Question

The arithmetic mean between $\dfrac{{x + a}}{x}$ and $\dfrac{{x - a}}{x}$ when $x \ne 0$, is (the symbol $\ne$ means "not equal to"):
A ) 2, if $a \ne 0$
B ) 1
C ) 1, only if $a = 0$
D ) $\dfrac{a}{x}$
E ) $x$

Answer 1

Hint:- We will proceed by using the formula of arithmetic mean between any two real numbers which is equal to the arithmetic average of the two numbers.

Complete step-by-step answer:
Since $x \ne 0$, both the numbers $\dfrac{{x + a}}{x}$ and $\dfrac{{x - a}}{x}$ exist.
If we have a data set consisting of the values ${a_1},{a_2},...,{a_n}$ , then the arithmetic mean $A.M.$ is defined by the formula:
$A = \dfrac{1}{n}\sum\limits_{i = 1}^n {{a_i}} = \dfrac{{{a_1} + {a_2} + ... + {a_n}}}{n}$
Thus, we know that in between any two real numbers $m$ and $n$ , the arithmetic mean is given by $\dfrac{{m + n}}{2}$ Using this result, we can now find out the arithmetic mean between the two given numbers $\dfrac{{x + a}}{x}$ and $\dfrac{{x - a}}{x}$ when $x \ne 0$
The arithmetic mean (A.M.) is equal to
$A.M. = \dfrac{{\dfrac{{x + a}}{x} + \dfrac{{x - a}}{x}}}{2}$
On simplifying we get
$A.M. = \dfrac{{x + a + x - a}}{{2x}}$
Which implies that $A.M. = \dfrac{{2x}}{{2x}} = 1$
Thus, the arithmetic mean comes out to be equal to 1.

Hence, the correct answer is (B)

Note:- In these types of questions, it is important to remember the formula of arithmetic mean between any given set of numbers. The arithmetic mean is the most commonly used and readily understood measure of central tendency in a data set. In statistics, the term average refers to any of the measures of central tendency. The arithmetic mean of a set of observed data is defined as being equal to the sum of the numerical values of each and every observation, divided by the total number of observations