A man takes twice as long to row a distance against the stream as to row the same distance in favour of the stream. The ratio of the speed of the boat (in still water) and the stream is :

Options

A. `2 : 1`
B. `3 : 1`
C. `3 : 2`
D. `4 :3`

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Define Variables
Let:
- \( v_b \) = speed of the boat in still water
- \( v_s \) = speed of the stream
- \( d \) = distance to be rowed

### Step 2: Determine Time Taken
The time taken to row a distance \( d \) in favor of the stream is given by:
\[ T_f = \frac{d}{v_b + v_s} \]
The time taken to row the same distance against the stream is given by:
\[ T_a = \frac{d}{v_b - v_s} \]

### Step 3: Set Up the Relationship
According to the problem, the time taken against the stream is twice the time taken in favor of the stream:
\[ T_a = 2 T_f \]
Substituting the expressions for \( T_a \) and \( T_f \):
\[ \frac{d}{v_b - v_s} = 2 \left(\frac{d}{v_b + v_s}\right) \]

### Step 4: Simplify the Equation
Since \( d \) is common on both sides, we can cancel it out (assuming \( d \neq 0 \)):
\[ \frac{1}{v_b - v_s} = \frac{2}{v_b + v_s} \]

### Step 5: Cross-Multiply
Cross-multiplying gives us:
\[ v_b + v_s = 2(v_b - v_s) \]

### Step 6: Expand and Rearrange
Expanding the right side:
\[ v_b + v_s = 2v_b - 2v_s \]
Rearranging the equation:
\[ v_b + v_s + 2v_s = 2v_b \]
This simplifies to:
\[ 3v_s = 2v_b - v_b \]
So,
\[ 3v_s = v_b \]

### Step 7: Find the Ratio
Now, we can express the ratio of the speed of the boat in still water to the speed of the stream:
\[ \frac{v_b}{v_s} = 3 \]
Thus, the ratio of the speed of the boat in still water to the speed of the stream is:
\[ \frac{v_b}{v_s} = 3:1 \]

### Final Answer
The ratio of the speed of the boat in still water to the speed of the stream is \( 3:1 \).

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