Step-by-step Solution:

Concept:

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

⇒ A = \(\rm \dfrac {x + y}{2}\)

⇒ \( G^2 = xy\)

⇒ \(\rm H = \dfrac {2xy}{x+y}\)

Calculations:

Consider, the two numbers are x and y.

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

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

⇒ \( G^2 = xy\) ....(2)

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

⇒ \(\rm \dfrac {2xy}{x+y}= 4\)

⇒ 2xy = 4(x + y)

⇒ \(\rm xy = 2(x+y)\)

⇒ \(G^2 = 4A\) (since x + y = 2A)

⇒ \(G^2 = 4A\) ....(3)

Given, their arithmetic mean A and the geometric mean G satisfy the relation:

2A +\( G^2\) = 27

⇒ 2A + \(G^2\) = 27

⇒ 6A = 27

⇒ A = \(\rm \dfrac 9{2}\)

From equations (1), (2), and (3), we have:

x + y = 9 and xy = 18

⇒ x = 6 and y = 3

Hence, the harmonic mean of two numbers is 4, and their arithmetic mean A and the geometric mean G satisfy the relation 2A + \(G^2\) = 27, then the two numbers are 6 and 3.