A man rows to a place 35 km in distant and back in 10 hours 30 minutes .He found that he could row 5 km with the stream in the same times as he can row 4 km against the stream .Find the rate of flow of the stream :

Options

A. 1km /hr
B. 0.5 km/hr
C. 0.75 km/hr
D. `1.5 km /hr

Answer

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

### Step 1: Understand the problem
A man rows to a place 35 km away and back in a total time of 10 hours and 30 minutes. He can row 5 km downstream in the same time it takes him to row 4 km upstream. We need to find the rate of flow of the stream.

### Step 2: Convert the total time into hours
The total time is given as 10 hours and 30 minutes. We convert this into hours:
\[ 10 \text{ hours} + 30 \text{ minutes} = 10 + \frac{30}{60} = 10.5 \text{ hours} \]

### Step 3: Set up the equations for speed
Let:
- \( u \) = speed of the man in still water (km/h)
- \( v \) = speed of the stream (km/h)

When rowing downstream (with the stream), the effective speed is \( u + v \) and when rowing upstream (against the stream), the effective speed is \( u - v \).

### Step 4: Use the information about rowing distances
From the problem, we know:
- Time to row 5 km downstream = Time to row 4 km upstream

Using the formula for time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
We can write:
\[ \frac{5}{u + v} = \frac{4}{u - v} \]

### Step 5: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 5(u - v) = 4(u + v) \]
Expanding both sides:
\[ 5u - 5v = 4u + 4v \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ 5u - 4u = 5v + 4v \]
\[ u = 9v \]

### Step 7: Set up the total time equation for the round trip
The total distance for the round trip is \( 35 + 35 = 70 \) km. The time taken for the round trip can be expressed as:
\[ \text{Total time} = \frac{35}{u - v} + \frac{35}{u + v} = 10.5 \]

### Step 8: Substitute \( u = 9v \) into the total time equation
Substituting \( u = 9v \):
\[ \frac{35}{9v - v} + \frac{35}{9v + v} = 10.5 \]
This simplifies to:
\[ \frac{35}{8v} + \frac{35}{10v} = 10.5 \]

### Step 9: Find a common denominator and solve
The common denominator is \( 40v \):
\[ \frac{35 \cdot 5}{40v} + \frac{35 \cdot 4}{40v} = 10.5 \]
\[ \frac{175 + 140}{40v} = 10.5 \]
\[ \frac{315}{40v} = 10.5 \]

### Step 10: Cross-multiply to solve for \( v \)
Cross-multiplying gives:
\[ 315 = 10.5 \cdot 40v \]
\[ 315 = 420v \]
\[ v = \frac{315}{420} = \frac{3}{4} \text{ km/h} \]

### Conclusion
The rate of flow of the stream is \( \frac{3}{4} \) km/h or 0.75 km/h.

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