The proof of comparing the fractions.

Question

Hi I came up with this method but couldn't find any proof can you provide one thanks

To compare $\frac{a}{b}$ with $\frac{c}{d}$ , compare the cross products as follows:

If ad > bc, then: $\frac{a}{b}$>$\frac{c}{d}$

If ad < bc, then: $\frac{a}{b}$>$\frac{c}{d}$

If ad = bc, then: $\frac{a}{b}$ = $\frac{c}{d}$

I mean how can we conclude the bigger or smaller ones.

Answer 1

It is easy applying the basic rule for use of inequalities:

You can multiply an inequality by any number $C$ without reversing the inequality if $C>0$, reversing it if $C<0$.

So here, starting from $\frac ab<\frac cd$, if you multiply by $bd$,

• If $b,d$ have the same sign, $bd>0$ and therefore $$\frac a{\not b} \cdot \not bd <\frac c{\not d}d\cdot b\not d\iff ad<bc.$$
• If $b,d$ have different signs, $bd<$ so $$\frac a{\not b} \cdot \not bd >\frac c{\not d}d\cdot b\not d\iff ad>bc.$$

Answer 2

Simply note that $\dfrac ab$ is greater than $\dfrac cd$ if $$\dfrac ab -\dfrac cd >0 \implies \dfrac{ad-bc}{bd} >0$$

Now, your method only works if $bd>0$. Similarly reverse the inequality for the other case.