Raul Walker

2022-07-10

Inequality relationship when additing a constant to the denominators
When $\frac{a}{b}>\frac{c}{d}$ where $a,b,c,$ and $d$ are positive real numbers, is $\frac{a}{b+1}>\frac{c}{d+1}$ true?

Jenna Farmer

Expert

No: $\frac{1}{2}>\frac{2}{5},$ but $\frac{1}{3}=\frac{2}{6}.$

Pattab

Expert

$\frac{a}{b}>\frac{c}{d}\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}\frac{ad-bc}{bd}>0\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}ad>bc$
Now
$\frac{a}{b+1}-\frac{c}{d+1}=\frac{ad+a-\left(bc+c\right)}{\left(b+1\right)\left(d+1\right)}$
will be $<0$ if $ad+a-\left(bc+c\right)<0\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}ad-bc

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