Ram travels 12 miles downstream and upstream. It takes 6 hours more to return than to go downstream. If he doubles his speed and covers the same distance of 12 miles downstream and upstream then it takes 1 hour more to go upstream than to go down stream. Find the speed of stream.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define Variables
Let:
- \( x \) = speed of Ram in still water (in miles per hour)
- \( y \) = speed of the current (in miles per hour)

### Step 2: Establish Speeds
The speeds for upstream and downstream are:
- Upstream speed = \( x - y \)
- Downstream speed = \( x + y \)

### Step 3: Set Up Equations from the First Scenario
According to the problem, Ram travels 12 miles downstream and upstream. It takes 6 hours more to return than to go downstream.

Let:
- Time taken to go downstream = \( t_d = \frac{12}{x + y} \)
- Time taken to go upstream = \( t_u = \frac{12}{x - y} \)

From the problem statement, we have:
\[ t_u - t_d = 6 \]
Substituting the expressions for \( t_u \) and \( t_d \):
\[ \frac{12}{x - y} - \frac{12}{x + y} = 6 \]

### Step 4: Solve the First Equation
To solve the equation:
\[ \frac{12}{x - y} - \frac{12}{x + y} = 6 \]
Multiply through by \( (x - y)(x + y) \):
\[ 12(x + y) - 12(x - y) = 6(x^2 - y^2) \]
Simplifying gives:
\[ 12y + 12y = 6(x^2 - y^2) \]
\[ 24y = 6(x^2 - y^2) \]
Dividing by 6:
\[ 4y = x^2 - y^2 \]
Rearranging gives:
\[ x^2 = y^2 + 4y \quad \text{(Equation 1)} \]

### Step 5: Set Up Equations from the Second Scenario
Now, if Ram doubles his speed, his new speed in still water becomes \( 2x \). The new speeds are:
- New upstream speed = \( 2x - y \)
- New downstream speed = \( 2x + y \)

According to the problem, it takes 1 hour more to go upstream than to go downstream:
\[ t_u' - t_d' = 1 \]
Where:
- \( t_d' = \frac{12}{2x + y} \)
- \( t_u' = \frac{12}{2x - y} \)

Substituting gives:
\[ \frac{12}{2x - y} - \frac{12}{2x + y} = 1 \]

### Step 6: Solve the Second Equation
Multiply through by \( (2x - y)(2x + y) \):
\[ 12(2x + y) - 12(2x - y) = (2x - y)(2x + y) \]
Simplifying gives:
\[ 12y + 12y = 4x^2 - y^2 \]
\[ 24y = 4x^2 - y^2 \]
Rearranging gives:
\[ 4x^2 = y^2 + 24y \quad \text{(Equation 2)} \]

### Step 7: Solve the System of Equations
Now we have two equations:
1. \( x^2 = y^2 + 4y \)
2. \( 4x^2 = y^2 + 24y \)

Substituting Equation 1 into Equation 2:
\[ 4(y^2 + 4y) = y^2 + 24y \]
Expanding gives:
\[ 4y^2 + 16y = y^2 + 24y \]
Rearranging gives:
\[ 3y^2 - 8y = 0 \]
Factoring out \( y \):
\[ y(3y - 8) = 0 \]
Thus, \( y = 0 \) or \( y = \frac{8}{3} \).

### Step 8: Conclusion
Since \( y = 0 \) does not make sense in this context, we have:
\[ y = \frac{8}{3} \text{ miles per hour} \]
Thus, the speed of the stream is \( \frac{8}{3} \) miles per hour.