Question

A diver rowing at the speed of 3 km/h in still water takes double the time going 50 km upstream compared to going 50 km downstream. The speed of the diver downstream is

• A. 3 km/h
• B. 6 km/h
• C. 4 km/h
• D. 5 km/h

Hint:

If the time taken upstream is $k$ times the time taken downstream over the same distance, then:
\[ \frac{u}{v} = \frac{k+1}{k-1} \]
Here, $k = 2$ and $u = 3$:
\[ \frac{3}{v} = \frac{2+1}{2-1} = 3 \implies v = 1\text{ km/h} \]
Downstream Speed = \( u + v = 3 + 1 = 4\text{ km/h} \).
This general formula is extremely useful for exams.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This problem is from the topic of Boats and Streams.
We are given the speed of a diver in still water and a scenario comparing the travel times over the same distance ($50\text{ km}$) upstream and downstream.
The speed of the current affects the overall upstream and downstream speeds.
We need to determine this stream speed first and then calculate the final downstream speed of the diver.

Step 2: Key Formula or Approach:

• Let the speed of the diver in still water be $u = 3\text{ km/h}$.
  
• Let the speed of the water current be $v\text{ km/h}$.
  
• Downstream speed ($S_{\text{down}}$) is given by: \( u + v = 3 + v \).
  
• Upstream speed ($S_{\text{up}}$) is given by: \( u - v = 3 - v \).
  
• Since Time = $\frac{\text{Distance}}{\text{Speed}}$, the times taken are: \( T_{\text{up}} = \frac{d}{u-v} \) and \( T_{\text{down}} = \frac{d}{u+v} \).
  
• The problem states: \( T_{\text{up}} = 2 \times T_{\text{down}} \).
  

Step 3: Detailed Explanation:

• Let the speed of the water current be $v\text{ km/h}$.
  
• Write down the expressions for the speeds:
  
• Downstream speed = $(3 + v)\text{ km/h}$
  
• Upstream speed = $(3 - v)\text{ km/h}$
  
• Write down the expressions for the travel times for a distance of $50\text{ km}$:
  \[ T_{\text{up}} = \frac{50}{3 - v}\text{ hours} \]
  \[ T_{\text{down}} = \frac{50}{3 + v}\text{ hours} \]
  
• According to the given condition, the upstream time is twice the downstream time:
  \[ T_{\text{up}} = 2 \times T_{\text{down}} \]
  
• Substitute the time expressions into the condition:
  \[ \frac{50}{3 - v} = 2 \times \frac{50}{3 + v} \]
  
• Divide both sides by $50$ to simplify:
  \[ \frac{1}{3 - v} = \frac{2}{3 + v} \]
  
• Cross-multiply to solve for $v$:
  \[ 1 \times (3 + v) = 2 \times (3 - v) \]
  \[ 3 + v = 6 - 2v \]
  
• Rearrange the terms to group $v$ on one side:
  \[ v + 2v = 6 - 3 \]
  \[ 3v = 3 \]
  \[ v = 1\text{ km/h} \]
  
• Now, determine the downstream speed of the diver:
  \[ S_{\text{down}} = u + v = 3 + 1 = 4\text{ km/h} \]
  
• Therefore, the downstream speed of the diver is $4\text{ km/h}$.
  

Step 4: Final Answer:
The downstream speed of the diver is $4\text{ km/h}$, which matches Option (C).