A train covered a certain distance at a uniform speed. If the train would have been 6 m/hr. faster, it would have taken 4 hours less than the scheduled time and if the train would have slowed down by 6 km/hr, it would have taken 6 hours more than scheduled time. Find the length of the journey.

Answer

To solve the problem step by step, we will define the variables and set up the equations based on the information provided in the question.

### Step 1: Define Variables
Let:
- \( x \) = speed of the train in km/hr
- \( y \) = time taken to cover the distance in hours

### Step 2: Write the Distance Formula
The distance covered by the train can be expressed as:
\[ \text{Distance} = \text{Speed} \times \text{Time} = x \cdot y \]

### Step 3: Set Up the First Condition
According to the first condition, if the speed increases by 6 km/hr, the time taken decreases by 4 hours. This gives us the equation:
\[ \text{New Speed} = x + 6 \quad \text{and} \quad \text{New Time} = y - 4 \]
So, the distance can also be expressed as:
\[ x \cdot y = (x + 6)(y - 4) \]
Expanding the right side:
\[ xy = xy - 4x + 6y - 24 \]
Cancelling \( xy \) from both sides:
\[ 0 = -4x + 6y - 24 \]
Rearranging gives us:
\[ 4x - 6y = -24 \quad \text{(Equation 1)} \]

### Step 4: Set Up the Second Condition
According to the second condition, if the speed decreases by 6 km/hr, the time taken increases by 6 hours. This gives us the equation:
\[ \text{New Speed} = x - 6 \quad \text{and} \quad \text{New Time} = y + 6 \]
So, the distance can also be expressed as:
\[ x \cdot y = (x - 6)(y + 6) \]
Expanding the right side:
\[ xy = xy + 6x - 6y - 36 \]
Cancelling \( xy \) from both sides:
\[ 0 = 6x - 6y - 36 \]
Rearranging gives us:
\[ 6x - 6y = 36 \quad \text{(Equation 2)} \]

### Step 5: Simplify the Equations
From Equation 1:
\[ 2x - 3y = -12 \quad \text{(Dividing by 2)} \]
From Equation 2:
\[ x - y = 6 \quad \text{(Dividing by 6)} \]

### Step 6: Solve the System of Equations
Now we have:
1. \( 2x - 3y = -12 \)
2. \( x - y = 6 \)

From the second equation, we can express \( x \) in terms of \( y \):
\[ x = y + 6 \]
Substituting this into the first equation:
\[ 2(y + 6) - 3y = -12 \]
Expanding:
\[ 2y + 12 - 3y = -12 \]
Combining like terms:
\[ -y + 12 = -12 \]
Subtracting 12 from both sides:
\[ -y = -24 \]
Thus, we find:
\[ y = 24 \]

### Step 7: Find \( x \)
Substituting \( y = 24 \) back into the equation \( x - y = 6 \):
\[ x - 24 = 6 \]
So:
\[ x = 30 \]

### Step 8: Calculate the Distance
Now we can find the distance:
\[ \text{Distance} = x \cdot y = 30 \cdot 24 = 720 \text{ km} \]

### Final Answer
The length of the journey is:
\[ \boxed{720 \text{ km}} \]