The harmonic mean of two number is 4, Their arithmetic mean A and the geometric mean G satisfy the relation 2A + G² = 27, then the two numbers are

Concept:

Let x and y be the two numbers. The the arithmetic mean A,  geometric mean G and the harmonic mean H of x and y is given by, 

⇒ A = \(\rm \dfrac {x + y}{2}\) 

⇒ G2 = xy

⇒  \(\rm H = \dfrac {2xy}{x+y}\)

Calculations:

Consider, the two numbers are x and y.

Given, the arithmetic mean and geometric mean of the x and y is A and G.

⇒ A = \(\rm \dfrac {x + y}{2}\)           ....(1)

⇒ G² = xy               ....(2)

The harmonic mean of two number x and y is 4.

⇒  \(\rm \dfrac {2xy}{x+y}= 4\)

⇒ 2xy = 4(x + y)

⇒  \(\rm xy = 2(x+y)\)

⇒ G2 = 4A                   (∵ x + y = 2A)

⇒ G2 = 4A            ....(3)

Given, Their arithmetic mean A and the geometric mean G satisfy the relation 2A + G2 = 27.

⇒2A + G2 = 27

⇒ 6A = 27

⇒ A = \(\rm \dfrac 9{2}\)

From equation (1), (2) and (3), we have

 x + y = 9 and xy = 18

⇒ x = 6 and y = 3

Hence, the harmonic mean of two number is 4, Their arithmetic mean A and the geometric mean G satisfy the relation 2A + G2 = 27, then the two numbers are 6 and 3.