the 90 members of a sports club play at least one of the games tennis, football, volleyball. 10 play tennis and football, 19 play football and volleyball and 29 play tennis and volleyball.n people play all three games. 2n people each play only one game, mathematics sets solution
Draw the Venn diagram.
6n + 58-3n + n = 90
n = 8
Now you can answer any questions
To solve this problem using mathematics sets, we can represent each game as a set and use set operations to find the desired values. Let's define the sets first:
Let A represent the set of members who play tennis.
Let B represent the set of members who play football.
Let C represent the set of members who play volleyball.
From the given information, we can set up the following equations:
|A ∩ B| = 10 (10 members play tennis and football)
|B ∩ C| = 19 (19 members play football and volleyball)
|A ∩ C| = 29 (29 members play tennis and volleyball)
|A ∩ B ∩ C| = n (n members play all three games)
We also know that the total number of members is 90. Therefore,
|A ∪ B ∪ C| = 90 (the total number of members who play at least one game)
Using these equations, we can find the values of n and 2n.
To solve these equations, you can use the Inclusion-Exclusion Principle, which states that:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|
Plugging in the given information into the above equation:
90 = |A| + |B| + |C| - 10 - 19 - 29 + n
We also know that |A ∩ B ∩ C| = n.
Next, we can consider the members playing only one game. This can be calculated using the formula:
|A ∪ B ∪ C| = 2n + |A ∩ B ∩ C|
Plugging in the given information into the above equation:
90 = 2n + n
Simplifying this equation:
90 = 3n
Dividing both sides by 3, we find:
n = 30
Now that we have found the value of n, we can substitute it back into the first equation:
90 = |A| + |B| + |C| - 10 - 19 - 29 + 30
Simplifying this equation:
90 = |A| + |B| + |C| - 28
Rearranging, we get:
|A| + |B| + |C| = 118
This means that the total number of members playing tennis, football, or volleyball is 118.
To find the number of members playing only one game, we can substitute the value of n back into the equation:
2n = 2 * 30
Therefore, 2n = 60. This means that 60 members play only one game.
This is how you can use mathematical sets to solve the problem and find the values of n and 2n.