The problem is to evaluate the expression $12\sqrt{12} \div 16\sqrt{18}$.

1. Problem Description

The problem is to evaluate the expression $12\sqrt{12} \div 16\sqrt{18}$.

2. Solution Steps

First, rewrite the expression as a fraction:

$\frac{12\sqrt{12}}{16\sqrt{18}}$.

Simplify the numbers outside the square roots:

$\frac{12}{16} = \frac{3}{4}$.

Now, simplify the square roots:

$\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4}\sqrt{3} = 2\sqrt{3}$

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

Substitute these simplified square roots into the expression:

$\frac{3}{4} \cdot \frac{2\sqrt{3}}{3\sqrt{2}} = \frac{3 \cdot 2 \cdot \sqrt{3}}{4 \cdot 3 \cdot \sqrt{2}} = \frac{6\sqrt{3}}{12\sqrt{2}}$.

Simplify the fraction:

$\frac{6}{12} = \frac{1}{2}$

So the expression becomes:

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

To rationalize the denominator, multiply both the numerator and denominator by $\sqrt{2}$:

$\frac{\sqrt{3}}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{3}\sqrt{2}}{2\sqrt{2}\sqrt{2}} = \frac{\sqrt{6}}{2 \cdot 2} = \frac{\sqrt{6}}{4}$.

3. Final Answer

The final answer is $\frac{\sqrt{6}}{4}$.