A boat goes 30 km upstream and 44 km downstream in 13 hours it can go 14 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.

Hint: Start by considering the speed of boat and stream as some variable , remember for upstream the net speed is the difference between boat and stream whereas in downstream it is the sum . Use the speed – time relation to find the time taken in different cases given , using the same solve for the speed of boat and stream.

Complete step-by-step answer:
Let the speed of the boat in still water ${v_{be}}$ be km/h and speed of stream be ${v_r}$km/h
Case 1:-
Boat takes 10 hours to cover 30 km upstream and 44 km downstream
Time taken by boat to cover upstream distance= ${t_u} = \dfrac{{upstream{\text{ }}distance{\text{ }}cover{\text{ }}by{\text{ }}boat}}{{speed{\text{ }}in{\text{ }}upstream}}$
$ \Rightarrow {t_u} = \dfrac{{30}}{{{v_{be}} - {v_r}}}$
Time taken by boat to cover downstream distance) = ${t_d} = \dfrac{{downstream{\text{ }}distance{\text{ }}cover{\text{ }}by{\text{ }}boat}}{{speed{\text{ }}in{\text{ down}}stream}}$
$ \Rightarrow {t_d} = \dfrac{{44}}{{{v_{be}} + {v_r}}}$
So, The Total time taken = ${t_u} + {t_d} = 10$
$ \Rightarrow \dfrac{{30}}{{{v_{be}} - {v_r}}} + \dfrac{{44}}{{{v_{be}} + {v_r}}} = 10 \to eqn.1$
Similarly,
Case 2:-
Boat takes 13 hours to cover 40 km upstream and 55 km downstream, using the same formula as in the above case , we get
$ \Rightarrow {t_u} = \dfrac{{40}}{{{v_{be}} - {v_r}}}$ and $ \Rightarrow {t_d} = \dfrac{{55}}{{{v_{be}} + {v_r}}}$
So, The Total time taken, ${t_u} + {t_d} = 13$
$ \Rightarrow \dfrac{{40}}{{{v_{be}} - {v_r}}} + \dfrac{{55}}{{{v_{be}} + {v_r}}} = 13 \to eqn.2$
Now , We have to solve equations 1 and 2
So , For easy calculation let \[x = \dfrac{1}{{{v_{be}} - {v_r}}}\] and \[y = \dfrac{1}{{{v_{be}} + {v_r}}}\]
Now convert (1) and (2) equation in x and y
$30x{\text{ }} + {\text{ }}44y{\text{ }} = {\text{ }}10\ and\ 40x{\text{ }} + {\text{ }}55y{\text{ }} = 13$
Solving for x and y ,we get
$x = \dfrac{1}{5}$ and $y = \dfrac{1}{{11}}$
Which means ,${v_{be}} - {v_r} = 5$ and ${v_{be}} + {v_r} = 11$
After solving them , we get
${v_{be}} = 8$ and ${v_r} = 3$
Therefore, The Speed of boat in still water = 8km/h and the speed of stream = 3km/h

Note: Similar questions can be asked involving two boats starting simultaneously , one moving downstream and the other upstream, And can ask to locate the point where they meet or cross each other. In that case take the distance of the meeting point be x from one end y from the other , convert all the equations and relation in speed – time relation. Remember that upstream speed is always less than downstream speed.