Inequality relationship when additing a constant to the denominators

Question

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?

Answer 1

No: $\dfrac12>\dfrac25,$ but $\dfrac13=\dfrac26.$

Answer 2

$$\dfrac ab>\dfrac cd\iff\dfrac{ad-bc}{bd}>0\iff ad>bc$$

Now $$\dfrac a{b+1}-\dfrac c{d+1}=\dfrac{ad+a-(bc+c)}{(b+1)(d+1)}$$ will be $<0$ if $ad+a-(bc+c)<0\iff ad-bc<c-a$