Question

Find the harmonic mean of two numbers whose geometric mean and arithmetic mean is 5 and 8

Answer 1

Solution of this sum is as follows

Answer 2

Given :

For any two numbers ,

Geometric mean = G.M = 5

Arithmetic mean = A.M = 8

To Find :

The Harmonic mean of two numbers

Solution :

Let The two numbers = a , b

Since, Arithmetic mean =  $\dfrac{a+b}{2}$  

i.e        A . M = $\dfrac{a+b}{2}$                               .............`1

The Geometric mean = $\sqrt{ab}$

i.e         G . M = $\sqrt{ab}$                                ..............2

The Harmonic mean = $\dfrac{2ab}{a+b}$

i.e        H . M =  $\dfrac{2ab}{a+b}$                             .................3

Now,

The product of Arithmetic mean and Harmonic mean = A .M × H.M

i.e  A .M × H.M  =  ($\dfrac{a+b}{2}$ )  ( $\dfrac{2ab}{a+b}$ )

Or,   A .M × H.M  = ab

∵     G . M = $\sqrt{ab}$

So,   ab = ( G . M)²

Or,     A .M × H.M  =  ( G . M)²

A/Q  ,  G.M = 5     and  A.M = 8

So,     8 × H . M = 5 ²

Or,      8 × H . M = 25

∴           H . M = $\dfrac{25}{8}$    = 3.125

Hence, The Harmonic mean of the two numbers is 3.125   Answer