A fraction proof problem

Question

Known: $$N=\frac{A+B}{A+C}$$ $$A>1$$ $$\frac{B}{C}>1$$ Does $N$ increase/decrease with respect to $A$ (In other words, will increase $A$ always decrease $N$?

Answer 1

Assume both $B$ and $C$ are positive. Then $B/C>1$ implies $B>C$, so we can write $B=C+D$ for some $D>0$. Then: $$N=\frac{A+B}{A+C}=\frac{A+C+D}{A+C}=1+\frac{D}{A+C},$$ which clearly decreases with respect to $A$ (since $C$ and $D$ are positive constants). Similarly, you can handle the other case, when both $B$ and $C$ are negative.