#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}#