Dividing summations that have existing properties in each element

Question

If $a_i/c_i > B$ for all $1 \le i \le k$, is it fair to assume that $(a_1 + a_2 + \cdots + a_k)/(c_1 + c_2 + \cdots + c_k) > B$ ? Is there a way to prove this? Thanks!

Answer 1

It holds if $c_i>0$ for each $i$.

$a_i>Bc_i$ for each $i$, so $\sum a_i>\sum Bc_i=B\sum c_i$. Now simply divide by $\sum c_i$.