I have the following sum listed in Algebra Demystified:
$\frac{71}{84} - \frac{13}{30x} = \frac{71}{84}\cdot \frac{5x}{5x} - \frac{13}{30x} \cdot \frac{14}{14} = \frac{355x}{420x} - \frac{182}{420x} = \frac{355x-182}{420x}$
However, I previously understood it as follows:
$\frac{a}{b} - \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{d} - \frac{c}{d} \cdot \frac{b}{b}= \frac{ad-bc}{bd}$
In which case, why would
$\frac{71}{84}\cdot \frac{5x}{5x}$
shouldn't it be $\frac{14}{14}$?
No. Because you need to match the denominators. That's it.