I can prove that if $\frac{a_{i}}{b_{i}} \geq \frac{a_{j}}{b_{j}} $, it follows that $\frac{a_{i}}{b_{i}} \geq \frac{a_{i} + a_{j}}{b_{i} + b_{j}} $. However, I'd like to generalise this result by proving that if $$ \frac{\sum_{i \in S} a_{i}}{\sum_{i \in S} b_{i}} \geq \frac{\sum_{i \in T} a_{i}}{\sum_{i \in T} b_{i}},$$ it follows that $$ \frac{\sum_{i \in S} a_{i}}{\sum_{i \in S} b_{i}} \geq \frac{\sum_{i \in S \cup T} a_{i}}{\sum_{i \in S \cup T} b_{i}} $$ for any pair of sets $S$ and $T$.
Do you know how to prove the latter statement?