Speed of a faster train is 100 km/hr and it takes 3 minutes rest after covering each 75 km distance while the slower train is running at the speed of 50 km/hr and it takes 1 minute rest after covering each 25 km distance. Find the distance travelled by the slower train when the faster train travel 600 km distance?

Answer

To solve the problem, we need to calculate the distance traveled by the slower train when the faster train travels 600 km. Let's break this down step by step.

### Step 1: Calculate the time taken by the faster train to travel 600 km.
The speed of the faster train is 100 km/hr. We can use the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
So, for 600 km:
\[ \text{Time} = \frac{600 \text{ km}}{100 \text{ km/hr}} = 6 \text{ hours} \]

### Step 2: Determine the number of intervals and total time including rest for the faster train.
The faster train takes a rest of 3 minutes after every 75 km. First, we need to find out how many 75 km intervals are there in 600 km:
\[ \text{Number of intervals} = \frac{600 \text{ km}}{75 \text{ km}} = 8 \]
Now, the total time taken for each interval (travel time + rest time):
- Travel time for 75 km:
\[ \text{Time} = \frac{75 \text{ km}}{100 \text{ km/hr}} = 0.75 \text{ hours} = 45 \text{ minutes} \]
- Total time for one interval (including rest):
\[ \text{Total time for one interval} = 45 \text{ minutes} + 3 \text{ minutes} = 48 \text{ minutes} \]
Now, for 8 intervals:
\[ \text{Total time} = 8 \times 48 \text{ minutes} = 384 \text{ minutes} \]

### Step 3: Adjust the total time for the last interval.
Since the faster train does not take a rest after completing the 600 km, we subtract the last rest time (3 minutes):
\[ \text{Adjusted total time} = 384 \text{ minutes} - 3 \text{ minutes} = 381 \text{ minutes} \]

### Step 4: Calculate the distance traveled by the slower train in 381 minutes.
The speed of the slower train is 50 km/hr. First, we need to find out how long it takes the slower train to travel 25 km:
\[ \text{Time for 25 km} = \frac{25 \text{ km}}{50 \text{ km/hr}} = 0.5 \text{ hours} = 30 \text{ minutes} \]
The slower train takes a rest of 1 minute after every 25 km, so the total time for one interval (25 km + rest) is:
\[ \text{Total time for one interval} = 30 \text{ minutes} + 1 \text{ minute} = 31 \text{ minutes} \]

### Step 5: Find how many intervals fit into 381 minutes.
Let \( x \) be the number of intervals:
\[ 31x \leq 381 \]
Calculating \( x \):
\[ x = \left\lfloor \frac{381}{31} \right\rfloor = 12 \]
So, the slower train completes 12 intervals of 25 km each:
\[ \text{Distance covered in 12 intervals} = 12 \times 25 \text{ km} = 300 \text{ km} \]

### Step 6: Calculate the remaining time after 12 intervals.
Total time used for 12 intervals:
\[ \text{Time used} = 12 \times 31 \text{ minutes} = 372 \text{ minutes} \]
Remaining time:
\[ \text{Remaining time} = 381 \text{ minutes} - 372 \text{ minutes} = 9 \text{ minutes} \]

### Step 7: Calculate the distance covered in the remaining 9 minutes.
In 9 minutes, the distance covered by the slower train is:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 50 \text{ km/hr} \times \left(\frac{9}{60} \text{ hr}\right) = 50 \times 0.15 = 7.5 \text{ km} \]

### Step 8: Total distance traveled by the slower train.
Adding the distance covered in the intervals and the remaining distance:
\[ \text{Total distance} = 300 \text{ km} + 7.5 \text{ km} = 307.5 \text{ km} \]

### Final Answer:
The distance traveled by the slower train when the faster train travels 600 km is **307.5 km**.

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