Solution
Given:We are given that,
• \(\frac{1}{4}^{th}\) of the players in a football club are capped players, and the remaining players are uncapped players
o Half of the capped players are defenders
• The number of uncapped players in the club = 36
To find:• The number of players, in the club, who are not defenders
Approach and Working:• Total number of players are divided as capped and uncapped, and all the players of these two groups are either defenders are non-defenders.
Now, let’s represent the above structure in the form a tree diagram.
Now, let’ see what we need to find : The number of players, in the club, who are not defenders = \(C_O + U_O\)
• Thus, we need to find the values of \(U_O\) and \(C_O\), to answer this question
Let us assume that the total number of players in the club = T
• From the given information , we can say that the number of capped players = 25% of T = \(\frac{T}{4}\)
• Which implies, that the number of uncapped players = \(T – \frac{T}{4} = \frac{3T}{4}\)
Now, from the diagram we can see that, total capped players = \(C_D + C_O\)
• Thus, the equation will be \(C_D + C_O = \frac{T}{4}\)
• And, from the given information, we get, \(C_D = (\frac{1}{2}) * (\frac{T}{4}) = \frac{T}{8}\)
o Thus, \(C_O = \frac{T}{4} – \frac{T}{8} = \frac{T}{8}\) ……………. (1)
We are also given that all uncapped players are defenders
• Thus, \(U_D = \frac{3T}{4}\) and \(U_O = 0\) ………………….…… (2)
So, from (1) and (2), we get, \(U_O + C_O = 0 + \frac{T}{8} = \frac{T}{8}\) ………………. (3)
• To find T, we have information that, number of uncapped players, \(U = \frac{3T}{4} = 36\)
o This implies, \(T = 36 * \frac{4}{3} = 48\)
• Substituting this value of T in (3), we get
o \(U_O + C_O = \frac{48}{8} = 6\)
Therefore, the number of players, in the club, who are not defenders = 6
Hence, the correct answer is option B.
Answer: B _________________