A boat goes upstream at \(\displaystyle\frac{\mathrm{b}}{2}\) km/hr and downstream at \(\displaystyle\frac{\mathrm{a}}{2}\) km/hr. The speed of boat in still water in km/hr is :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\left(\displaystyle\frac{\mathrm{a}+\mathrm{b}}{4}\right)\) km/hr
Let u be the speed of the boat in still water and v the speed of the stream. Then
\(\text{downstream speed} = u + v = \frac{a}{2}\)
\(\text{upstream speed} = u - v = \frac{b}{2}\)
Adding the two equations removes v, so the speed in still water is the average of the downstream and upstream speeds:
\( \displaystyle u = \frac{1}{2}\left(\frac{a}{2} + \frac{b}{2}\right) \)
\( \displaystyle = \frac{a + b}{4} \)
Why the other options are incorrect
Option (1) \(\left(\displaystyle\frac{\mathrm{b}-\mathrm{a}}{2}\right)\) km/hr — Incorrect. This is the difference of the two speeds; also b < a here, so this value is negative.
Option (2) \(\left(\displaystyle\frac{\mathrm{b}-\mathrm{a}}{4}\right)\) km/hr — Incorrect. Half the difference of the speeds gives the speed of the stream (with the sign reversed), not the speed of the boat.
Option (3) \(\left(\displaystyle\frac{\mathrm{a}-\mathrm{b}}{4}\right)\) km/hr — Incorrect. This is the speed of the stream, \(v = \frac{1}{2}\left(\frac{a}{2} - \frac{b}{2}\right)\).