Was learning a bit about discrete math and was trying to get an expression for when the above equality is true/false.
I have this so far: For any real numbers $x$, $y$: (I know I should avoid division by zero here, but otherwise I am unsure if I need to narrow the domain to just positive numbers, et cetera)
Separating $x$ and $y$ into an integer and decimal part means that $\displaystyle \left\lfloor \frac{\lfloor \operatorname{int}_x + \operatorname{decimal}_x \rfloor}{\lfloor \operatorname{int}_y + \operatorname{decimal}_y \rfloor} \right\rfloor$ will simplify to $\displaystyle \left\lfloor \frac{\operatorname{int}_x}{\operatorname{int}_y} \right\rfloor$, since the floor function would remove the decimal parts.
So now I am trying to simplify $\displaystyle \left\lfloor \frac{\lfloor \operatorname{int}_x + \operatorname{decimal}_x \rfloor}{\lfloor \operatorname{int}_y + \operatorname{decimal}_y \rfloor} \right\rfloor$ to $\displaystyle \left\lfloor \frac{\operatorname{int}_x}{\operatorname{int}_y} \right\rfloor$, but got stuck there.
Thanks for the help!