Can adding 1 to the numerator and denominator on both sides of an inequation change the outcome

Question

I'm computing a fraction from a database when both numerator and denominator can be zero. To solve this problem I thought of adding 1 to each.

I know I can add 1 only to the denominator, but this is for optimization of resources and adding 1 to the denominator favors tasks which have a low denominator.

Because 3/3 == 4/4, but 3/4 > 4/5 and thus the task with 4 will get the resources because the program will think it has more to complete.

This brings me to my question:

If I know that

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

Can $\frac{a+1}{b+1} < \frac {c+1}{d+1}$ happen, even once?

The above formula translates to

$a+d > c+b+(bc-ad)$

and this is where I'm stuck.

Answer 1

$\frac{7}{10}\gt\frac{2}{3}$ but $\frac{8}{11}\lt \frac{3}{4}$