A boat takes 4 hrs to travel certain distance in a river in down stream and it takes 6 hrs to travel the same distance in upstream. Then the time taken by the boat to travel the same distance in still water is

Options

A. 4.8 hrs
B. 9.8 hrs
C. 24 hrs
D. 10 hrs

Answer

To solve the problem, we need to determine the time taken by a boat to travel a certain distance in still water, given its travel times in downstream and upstream conditions. Let's break it down step by step.

### Step 1: Define Variables
Let:
- \( L \) = distance traveled by the boat (in kilometers)
- \( V_b \) = speed of the boat in still water (in km/hr)
- \( V_s \) = speed of the river current (in km/hr)

### Step 2: Write Down the Equations for Downstream and Upstream
1. **Downstream**: The effective speed of the boat when going downstream is the sum of the boat's speed and the river's speed:
\[ V_{net, downstream} = V_b + V_s \]
The time taken to travel distance \( L \) downstream is given as 4 hours:
\[ \frac{L}{V_b + V_s} = 4 \quad \text{(1)} \]

2. **Upstream**: The effective speed of the boat when going upstream is the difference of the boat's speed and the river's speed:
\[ V_{net, upstream} = V_b - V_s \]
The time taken to travel distance \( L \) upstream is given as 6 hours:
\[ \frac{L}{V_b - V_s} = 6 \quad \text{(2)} \]

### Step 3: Rearranging the Equations
From equation (1):
\[ L = 4(V_b + V_s) \quad \text{(3)} \]
From equation (2):
\[ L = 6(V_b - V_s) \quad \text{(4)} \]

### Step 4: Set Equations (3) and (4) Equal to Each Other
Since both equations equal \( L \), we can set them equal to each other:
\[ 4(V_b + V_s) = 6(V_b - V_s) \]

### Step 5: Expand and Simplify
Expanding both sides:
\[ 4V_b + 4V_s = 6V_b - 6V_s \]
Rearranging gives:
\[ 4V_s + 6V_s = 6V_b - 4V_b \]
\[ 10V_s = 2V_b \]
Thus, we find:
\[ V_b = 5V_s \quad \text{(5)} \]

### Step 6: Substitute Equation (5) Back into One of the Original Equations
Substituting \( V_b = 5V_s \) into equation (3):
\[ L = 4(5V_s + V_s) = 4(6V_s) = 24V_s \quad \text{(6)} \]

### Step 7: Find the Time Taken in Still Water
Now, we can find the time taken to travel distance \( L \) in still water:
\[ \text{Time in still water} = \frac{L}{V_b} = \frac{24V_s}{5V_s} = \frac{24}{5} = 4.8 \text{ hours} \]

### Final Answer
The time taken by the boat to travel the same distance in still water is **4.8 hours**.