Question

Arithmetic mean between two numbers is 5 and Geometric mean bets
them is 4. Find the Harmonic mean between the numbers.​

Answer 1

Given:

The arithmetic mean between the two numbers is 5

The geometric mean between the two numbers is 4

To find:

Find the Harmonic mean between the numbers.​

Solution:

We know If a and b are two positive numbers then,

$\boxed{Arithmatic \:Mean \:(A.M.) = \frac{a+b}{2} }\\\\\boxed{Geometric \:Mean \:(G.M.) = \sqrt{ab}}\\\\\boxed{Harmonic \:Mean \:(A.M.) = \frac{2ab}{a+b} }$

Therefore, we can conclude the relationship between A.M., G.M. & H.M. is:

$\boxed{\bold{G.M. = \sqrt{A.M. \times H.M.} }}$

Now, substituting A.M. = 5 & G.M. = 4, we get

$4 = \sqrt{5 \times H.M.}$

squaring both sides

$\implies (4)^2 = (\sqrt{5 \times H.M.})^2$

$\implies 16 = 5 \times H.M.$

$\implies H.M. = \frac{16}{5}$

$\implies \bold{H.M. = 3.2}$

Thus, the Harmonic mean between the two numbers is 3.2.

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