Let the two number be a and b. According to the question, we have
2ab
a+b
= 4 and 2A + G2 = 27 [ A =
a+b
2
, G = √ab]
∴ ab = 2 (a + b) ... (i) and a + b + ab = 27 ... (ii) ⇒ a +b + 2 (a + b) = 27 ⇒ 3 (a + b) = 27 ⇒ a + b = 9 ... (iii) From Eqs. (i) and (iii), we have ab = 18 ... (iv) Putting, b =
18
a
from Eqs, (iv) in Eqs. (iii), we get a +
18
a
= 9 ⇒ a2 - 9a + 18 = 0 ⇒ (a + 3) (a - 6) = 0 ⇒ a = 3 , 6 ∴ b = 6 , 3 ∴ Required numbers are 3 and 6.