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A poll was taken of 100 students at a commuter campus to find out how they got to campus. The results were as follows: 31 said they drove alone, 39 rode in carpool, 35 rode public transportation, 10 used both carpool and public transportation, 7 used both a carpool and sometimes their own cars, 9 used buses as well as their own cars, 5 used all three methods. how many used none of the above-mentioned means of transportation?

  • Algebra -

    Make a Venn diagram, showing three overlapping circles, label them
    A for driving alone
    C for carpool
    P for public trans

    put 5 in the intersection of all 3 circles
    10 used C and P, so 10 goes in that intersection, BUT 5 are already counted, so place 5 in the outer part of the intersection of C and P
    in the same way, place 4 in the outer part of A and P
    in the same way , place 2 in the outer part of A and C
    now each circle itself
    A should have a total of 31, but I already have 11 counting in the A circle, leaving 20 in the unused part of A
    C should have 39, but I already have 12 filled in, leaving 27 for the rest of circle C
    P should have 35, but I already have 14 filled in, leaving 21 for the rest of circle P

    Adding up all the numbers I see filled in, I get 84
    so from the 100 students, 16 do not use any of the 3 methods.

    check my arithmetic

  • OR - Algebra -

    N(A and C and P)
    = N(A) + N(C) + N(P) - N(A and C) - N(A and P) - N(C and P) + N(A and C and P)
    = 31 + 39 + 35 - 10 - 7 - 9 + 5
    = 84

    so 100 - 84 are unaccounted for
    16 don't use any of the given ways

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