A motor boat covers a certain distance downstream in a river in five hours. It covers the same distance upstream in five hours and a half. The speed of the water is 1.5km.hr. The speed of the boat in still water is ___________.

A. 30km/hr
B. 30.5km/hr
C. 31.5km/hr
D. 20km/hr

Hint: An important formula that would be required in this question would be
\[distance~=~speed\times time\]
Another important thing that is to be known is that while moving upstream, the net speed of the boat is increased and while moving downstream, the net speed of the boat is reduced which means when the boat moves upstream then the boat will move slower as compared to when it will be moving downstream as the flowing water will help the motion of the boat when it is moving downstream whereas the flowing river is going to obstruct the motion of boat when it is moving upstream.

Complete step-by-step answer:
As mentioned in the question, we have to find the speed of the boat in still water.

Now, let’s the distance covered be ‘y’ and speed of boat in still water be x km/hr.

As mentioned in the hint, we know that

\[distance~=~speed\times time\]
Now, we also know that flowing water helps in moving the boat if the boat moves upstream and as given in the question, time taken for the boat to go downstream is 5 hrs.
Hence, the net speed of boat in downstream is (x+1.5)      (∵ water helps in moving)

\[y=\left( x+1.5 \right)\times 5\ \ \ \ \ \ ...\left( i \right)\]
Similarly, it is also given that the time taken for upstream is 5.5 hrs.
Hence, the net speed of boat in downstream is (x-1.5)      (∵ water retard the motion)

\[y=\left( x-1.5 \right)\times 5\ \ \ \ \ \ ...\left( ii \right)\]

Solving equations (i) and (ii), we get
\[\begin{array}{*{35}{l}}
   y=\left( 5x+7.5 \right)=\left( 5.5x-8.25 \right) \\
   5x+7.5=5.5x-8.25 \\
   5.5x-5x=7.5+8.25 \\
   0.5x=15.75 \\
\end{array}\]
∴x=0.515.75​=31.5

Hence, the speed of a boat in still water is 31.5 km/hr.


NOTE: -
The students can make an error if they don’t know about the information which is given in the hint as while moving upstream, the net speed of the boat is increased and while moving downstream, the net speed of the boat is reduced as the flowing water will help the motion of the boat when it is moving downstream whereas the flowing river is going to obstruct the motion of boat when it is moving upstream.