Three boats, A, B and C, start from the same jetty and move in the same direction along a river. Their speeds in still water are…

Question

Three boats, A, B and C, start from the same jetty and move in the same direction along a river. Their speeds in still water are 18 km/h, 24 km/h and 36 km/h, respectively. Boat B starts 1 hour after Boat A. If Boat B and Boat C overtake Boat A at the same time, how many hours after did Boat C start?

1. A. 1 hour (Correct answer)
2. B. 2 hours
3. C. 3 hours
4. D. Cannot be determined

Correct Answer

Option A — 1 hour

Detailed Solution & Explanation

The correct answer is 1 hour.

Key Points

• All three boats travel downstream in the same direction along the same river, so the current adds equally to each speed and cancels out in every relative-speed calculation. The still-water speeds can be used directly.
• Step 1 — when does B overtake A? B starts 1 hour after A, by which time A is $18 \times 1 = 18$ km ahead.
  • Relative speed of B with respect to A $= 24 - 18 = 6$ km/h
  • Time for B to close the gap $= \frac{18}{6} = 3$ hours after B starts
  • So B catches A at 4 hours after A's start.
• Step 2 — when must C start? Let C start $t$ hours after A. A is then $18t$ km ahead.
  • Relative speed of C with respect to A $= 36 - 18 = 18$ km/h
  • Time for C to close the gap $= \frac{18t}{18} = t$ hours
  • So C catches A at $t + t = 2t$ hours after A's start.
• Step 3 — equate the two overtaking times. Both overtake A simultaneously:
  • $2t = 4 \;\Rightarrow\; t = 2$
• C therefore starts 2 hours after A, while B started 1 hour after A — so C starts 1 hour after B.

Shortcut Trick

• Notice that C is exactly twice as fast as A ($36 = 2 \times 18$). A boat travelling at double the speed always takes as long to catch up as the head start it was given — so C's total journey time to the meeting point is half of A's. Since the meeting happens 4 hours after A starts, C must have set out at the 2-hour mark.

Additional Information

• Why the current is irrelevant here: if the stream speed is $s$, the boats travel at $18+s$, $24+s$ and $36+s$. Every relative speed is a difference, so $s$ cancels: $(24+s)-(18+s) = 6$. This is worth recognising quickly, since the question supplies still-water speeds precisely to see whether candidates waste time worrying about the current.
• Standard boats-and-streams formulae:
  • Downstream speed $= u + v$; upstream speed $= u - v$
  • Speed in still water $u = \dfrac{\text{Downstream} + \text{Upstream}}{2}$
  • Stream speed $v = \dfrac{\text{Downstream} - \text{Upstream}}{2}$
• Read the final question carefully — it asks how long after Boat B that C started, not how long after A. The answer to the latter would be 2 hours, which is offered as option B.