smc

If we want $\frac{A+B}{C+D}$ to be as large as possible, we want to try to maximize the numerator $A+B$ and minimize the denominator $C+D$. Picking $A=9$ and $B=8$ will maximize the numerator, and picking $C=0$ and $D=1$ will minimize the denominator. Checking to make sure the fraction is an integer, $\frac{A+B}{C+D}=\frac{17}{1}=17$, and so the values are correct, and $A+B=17$, giving the answer $\boxed{17}$.