A boatman rows to a place 45 km distant and back in 20 hours. He finds that he can row 12 km with the stream in same time as 4 km against the stream . Find the speed of the stream.

Options

A. 3 km/hr
B. 2.5 km/hr
C. 4 km/hr
D. cannot be determined

Answer

To solve the problem step by step, we need to find the speed of the stream based on the information provided.

### Step 1: Understand the Problem
The boatman rows to a place 45 km away and back in a total of 20 hours. He also finds that he can row 12 km with the stream in the same time it takes him to row 4 km against the stream.

### Step 2: Define Variables
Let:
- \( S \) = speed of the boat in still water (in km/h)
- \( R \) = speed of the stream (in km/h)

### Step 3: Set Up Equations for Time
1. **Time taken to row downstream (with the stream)**:
\[ \text{Speed downstream} = S + R \]
Time taken to row 12 km downstream:
\[ T_{down} = \frac{12}{S + R} \]

2. **Time taken to row upstream (against the stream)**:
\[ \text{Speed upstream} = S - R \]
Time taken to row 4 km upstream:
\[ T_{up} = \frac{4}{S - R} \]

3. **Since both times are equal**:
\[ \frac{12}{S + R} = \frac{4}{S - R} \]

### Step 4: Cross Multiply and Simplify
Cross-multiplying gives us:
\[ 12(S - R) = 4(S + R) \]
Expanding both sides:
\[ 12S - 12R = 4S + 4R \]
Rearranging the equation:
\[ 12S - 4S = 12R + 4R \]
\[ 8S = 16R \]
Simplifying:
\[ S = 2R \]

### Step 5: Use Total Time for Round Trip
The total distance for the round trip is \( 45 + 45 = 90 \) km, and the total time is 20 hours.

The time taken for the round trip can be expressed as:
\[ T_{total} = \frac{45}{S + R} + \frac{45}{S - R} = 20 \]

### Step 6: Substitute \( S = 2R \) into the Total Time Equation
Substituting \( S \) in the total time equation:
\[ \frac{45}{2R + R} + \frac{45}{2R - R} = 20 \]
This simplifies to:
\[ \frac{45}{3R} + \frac{45}{R} = 20 \]
Combining the fractions:
\[ \frac{15}{R} + \frac{45}{R} = 20 \]
\[ \frac{60}{R} = 20 \]

### Step 7: Solve for \( R \)
Cross-multiplying gives:
\[ 60 = 20R \]
Thus:
\[ R = \frac{60}{20} = 3 \text{ km/h} \]

### Conclusion
The speed of the stream is **3 km/h**.

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