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Math Help - The answer is easy if you know it....

  1. #1
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    The answer is easy if you know it....

    There are 208 geocachers at a convention. Each of them was asked if they liked traditional caches, multi-caches and mystery caches.
    4 of them said they liked all three types of cache.
    48 of them said they liked traditional caches.
    There were twice as many who said they liked both traditional and mystery caches (but not multi-caches) as who said they liked both traditional and multi-caches (but not mystery caches), and 4 times as many who said they liked both traditional and mystery caches (but not multi-caches) as who liked all three types.
    124 geocachers said they liked mystery caches.
    27 geocachers said they did not like any of these three types of cache.
    The group of geocachers who said they liked both traditional and mystery caches (but not multi-caches) was exactly the same size as the group who said they liked both mystery and multi-caches.
    A = The number of geocachers who liked only traditional caches.
    B = The number of geocachers who liked both traditional and multi-caches but not mystery caches.
    C = The number of geocachers who liked both mystery and multi-caches but not traditional caches.
    D = The number of geocachers who liked only mystery caches.
    E = The number of geocachers who liked both mystery and traditional caches but not multi-caches.
    F = The number of geocachers who liked only multi-caches.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Venn View Post
    There are 208 geocachers at a convention. Each of them was asked if they liked traditional caches, multi-caches and mystery caches.
    4 of them said they liked all three types of cache.
    48 of them said they liked traditional caches.
    There were twice as many who said they liked both traditional and mystery caches (but not multi-caches) as who said they liked both traditional and multi-caches (but not mystery caches), and 4 times as many who said they liked both traditional and mystery caches (but not multi-caches) as who liked all three types.
    124 geocachers said they liked mystery caches.
    27 geocachers said they did not like any of these three types of cache.
    The group of geocachers who said they liked both traditional and mystery caches (but not multi-caches) was exactly the same size as the group who said they liked both mystery and multi-caches.
    A = The number of geocachers who liked only traditional caches.
    B = The number of geocachers who liked both traditional and multi-caches but not mystery caches.
    C = The number of geocachers who liked both mystery and multi-caches but not traditional caches.
    D = The number of geocachers who liked only mystery caches.
    E = The number of geocachers who liked both mystery and traditional caches but not multi-caches.
    F = The number of geocachers who liked only multi-caches.
    The clue is in your name - draw a Venn diagram.

    Are you asking for help, or just posing a question? If you are asking for help, what have you done already? Have you tried drawing a Venn diagram?
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  3. #3
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    Help

    I am just trying to get it solved, I have tried a venn diagram but just can't get my head round it... :-(
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Venn View Post
    I am just trying to get it solved, I have tried a venn diagram but just can't get my head round it... :-(
    What I did was this:

    start by drawing the normal three circles all intersecting. Label them Trad. Multi. and Myst.

    Immediately, a 4 goes in the very middle, and write 48 on the outside of Trad. (the three sections in that circle sum to 48).

    Now, the tricky line is,

    There were twice as many who said they liked both traditional and mystery caches (but not multi-caches) as who said they liked both traditional and multi-caches (but not mystery caches), and 4 times as many who said they liked both traditional and mystery caches (but not multi-caches) as who liked all three types.

    Work backwards. |(trad. \cap myst.) \setminus multi.|=4 \times 4=16 is from the second half, while you can apply this to |(trad. \cap multi.) \setminus myst.| to get 8.

    Can you take it from there?

    That is,

    B=8
    E=8
    G=4=people who liked all three (you missed this one in your list)
    and
    A+B+E+G=48

    But don't do it algebraically...
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  5. #5
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    I have come up with

    A=20
    B=8
    C=16
    D=88
    E=16
    F=29
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