In a survey, [tex]$\frac{1}{3}$[/tex] of the total players play volleyball only, [tex]$\frac{1}{5}$[/tex] of the total players play football only, [tex]$0130 \frac{3}{5}$[/tex] of total players play football, but 50 play neither of them. Find the number of players who play only one game.
Answer (1)
Define x as the total number of players and express the number of players in each category (volleyball only, football only, both, and neither) in terms of x .
Set up an equation for the total number of players: x = 3 1 x + 5 1 x + 5 2 x + 50 .
Solve for x to find the total number of players: x = 750 .
Calculate the number of players who play only one game: 3 1 ( 750 ) + 5 1 ( 750 ) = 400 .
The number of players who play only one game is 400 .
Explanation
Problem Analysis Let's analyze the problem. We are given fractions of players who play volleyball only and football only, a fraction of players who play football (including those who play both), and the number of players who play neither sport. We need to find the number of players who play only one sport.
Defining Variables Let x be the total number of players. Then:
Number of players who play volleyball only: 3 1 x
Number of players who play football only: 5 1 x
Number of players who play football: 5 3 x
Number of players who play neither: 50
Players Playing Both Sports Let B be the number of players who play both football and volleyball. Since 5 3 x play football in total, and 5 1 x play football only, we have: 5 1 x + B = 5 3 x Solving for B :
B = 5 3 x − 5 1 x = 5 2 x
Total Number of Players The total number of players x is the sum of those who play volleyball only, football only, both sports, and neither sport: x = 3 1 x + 5 1 x + 5 2 x + 50
Solving for Total Players Now, we solve for x :
x = 3 1 x + 5 3 x + 50 x − 3 1 x − 5 3 x = 50 15 15 x − 5 x − 9 x = 50 15 x = 50 x = 750
Players Playing Only One Sport The number of players who play only one game is the sum of those who play volleyball only and those who play football only: 3 1 x + 5 1 x = 15 5 x + 3 x = 15 8 x
Calculating the Final Answer Substitute x = 750 :
15 8 ( 750 ) = 8 ( 50 ) = 400
Conclusion Therefore, the number of players who play only one game is 400.
Examples
Imagine you're organizing a sports club and need to understand how many members focus on just one sport to plan training sessions effectively. Knowing the fractions of members playing only volleyball or football helps you allocate resources and schedule practices that cater specifically to these groups. This type of analysis is useful in resource allocation, understanding preferences, and tailoring programs to meet specific needs within a community or organization. By understanding the distribution, you can better manage and support the different sporting interests within the club.
