Question

Find the AM, GM and HM between the numbers 12 and 30.

Answer 1

Hint: The formulas to determine arithmetic, geometric and harmonic mean between two numbers $a$ and $b$ are:
$ \Rightarrow AM = \dfrac{{a + b}}{2},{\text{ }}GM = {\left( {ab} \right)^{\dfrac{1}{2}}}$ and $HM = \dfrac{{2ab}}{{a + b}}$.
Use these formulas and put values of the given numbers in place of $a$ and $b$ to get the required means.

Complete step-by-step answer:
According to the question, the given two numbers are 12 and 30.
We know that the formula for finding the arithmetic mean between two numbers $a$ and $b$ is given as:
\[ \Rightarrow AM = \dfrac{{a + b}}{2}\]
Putting the given numbers in place of $a$ and $b$, we’ll get:
\[
   \Rightarrow AM = \dfrac{{12 + 30}}{2} \\
   \Rightarrow AM = \dfrac{{42}}{2} \\
   \Rightarrow AM = 21
 \]
Further, the formula for finding the geometric mean between two numbers $a$ and $b$ is given as:
$ \Rightarrow GM = {\left( {ab} \right)^{\dfrac{1}{2}}} = \sqrt {ab} $
Again putting the given numbers in place of $a$ and $b$, we’ll get:
$
   \Rightarrow GM = \sqrt {12 \times 30} \\
   \Rightarrow GM = \sqrt {6 \times 2 \times 6 \times 5} \\
   \Rightarrow GM = 6\sqrt {10}
 $
And the formula for finding the harmonic mean between two numbers $a$ and $b$ is given as:
$ \Rightarrow HM = \dfrac{{2ab}}{{a + b}}$
Putting the given numbers in place of $a$ and $b$, we’ll get:
$
   \Rightarrow HM = \dfrac{{2 \times 12 \times 30}}{{12 + 30}} \\
   \Rightarrow HM = \dfrac{{24 \times 6 \times 5}}{{42}} \\
   \Rightarrow HM = \dfrac{{120}}{7}
 $

Thus the AM, GM and HM between the numbers 12 and 30 are $20,{\text{ 6}}\sqrt {10} $ and $\dfrac{{120}}{7}$ respectively.

Note: The formulas used above can be extended for more than two numbers also. If ${a_1}{\text{, }}{a_2},....,{a_n}$ are $n$ different numbers, then the formula for arithmetic mean of these numbers will be:
$ \Rightarrow AM = \dfrac{{{a_1} + {a_2} + ..... + {a_n}}}{n}$
Similarly, the formula for geometric mean of these numbers will be:
$ \Rightarrow GM = {\left( {{a_1}.{a_2}....{a_n}} \right)^{\dfrac{1}{n}}}$
And, the formula for harmonic mean of these numbers will be:
$ \Rightarrow HM = \dfrac{n}{{\dfrac{1}{{{a_1}}} + \dfrac{1}{{{a_2}}} + .... + \dfrac{1}{{{a_n}}}}}$