Question

If $\dfrac{1}{a} + \dfrac{1}{b} = \dfrac{1}{c}$ and $ab = c$, what is the average (arithmetic mean) of $a$and $b$?
A.$0$
B.$\dfrac{1}{2}$
C.$1$
D.$\dfrac{{a + b}}{{2c}}$

Answer 1

Here, we will find the average of the given two variables. First, we will solve the given conditions to find the sum of the given variables. Then by using the average formula, we will find the arithmetic mean of the variables. The arithmetic mean is defined as the ratio between the sum of the observations to the number of observations.

Formula Used:
Average of the observations is given by the formula $\overline X = \dfrac{{\sum x }}{n}$ where $\sum x$ is the sum of the observations, $n$ is the number of observation and $\overline X$ is the average.

Complete step-by-step answer:
We are given that
$\dfrac{1}{a} + \dfrac{1}{b} = \dfrac{1}{c}$……………………………..$\left( 1 \right)$
$ab = c$…………………………………..$\left( 2 \right)$
By cross-multiplying on LHS of the equation $\left( 1 \right)$ , we get
$\Rightarrow \dfrac{1}{a} \times \dfrac{b}{b} + \dfrac{1}{b} \times \dfrac{a}{a} = \dfrac{1}{c}$
By multiplying the terms, we get
$\Rightarrow \dfrac{b}{{ab}} + \dfrac{a}{{ab}} = \dfrac{1}{c}$
By adding the like terms, we get
$\Rightarrow \dfrac{{a + b}}{{ab}} = \dfrac{1}{c}$
Substituting $ab = c$ in the above equation, we get
$\Rightarrow \dfrac{{a + b}}{c} = \dfrac{1}{c}$
By cancelling out the denominators, we get
$\Rightarrow a + b = 1$………………………………………………………………………………..$\left( 3 \right)$
Now, we will find the average of $a$ and $b$.
Average of the observations is given by the formula $\overline X = \dfrac{{\sum x }}{n}$. So,
Average of $a$ and b$$$$ = \dfrac{{a + b}}{2}
Substituting equation $\left( 3 \right)$ in the above equation, we get
$\Rightarrow$ Average of $a$ and b$$$$ = \dfrac{1}{2}
Therefore, the average or arithmetic mean of $a$ and $b$ is $\dfrac{1}{2}$ .
Thus option (B) is the correct answer.

Note: We should note that at first, we will find the sum of the variables by using the condition. We have three different types of mean which include Arithmetic mean, Geometric mean, and Harmonic mean. Geometric mean is defined as the ${n^{th}}$ root of the product of the variables, Harmonic mean is defined as the average of ratios. Thus, the average and arithmetic mean are the same.