Question 1180850
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Draw a Venn diagram to follow the analysis below....<br>
Let x be the number who go to all three activities<br>
let a be the number who go to baseball games and the opera but not the theater
let b be the number who go to baseball games and the theater but not the opera
let c be the number who go to the opera and the theater but not to baseball games<br>
Then the given numbers give us these equations:<br>
33 go to baseball games: 14+a+b+x=33; a+b+x=19 [1]
35 go to the opera: 15+a+c+x=35; a+c+x=20 [2]
39 go to the theater: 17+b+c+x=39; b+c+x=22 [3]<br>
From [1] and [2] we can conclude
b=a+2 [4]
c=a+3 [5]<br>
86-11=75 go to at least one of the three activities:
(14+15+17)+(a+a+2+a+3)+x=75; 3a+x=24 [6]<br>
Again, 35 go to the opera:
15+a+a+3+x=35; 2a+x=17 [7]<br>
Then [6] and [7] give us
a=7
x=3<br>
From which we last get
b=9
c=10<br>
ANSWERS:
a. go to all three: x=3
b. go to baseball games and the opera but not the theater: a=7
c. go to at least two activities: a+b+c+x=29<br>
Of course, answer c can also be obtained as (number who go to at least one activity) minus (number who go to only one of the three activities):
75-(14+15+17)=29<br>