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?
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
To find the number of people who play both chess and tennis, we need to analyze the given information step by step.
Let's start by representing the total number of people in the club with the variable 'Total = 47'.
Next, we have the number of people who play chess, which is represented by 'Chess = 18'.
Similarly, the number of people who play tennis is represented by 'Tennis = 13'.
Now, we're told that the number of people who play neither chess nor tennis is 3 times the number of people who play both. Let's represent this as an equation:
Neither = 3 * (Both)
Since the total number of people in the club is equal to the sum of those who play chess, tennis, both, and neither, we can express this as an equation as well:
Total = Chess + Tennis + Both + Neither
Now, let's substitute the given values into the equations:
47 = 18 + 13 + Both + Neither
Neither = 3 * Both
To solve these equations, we'll use substitution. We'll substitute the second equation into the first equation to express the Total in terms of Both:
47 = 18 + 13 + Both + (3 * Both)
Simplifying the equation, we get:
47 = 31 + 4 * Both
Now, let's isolate the variable "Both" by subtracting 31 from both sides:
16 = 4 * Both
Finally, divide both sides of the equation by 4 to solve for Both:
Both = 16 / 4
Both = 4
Therefore, there are 4 people who play both chess and tennis.