For $a_1,a_2$, $b_1,b_2$ $\in\mathbb{R}^+$, if $a_1<b_1$ , then for any perturbation $\epsilon\in \mathbb{R}^+$, $$r_1=\frac{a_1+\epsilon}{b_1+\epsilon}>\frac{a_1}{b_1} $$ and if $a_2>b_2$, $$r_2=\frac{a_2+\epsilon}{b_2+\epsilon}<\frac{a_2}{b_2} $$
If $\epsilon$ is generated from a continuous function as $\epsilon={f_\epsilon(t)}$. What would be a way to characterize $r_1-r_2$ in terms of $f_\epsilon(t)?$
I don't understand the problem. If $r_1=(a_1+\epsilon)/(b_1+\epsilon)$ and $r_2=(a_2+\epsilon)/(b_2+\epsilon)$ and $\epsilon=f_{\epsilon}(t)$, then I can express $r_1-r_2$ in terms of $f_{\epsilon}(t)$ as $$r_1-r_2={a_1b_2-a_2b_1+(a_1-a_2-b_1+b_2)f_{\epsilon}(t)\over(b_1+f_{\epsilon}(t))(b_2+f_{\epsilon}(t))}$$ but I don't know what it means to characterize $r_1-r_2$ in terms of $f_{\epsilon}(t)$.