Question

The average of 5 teachers is 28 years. If one teacher is excluded the mean gets reduced by 2 years, then find the age of the excluded teacher?
(a) 26 years
(b) 33 years
(c) 36 years
(d) 35 years

Answer 1

We start solving the problem by assigning the variables for the ages of all teachers. We then recall the definition of mathematical average of given ‘n’ numbers as $\dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}+......+{{y}_{n}}}{n}$. We then use this definition for the condition that the average of 5 teachers is 28 years. We then exclude the age of one teacher and apply the same definition for ages of remaining teachers. We then compare both equations and make necessary calculations to get the required age of the excluded teacher.

Complete step by step answer:
According to the problem, we are given that the average of 5 teachers is 28 years and the mean gets reduced by 2 years if one teacher is excluded from them. We need to find the age of the excluded teacher.
Let us assume the age of excluded teacher be ‘x’ years and the other teacher’s age be ${{a}_{1}}$, ${{a}_{2}}$, ${{a}_{3}}$ and ${{a}_{4}}$.
According to the problem, we are given that the average of x, ${{a}_{1}}$, ${{a}_{2}}$, ${{a}_{3}}$ and ${{a}_{4}}$ is 28 years.
We know that the mathematical average for numbers ${{y}_{1}}$, ${{y}_{2}}$, ……, ${{y}_{n}}$ is $\dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}+......+{{y}_{n}}}{n}$.
So, we get $\dfrac{x+{{a}_{1}}+{{a}_{2}}+{{a}_{3}}+{{a}_{4}}}{5}=28$.
$\Rightarrow x+{{a}_{1}}+{{a}_{2}}+{{a}_{3}}+{{a}_{4}}=140$.
$\Rightarrow {{a}_{1}}+{{a}_{2}}+{{a}_{3}}+{{a}_{4}}=140-x$ ---(1).
From the problem, we are given that the average of ${{a}_{1}}$, ${{a}_{2}}$, ${{a}_{3}}$ and ${{a}_{4}}$ is $\left( 28-2 \right)=26$ years.
So, we get $\dfrac{{{a}_{1}}+{{a}_{2}}+{{a}_{3}}+{{a}_{4}}}{4}=26$.
$\Rightarrow {{a}_{1}}+{{a}_{2}}+{{a}_{3}}+{{a}_{4}}=104$.
From equation (1), we get $140-x=104$.
$\Rightarrow x=140-104$.
$\therefore x=36$ years.
So, we have found the age of the excluded teacher as 36 years.

The correct option for the given problem is (c).

Note:
Here we should consider the mean as an arithmetic mean which is equal to the mathematical average while taking the average of the ages of the remaining 4 teachers. We should not make calculation mistakes while solving this problem. We can also find the age of the remaining teachers if the information about them is also given in the problem. We can expect problems to find the average if a teacher of age 36 years is added to the existing teachers.