Ramesh can row a certain distance downstream in 6 hours and return the same distance in 9 hours. If the stream flows at the rate of 3
km per hour find the speed of Ramesh in still water.

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step-by-Step Solution:

1. **Define Variables**:
Let \( X \) be the speed of Ramesh in still water (in km/h).

2. **Determine Relative Speeds**:
- Downstream speed (speed of Ramesh + speed of stream): \( X + 3 \) km/h
- Upstream speed (speed of Ramesh - speed of stream): \( X - 3 \) km/h

3. **Calculate Distance**:
Since distance is the same for both downstream and upstream, we can express the distance in terms of speed and time:
- Distance downstream = Speed downstream × Time downstream
- Distance upstream = Speed upstream × Time upstream

Therefore:
- Distance downstream = \( (X + 3) \times 6 \) km
- Distance upstream = \( (X - 3) \times 9 \) km

4. **Set Up the Equation**:
Since both distances are equal, we can set the two expressions equal to each other:
\[ (X + 3) \times 6 = (X - 3) \times 9 \]

5. **Expand the Equation**:
Expanding both sides gives:
\[ 6X + 18 = 9X - 27 \]

6. **Rearrange the Equation**:
To isolate \( X \), we can rearrange the equation:
\[ 6X + 18 + 27 = 9X \]
\[ 45 = 9X - 6X \]
\[ 45 = 3X \]

7. **Solve for \( X \)**:
Dividing both sides by 3 gives:
\[ X = 15 \text{ km/h} \]

### Final Answer:
The speed of Ramesh in still water is **15 km/h**.

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