SOLUTION: A group of faculty members at a small college operate a carpool to three kinds of activities; baseball games, the opera, and the theatre. Suppose there are 86 families in the carpo

Question 1180850: A group of faculty members at a small college operate a carpool to three kinds of activities; baseball games, the opera, and the theatre. Suppose there are 86 families in the carpool and that in a given month,
11 families attend none of the activities 33 families go to the baseball games 35 families go to the opera
39 families go to the theatre
14 families go to just baseball games 17 families go to just the theatre
15 families go to just opera
a. How many families go to all three activities?
b. How many families go to baseball games and the opera but not to the theatre?
c. How many families go to at least two activities?
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Draw a Venn diagram to follow the analysis below....

Let x be the number who go to all three activities

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

Then the given numbers give us these equations:

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]

From [1] and [2] we can conclude
b=a+2 [4]
c=a+3 [5]

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]

Again, 35 go to the opera:
15+a+a+3+x=35; 2a+x=17 [7]

Then [6] and [7] give us
a=7
x=3

From which we last get
b=9
c=10

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

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

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