A man swimming in a stream which flows `1(1)/(2)` km/hr finds that in a given time he can swim twice as far with the stream as he can against it. At what rate does he swim?

Options

A. `4(1)/(2)km//h`
B. `5(1)/(2)km//h`
C. `7(1)/(2)km//h`
D. `8(1)/(2)km//h`

Answer

To solve the problem step-by-step, we need to determine the swimming speed of the man in still water (let's denote this speed as \( S \) km/hr). The stream flows at a speed of \( 1.5 \) km/hr.

### Step 1: Define the speeds
- Speed of the man in still water = \( S \) km/hr
- Speed of the stream = \( 1.5 \) km/hr

### Step 2: Determine the effective speeds
- When swimming upstream (against the stream), the effective speed is:
\[ S - 1.5 \text{ km/hr} \]
- When swimming downstream (with the stream), the effective speed is:
\[ S + 1.5 \text{ km/hr} \]

### Step 3: Set up the relationship based on the problem statement
According to the problem, the distance the man swims downstream is twice the distance he swims upstream in the same amount of time. Let's denote the distance he swims upstream as \( D \) km. Therefore, the distance he swims downstream is \( 2D \) km.

### Step 4: Write the time equations
The time taken to swim upstream can be expressed as:
\[ \text{Time upstream} = \frac{D}{S - 1.5} \]
The time taken to swim downstream can be expressed as:
\[ \text{Time downstream} = \frac{2D}{S + 1.5} \]

Since the time taken is the same for both upstream and downstream, we can set the two equations equal to each other:
\[ \frac{D}{S - 1.5} = \frac{2D}{S + 1.5} \]

### Step 5: Simplify the equation
We can cancel \( D \) from both sides (assuming \( D \neq 0 \)):
\[ \frac{1}{S - 1.5} = \frac{2}{S + 1.5} \]

Cross-multiplying gives us:
\[ S + 1.5 = 2(S - 1.5) \]

### Step 6: Expand and solve for \( S \)
Expanding the right side:
\[ S + 1.5 = 2S - 3 \]

Rearranging the equation:
\[ 1.5 + 3 = 2S - S \]
\[ 4.5 = S \]

### Step 7: Conclusion
The speed of the man in still water is:
\[ S = 4.5 \text{ km/hr} \]