v_b : Speed of the boat in still water,
v_r : Speed of the river current.
v_{\uarr} : Speed of the boat upstream,
v_{\darr} : Speed of the boat downstream.
v_{\uarr} = v_b - v_r; \qquad v_{\darr} = v_b + v_r;
S_{\uarr} : Distance travelled upstream in time \Delta t
S_{\darr} : Distance travelled downstream in time \Delta t
Given: \qquad v_b = 2 km.hr^{-1}; \qquad S_{\uarr} = 3 km; \qquad S_{\darr} = 6 km
For the same time interval, calculate the distance travelled upstream and downstream,
Upstream: \qquad S_{\uarr} = v_{uarr}.\Delta t;
\Delta t = \frac{S_{\uarr}}{v_{\uarr}} = S_{\uarr}/(v_b - v_r) ...... (1)
Downstream:\qquad S_{\darr} = v_{darr}.\Delta t;
\Delta t = \frac{S_{\darr}}{v_{\darr}} = S_{\darr}/(v_b + v_r) ...... (2)
Comparing (1) and (2) we get,
\frac{S_{\uarr}}{v_b - v_r} = \frac{S_{\darr}}{v_b + v_r}; \qquad (v_b + v_r) = (\frac{S_{\darr}}{S_{\uarr}}) (v_b - v_r)
S_{\uarr} = 3 km; \qquad S_{\darr} = 6 km; \qquad \frac{S_{\darr}}{S_{\uarr}} = (6 km)/(3 km) = 2
(v_b + v_r) = 2 (v_b - v_r);
v_r = v_b/3 = (2 km.hr^{-1})/3 = 2/3 km.hr^{-1}
