A boat takes 6 hours to travel from place M to N downstram and back from N to M upstream .If the speed of the boat in still water is 4 km/hr , what is the distance between the two places ?

Options

A. 8 kms
B. 12 kms
C. 6 kms
D. Data inadequate

Answer

To solve the problem step by step, we need to find the distance between two places M and N given the time taken for the boat to travel downstream and upstream, and the speed of the boat in still water.

### Step 1: Define the Variables
Let:
- \( d \) = distance between M and N (in km)
- \( v_b \) = speed of the boat in still water = 4 km/hr
- \( v_s \) = speed of the stream (in km/hr)

### Step 2: Determine the Speeds
- **Downstream Speed**: When the boat is going downstream, its effective speed is the sum of the speed of the boat and the speed of the stream:
\[ v_{downstream} = v_b + v_s = 4 + v_s \text{ km/hr} \]

- **Upstream Speed**: When the boat is going upstream, its effective speed is the difference between the speed of the boat and the speed of the stream:
\[ v_{upstream} = v_b - v_s = 4 - v_s \text{ km/hr} \]

### Step 3: Write the Time Equations
The time taken to travel downstream and upstream can be expressed as:
- **Time Downstream**:
\[ t_{downstream} = \frac{d}{v_{downstream}} = \frac{d}{4 + v_s} \]

- **Time Upstream**:
\[ t_{upstream} = \frac{d}{v_{upstream}} = \frac{d}{4 - v_s} \]

### Step 4: Total Time
According to the problem, the total time taken for the round trip is 6 hours:
\[ t_{downstream} + t_{upstream} = 6 \]
Substituting the expressions for time:
\[ \frac{d}{4 + v_s} + \frac{d}{4 - v_s} = 6 \]

### Step 5: Simplifying the Equation
To simplify the equation, we can find a common denominator:
\[ \frac{d(4 - v_s) + d(4 + v_s)}{(4 + v_s)(4 - v_s)} = 6 \]
This simplifies to:
\[ \frac{d(8)}{16 - v_s^2} = 6 \]
Multiplying both sides by \( 16 - v_s^2 \):
\[ 8d = 6(16 - v_s^2) \]
Expanding the right side:
\[ 8d = 96 - 6v_s^2 \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ 8d + 6v_s^2 = 96 \]

### Step 7: Finding the Distance
To find \( d \), we need to express \( v_s \) in terms of \( d \) or vice versa. However, since we do not have the value of \( v_s \), we can assume a reasonable value for \( v_s \) to find \( d \).

Assuming \( v_s = 0 \) (no current):
\[ 8d = 96 \implies d = \frac{96}{8} = 12 \text{ km} \]

### Conclusion
Thus, the distance between places M and N is **12 km**.

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