A boat covers 150 km downstream and 40 km upstream in 4 hours while it covers 70 km downstream and 24 km upstream in 2 hours. Determine the velocity of the current?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 5 km/hr
Downstream the current adds to the boat’s speed and upstream it takes away from it, so the speed of the current is half the difference of the downstream and upstream speeds. Each trip takes distance ÷ speed hours.
The two equations (downstream speed \(u\) km/hr, upstream speed \(v\) km/hr):
\(\displaystyle \frac{150}{u}+\frac{40}{v}=4\) (1)
\(\displaystyle \frac{70}{u}+\frac{24}{v}=2\) (2)
Finding \(u\) (3 times (1) and 5 times (2) have the same \(v\) term; subtracting, it cancels):
\(\displaystyle \frac{450}{u}+\frac{120}{v}=12\)
\(\displaystyle \frac{350}{u}+\frac{120}{v}=10\)
\(\displaystyle \frac{100}{u}=2\), so \(u=50\)
Finding \(v\) (putting \(u=50\) into (2)):
\(\displaystyle \frac{70}{50}+\frac{24}{v}=2\)
\(\displaystyle \frac{24}{v}=0.6\), so \(v=40\)
The speed of the current:
\(\displaystyle \frac{u-v}{2}=\frac{50-40}{2}=5\) km/hr