All 3 = 90

We also have the following formula:

Total = #Exactly one sport + #Exactly two sports + #All 3 + #None

#Exactly one sport + #Exactly two sports + #All 3 + #None = 40

Notice that as the number of students who play none of the sports increases, the number of overlaps between the given sports will also increase; therefore, to maximize #All 3, we must assume #None = 0. Thus:

#Exactly one sport + #Exactly two sports + #All 3 = 40

Multiplying this equality by 2, we obtain:

2 * #Exactly one sport + 2 * #Exactly two sports + 2 * #All 3 = 80

Subtracting this equality from #Exactly one sport + 2*(#Exactly two sports) + 3 * #All 3 = 90, we obtain:

#All 3 - #Exactly one sport = 10

We see that the minimum value of #All 3 is 10 and that happens when #Exactly one sport is 0. We just need to show that it is possible to make the number of students who play only one sport zero. If we take #Basketball only = #Football only = #Baseball only = 0, #Basketball and Football only = 18, #Basketball and Baseball only = 7, #Football and Baseball only = 5 and #All 3 = 10, we get:

#Basketball = #Basketball only + #Basketball & Football only + #Basketball & Baseball only + #All 3 = 0 + 18 + 7 + 10 = 35,

#Football = #Football only + #Basketball & Football only + #Football & Baseball only + #All 3 = 0 + 18 + 5 + 10 = 33,

#Baseball = #Baseball only + #Basketball & Baseball only + #Football & Baseball only + #All 3 = 0 + 7 + 5 + 10 = 22,

and

Total = #Exactly one sport + #Exactly two sports + #All 3 + #None = 0 + (18 + 7 + 5) + 10 + 0 = 40

So, the minimum value of #All 3 is 10. The difference between the maximum and minimum values of #All 3 is 22 - 10 = 12.

Answer: B

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There are 40 students at Boston College. 35 students play Basketball

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