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Math Help - 3 set venn diagram

  1. #1
    Senior Member sfspitfire23's Avatar
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    3 set venn diagram

    Hi guys,

    A group of 15-18 year old kids were studied. 68% have a television, 92% have a CD/tape player, and 32% have a computer. Only 4% have none of the above. 9% have a computer and a CD/tape player but no television 44% have a TV and a CD/tape player but no computer, and 1% have a TV and computer but no CD/tape player.

    How would I fill out this venn diagram?

    Work:

    Let A= TV, B= CD/tape, C= computer.
    (Let u= union and n=intersection)
    So, we know that (AnB)= .44, (AnC)= .01, and (BnC)=.09.
    We must find (AnBnC).

    Formula:
    Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps)+(all three overlap) + Neither.

    So, 1=.68+.92+.32-(.54)+X+.04

    We get (all three overlap)= -.42. This does not make sense...

    I know these formulas but am unsuer how to apply them:
    Exactly 2 sets=(sum of 2-group overlaps)-3(all three overlap)
    More than 3 sets=(sum of 2-group overlaps)-2(all three overlap).

    Am I way off here? What formulas can I use for this?
    Thanks
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  2. #2
    Super Member

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    Hello, sfspitfire23!

    This is a tricky one . . .


    \text{A group of youths was studied. 68\% have a television,} . . \text{92\% have a CD/tape player, and 32\% have a personal computer.}

    \text{Only 4\% have none of the above.}
    \text{9\% have a PCr and a CD player but no TV.}
    \text{44\% have a TV and a CD/ player but no PC.}
    \text{1\% have a TV and PC but no CD player.}

    \text{How would I fill out the Venn diagram?}

    Construct the Venn diagram and fill in the known quantities.


    Code:
          o---------------------------------------------------o
          |                                                   |
          |     o-----------------------o                     |
          |     | TV                    |                     |
          |     |       b       o-------+---------------o     |
          |     |               |       |            CD |     |
          |     |               |  44   |       c       |     |
          |     |               |       |               |     |
          |     |       o-------+-------+-------o       |     |
          |     |       |       |       |       |       |     |
          |     |       |   1   |   a   |       |       |     |
          |     |       |       |       |       |       |     |
          |     o-------+-------+-------o       |       |     |
          |             |       |           9   |       |     |
          |             |       |               |       |     |
          |             |       o---------------+-------o . . |
          |             |     d                 |             |
          |             | PC                    |             |
          |   4         o-----------------------o             |
          |                                                   |
          o---------------------------------------------------o

    Use a,b,c,d in the unknown spaces.


    In the TV-circle, the total is 68.
    . . a + b +45 \:=\:68 \quad\Rightarrow\quad a + b \:=\:23 \;\;[1]

    In the CD-circle, the total is 92.
    . . a + c + 53 \:=\:92 \quad\Rightarrow\qud a + c \:=\:39 \;\;[2]

    In the PC-circle, the total is 32.
    . . a + d + 10 \:=\:32 \quad\Rightarrow\quad a + d \:=\:22\;\;[3]

    In the three circles, the total is 96.
    . . a + b + c + d + 54 \:=\:96 \quad\Rightarrow\quad a + b + c + d \:=\:42 \;\;[4]


    Subtract [2]-[1]: . c - b \:=\:16\;\;[5]

    Subtract [2]-[3]: . c - d \:=\:17\;\;[6]

    Subtract [4]-[1]: . c + d \:=\:19\;\;[7]


    Add [6]+[7]: . 2c \:=\:36 \quad\Rightarrow\quad \boxed{c \,=\,18}

    Substitute into [7]: . 18 + d \:=\:19 \quad\Rightarrow\quad \boxed{d \,=\,1}

    Substitute into [5]: . 18 - b \:=\:16 \quad\Rightarrow\quad \boxed{b \,=\,2}

    Substitute into [1]: . a + 16 2 \:=\:23 \quad\Rightarrow\quadn \boxed{a \,=\,21}


    Now you can fill in the rest of the Venn diagram . . .

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  3. #3
    Senior Member sfspitfire23's Avatar
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    Ah! Thanks! I see now. thanks for the detailed explanation.
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