After travelling a distance of 50 km train meets with an accident and its speed becomes 3/4th of its actual speed and reaches 35 minutes late. If this accident had occurred after travelling 24 km more train would have reached the station 25 minutes late. Find out the distance and speed of train.

Answer

To solve the problem step by step, we will break down the information given and use it to find the speed and total distance of the train.

### Step 1: Define Variables
Let:
- \( S \) = actual speed of the train (in km/h)
- \( D \) = total distance to the station (in km)

### Step 2: Analyze the First Scenario
The train travels 50 km before the accident occurs. After the accident, its speed becomes \( \frac{3}{4}S \). It reaches the station 35 minutes late.

- Distance traveled before the accident = 50 km
- Remaining distance = \( D - 50 \) km
- Time taken to travel the remaining distance at reduced speed = \( \frac{D - 50}{\frac{3}{4}S} = \frac{4(D - 50)}{3S} \)

The time taken at actual speed for the remaining distance would have been:
- Time taken at actual speed = \( \frac{D - 50}{S} \)

The difference in time due to the accident is 35 minutes (which is \( \frac{35}{60} \) hours):
\[ \frac{4(D - 50)}{3S} - \frac{D - 50}{S} = \frac{35}{60} \]

### Step 3: Simplify the Equation
To simplify, we can multiply through by \( 3S \):
\[ 4(D - 50) - 3(D - 50) = \frac{35}{60} \cdot 3S \]
\[ (D - 50) = \frac{35S}{60} \]
\[ D - 50 = \frac{7S}{12} \]
\[ D = 50 + \frac{7S}{12} \quad \text{(Equation 1)} \]

### Step 4: Analyze the Second Scenario
If the accident had occurred after traveling 24 km more (i.e., after 74 km), it would have reached the station 25 minutes late.

- Distance traveled before the accident = 74 km
- Remaining distance = \( D - 74 \) km
- Time taken to travel the remaining distance at reduced speed = \( \frac{D - 74}{\frac{3}{4}S} = \frac{4(D - 74)}{3S} \)

The time taken at actual speed for the remaining distance would have been:
- Time taken at actual speed = \( \frac{D - 74}{S} \)

The difference in time due to the accident is 25 minutes (which is \( \frac{25}{60} \) hours):
\[ \frac{4(D - 74)}{3S} - \frac{D - 74}{S} = \frac{25}{60} \]

### Step 5: Simplify the Second Equation
Multiply through by \( 3S \):
\[ 4(D - 74) - 3(D - 74) = \frac{25}{60} \cdot 3S \]
\[ (D - 74) = \frac{25S}{60} \]
\[ D - 74 = \frac{5S}{12} \]
\[ D = 74 + \frac{5S}{12} \quad \text{(Equation 2)} \]

### Step 6: Set Equations Equal
Now we have two equations for \( D \):
1. \( D = 50 + \frac{7S}{12} \)
2. \( D = 74 + \frac{5S}{12} \)

Set them equal to each other:
\[ 50 + \frac{7S}{12} = 74 + \frac{5S}{12} \]

### Step 7: Solve for \( S \)
Rearranging gives:
\[ \frac{7S}{12} - \frac{5S}{12} = 74 - 50 \]
\[ \frac{2S}{12} = 24 \]
\[ 2S = 288 \]
\[ S = 144 \text{ km/h} \]

### Step 8: Find Total Distance \( D \)
Substituting \( S \) back into Equation 1:
\[ D = 50 + \frac{7 \times 144}{12} \]
\[ D = 50 + 84 = 134 \text{ km} \]

### Final Answer
- Speed of the train = 144 km/h
- Total distance = 134 km