First of all, my math skills are very basic. Well, reviewing some basic arithmetic (from school), I find that I don't really understand why comparing fractions by multiplying the numerators and denominators works.

Given $$ \frac{a}{b}, \frac{c}{d} $$

I can compare fractions by doing $$ a \cdot b, \ c \cdot d$$

And comparing the results of the products to determine the relationship $ =, <, >$.

Okay, now, how should I interpret the product of the numerator of the first fraction and the denominator of the second fraction, $ad$ and $bc$? What am I actually doing?