How did my book simplify this? How did my book go from:
$\frac{4}{5}=\frac{x}{30}$
to
$\frac{4}{1}=\frac{x}{6}$
I understand that I could have cross multiplied it in the first place but what I don't understand is why my book changes the denominators without changing the numerators when simplifying the problem.

You can rewrite 30 as $5\times 6$:
$$\frac{4}{5}=\frac{x}{5\times 6}\;\; \text{ multiply both sides by 5 } \;\;5\times \frac{4}{5}=5\times \frac{x}{5 \times 6}$$
if you then do some rearrangement you can get:
$$ \frac{5}{5}\times \frac{4}{1}=\frac{5}{5}\times \frac{x}{6}$$
Then replacing $5/5$ with 1 gives your result:
$$ \frac{4}{1}=\frac{x}{6} $$
