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

Question

The harmonic mean of two numbers is 4 and the arithmetic and geometric mean satisfy the relation 2 A + G2 =27 , the numbers are (A) 6 , 3 (B) 5 , 4 (C) 5 , -2.5 (D) -3 , 1

Tardigrade Admin · Accepted Answer

Let numbers be x and y. Then, A=(1/2)(x+y), √x y=G or G2=x y and (2 x y/x+y)=4 ⇒ G2=4 A Also, 2 A+G2=2 A+4 A=27 ⇒ A=(9/2) So, x+y=9, x y=18 Hence. numbers are 6 and 3 .

Q. The harmonic mean of two numbers is $4$ and the arithmetic and geometric mean satisfy the relation $2 A + G^{2} = 27$ , the numbers are

6 , 3

5 , 4

5 , -2.5

-3 , 1

Let numbers be $x$ and $y$ .
Then, $A = \frac{1}{2} (x + y), x y = G$ or $G^{2} = x y$
and $\frac{2 x y}{x + y} = 4$
$\Rightarrow G^{2} = 4 A$
Also, $2 A + G^{2} = 2 A + 4 A = 27 \Rightarrow A = \frac{9}{2}$
So, $x + y = 9, x y = 18$
Hence. numbers are $6$ and $3$ .

Q. The harmonic mean of two numbers is 4 and the arithmetic and geometric mean satisfy the relation 2A+G2=27 , the numbers are