A man can row at a speed of `4(1)/(2)` km /hr in still water . If he takes 2 times as long to row a distance upstream as to row the same distance downstream . Then , the speed of stream (in km/hr) is

Options

A. 1
B. 1.5
C. 2
D. 2.5

Answer

To solve the problem step by step, we need to determine the speed of the stream based on the information provided.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Speed of the man in still water = \(4 \frac{1}{2}\) km/hr = \(4.5\) km/hr
- Let the speed of the stream be \(x\) km/hr.

2. **Understand the Time Relationship:**
- The man takes twice as long to row upstream than he does downstream.
- Let the time taken to row downstream be \(t\) hours.
- Then, the time taken to row upstream is \(2t\) hours.

3. **Set Up the Speed Equations:**
- The speed of the boat downstream = Speed of the man + Speed of the stream = \(4.5 + x\) km/hr.
- The speed of the boat upstream = Speed of the man - Speed of the stream = \(4.5 - x\) km/hr.

4. **Use the Formula for Time:**
- Time = Distance / Speed
- Let the distance be \(d\) km (the same for both upstream and downstream).
- Time taken downstream:
\[ t = \frac{d}{4.5 + x} \]
- Time taken upstream:
\[ 2t = \frac{d}{4.5 - x} \]

5. **Set Up the Equation Based on Time:**
- Since \(2t = \frac{d}{4.5 - x}\), we can substitute \(t\):
\[ 2 \left(\frac{d}{4.5 + x}\right) = \frac{d}{4.5 - x} \]

6. **Eliminate \(d\) from the Equation:**
- Assuming \(d \neq 0\), we can divide both sides by \(d\):
\[ 2 \cdot \frac{1}{4.5 + x} = \frac{1}{4.5 - x} \]

7. **Cross-Multiply to Solve for \(x\):**
- Cross-multiplying gives:
\[ 2(4.5 - x) = 1(4.5 + x) \]
- Expanding both sides:
\[ 9 - 2x = 4.5 + x \]

8. **Rearranging the Equation:**
- Bring all terms involving \(x\) to one side and constant terms to the other:
\[ 9 - 4.5 = 2x + x \]
\[ 4.5 = 3x \]

9. **Solve for \(x\):**
- Divide both sides by 3:
\[ x = \frac{4.5}{3} = 1.5 \text{ km/hr} \]

### Final Answer:
The speed of the stream is **1.5 km/hr**.