This came up in statistics homework:
$$ \left(\frac{a_1}{b_1}\right)^{r_1} \cdot \left(\frac{a_2}{b_2}\right) ^{r_2} \cdot \ldots \cdot \left(\frac{a_s}{b_s}\right)^{r_s} = \frac{a_1+a_2+\ldots+a_s}{b_1+b_2+\ldots+b_s}, $$
or, equivalently
$$ \prod_{i=1}^{s}{\left[\left(\frac{a_i}{b_i}\right)^{r_i}\right]} = \frac{\sum_1^s a_i}{\sum_1^s b_i}. $$
$r_i>0$ are rationals that add up to 1, i.e. $\sum r_i=1$. $a_i,b_i$ are reals.
If $r_i$ are given, can $a_i, b_i$ be found that satisfy this equation? The case for $s=1$ is trivial, and some numerical explorations seem to be able to find answers for $s=2$, so my hunch is that this should be true. But how to show it?