Question

The harmonic mean of two numbers is 4. The arithmetic mean A & the geometric mean G satisfy the relation 2A + G²= 27. Find the sum of two numbers.

• A. 7
• B. 8
• C. 9
• D. 10

Answer 1

Answer: (C)

9

Answer 2

Let the two numbers be $x$ and $y$.

Given:

• Harmonic Mean (H) = 4, so $\frac{2xy}{x+y} = 4$
• $2A + G^2 = 27$, where A is the arithmetic mean and G is the geometric mean.

We have $A = \frac{x+y}{2}$ and $G = \sqrt{xy}$. Substituting these into the second equation:

$2(\frac{x+y}{2}) + (\sqrt{xy})^2 = 27$ $x + y + xy = 27$

From the harmonic mean equation, we have $2xy = 4(x+y)$, which simplifies to $xy = 2(x+y)$.

Substitute $xy = 2(x+y)$ into $x + y + xy = 27$:

$x + y + 2(x+y) = 27$ $3(x+y) = 27$ $x+y = 9$

Therefore, the sum of the two numbers is 9.

To verify:

If $x+y = 9$, then $xy = 2(9) = 18$. The quadratic equation with roots x and y is $t^2 - 9t + 18 = 0$. Factoring gives $(t-3)(t-6) = 0$, so $x = 3$ and $y = 6$ (or vice versa).

Check:

• $H = \frac{2(3)(6)}{3+6} = \frac{36}{9} = 4$
• $A = \frac{3+6}{2} = \frac{9}{2}$
• $G = \sqrt{3 \times 6} = \sqrt{18}$
• $2A + G^2 = 2(\frac{9}{2}) + (\sqrt{18})^2 = 9 + 18 = 27$

Both conditions are satisfied.