A boat can travel 78 km upstream and back in a total of 32 hours. It can travel 15 km upstream and 52 km downstream in a total of 9 hours. How much distance will the boat cover in 12 hours in still water?
Show answer & explanation
Correct answer: B
Explanation
The correct answer is option 2 i.e. 96 km.
Let:
b: Speed of the boat in still water (km/h).
c: Speed of the current (km/h).
Upstream speed = b − c,
Downstream speed = b + c
The boat travels 78 km upstream and back (78 km downstream), taking a total of 32 hours.
The time taken = Time upstream + Time downstream = 32
⇒ 78/(b - c) + 78/(b + c) = 32
⇒(78(b + c) + 78(b − c))/((b − c)(b + c)) = 32
⇒ 156b/(b2 - c2) = 32
⇒ 156b = 32(b2 − c2)
⇒ 32b2 − 156b − 32c2 = 0
The boat travels 15 km upstream and 52 km downstream, taking a total of 9 hours.
The time taken = 15/(b − c) + 52/(b + c) = 9
⇒ (15(b + c) + 52(b − c))/(b2 − c2)=9,
⇒ (15b + 15c + 52b − 52c)/(b2 − c2) = 9,
⇒ (67b - 37c)/(b2 - c2) = 9
⇒ 67b − 37c = 9(b2 − c2).
⇒ 9b2 − 9c2 − 67b + 37c = 0 .............(2)
From equations (1) and (2), solve for b and c.
b = 8 km/h
c = 2 km/h
In still water, the boat’s speed is b = 8 km/h. In 12 hours, the boat will travel:
Distance = Speed × Time = 8 × 12 = 96 km.