set theory

Question

a survey by members of a school house reviewed the following assignment, 60% like football,56% like hockey, 50% like athletics,26% like football and hockey 20% like hockey and athletics 30% like football and athletics, 10% like all the three sports (a) what percent of the house like football and hockey but not athletics? (b)what percent like exactly two out of the three sports? (c) what percent did not like any of the three sports?

Answer 1

Draw three interlocking circles. This is a Venn diagram, and with it the problem is easier to solve. Surrounding the interlocking circles, draw a large circle. The four circles produce 8 regions A-H representing percentages:

A. Athletics members only

B. Football members only 

C. Hockey members only

D. Athletics and hockey members (not football)

E. Hockey and football members (not athletics)

F. Athletics and football members (not hockey)

G. All three

H. None

Now we can put in some figures:

A+B+C+D+E+F+G+H=100

B+E+F+G=60

C+D+E+G=56

A+D+F+G=50

G+E=26

D+G=20

F+G=30

G=10, so E=16, D=10, F=20.

And A=50-(D+F+G)=10; C=56-(D+E+G)=20; B=60-(E+F+G)=14

H=100-(A+B+C+D+E+F+G)=100-(10+14+20+10+16+20+10)=0

(a) B+E=14+16=30%

(b) D+E+F=10+16+20=46%

(c) H=0%