[ Motion problems ]  $[$ Average values ]

Two identical boats, having the same speed in still water, travel the same distance (downstream) along two different rivers and return back (upstream). In which river will this trip take more time: in the river with fast current or in the river with slow current?
Let the speed of the boats be $v$ km/h, the speed of the current in the first river be $v_{1}$ km/h, and the speed of the current in the second river be $v_{2}$ km/h. Let $v_{1}>v_{2}$. If the distance traveled in one direction by the boats is denoted as $S$, then the time spent by the first boat on the entire journey,

$t_{1}=\frac{S}{v+v_{1}}+\frac{S}{v-v_{1}}=\frac{2 S v}{v^{2}-v_{1}^{2}}$, and the time spent by the second boat, $t_{2}=\frac{2 S v}{v^{2}-v_{2}^{2}}$.

Since the numerators of both expressions are the same, the fraction with the smaller denominator will be larger, and since the denominators are differences with equal minuends, the denominator is smaller in the first fraction, where the subtrahend $v_{1}^{2}$ is larger.

## Answer

In the river with the faster current.