The boat passes between two points on the river downstream in 8 hours, and back in 12 hours.

The boat passes between two points on the river downstream in 8 hours, and back in 12 hours. Considering the speed of the boat and the speed of the current constant, find the time for which the boat would have covered such a distance in the still water of the lake.

We denote:
Tpo – time downstream;
Tpr – time upstream;
Tk – time in stagnant water;
S – path;
Vk is the speed of the boat;
Vt is the current velocity.

Let’s make the equations:
Tpo = S / (Vk + Vt);
Tpr = S / (Vk – Vt);
Tk = S / Vk.

Let us add the first two equations
Tpo + Tpr = S / Vk + S / Vt + S / Vk – S / Vt;
Tpo + Tpr = 2 * S / Vk;
Vk = 2 * S / (Tpo + Tpr).

Substitute Vk into the third equation
Tk = S / ((2 * S) / (Tpo + Tpr)) = (Tpo + Tpr) * S / 2S =
= (Tpo + Tpr) / 2 = (8 + 12) / 2 = 10 hours.

Answer: 10 hours