Question

The arithmetic mean of two observation is 127.5 and their geometric mean is 60. Find their harmonic mean also obtain the observation

Answer 1

Step-by-step explanation:

This is You Answer ......................

Answer 2

Given:

Arithmetic mean $= 127.5$

Geometric mean $=60$

To find :

The two observations and their harmonic mean.

Explanation:

Let the two observations as asked in the question be a and b.

Their Arithmetic mean is given by $A.M. = \frac{a+b}{2}$

The geometric mean will be $G.M. = \sqrt{ab}$   $;$ $and$

The harmonic mean is given by $H.M. = \frac{2ab}{a+b}$

Solution:

Step 1

We have the given values of arithmetic and geometric mean therefore if we substitute the given values in the equations, we get

$a+b=255$   $(i)$

$ab = 3600$     $(ii)$

From (ii) , we get

$b=\frac{3600}{a}$       $(iii)$

Substituting this value in (i), we get

$a+\frac{3600}{a} = 255\\a^{2}-255a+3600=0\\a^{2}-15a-240a+3600=0\\a(a-15)-240(a-15)=0\\(a-240)(a-15)=0$

$or$

$a=15$ $or$ $a=240$

Hence, substituting this in equation (iii) , we get

$b=240$  $or$  $a=15$

Now,

Step 2

Substituting values of (i) and (ii) in the equation of H.M.

We get,

$H.M. = \frac{2(3600)}{255} = 28.2$

Final answer:

Hence, the observations (a,b) are (15,240) or (240,15) and their harmonic mean is 28.2 .