jmc

Let the two chosen numbers be $a$ and $b$. To maximize the quotient, we first have either $a,b>0$ or $a,b<0$, and from there we maximize $|a|$ and minimize $|b|$. For the case $a,b<0$, we have $a=-24$ and $b=-2$, which gives us $(-24)/(-2)=12$. For the case $a,b>0$, we have $a=8$ and $b=1$, which gives us $8/1=8$. Since $12>8$, our answer is $\boxed{12}$.