Let $a_1, a_2, \ldots a_n$ and $b_1, b_2, \ldots b_n$ be two sequences of $n\gg 1$ real numbers such that, for all $1\le i\le n$, we have $0<a_i \le b_i\le 1$. Let the ratio $R$ defined as follows:

$$R:=\frac{\sum_{1\le i\le n} a_i}{\sum_{1\le i\le n} b_i }~.$$

**Question:** How can we set values $x_1, x_2, \ldots, x_n$ *as functions of the ratios $r_i:=\frac{a_i}{b_i}$* in such a way that $R':=\sum_{1\le i\le n} (x_i\cdot r_i)$ is a *tight* lower bound for $R$?