A person can row `7"" (1)/(2)` km an hour in still water. He finds that It takes twice the time to row upstream than the time to row downstream. The speed of the stream is

Options

A. 2 km/hour
B. 2.5 km/hour
C. 3 km/hour
D. 4 km/hour

Answer

To solve the problem, we need to find the speed of the stream given that a person can row at a speed of \(7 \frac{1}{2}\) km/h in still water and that it takes him twice as long to row upstream as it does to row downstream.

### Step-by-Step Solution:

1. **Convert the speed in still water**:
\[ 7 \frac{1}{2} = \frac{15}{2} \text{ km/h} \]
Let the speed of the stream be \(y\) km/h.

2. **Determine the speeds for upstream and downstream**:
- **Downstream speed**:
\[ \text{Speed downstream} = \text{Speed in still water} + \text{Speed of stream} = \frac{15}{2} + y \text{ km/h} \]
- **Upstream speed**:
\[ \text{Speed upstream} = \text{Speed in still water} - \text{Speed of stream} = \frac{15}{2} - y \text{ km/h} \]

3. **Set up the time relationship**:
Let the distance covered be \(x\) km. According to the problem, the time taken to row upstream is twice the time taken to row downstream:
\[ \text{Time upstream} = \frac{x}{\frac{15}{2} - y} \]
\[ \text{Time downstream} = \frac{x}{\frac{15}{2} + y} \]
Given that:
\[ \frac{x}{\frac{15}{2} - y} = 2 \cdot \frac{x}{\frac{15}{2} + y} \]

4. **Eliminate \(x\)**:
Since \(x\) is common in both terms, we can cancel it out (assuming \(x \neq 0\)):
\[ \frac{1}{\frac{15}{2} - y} = 2 \cdot \frac{1}{\frac{15}{2} + y} \]

5. **Cross-multiply**:
\[ \frac{15}{2} + y = 2 \left(\frac{15}{2} - y\right) \]
Expanding the right side:
\[ \frac{15}{2} + y = 15 - 2y \]

6. **Rearranging the equation**:
\[ y + 2y = 15 - \frac{15}{2} \]
\[ 3y = 15 - 7.5 \]
\[ 3y = 7.5 \]

7. **Solve for \(y\)**:
\[ y = \frac{7.5}{3} = 2.5 \text{ km/h} \]

### Conclusion:
The speed of the stream is \(2.5\) km/h.