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?

Answer 1

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