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 will denote the following:

- Let the speed of the boat in still water be \( b \).
- Let the speed of the stream be \( s \).

### Step 1: Understand the time taken to row upstream and downstream
According to the problem, the man takes twice as long to row a distance against the stream (upstream) as he does to row the same distance in favor of the stream (downstream).

Let the time taken to row downstream be \( t \). Then, the time taken to row upstream will be \( 2t \).

### Step 2: Relate time to speed and distance
Since the distance is the same, we can use the relationship between speed, distance, and time:
- Time = Distance / Speed

Let the distance be \( d \).

For downstream:
\[ t = \frac{d}{b+s} \]

For upstream:
\[ 2t = \frac{d}{b-s} \]

### Step 3: Set up the equations
From the above relationships, we can express \( t \) in terms of \( d \), \( b \), and \( s \):

1. From downstream:
\[ t = \frac{d}{b+s} \]

2. From upstream:
\[ 2t = \frac{d}{b-s} \]

### Step 4: Substitute \( t \) from the first equation into the second
Substituting \( t \) from the first equation into the second gives:
\[ 2 \left( \frac{d}{b+s} \right) = \frac{d}{b-s} \]

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

Cross-multiplying gives:
\[ 2(b - s) = 1(b + s) \]

### Step 6: Expand and rearrange
Expanding both sides:
\[ 2b - 2s = b + s \]

Rearranging the equation:
\[ 2b - b = 2s + s \]
\[ b = 3s \]

### Step 7: Find the ratio of the speed of the boat to the speed of the stream
The ratio of the speed of the boat \( b \) to the speed of the stream \( s \) is:
\[ \frac{b}{s} = \frac{3s}{s} = 3 \]

Thus, the ratio of the speed of the boat in still water 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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