Calculating Distance Using Speed and Time Data

Understanding the relationship between speed, distance, and time is crucial in various fields, including transportation, logistics, and even everyday travel. This article will walk through a specific problem where a person travels from point A to point B at a slower speed and from B to A at a faster speed. We will demonstrate how to solve for the distance between A and B, providing a clear and comprehensive solution.

Case Study: A Man's Round Trip Journey

Imagine a scenario where a man travels from point A to point B at a speed of 4 kilometers per hour (km/h) and then returns at a speed of 6 km/h. The total journey takes 45 minutes, which is equivalent to 0.75 hours. The question is: how far apart are points A and B?

Mathematical Formulation

We can denote the distance between A and B as d kilometers. The time taken to travel from A to B is given by:

TimeA to B d/4 hours TimeB to A d/6 hours

The total time for the round trip is 0.75 hours. Therefore, we can set up the following equation:

d/4 d/6 0.75

Let's solve this equation step by step.

Step-by-Step Solution

Find a common denominator: The least common multiple of 4 and 6 is 12. Rewrite the equation with a common denominator: (3d/12 2d/12) 0.75 Simplify the expression: 5d/12 0.75 Solve for d by eliminating the fraction: 5d 12 * 0.75 5d 9 d 9/5 1.8 km

Thus, the distance between points A and B is 1.8 kilometers.

The Answer

The distance from A to B is 1.8 kilometers, which can also be expressed as 1 kilometer and 800 meters. This solution demonstrates the importance of converting time units correctly and using algebra to solve real-world problems.

Alternative Steps

Here's another way to approach the problem:

Use the formula 4tA 6tB tA - tB 45 minutes 0.75 hours tA 0.75 - tB 4(0.75 - tB) 6tB 3 - 4tB 6tB 10tB 3 tB 3/10 0.3 hours x 6 * 0.3 1.8 km

This confirms the initial solution and provides another method to arrive at the same result.

Final Answer

The distance from A to B is 1.8 kilometers.

Conclusion

Understanding the relationship between speed, distance, and time is essential for solving real-world problems. This article has demonstrated how to calculate the distance between two points given their travel speeds and the total time taken for a round trip. By following the steps outlined here, you can apply this method to similar problems in various contexts.