Understanding Train Durations from Railway Timetable Data

The question asks us to find the difference in travel duration between the train that takes the longest time and the train that takes the shortest time, based on the provided railway timetable data.

The duration of a train journey can be calculated using the fundamental formula relating distance, speed, and time:

$$\text{Duration} = \frac{\text{Distance}}{\text{Speed}}$$$

The table contains the necessary information for each train: Distance (in km) and Speed (in km/h). Let's first present the data from the table:

Calculating Duration for Each Train

We will now calculate the duration for each train using the formula $\text{Duration} = \frac{\text{Distance}}{\text{Speed}}$.

• Train 1001: Duration = $\frac{1200 \text{ km}}{130 \text{ km/h}} = \frac{120}{13} \text{ hours} \approx 9.2308 \text{ hours}$
• Train 1002: Duration = $\frac{1080 \text{ km}}{160 \text{ km/h}} = \frac{108}{16} = \frac{27}{4} = 6.75 \text{ hours}$
• Train 1003: Duration = $\frac{1280 \text{ km}}{155 \text{ km/h}} = \frac{128}{15.5} = \frac{256}{31} \text{ hours} \approx 8.2581 \text{ hours}$
• Train 1004: Duration = $\frac{1250 \text{ km}}{130 \text{ km/h}} = \frac{125}{13} \text{ hours} \approx 9.6154 \text{ hours}$
• Train 1005: Duration = $\frac{1180 \text{ km}}{125 \text{ km/h}} = \frac{1180}{125} = \frac{236}{25} = 9.44 \text{ hours}$

Identifying Maximum and Minimum Durations

Let's compare the calculated durations to find the maximum and minimum values:

• Train 1001: $\approx 9.2308$ hours
• Train 1002: $6.75$ hours
• Train 1003: $\approx 8.2581$ hours
• Train 1004: $\approx 9.6154$ hours
• Train 1005: $9.44$ hours

The maximum duration is approximately $9.6154$ hours, taken by Train 1004.

The minimum duration is exactly $6.75$ hours, taken by Train 1002.

Calculating the Time Difference

Now, we calculate the difference between the maximum and minimum durations:

$$\text{Time Difference} = \text{Maximum Duration} - \text{Minimum Duration}$$$

$$\text{Time Difference} = \frac{1250}{130} \text{ hours} - \frac{1080}{160} \text{ hours}$$$

$$\text{Time Difference} = \frac{125}{13} - \frac{27}{4}$$$

To subtract these fractions, we find a common denominator, which is $13 \times 4 = 52$.

$$\text{Time Difference} = \frac{125 \times 4}{13 \times 4} - \frac{27 \times 13}{4 \times 13}$$$

$$\text{Time Difference} = \frac{500}{52} - \frac{351}{52}$$$

$$\text{Time Difference} = \frac{500 - 351}{52} = \frac{149}{52}$$$

Now, we convert the fraction $\frac{149}{52}$ to a decimal:

$$\frac{149}{52} \approx 2.86538 \text{ hours}$$$

Rounding the result to two decimal places, we get approximately $2.87$ hours. Comparing this with the given options, the closest value is $2.86$ hours.

Final Answer

The time difference between the train taking the maximum duration and the train taking the minimum duration is approximately $2.86$ hours.

Revision Table: Train Journey Analysis

Maximum Duration $\approx 9.6154$ hours (Train 1004)

Minimum Duration $= 6.75$ hours (Train 1002)

Difference $\approx 9.6154 - 6.75 = 2.8654$ hours

Additional Information: Understanding Speed, Distance, and Time

The relationship between speed, distance, and time is a fundamental concept in physics and is crucial for solving problems involving travel. The formulas are:

• Distance = Speed $\times$ Time: If you know how fast something is moving and for how long, you can find out how far it travelled.
• Speed = Distance / Time: If you know the distance travelled and the time taken, you can calculate the average speed.
• Time = Distance / Speed: If you know the distance and the speed, you can calculate the time taken for the journey, as used in this problem.

It's important that the units are consistent. In this problem, distance is in kilometers (km) and speed is in kilometers per hour (km/h), which results in time being calculated in hours.