A train covers distance between two stations A and B in 2 h. If the speed of train is re- duced by 6 km/h, then it travels the same distance in 3 h. Calculate the distance between two stations and speed of the train.

Answer

To solve the problem, we will use the relationship between distance, speed, and time. The formula we will use is:

\[ \text{Distance} = \text{Speed} \times \text{Time} \]

Let the speed of the train be \( x \) km/h.

### Step 1: Set up the equations

1. The distance between stations A and B can be expressed using the speed and time taken in the first scenario:
\[ \text{Distance} = x \times 2 \]
Thus, the distance \( D \) is:
\[ D = 2x \quad \text{(1)} \]

2. In the second scenario, the speed is reduced by 6 km/h, and the time taken is 3 hours:
\[ \text{Distance} = (x - 6) \times 3 \]
Thus, the distance \( D \) can also be expressed as:
\[ D = 3(x - 6) \quad \text{(2)} \]

### Step 2: Equate the two expressions for distance

From equations (1) and (2), we can set them equal to each other:
\[ 2x = 3(x - 6) \]

### Step 3: Solve for \( x \)

Now, we will solve the equation:
\[ 2x = 3x - 18 \]
Rearranging gives:
\[ 2x - 3x = -18 \]
\[ -x = -18 \]
\[ x = 18 \quad \text{(speed of the train)} \]

### Step 4: Calculate the distance

Now that we have the speed, we can find the distance using equation (1):
\[ D = 2x = 2 \times 18 = 36 \quad \text{(distance between stations A and B)} \]

### Final Results

- Speed of the train: \( 18 \) km/h
- Distance between stations A and B: \( 36 \) km