Question

The mean proportion of $\frac{a^2}{b^3} and \frac{9b^2}{4a^3}$ is _________.

• A. $$\frac{9}{4\sqrt{ab}}$$
• B. $$\frac{3}{2\sqrt{ab}}$$
• C. $$\frac{3}{2(ab)}$$
• D. $$\frac{9}{4(ab)}$$

Answer 1

Answer:

$$\frac{3}{2\sqrt{ab}}$$

Mean proportional (geometric mean) of two numbers (x) and (y) is:
$\sqrt{xy}$

Given:
$x = \frac{a^2}{b^3}, \quad y = \frac{9b^2}{4a^3}$

Multiply

$xy = \frac{a^2}{b^3} \cdot \frac{9b^2}{4a^3}$

$= \frac{9a^2 b^2}{4a^3 b^3}$

Simplify powers:
$= \frac{9}{4} \cdot \frac{1}{ab}$

Take square root

$\sqrt{xy} = \sqrt{\frac{9}{4ab}} = \frac{3}{2\sqrt{ab}}$