Solving the Problem of Two Drivers Meeting on a Road: A Comprehensive Guide

When it comes to solving mathematical problems related to road travel, many problems can appear daunting at first glance. However, with a bit of algebra and a clear approach, these problems can be easily tackled. One such classic example involves two drivers traveling in opposite directions. In this article, we will walk through the process of determining the exact meeting point and time for two drivers, providing both the algebraic solution and a simplified approach.

The Problem Restated

Here is the original problem: two drivers, Mr. X and Mr. Y, are driving towards each other from two different points, A and B. Mr. X leaves point A with an average speed of 90 km/h, and Mr. Y leaves point B with an average speed of 124 km/h. There is a 15-minute offset between their departures. We need to find out when and where they will meet.

The Algebraic Solution

To solve this problem, we will start by setting up the equations and then solving them step-by-step.

Step 1: Define Variables and Set Up Equations

Let t be the time (in hours) after Mr. X's departure when the two drivers meet. Let z be the distance (in km) from point A to the meeting point.

The first equation is based on Mr. X's speed and distance:

[ frac{z}{t} 90 ]

The second equation is based on Mr. Y's speed and distance:

[ frac{87 - z}{t - 0.25} 124 ]

Note that 15 minutes (the offset) is 0.25 hours.

Step 2: Solve for t and z

From the first equation, solve for z:

[ z 90t ]

Substitute z into the second equation:

[ frac{87 - 90t}{t - 0.25} 124 ]

Solve for t:

[ 87 - 90t 124(t - 0.25) ]

[ 87 - 90t 124t - 31 ]

[ 118 214t ]

[ t frac{118}{214} 0.5514 text{ hours} ]

Calculate z:

[ z 90 times 0.5514 49.6 text{ kilometers} ]

So, the cars will meet 49.6 kilometers from point A after 0.5514 hours.

A Simplified Approach

While the algebraic solution is precise, it can be complex. A simpler approach can be taken if we ignore the 15-minute offset for the initial calculation. Here is how it works:

Step 1: Eliminate the Offset

Calculate the distance Mr. X covers in the 15 minutes before Mr. Y starts driving:

[ 90 times frac{15}{60} 22.5 text{ kilometers} ]

The distance between the drivers when they start moving is:

87 - 22.5 64.5 kilometers.

Step 2: Calculate Combined Speed and Time to Meet

Add their speeds:

90 km/h 124 km/h 214 km/h.

Calculate the time to cover the 64.5 kilometers:

[ frac{64.5}{214} 0.3 text{ hours (18 minutes)} ]

Now, add back the 15 minutes offset to get the total time from Mr. X's departure:

15 18 33 minutes.

Step 3: Validate the Solution

To verify the solution:

Mr. X travels 49.5 kilometers in 33 minutes:

[ 33 div 60 times 90 49.5 text{ kilometers} ]

Mr. Y travels 37.2 kilometers in 18 minutes:

[ 18 div 60 times 124 37.2 text{ kilometers} ]

Together, they cover:

49.5 37.2 86.7 kilometers, which is close to 87 kilometers.

Conclusion

This problem demonstrates the importance of simplifying complex scenarios, understanding algebraic methods, and verifying solutions. Whether using the precise algebraic method or the simplified approach, both lead to the same result, making it clear that Mr. X and Mr. Y will meet 33 minutes after Mr. X's departure, approximately 49.6 kilometers from point A.