help

If the arithmetic mean of a two numbers is 10 and the harmonic mean is 360. Find the geometric mean of the numbers.

 

SInce there are two numbers, and the arithmetic mean is 10, then the sum of the two numbers is 20.

 

The harmonic mean is \(2/(1/a+1/b)\) where a and b are the two numbers. Since it equals 360, then 1/a + 1/b must equal 1/180.

We can rewrite 1/a + 1/b to get \((a+b)/ab\)  and since we know a+b = 20, we get 20/ab = 1/180. Therefore ab = 3600.

The geometric mean is \(\sqrt{ab}\) so we get an answer of 60. 

I hope this helped.

Definitions: 

\(\text{Arithmetic mean} = \dfrac{a + b}2\\ \text{Geometric mean} = \sqrt{ab}\\ \text{Harmonic mean} = \dfrac2{\dfrac1a + \dfrac1b}\)

 

We get, by using the definitions,

\(\begin{cases}a + b = 20\\\dfrac1a + \dfrac1b = \dfrac1{180}\end{cases}\)

 

From equation (2), 

\(\dfrac{a + b}{ab} = \dfrac1{180}\\ \dfrac{20}{ab} = \dfrac1{180}\\ ab = 3600\\ \text{Geometric mean} = \sqrt{ab} = \sqrt{3600} = \boxed{60}\)