Understanding Distance and Speed with a Real-World Problem

Have you ever encountered a problem where finding the relationship between distance, speed, and time becomes crucial? In this article, we will explore such a scenario and solve it using basic algebra and problem-solving techniques.

Problem Statement

Li leaves her house every morning for work. If she travels at a speed of 60 km/h, she arrives 3 minutes late for work. If she travels at a speed of 80 km/h, she arrives 15 minutes early. Using this information, can we figure out the distance from her house to her workplace?

Step-by-Step Solution

Let's denote the distance from Li's house to her workplace as x kilometers. We know that time is inversely proportional to speed, so the relationship can be expressed as follows:

Given speeds: v1 60 km/h and v2 80 km/h

Using the inverse proportionality, we find that:

t1 : t2 ~ v2 : v1 ~ 80 : 60 ~ 4 : 3

The time ratio can be written as:

t1 : t2 : t1 - t2 4 : 3 : 1

This implies that the difference in time between arriving 3 minutes late and 15 minutes early is 18 minutes (3 15 18).

We can express the total time in each scenario as:

t1 4 parts, t2 3 parts, t1 - t2 1 part 18 minutes

Hence, converting 18 minutes to hours, we get:

t1 4 parts * 18 minutes 72 minutes 72/60 1.2 hours

The distance can be calculated using the formula: Distance Speed * Time. Therefore, for the first scenario where she travels at 60 km/h:

D 60 km/h * 1.2 hours 72 km

Alternative Methods

Another way to solve this problem is by setting up an equation based on the difference in time:

x/60 - 3/60 x/80 * 15/60

By simplifying this equation, we get:

x/60 - 3/60 x/80 * 15/60

x/60 - x/80 1/20 1/4 6/20

x/240 18/60

x 18 * 240/60 72 km

A third method involves directly solving the equations:

x/60 - 3/60 x/80 * 15/60

x/60 - x/80 1/20 1/4 3/10

4x - 3x/240 3/10

x 3240/10 72 km

Thus, we arrive at the same conclusion that the distance from Li's house to her workplace is 72 kilometers.

Conclusion

Solving real-world problems involving distance, speed, and time not only enhances our understanding of these concepts but also sharpens our problem-solving skills. By using various methods and logical reasoning, we can find the solution to complex problems efficiently.