Prove that $\frac{\sum_i a_i}{\sum_i b_i} \leq \arg\max_i \frac{a_i}{b_i}$

Asked May 26 '20 at 16:20

Active May 26 '20 at 16:29

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I am pretty sure that I have seen this inequality before but I don't remember ever seeing a proof of it, and I can't seem to prove it.

edited May 26 '20 at 16:29

Bernard

asked May 26 '20 at 16:20

SPV

What is the “arg” on the right-hand side? Do you mean this: https://math.stackexchange.com/q/704411/42969 ? – Martin R May 26 '20 at 16:24
Yes! Thank you so much! – SPV May 26 '20 at 16:26
You need to deman $b_{i} >0$. If not, take $a_{1} = -1$, $a_{2}=1$, $b_{1}=2$ and $b_{2}=-1$. You would have $\frac{a_{1}+a_{2}}{b_{1}+b_{2}}=0$, $\frac{a_{1}}{b_{1}}=-\frac{1}{2}$ and $\frac{a_{2}}{b_{2}} = -1$. – JustWannaKnow May 26 '20 at 16:27

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