An express train travelled at an average speed of 100 km/hr, stopping for 3 minutes after every 75 km. How long it take to reach its destination 600 km from the starting point?
6 hr 21 min
Understanding the Express Train Travel Problem
This problem asks us to calculate the total time an express train takes to cover a certain distance, considering both its travel speed and the time spent on scheduled stops along the route.
We are given the following information:
- Total distance to be covered: 600 km
- Average speed of the train while moving: 100 km/hr
- Stop duration after every 75 km: 3 minutes
To find the total time, we need to calculate two components:
- The time the train spends actually travelling (without stops).
- The total time the train spends stopping.
Calculating Travel Time Without Stops
The formula relating distance, speed, and time is:
\( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)
Using the given values:
\( \text{Travel Time} = \frac{600 \text{ km}}{100 \text{ km/hr}} \)
\( \text{Travel Time} = 6 \text{ hours} \)
So, the train spends 6 hours travelling if there were no stops.
Determining the Number of Stops
The train stops after every 75 km. The total distance is 600 km. To find the number of times the train stops before reaching its final destination, we first determine how many segments of 75 km are there in 600 km.
Number of segments = \( \frac{\text{Total Distance}}{\text{Stopping Interval}} = \frac{600 \text{ km}}{75 \text{ km}} \)
Number of segments = 8
This means the train completes 8 segments of 75 km each to cover 600 km. The stops occur *after* each segment is completed. However, the train stops after 75 km, 150 km, 225 km, 300 km, 375 km, 450 km, and 525 km. It reaches the destination at 600 km and does not stop *after* reaching the destination.
Therefore, the number of stops is one less than the number of segments.
Number of stops = Number of segments - 1
Number of stops = 8 - 1 = 7 stops
Calculating Total Stop Time
Each stop lasts for 3 minutes. We know there are 7 stops.
Total Stop Time = Number of stops \(\times\) Duration per stop
Total Stop Time = 7 \(\times\) 3 minutes
Total Stop Time = 21 minutes
Calculating Total Journey Time
The total time for the journey is the sum of the travel time without stops and the total time spent stopping.
Total Journey Time = Travel Time + Total Stop Time
Total Journey Time = 6 hours + 21 minutes
Total Journey Time = 6 hours 21 minutes
Let's summarize the components:
| Component | Calculation | Time |
|---|---|---|
| Travel Time (without stops) | \( \frac{600 \text{ km}}{100 \text{ km/hr}} \) | 6 hours |
| Number of Stops | \( \frac{600}{75} - 1 \) | 7 stops |
| Total Stop Time | \( 7 \times 3 \text{ minutes} \) | 21 minutes |
| Total Journey Time | Travel Time + Total Stop Time | 6 hours 21 minutes |
The total time taken by the express train to reach its destination 600 km away, including the stops, is 6 hours and 21 minutes.
Revision Table: Key Concepts for Train Travel Problems
| Concept | Explanation | Formula |
|---|---|---|
| Travel Time | Time spent actively moving, excluding stops. | \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \) |
| Number of Stops | Calculated based on the total distance and the stopping interval. Remember the stop at the destination is usually not counted. | \( \text{Number of Stops} = \left(\frac{\text{Total Distance}}{\text{Stopping Interval}}\right) - 1 \) (if stop is after interval and not at the end) |
| Total Stop Time | Sum of durations of all stops. | \( \text{Total Stop Time} = \text{Number of Stops} \times \text{Duration per Stop} \) |
| Total Journey Time | The overall time from start to end, including travel and stops. | Total Travel Time + Total Stop Time |
Additional Information: Understanding Speed, Distance, and Time
Speed, distance, and time are fundamental concepts in physics and mathematics, often used to describe motion. Understanding their relationship is crucial for solving problems like this express train scenario.
- Speed: Speed is the rate at which an object covers a distance. It is typically measured in units like kilometers per hour (km/hr) or meters per second (m/s).
- Distance: Distance is the total length covered by an object during its motion. It is a scalar quantity.
- Time: Time is the duration over which the motion occurs.
These three are related by the formula: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \). This formula can be rearranged to find any one variable if the other two are known.
- To find Distance: \( \text{Distance} = \text{Speed} \times \text{Time} \)
- To find Time: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)
In problems involving stops, it's important to differentiate between the time spent moving and the total time elapsed for the journey. The total journey time includes both the travel time and the time spent at rest during stops.
Important Questions from Average Speed
A person covers a certain distance at the speed of 60 kmph and returns to the starting point at a speed of 40 kmph . Find the average speed (in km/hour) of the person for the whole journey.
A train runs at a speed of 28 kmph for 4 hours and 30 kmph for 5 hours and the remaining 40 kms in one hour. What is the average speed per hour?
Kapil travels for 4.5 hours at a speed of 50 km / h and 7.5 hours at a speed of 70 km / h. At the end of it, he finds that he covered only 6/7 of the total distance. At what average speed should he travel so that the remaining distance traveled in 5 hours?
A car has to cover 125 kms in 5 hours. What will be the average speed of the car if it has covered 90 kms in the first 3 hours?
A scooter from P to Q travels at 40 km/h and from Q to P at 30 km/h. What is the average speed of the scooter?
