Let $a,b,c,d$ be positive numbers. To compare the fractions $\frac{a}{b}$ and $\frac{c}{d}$, we first write

$\frac{a}{b} ? \frac{c}{d}$

since we do not yet know whether the correct relation is $=$, $<$, or $>$ we will use "?" for now.

Because $b>0$ and $d>0$, their product $bd$ is also positive. Multiplying both sides by a positive number does not change the inequality sign. Therefore,

$\frac{a}{b}$ ? $\frac{c}{d}$ $\Longleftrightarrow \frac{a}{b}\cdot bd \ ? \frac{c}{d}\cdot bd$

Simplifying both sides gives

$ad$ ? $bc$.

Now $ad$ and $bc$ are positive numbers, so they can be compared directly. If $ad=bc$, then $\frac{a}{b}=\frac{c}{d}$; if $ad<bc$, then $\frac{a}{b}<\frac{c}{d}$; and if $ad>bc$, then $\frac{a}{b}>\frac{c}{d}$.

Thus, cross multiplication works because multiplying both sides of the comparison by the positive number $bd$ converts the comparison of fractions into a comparison of ordinary positive numbers, while preserving the inequality.

Let $a,b,c,d$ be positive numbers. To compare the fractions $\frac{a}{b}$ and $\frac{c}{d}$, we first write

$\frac{a}{b} ? \frac{c}{d}$

since we do not yet know whether the correct relation is $=$, $<$, or $>$ we will use "?" for now.

Because $b>0$ and $d>0$, their product $bd$ is also positive. Multiplying both sides by a positive number does not change the inequality sign (this is what basically cross multilication does). Therefore,

$\frac{a}{b}$ ? $\frac{c}{d}$ $\Longleftrightarrow \frac{a}{b}\cdot bd \ ? \frac{c}{d}\cdot bd$

Simplifying both sides gives

$ad$ ? $bc$.

Now $ad$ and $bc$ are positive numbers, so they can be compared directly. If $ad=bc$, then $\frac{a}{b}=\frac{c}{d}$; if $ad<bc$, then $\frac{a}{b}<\frac{c}{d}$; and if $ad>bc$, then $\frac{a}{b}>\frac{c}{d}$.

Thus, cross multiplication works because multiplying both sides of the comparison by the positive number $bd$ converts the comparison of fractions into a comparison of ordinary positive numbers, while preserving the inequality.