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 need to analyze the situation involving rowing against and with the stream. Let's denote:

- \( V_b \) = speed of the boat in still water
- \( V_s \) = speed of the stream

### Step 1: Define the time taken for rowing
Let the distance to be rowed be \( d \).

- Time taken to row upstream (against the stream) = \( t_1 \)
- Time taken to row downstream (with the stream) = \( t_2 \)

According to the problem, the man takes twice as long to row upstream as he does downstream:

\[ t_1 = 2t_2 \]

### Step 2: Express time in terms of speed
The time taken to row a distance \( d \) can be expressed as:

\[ t_1 = \frac{d}{V_b - V_s} \quad \text{(upstream)} \]
\[ t_2 = \frac{d}{V_b + V_s} \quad \text{(downstream)} \]

### Step 3: Substitute the time relationship
Substituting the expressions for \( t_1 \) and \( t_2 \) into the time relationship:

\[ \frac{d}{V_b - V_s} = 2 \cdot \frac{d}{V_b + V_s} \]

### Step 4: Simplify the equation
Cancel \( d \) from both sides (assuming \( d \neq 0 \)):

\[ \frac{1}{V_b - V_s} = \frac{2}{V_b + V_s} \]

Cross-multiplying gives:

\[ V_b + V_s = 2(V_b - V_s) \]

### Step 5: Expand and rearrange
Expanding the right side:

\[ V_b + V_s = 2V_b - 2V_s \]

Rearranging the equation:

\[ V_b + V_s + 2V_s = 2V_b \]
\[ V_b + 3V_s = 2V_b \]

### Step 6: Isolate \( V_b \)
Rearranging to isolate \( V_b \):

\[ 3V_s = 2V_b - V_b \]
\[ 3V_s = V_b \]

### Step 7: Find the ratio
Now we can find the ratio of the speed of the boat in still water to the speed of the stream:

\[ \frac{V_b}{V_s} = \frac{3V_s}{V_s} = 3 \]

Thus, the ratio of the speed of the boat to the speed of the stream is:

\[ \text{Ratio} = 3:1 \]

### 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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