Let $p_1,\ldots,p_n$ be positive real numbers ("weights") and $a_1,\ldots,a_n$ be given real numbers. Then the weighted arithmetic mean can be approximated using the following inequality:
$$\min(a_1,\ldots,a_n)\le \frac {p_1a_1+\cdots+p_na_n}{p_1+\cdots+p_n} \le\max(a_1,\ldots,a_n),$$ where $\min$ and $\max$ denote the minimum and the maximum of the numbers $a_1,\ldots,a_n.$
Proofs: 1
Proofs: 1