Let $a_i , b_i \geqslant 0$  $\forall i \in \{1 , 2 , 3 , .... , n\}$ Prove that $$min\{\frac{a_i}{b_i}\} \leqslant \frac{\sum_{i=1}^n a_i}{\sum_{i=1}^n b_i} \leqslant max\{\frac{a_i}{b_i}\}$$ Given that $min\{...\}$ and $max\{...\}$ equals to the minimum and the maximum element in the set.  I don’t know how to deal with minimum and maximum. I would like some hints (or solution) please. Thank you.
Show $\frac{a_{1}+...+a_{n}}{b_{1}+...+b_{n}}$ is between the smallest and largest fraction $\frac{a_{i}}{b_{i}}$, where $b_{i}&gt;0$.  Attempt  Assume the largest is $\frac{a_{n}}{b_{n}}\Rightarrow$  $\frac{a_{n}}{b_{n}}-\frac{a_{1}+...+a_{n}}{b_{1}+...+b_{n}}\Rightarrow $  $\frac{b_{1}+...+b_{n-1}}{{b_{1}+...+b_{n}}}[\frac{a_{n}}{b_{n}}-\frac{a_{1}}{b_{1}+...+b_{n-1}}-...-\frac{a_{n-1}}{b_{1}+...+b_{n-1}}]\Rightarrow $  if $a_{1}&lt;0$, $w=\frac{a_{n}}{b_{n}}-\frac{a_{1}}{b_{1}+...+b_{n-1}}-...-\frac{a_{n-1}}{b_{1}+...+b_{n-1}}&gt;\frac{a_{n}}{b_{n}}-\frac{a_{2}}{b_{1}+...+b_{n-1}}-...-\frac{a_{n-1}}{b_{1}+...+b_{n-1}}$  any hints or solutions?
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