How to Solve the Train and Walking Distance Problem Using Speed and Time Optimization

In this article, we will explore the solution to a classic problem involving a person walking at two different speeds and their respective impacts on the train schedule. This problem is often used in competitive exams and mathematical puzzles. We will use various methods to find the distance to the station for a given set of conditions. This article is tailored for SEO (Search Engine Optimization) and follows Google's content standards for ranking in search engines.

Problem Statement

The problem states that if a person walks at 3 km/h, they miss the train by 2 minutes. If they walk at 4 km/h, they reach the station 2 minutes before the train arrives. The objective is to find the distance to the station.

Method 1: Algebraic Approach

We will denote the distance to the station as d kilometers.

Case 1: Walking at 3 km/h

The time taken to walk the distance is d / 3 hours. You miss the train by 2 minutes, which is 2/60 1/30 hours. Therefore, d / 3 t - 1/30 where t is the time it takes for the train to arrive.

Case 2: Walking at 4 km/h

The time taken to walk the distance is d / 4 hours. You reach the station 2 minutes before the train arrives, which is 2/60 1/30 hours. Therefore, d / 4 t - 1/30.

By solving the two equations:

d / 3 t - 1/30 d / 4 t - 1/30

Subtracting the second equation from the first:

d / 3 - d / 4 t - 1/30 - (t - 1/30) 4d - 3d / 12 0 d 3 km

Therefore, the distance you need to walk to reach the station is 3 km.

Method 2: Algebraic Calculation

If T is the opt time and S is the distance to walk:

S T / (10/60) 4 T - (5/60) 5 T (40 / 25) / 60 65 / 60 S (65 / 60) / (10 / 60) 4 300 / 60 5 km

Method 3: Time Ratio and Speed

The speeds are v5 km/hr and v’6 km/hr. The times t and t’ are in the inverse ratio of speeds: t : t’ ~6:5.

t : t’ : t - t’ ~6:5:1 The difference in times is 1 part which from the data is 7 - 5 12 min. t’ 5 parts 60 min 1 hr D v t’ 6 km/hr 1 hr 6 km

Method 4: Direct Distance Calculation

Let the distance be x km

At 5 km/h, the time is x / 5 hours. At 6 km/h, the time is x / 6 hours. x/5 - x/6 57 / 60 1/5 hr x/30 1/5 x 30/5 6 km

Thus, the distance to the station is 6 km.

Method 5: Difference in Time and Distance

Let d km be the distance and t min require to reach the station in time

5 km/h 1/12 km/min 6 km/h 1/10 km/min hence 12d t 7 t 12d - 7 and 10d t - 5 t 10d - 5

Therefore, d 6 km

Conclusion

The distance to the station, given the conditions, is 6 km. All methods provide consistent results, using algebraic manipulation and logical reasoning to solve the problem effectively. This approach is crucial for competitive exams and mathematical assessments.

Related Keywords

train schedule optimization walking distance problem speed and time optimization