A boat goes 8 km upstream and returns. If the speed of the current is 1 km/h and the total time taken to cover the to and fro distance is 4 hours 16 minutes, then what is the speed of the boat in still water?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 4 km/h
Against the current (upstream) the boat’s speed is its own speed minus the current’s; with the current (downstream) it is its own speed plus the current’s. The times for the two 8 km trips add up to 4 h 16 min.
Let the speed of the boat in still water be \(s\) km/h:
Upstream speed \(=s-1\), downstream speed \(=s+1\)
Total time \(\displaystyle =4\frac{16}{60}=4\frac{4}{15}=\frac{64}{15}\) h
Time upstream + time downstream = total time: \(\displaystyle \frac{8}{s-1}+\frac{8}{s+1}=\frac{64}{15}\)
So \(\displaystyle \frac{16s}{s^2-1}=\frac{64}{15}\), which becomes \(4s^2-15s-4=0\)
\((s-4)(4s+1)=0\), and a speed cannot be negative, so \(s=4\) km/h
Shortcut: Test the options; the right speed must make the two times add up to 4 h 16 min.
\(s=5\): upstream \(8\div4\) h = 2 h, downstream \(8\div6\) h = 1 h 20 min, total 3 h 20 min, too short; a faster boat (6 or 8 km/h) takes even less time
\(s=4\): upstream \(8\div3\) h = 2 h 40 min, downstream \(8\div5\) h = 1 h 36 min, total 4 h 16 min