The problem asks to simplify the expression $\frac{\sqrt{18}}{\sqrt{8}} \times \frac{\sqrt{20}}{\sqrt{30}}$.

1. Problem Description

The problem asks to simplify the expression $\frac{\sqrt{18}}{\sqrt{8}} \times \frac{\sqrt{20}}{\sqrt{30}}$.

2. Solution Steps

First, simplify each square root:

$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$

$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$

$\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}$

$\sqrt{30} = \sqrt{30}$

Substitute the simplified square roots into the original expression:

$\frac{3\sqrt{2}}{2\sqrt{2}} \times \frac{2\sqrt{5}}{\sqrt{30}}$

Simplify the first fraction by cancelling out the $\sqrt{2}$ terms:

$\frac{3}{2} \times \frac{2\sqrt{5}}{\sqrt{30}}$

Simplify the second fraction by cancelling out the 2:

$\frac{3}{1} \times \frac{\sqrt{5}}{\sqrt{30}} = 3 \times \frac{\sqrt{5}}{\sqrt{30}}$

Combine the fractions:

$3 \times \sqrt{\frac{5}{30}} = 3 \times \sqrt{\frac{1}{6}}$

Simplify the square root:

$3 \times \frac{\sqrt{1}}{\sqrt{6}} = 3 \times \frac{1}{\sqrt{6}} = \frac{3}{\sqrt{6}}$

Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{6}$:

$\frac{3}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{3\sqrt{6}}{6}$

Simplify the fraction:

$\frac{3\sqrt{6}}{6} = \frac{\sqrt{6}}{2}$

3. Final Answer

$\frac{\sqrt{6}}{2}$