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Math Help - Venn Diagram Problem

  1. #1
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    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.
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  2. #2
    Junior Member atalay's Avatar
    Joined
    Apr 2010
    Posts
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    We have

    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.

    x=4.

    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)
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