Venn Diagram Problem
A local radio station asked listeners which of the following types of music did they like: Pop, Rock or Rap.
Of the 44 listeners who responded, 18 said they like Rap.
9 said they like pop and rock only, 7 said they like pop and rap only.
While 1 person likes rock and rap only.
Three times as many said they like pop only as said they like all 3 types of music.
Twice as many said they like pop as said they like Rock.
3 people said they do not like any of this music.
Let x represent those listeners who like all 3 types of music.
i) Represent this on a venn diagram
ii) Calculate x
iii) Calculate the percentage of listeners who like one type of music only. Give your answer, correct to 1 decimal place.
s(Rap)=18, s(Pop∩Rock)=9, s(Pop∩Rap)=7, s(Rap∩Rock)=1, 3s(Rap∩Rock∩Pop)=s(Pop|RockURap) and 2s(Rock)=s(Pop)
Assume s(Rap∩Rock∩Pop)=x, So s(Pop|RockURap)=3x, s(Rap|RockUPop)=10-x and s(Rock|PopURap)=x-2.
If 3 people said they do not like any of this music, s(RockURapUPop)=41.
only rock=2, only rap=8 only pop=12, therefore 22/44=1/2 (the percentage of listeners who like one type of music only)