Math Problem: Cycling Distance Calculation with Speed and Time

Understanding the relationship between speed, time, and distance is crucial in solving practical problems. In this article, we will tackle a real-world scenario where a person cycles from the hostel to the college at two different speeds and encounters different arrival times. Let's break down the problem and find the distance between the hostel and the college.

Problem Statement

A person cycles from the hostel to the college at a speed of 20 km/h and arrives 16.5 minutes late. At a speed of 24 km/h, the person arrives 16.5 minutes early. We need to find the distance between the hostel and the college.

Mathematical Formulation

Let's define the distance from the hostel to the college as d km and the time required to reach the college on time as t hours.

Situation 1: Cycles at 20 km/h and Arrives 16.5 Minutes Late

When the speed is 20 km/h, the time taken to cover the distance is:

[frac{d}{20} t frac{16.5}{60} quad text{(1)}

Situation 2: Cycles at 24 km/h and Arrives 16.5 Minutes Early

When the speed is 24 km/h, the time taken to cover the distance is:

[frac{d}{24} t - frac{16.5}{60} quad text{(2)}

Solving the Equations

Subtract equation (2) from equation (1):

[frac{d}{20} - frac{d}{24} frac{16.5}{60} frac{16.5}{60}]

This simplifies to:

[frac{d}{120} frac{16.5 times 2}{60} frac{33}{120}

Further simplification:

[frac{d}{120} frac{11}{20}]

Solving for d:

[frac{d}{120} times 120 frac{11}{20} times 120]

[d frac{11 times 120}{20} 66 , text{km}

Conclusion

The distance between the hostel and the college is 66 km. However, this seems impractical given the context. If you have doubts about the problem statement or the given numbers, please recheck them.

Algebraic Insight and Real-World Application

This problem demonstrates the application of algebra to solve real-life scenarios. Such problems help in understanding the interplay between different variables in a given system. In the context of cycling or any other mode of transportation, understanding speed, time, and distance is crucial for efficient planning and execution.

Understanding and solving such problems can also help in:

Developing problem-solving skills in mathematics and algebra. Improving decision-making in transportation planning. Enhancing overall mathematical literacy.

Key Takeaways

The distance between the two points can be calculated using the given speed and time differences. Algebraic equations are powerful tools for solving real-world problems. Verifying the practicality of the solution is essential.