Question

A motor boat heads upstream a distance of $24km$ on a river whose current is running at $3kmph$. The trip up and back takes $6hours$.(Assuming that the motorboat maintained a constant speed) what was its speed in still water?

Answer 1

Hint: Before solving the question, lets know the concept of stream problems
When the motor boat is going against the flow of a river is called upstream and when the motor boat is going along the flow of a river is called downstream.

Complete step-by-step answer:
Let the speed of the boat is $U\,kmph$ and the speed of the river flow is $V\,kmph$
Now, Downstream speed of the boat is $(U + V)kmph$, because both are going in the same direction and the speed gets added.
Now, the Upstream speed of the boat is $(U - V)kmph$, because both are going in opposite directions to each other and the speed gets subtracted.
Speed of the boat in still water is $\dfrac{1}{2}\left( {x + y} \right)kmph$, where $x$ and $y$ are the downward and upward speed
Speed of stream is $\dfrac{1}{2}\left( {x - y} \right)kmph$, where $x$ and $y$ are the downward and upward speed
Let the speed of the boat be $S\,kmph$
So, Downstream speed= $(S + 3)\,kmph$
And upstream speed= $(S - 3)\,kmph$
So, it takes $24km$ for upstream as well as downstream too, so we know that $Speed = \dfrac{{Dis\tan ce}}{{Time}}$
So, \[{\left( {\dfrac{{Dis\tan ce}}{{Speed}}} \right)_{upstream}} + {\left( {\dfrac{{Dis\tan ce}}{{Speed}}} \right)_{downstream}} = Timetaken\]
\[
   \Rightarrow \dfrac{{24}}{{S - 3}} + \dfrac{{24}}{{S + 3}} = 6 \\
   \Rightarrow 24\left( {\dfrac{1}{{S - 3}} + \dfrac{1}{{S + 3}}} \right) = 6 \\
   \Rightarrow \left( {\dfrac{1}{{S - 3}} + \dfrac{1}{{S + 3}}} \right) = \dfrac{6}{{24}} \\
   \Rightarrow \left( {\dfrac{{S + 3 + S - 3}}{{(S - 3)(S + 3)}}} \right) = \dfrac{1}{4} \\
   \Rightarrow \dfrac{{2S}}{{{S^2} - 9}} = \dfrac{1}{4} \\
   \Rightarrow 8S = {S^2} - 9 \\
   \Rightarrow {S^2} - 8S - 9 = 0 \\
   \Rightarrow {S^2} - 9S + S - 9 = 0 \\
   \Rightarrow S(S - 9) + (S - 9) = 0 \\
   \Rightarrow (S + 1)(S - 9) = 0 \\
   \Rightarrow S = - 1,S = 9 \\
 \]
Speed is in positive value hence consider speed of the motor boat in still water as positive,
So, the speed of the motor boat in still water is $9kmph$.


Note: Take care of upstream speed and downstream speed, when the travel time is given then the ratio of distance/speed is to be considered for both upstream and downstream when for a trip up and back is given.