The problem asks us to express $\frac{3\sqrt{2}-\sqrt{3}}{2\sqrt{3}-\sqrt{2}}$ in the form $\frac{\sqrt{m}}{\sqrt{n}}$, where $m$ and $n$ are whole numbers. Then, we must choose the correct answer from the given options: (a) $\frac{\sqrt{6}}{\sqrt{10}}$, (b) $\frac{\sqrt{150}}{\sqrt{100}}$, (c) $\frac{2\sqrt{6}}{\sqrt{10}}$, (d) $\frac{5\sqrt{6}}{\sqrt{10}}$.

1. Problem Description

The problem asks us to express $\frac{3\sqrt{2}-\sqrt{3}}{2\sqrt{3}-\sqrt{2}}$ in the form $\frac{\sqrt{m}}{\sqrt{n}}$, where $m$ and $n$ are whole numbers. Then, we must choose the correct answer from the given options: (a) $\frac{\sqrt{6}}{\sqrt{10}}$, (b) $\frac{\sqrt{150}}{\sqrt{100}}$, (c) $\frac{2\sqrt{6}}{\sqrt{10}}$, (d) $\frac{5\sqrt{6}}{\sqrt{10}}$.

2. Solution Steps

First, let's rationalize the denominator of the given expression:

$$\frac{3\sqrt{2}-\sqrt{3}}{2\sqrt{3}-\sqrt{2}} = \frac{3\sqrt{2}-\sqrt{3}}{2\sqrt{3}-\sqrt{2}} \cdot \frac{2\sqrt{3}+\sqrt{2}}{2\sqrt{3}+\sqrt{2}}$$

$$= \frac{(3\sqrt{2}-\sqrt{3})(2\sqrt{3}+\sqrt{2})}{(2\sqrt{3})^2 - (\sqrt{2})^2}$$

$$= \frac{3\sqrt{2} \cdot 2\sqrt{3} + 3\sqrt{2} \cdot \sqrt{2} - \sqrt{3} \cdot 2\sqrt{3} - \sqrt{3} \cdot \sqrt{2}}{4 \cdot 3 - 2}$$

$$= \frac{6\sqrt{6} + 3 \cdot 2 - 2 \cdot 3 - \sqrt{6}}{12 - 2}$$

$$= \frac{6\sqrt{6} + 6 - 6 - \sqrt{6}}{10}$$

$$= \frac{5\sqrt{6}}{10} = \frac{\sqrt{6}}{2}$$

Now, we need to express $\frac{\sqrt{6}}{2}$ in the form $\frac{\sqrt{m}}{\sqrt{n}}$.

We can rewrite 2 as $\sqrt{4}$, so

$$\frac{\sqrt{6}}{2} = \frac{\sqrt{6}}{\sqrt{4}}$$

To obtain one of the answer options, we can multiply both the numerator and denominator by $\sqrt{\frac{5}{5}}$ to get

$$\frac{\sqrt{6}}{\sqrt{4}} \cdot \frac{\sqrt{\frac{5}{5}}}{1} = \frac{\sqrt{6}}{\sqrt{4}}$$

To match one of the forms of the given answers, we can write

$$\frac{\sqrt{6}}{2} = \frac{5\sqrt{6}}{10} = \frac{\sqrt{25 \cdot 6}}{\sqrt{100}} = \frac{\sqrt{150}}{\sqrt{100}}$$

Another way to compare the result $\frac{\sqrt{6}}{2}$ to the given options is to compare their squares:

$(\frac{\sqrt{6}}{\sqrt{10}})^2 = \frac{6}{10} = \frac{3}{5} = 0.6$

$(\frac{\sqrt{150}}{\sqrt{100}})^2 = \frac{150}{100} = \frac{3}{2} = 1.5$

$(\frac{2\sqrt{6}}{\sqrt{10}})^2 = \frac{4 \cdot 6}{10} = \frac{24}{10} = 2.4$

$(\frac{5\sqrt{6}}{\sqrt{10}})^2 = \frac{25 \cdot 6}{10} = \frac{150}{10} = 15$

$(\frac{\sqrt{6}}{2})^2 = \frac{6}{4} = \frac{3}{2} = 1.5$

Therefore, the correct answer is $\frac{\sqrt{150}}{\sqrt{100}}$.

3. Final Answer

(b) $\frac{\sqrt{150}}{\sqrt{100}}$