Question

The arithmetic mean of numbers a, b, c, d, e is M. What is the value of $(a-M)+(b-M)+(c-M)+(d-M)+(e-M)$?
(a) M
(b) $a+b+c+d+e$
(c) 0
(d) 5M

Answer 1

Hint: Arithmetic mean is calculated by adding the values of all the terms and dividing the sum by the total number of terms.

The arithmetic mean of numbers a, b, c, d, e is given to us as M. So, here it has been given the mean of five terms.
We know that the arithmetic mean is calculated by adding the value of all the terms and dividing the sum by the total number of terms. i.e. $=\dfrac{\sum\limits_{i=1}^{i=n}{{{x}_{i}}}}{n}$ .
The arithmetic mean is also known as the average number.
The arithmetic mean is used mostly in mathematics and statistics but it has some uses in the field of economics, anthropology, and history also.
Here in the question, arithmetic mean is given to us and we know that the total number of terms is 5.
So,
$\dfrac{a+b+c+d+e}{5}=M$
Now multiplying by $5$ on both sides of the equation.
$a+b+c+d+e=5M....\left( i \right)$
Now we need to find $(a-M)+(b-M)+(c-M)+(d-M)+(e-M)$ so, we solve the equation and open the bracket and add or subtract the like terms in the equation. Like terms are those terms that have the same variable in the equation.
And now, we will get
$\begin{align}
  & =(a-M)+(b-M)+(c-M)+(d-M)+(e-M) \\
 & =a+b+c+d+e-5M \\
\end{align}$
Using equation (i).
We can substitute the value from equation (i) and we will get
$=5M-5M$
$=0$
So, $(a-M)+(b-M)+(c-M)+(d-M)+(e-M)=0$
Our answer after solving the equation is 0.
So, the correct option is c.

Note: The arithmetic mean is simply considered as a form of average. So mean and average are the same with the same formula. And remember in solving this type of question, we must first use the arithmetic formula to get an equation that can be substituted in $(a-M)+(b-M)+(c-M)+(d-M)+(e-M)$.