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Posted by on Wednesday, August 31, 2011 at 3:44pm.

There are 47 in a club. 18 play chess and 13 play tennis. The number who play neither is 3 times the number who play both. How many play both?

  • Maths - , Wednesday, August 31, 2011 at 7:38pm

    Let x = number who play both
    then 3x = number who play neither.
    C = set of people who play chess
    T = set of people who play tennis
    U = total membership of the club

    By the principle of inclusion-exclusion,
    |C∪T| = |C| + |T| - |C∩T|

    or

    |U| - |C∪T| = |U| - (|C| + |T| - |C∩T|)

    3x = 47 - (18+13-x)
    2x = 16
    x = 8 (number of members who play both)
    3x = 24 (number of members who play neither)

    Total number of players
    = |C∪T|
    = |C| + |T| - |C∩T|
    = 18 + 13 - 8
    = 23

    Check:
    |U|=|C∪T|+24
    = 23+24
    =47 OK

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