Understanding the Train Stoppage Problem

This question asks us to determine how many minutes a train stops on average during each hour of its journey, given its average speeds with and without stoppages. The presence of stoppages reduces the overall average speed of the train. The difference in speed is directly related to the time the train spends stopped.

Key Information Provided

• Average speed of the train without any stoppages: 70 km/h
• Average speed of the train with stoppages included: 56 km/h

We need to find the average stoppage time per hour, expressed in minutes.

Calculating the Stoppage Time Per Hour

The reduction in the train's average speed is caused entirely by the time it spends stopped at stations or elsewhere. In one hour of travel, the train covers less distance when it stops compared to when it travels non-stop. The distance it fails to cover in one hour due to stopping is what we need to figure out.

The difference in speed represents the 'loss' in distance covered per hour because of the stoppages.

Difference in speed \( = \text{Speed without stoppage} - \text{Speed with stoppage} \)

Difference in speed \( = 70 \text{ km/h} - 56 \text{ km/h} \)

Difference in speed \( = 14 \text{ km/h} \)

This means that in one hour, the train covers 14 km less distance due to stopping. The time it takes to cover this 'lost' distance (14 km) at its non-stop speed (70 km/h) is the actual time it spent stopped during that hour.

Stoppage time per hour \( = \frac{\text{Distance lost per hour}}{\text{Speed without stoppage}} \)

Stoppage time per hour \( = \frac{14 \text{ km}}{70 \text{ km/h}} \)

Stoppage time per hour \( = \frac{14}{70} \text{ hours} \)

Simplifying the fraction \( \frac{14}{70} \):

\( \frac{14}{70} = \frac{14 \times 1}{14 \times 5} = \frac{1}{5} \text{ hours} \)

The question asks for the stoppage time in minutes, so we need to convert hours to minutes. There are 60 minutes in an hour.

Stoppage time in minutes \( = \text{Stoppage time in hours} \times 60 \)

Stoppage time in minutes \( = \frac{1}{5} \times 60 \)

Stoppage time in minutes \( = \frac{60}{5} \)

Stoppage time in minutes \( = 12 \text{ minutes} \)

Summary of Calculation

Therefore, the train stops for an average of 12 minutes per hour.

Conclusion

By calculating the difference in average speed and relating it to the non-stop speed, we determined the fraction of an hour lost due to stoppages. Converting this fraction to minutes gives us the average stoppage time per hour. The average stoppage time is 12 minutes per hour.

Revision Table: Train Stoppage Time Calculation

Review the key steps for calculating train stoppage time per hour:

• Identify the speed without stoppages.
• Identify the average speed with stoppages.
• Calculate the difference between the two speeds.
• Divide the difference in speed by the speed without stoppages. This gives the stoppage time as a fraction of an hour.
• Multiply the result by 60 to convert the time into minutes.

Additional Information: Average Speed Concepts

Average speed is calculated as the total distance covered divided by the total time taken. In this problem, the 'total time taken' includes both the time spent travelling and the time spent stopped when considering the average speed with stoppages. The 'total distance' remains the same whether we consider the journey with or without stops. The difference in average speed arises because the total time taken is longer when stoppages occur.

If a train travels for 1 hour without stopping, it covers 70 km. If its average speed over some period is 56 km/h, it means that for every hour of that period, it effectively covers only 56 km, despite moving at 70 km/h when it is not stopped. The 14 km difference (70 km - 56 km) is the distance it didn't cover because it was standing still. The time required to cover these 14 km at 70 km/h is the stoppage time in that hour.