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: $$ \frac{4}{1}=\frac{x}{6} $$