Question

The age of the captain of a cricket team of 11 players is 25 years and the wicket-keeper is 3 years older than the captain if the ages of these two are excluded, the average age of the remaining players of the team becomes 1 year less than the average age of the whole team. What is the average age of the whole team is
A) 21.5 years
B) 22 years
C) 22.5 years
D) 23 years

Answer 1

Hint: First assume the average age of the whole team. After that find the total age of the team excluding captain and wicket-keeper. Then, substitute the value in the formula of the average age of the whole team to find the exact value of the average age of the whole team.

Complete step-by-step answer:
Let the average age of the whole team be $x$.
The team consists of 11 players.
The age of the captain is 25 years.
Since the age of the wicket keeper is 3 years more than the age of the captain. So, the age of wicket-keeper is,
$ \Rightarrow 25 + 3 = 28$ years
The average age of the team excluding captain and wicket-keeper is 1 less than the average age of the team.
As we know,
$\bar x = \dfrac{{\sum {{x_i}} }}{n}$
Substitute the values,
$ \Rightarrow x - 1 = \dfrac{{\sum {{x_i}} }}{9}$
Cross-multiply the terms,
$ \Rightarrow \sum {{x_i}} = 9\left( {x - 1} \right)$..............…. (1)
Now, the average age of the whole team is given by,
$ \Rightarrow x = \dfrac{{\sum {{x_i}} + 25 + 28}}{{9 + 2}}$
Substitute the value from equation (1) and simplify,
$ \Rightarrow x = \dfrac{{9\left( {x - 1} \right) + 53}}{{11}}$
Cross-multiply the terms and open the bracket,
$ \Rightarrow 11x = 9x - 9 + 53$
Move variable on the left side and simplify,
$ \Rightarrow 2x = 44$
Divide both sides by 2,
$\therefore x = 22$
So, the average age of the whole team is 22 years.

Hence, option (B) is correct.

Note: Arithmetic Mean is the most common measurement of central tendency. According to the layman, the mean of data represents an average of the given collection of the data. It is equivalent to the sum of all the observations of a given data divided by the total number of observations.
The mean of data for n values in a set of data namely ${x_1},{x_2},{x_3}, \ldots ,{x_n}$ is given by,
$\bar x = \dfrac{{{x_1} + {x_2} + {x_3} + \ldots + {x_n}}}{n}$