A boat takes 3 hours to travel from place M to N downstream and back from N to M upstream. If the speed of the boat in still water is 4 km, what is the distance between the two places ?

Options

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

Answer

To solve the problem, we need to find the distance between two places M and N, given that a boat takes 3 hours to travel downstream and back upstream, with a speed of the boat in still water being 4 km/h.

### Step-by-Step Solution:

1. **Define Variables**:
- Let the speed of the current be \( x \) km/h.
- Let the distance between places M and N be \( d \) km.

2. **Determine Speeds**:
- Downstream speed = Speed of the boat in still water + Speed of the current = \( 4 + x \) km/h.
- Upstream speed = Speed of the boat in still water - Speed of the current = \( 4 - x \) km/h.

3. **Set Up the Time Equation**:
- Time taken to travel downstream from M to N = \( \frac{d}{4 + x} \) hours.
- Time taken to travel upstream from N to M = \( \frac{d}{4 - x} \) hours.
- Total time for the round trip = 3 hours.

Therefore, we can write the equation:
\[ \frac{d}{4 + x} + \frac{d}{4 - x} = 3 \]

4. **Combine the Fractions**:
- Find a common denominator, which is \( (4 + x)(4 - x) \):
\[ \frac{d(4 - x) + d(4 + x)}{(4 + x)(4 - x)} = 3 \]
- Simplifying the numerator:
\[ d(4 - x + 4 + x) = d(8) = 8d \]
- Thus, the equation becomes:
\[ \frac{8d}{(4 + x)(4 - x)} = 3 \]

5. **Cross-Multiply**:
\[ 8d = 3(4 + x)(4 - x) \]
- Expanding the right side:
\[ 8d = 3(16 - x^2) \]
- Rearranging gives:
\[ d = \frac{3(16 - x^2)}{8} \]

6. **Conclusion**:
- Since we have two variables \( d \) and \( x \), we cannot find a unique solution for \( d \) without additional information about the speed of the current \( x \). Thus, we conclude that the data is insufficient to determine the distance \( d \).

### Final Answer:
The distance between places M and N cannot be determined with the given information.