Ram goes 18 km downstream and then upstream. It takes 9 hours more to return than to go. If he doubles his speed then it takes 1 hour more to come upstream than to go downstream. Find the speed of the stream.

Answer

To solve the problem step by step, we will define variables for the speeds and use the information given in the question to set up equations.

### Step 1: Define Variables
Let:
- \( x \) = speed of Ram (in km/h)
- \( y \) = speed of the stream (in km/h)

### Step 2: Write the Time Equations
When Ram goes downstream, his effective speed is \( x + y \), and when he goes upstream, his effective speed is \( x - y \).

The time taken to go downstream (18 km) is:
\[ \text{Time downstream} = \frac{18}{x + y} \]

The time taken to go upstream (18 km) is:
\[ \text{Time upstream} = \frac{18}{x - y} \]

According to the problem, it takes 9 hours more to return than to go downstream:
\[ \frac{18}{x - y} - \frac{18}{x + y} = 9 \]

### Step 3: Simplify the Equation
To solve the equation, we will find a common denominator:
\[ \frac{18(x + y) - 18(x - y)}{(x - y)(x + y)} = 9 \]
This simplifies to:
\[ \frac{36y}{(x - y)(x + y)} = 9 \]

Cross-multiplying gives:
\[ 36y = 9(x^2 - y^2) \]
Dividing both sides by 9:
\[ 4y = x^2 - y^2 \]
Rearranging gives us:
\[ x^2 = y^2 + 4y \quad \text{(Equation 1)} \]

### Step 4: Analyze the Second Condition
When Ram doubles his speed, his new speed becomes \( 2x \). The time taken to go downstream and upstream with the new speed is:
\[ \text{Time downstream} = \frac{18}{2x + y} \]
\[ \text{Time upstream} = \frac{18}{2x - y} \]

According to the problem, it takes 1 hour more to come upstream than to go downstream:
\[ \frac{18}{2x - y} - \frac{18}{2x + y} = 1 \]

### Step 5: Simplify the Second Equation
Finding a common denominator:
\[ \frac{18(2x + y) - 18(2x - y)}{(2x - y)(2x + y)} = 1 \]
This simplifies to:
\[ \frac{36y}{(2x - y)(2x + y)} = 1 \]

Cross-multiplying gives:
\[ 36y = (2x - y)(2x + y) \]
Expanding the right side:
\[ 36y = 4x^2 - y^2 \]

### Step 6: Rearranging the Second Equation
Rearranging gives us:
\[ 4x^2 = 36y + y^2 \quad \text{(Equation 2)} \]

### Step 7: Substitute Equation 1 into Equation 2
From Equation 1, substitute \( x^2 \):
\[ 4( y^2 + 4y) = 36y + y^2 \]
Expanding gives:
\[ 4y^2 + 16y = 36y + y^2 \]
Rearranging gives:
\[ 3y^2 - 20y = 0 \]

### Step 8: Factor the Quadratic Equation
Factoring out \( y \):
\[ y(3y - 20) = 0 \]
Thus, \( y = 0 \) or \( 3y - 20 = 0 \).

### Step 9: Solve for \( y \)
Since \( y = 0 \) is not a valid solution (the speed of the stream cannot be zero), we solve:
\[ 3y = 20 \implies y = \frac{20}{3} \approx 6.67 \text{ km/h} \]

### Conclusion
The speed of the stream is \( \frac{20}{3} \) km/h or approximately 6.67 km/h.