Question

The average weight of $11$ players of Indian cricket team is increased by $1$ kg, when one player of the team weighing $55$ kg is replaced by a new player. The weight of the new player is-
A.$55$ kg
B.$64$ kg
C.$66$ kg
D.None of these

Answer 1

Hint: Let us assume the sum of weights of all the players of the team to be x kg. Then find the average weight using the formula-
$ \Rightarrow $ Average weight =$\dfrac{{{\text{Sum of the weights of the 11 players}}}}{{{\text{total number of players}}}}$
Now, it is given in the question that the average weight is increased by one kg so add one in the required average weight. Then assume the weight of the new player to be w and find the new average weight using the same average formula. Equate both the values of the new average weight and solve the given equation to find the answer.

Complete step-by-step answer:
Given, the total number of players=$11$
Let the sum of the weight of all the players be x kg. Then, we know the formula of average is given as-
$ \Rightarrow $ Average=$\dfrac{{{\text{Sum of the observations}}}}{{{\text{total number of observations}}}}$
According to the above formula, the average weight of the players of team can be given as-
$ \Rightarrow $ Average weight =$\dfrac{{{\text{Sum of the weights of the 11 players}}}}{{{\text{total number of players}}}}$
On putting the given values, we get-
$ \Rightarrow $ Average weight =$\dfrac{x}{{{\text{11}}}}$
Now, it is given in the question that when one player is replaced by a new player the average weight increases by $1$ kg then the average weight of the players will becomes $\dfrac{x}{{{\text{11}}}} + 1$
On solving, we get-
$ \Rightarrow $ New average weight =$\dfrac{{x + 11}}{{{\text{11}}}}$-- (i)
It is also given that the new player replaces the person weighing $55$ kg. Let the weight of the new player be w then we can write the new average weight as-
$ \Rightarrow $ New average weight =$\dfrac{{x + w - 55}}{{{\text{11}}}}$-- (ii)
From eq. (i) and (ii), we get-
$ \Rightarrow \;\dfrac{{x + 11}}{{{\text{11}}}} = \dfrac{{x + w - 55}}{{{\text{11}}}}$
Here, since the denominator both sides is same so it will get cancelled and we get-
$ \Rightarrow \;x + 11 = x + w - 55$
On solving, we get-
$ \Rightarrow \;11 = w - 55$
On rearranging, we get-
$ \Rightarrow \;w = 55 + 11$
On further solving, we get-
$ \Rightarrow \;w = 66$kg
Hence the correct answer is ‘option C’.

Note: Here, we have to remember that the person weighing $55$ kg is replaced by the person weighing w kg so if students forget to subtract the weight of the person being replaced from the new average weight then they will get the wrong answer.