Survey Analysis Determining Players In One Game Volleyball And Football Preferences

The challenge at hand is to dissect the survey data and extract the precise number of players who confine their athletic endeavors to a single sport. This requires a meticulous approach, carefully considering the fractions representing the players' preferences and the additional information regarding those who abstain from both sports. By employing mathematical tools such as fractions, variables, and equations, we can construct a framework to represent the survey results and systematically solve for the unknown quantity – the number of players who play only one game.

To embark on our mathematical journey, let's first define the variables that will serve as the building blocks of our analysis. Let 'T' represent the total number of players surveyed, a crucial piece of information that will serve as the foundation for our calculations. Additionally, let 'V' denote the number of players who exclusively play volleyball, 'F' represent the number of players who exclusively play football, and 'B' signify the number of players who play both volleyball and football. These variables will allow us to translate the survey results into mathematical expressions, paving the way for a clear and concise representation of the players' sporting preferences.

Now, let's translate the survey findings into mathematical equations, establishing the relationships between the variables we have defined. The survey reveals that one-third of the total players play volleyball only, which can be expressed as V = (1/3)T. Similarly, one-fifth of the total players play football only, represented by F = (1/5)T. Furthermore, three-fifths of the total players play football, implying that the sum of players who play football only and those who play both sports equals (3/5)T, which can be written as F + B = (3/5)T. Finally, we know that 50 players play neither volleyball nor football, providing us with an additional piece of information to incorporate into our analysis. These equations form the cornerstone of our mathematical framework, allowing us to systematically solve for the unknown variables and ultimately determine the number of players who play only one game.

With our variables defined and equations established, we now embark on the exciting phase of solving for the unknown – the number of players who play only one game. Our objective is to determine the combined number of players who exclusively play volleyball (V) and those who exclusively play football (F). To achieve this, we will employ a series of algebraic manipulations, carefully combining and rearranging the equations to isolate the desired variables.

First, let's express the total number of players (T) in terms of the individual groups. We know that the total number of players comprises those who play volleyball only (V), those who play football only (F), those who play both sports (B), and those who play neither sport (50). This can be represented as T = V + F + B + 50. Now, we can substitute the expressions for V and F from our earlier equations into this equation, resulting in T = (1/3)T + (1/5)T + B + 50. This equation effectively captures the relationship between the total number of players and the individual sporting preferences expressed in the survey.

Next, let's simplify the equation by combining the fractions and rearranging the terms. Multiplying both sides of the equation by 15 (the least common multiple of 3 and 5) eliminates the fractions, resulting in 15T = 5T + 3T + 15B + 750. Combining like terms, we get 7T = 15B + 750. This equation provides a crucial link between the total number of players (T) and the number of players who play both sports (B), allowing us to further refine our analysis.

Now, let's revisit the equation F + B = (3/5)T, which represents the total number of players who play football. Substituting F = (1/5)T into this equation, we get (1/5)T + B = (3/5)T. Subtracting (1/5)T from both sides, we find B = (2/5)T. This equation provides a direct relationship between the number of players who play both sports (B) and the total number of players (T), allowing us to express B in terms of T.

Substituting B = (2/5)T into the equation 7T = 15B + 750, we get 7T = 15(2/5)T + 750. Simplifying, we have 7T = 6T + 750. Subtracting 6T from both sides, we arrive at T = 750. This remarkable result reveals the total number of players surveyed – 750 individuals. With this crucial piece of information, we can now readily determine the number of players who play only volleyball and the number who play only football.

With the total number of players (T) firmly established at 750, we can now effortlessly calculate the number of players who play only one game. Recall that V = (1/3)T, which represents the number of players who exclusively play volleyball. Substituting T = 750, we find V = (1/3)(750) = 250. Thus, 250 players dedicate their athletic endeavors solely to the exhilarating sport of volleyball.

Similarly, F = (1/5)T represents the number of players who exclusively play football. Substituting T = 750, we get F = (1/5)(750) = 150. This reveals that 150 players immerse themselves in the beautiful game of football, focusing their sporting passion on this dynamic and captivating sport.

To determine the total number of players who play only one game, we simply add the number of volleyball-only players (V) and the number of football-only players (F). Therefore, the number of players who play only one game is V + F = 250 + 150 = 400.

In conclusion, our meticulous analysis of the survey data has successfully unveiled the number of players who engage in only one sport. Through the strategic application of mathematical principles, including fractions, variables, equations, and algebraic manipulations, we have deciphered the intricate web of sporting preferences and arrived at a definitive solution. We have determined that a total of 400 players dedicate their athletic endeavors to a single sport, with 250 players exclusively playing volleyball and 150 players exclusively playing football. This result showcases the power of mathematical reasoning in unraveling real-world scenarios, transforming raw data into meaningful insights. The survey results provide a valuable glimpse into the sporting landscape of the group, highlighting the diverse athletic interests and the dedication of individuals to their chosen sports. The successful resolution of this problem underscores the importance of mathematical literacy and its ability to illuminate complex situations, providing a deeper understanding of the world around us.

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In conclusion, this article has provided a comprehensive analysis of the survey data, successfully determining the number of players who play only one game. The solution process involved a blend of mathematical reasoning, algebraic techniques, and strategic keyword optimization. We hope this article has provided valuable insights into the world of mathematics and its applications in real-world scenarios.