A motor boat takes 12 hours to go downstreams and it takes 24 hours to returns the same distance.what is the time taken by boat in still water?

Options

A. 15 h
B. 16 h
C. 8 h
D. 20 h

Answer

To solve the problem of determining the time taken by a motorboat in still water, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables:**
- Let the speed of the boat in still water be \( A \) km/h.
- Let the speed of the stream be \( B \) km/h.
- Let the distance traveled downstream (and upstream) be \( X \) km.

2. **Set Up Equations:**
- The time taken to go downstream (with the current) is given as 12 hours. The speed downstream is \( A + B \) km/h. Therefore, we can write the equation:
\[ \frac{X}{A + B} = 12 \quad \text{(Equation 1)} \]
- The time taken to return upstream (against the current) is given as 24 hours. The speed upstream is \( A - B \) km/h. Thus, we can write the equation:
\[ \frac{X}{A - B} = 24 \quad \text{(Equation 2)} \]

3. **Express Distance in Terms of Speed and Time:**
- From Equation 1, we can express \( X \):
\[ X = 12(A + B) \]
- From Equation 2, we can express \( X \) as well:
\[ X = 24(A - B) \]

4. **Set the Two Expressions for Distance Equal:**
- Since both expressions equal \( X \), we can set them equal to each other:
\[ 12(A + B) = 24(A - B) \]

5. **Simplify and Solve for A and B:**
- Expanding both sides:
\[ 12A + 12B = 24A - 24B \]
- Rearranging gives:
\[ 12B + 24B = 24A - 12A \]
\[ 36B = 12A \]
- Dividing both sides by 12:
\[ 3B = A \quad \text{(Equation 3)} \]

6. **Substitute A in Terms of B:**
- Substitute \( A = 3B \) back into either Equation 1 or Equation 2. Using Equation 1:
\[ X = 12(3B + B) = 12(4B) = 48B \]

7. **Substitute X into Equation 2:**
- Using Equation 2:
\[ 48B = 24(3B - B) \]
\[ 48B = 24(2B) \]
\[ 48B = 48B \]
- This confirms our equations are consistent.

8. **Calculate Time Taken in Still Water:**
- The speed of the boat in still water is \( A = 3B \). The time taken to cover distance \( X \) in still water is:
\[ \text{Time} = \frac{X}{A} = \frac{48B}{3B} = 16 \text{ hours} \]

### Final Answer:
The time taken by the boat in still water is **16 hours**.