The arithmetic mean of $5$ numbers is $27$. If one of the numbers is excluded the mean of the remaining number is $25$. Find the excluded number.
A. $27$
B. $25$
C. $30$
D. $35$

Hint: For this problem we are going to use the relation between the average and the number of observations. We can define the average of $n$ number of observations as the ratio of the sum of the $n$ observations to the $n$. From this we can say that the sum of the $n$ observations is equal to the $n$ times of the mean. From this relation we will find the sum of the numbers whose mean is $27$. After that we have given that one number is excluded the mean, we will assume this number as $x$. If a number is excluded from the observations, then we need to subtract it from the total sum of the observations and decrease the number of variables from $n$ to $n-1$. So, we will subtract the value $x$ from the obtained sum of the numbers whose mean is $27$ and decrease the number of variables by one. From these values we will find the new mean of the observation after excluding the value $x$ and equate it to the given value of the new mean to find the value of $x$.

Complete step-by-step answer:
We have been given that,
Mean of $5$ numbers is $27$, hence we can write that
Number of observations is $n=5$
Mean of $n$ observations is $M=27$.
We can define the average of $n$ number of observations as the ratio of the sum of the $n$ observations to the $n$. So, from the relation between mean and number of observations, we can write the value of sum of $n$ observations as
$\begin{align}
  & s=n\times M \\
 & =5\times 27 \\
 & =135 \\
\end{align}$
Hence the sum of the $n={{n}_{1}}=5$ observations is ${{s}_{1}}=135$
We are given that a number is excluded from the observations. Let the excluded number be $x$. Then we will have
Number of observations after excluding the variable $x$ as
$\begin{align}
  & {{n}_{2}}={{n}_{1}}-1 \\
 & =5-1 \\
 & =4 \\
\end{align}$
The sum of the observations after excluding the variable as
$\begin{align}
  & {{s}_{2}}={{s}_{1}}-x \\
 & =135-x \\
\end{align}$
Now the mean of the observations after excluding the variable as
$\begin{align}
  & {{M}_{2}}=\dfrac{{{s}_{2}}}{{{n}_{2}}} \\
 & =\dfrac{135-x}{4} \\
\end{align}$
In the problem we have given that, the mean of the observations after excluding the variable $x$ is $25$. Hence
$\begin{align}
  & {{M}_{2}}=25 \\
 &\Rightarrow \dfrac{135-x}{4}=25 \\
 &\Rightarrow 135-x=100 \\
 &\Rightarrow x=135-100 \\
 &\Rightarrow x=35 \\
\end{align}$
Hence the excluded variable is $35$.

So, the correct answer is “Option d”.

Note: For this kind of problem students may forget to reduce the number of variables when some of the elements are excluded. If you do not reduce the number of variables then the result won’t be perfect. Some students might add the term x to 135 instead of subtracting it. This too can lead to a different answer, which the student might understand at the end after solving the entire question. So, such mistakes must be avoided.