Trains Traveling In Opposite Directions Problem

Let's dive into a classic math problem involving trains, distance, and speed! This is a common type of question you might encounter in math classes or even on standardized tests. We're going to break it down step by step so you can understand how to solve it. Our main goal is to figure out the speeds of two trains that are traveling in opposite directions.

Problem Statement

Okay, guys, here's the scenario: Imagine two train stations that are 124 kilometers apart. At the exact same time, two trains leave these stations and start heading towards each other. After 1 hour and 45 minutes of chugging along, the trains are now 369 kilometers apart. We also know that the ratio of the first train's speed to the second train's speed is 3:2. So, the big question is: How fast is each train traveling?

Understanding the Fundamentals

Before we jump into solving this, let's brush up on some key concepts. The most important one here is the relationship between distance, speed, and time. Remember the formula: Distance = Speed × Time. This simple formula is the backbone of solving most motion-related problems. We'll also need to understand how to work with ratios and how to convert time units (like minutes into hours).

Think of it this way: if you know how fast you're going (speed) and how long you travel (time), you can easily figure out the total distance you've covered. Similarly, if you know the distance and the time, you can calculate the speed. In this problem, we have some information about the distance and time, and we need to find the speeds.

Setting Up the Equations

Alright, let's get down to business. The first step in solving any word problem is to translate the words into mathematical equations. This might sound intimidating, but it's just about representing the information we have in a way we can work with.

Let's use some variables to represent the unknowns. Let's say:

• v1 = the speed of the first train (in kilometers per hour)
• v2 = the speed of the second train (in kilometers per hour)

We know that the ratio of their speeds is 3:2, which means we can write this as:

v1 / v2 = 3 / 2

This gives us one equation. Now we need another one. Think about the distances involved. The trains start 124 km apart and end up 369 km apart after 1 hour and 45 minutes. This means they've effectively covered a total distance that is the sum of their initial separation and the additional distance after the time has passed. So, the total distance covered by both trains combined is 124 km + 369 km = 493 km.

Now, let's convert the time into hours. 1 hour and 45 minutes is equal to 1 + (45/60) = 1.75 hours. Using the formula Distance = Speed × Time, we can write the equation for the total distance covered by both trains:

(v1 * 1.75) + (v2 * 1.75) = 493

This is because the first train covers a distance of v1 * 1.75 km, and the second train covers a distance of v2 * 1.75 km in the same time. The sum of these distances is the total distance they've moved apart from each other.

Solving the System of Equations

Now we have two equations:

1. v1 / v2 = 3 / 2
2. (v1 * 1.75) + (v2 * 1.75) = 493

This is a system of two equations with two unknowns, which we can solve using several methods. One common method is substitution. From the first equation, we can express v1 in terms of v2:

v1 = (3 / 2) * v2

Now, substitute this expression for v1 into the second equation:

(((3 / 2) * v2) * 1.75) + (v2 * 1.75) = 493

Simplify this equation:

(2.625 * v2) + (1.75 * v2) = 493

Combine the terms with v2:

4.375 * v2 = 493

Now, solve for v2:

v2 = 493 / 4.375

v2 = 112.6857... (approximately)

So, the speed of the second train is approximately 112.69 km/h. Now, we can plug this value back into the equation v1 = (3 / 2) * v2 to find v1:

v1 = (3 / 2) * 112.6857

v1 = 169.02855 (approximately)

So, the speed of the first train is approximately 169.03 km/h.

Putting It All Together

To recap, we started with a word problem about two trains moving in opposite directions. We identified the key information, defined variables, and set up a system of equations. Then, we used substitution to solve for the speeds of the two trains. We found that the first train is traveling at approximately 169.03 km/h, and the second train is traveling at approximately 112.69 km/h.

This type of problem demonstrates the power of using algebra to solve real-world scenarios. By breaking down the problem into smaller steps and translating the information into equations, we can find the solutions. Remember guys, practice makes perfect, so try solving similar problems to sharpen your skills!

Checking Our Work

It's always a good idea to check your work, especially in math problems. Let's see if our answers make sense in the context of the original problem. We found that the first train travels at approximately 169.03 km/h and the second train travels at approximately 112.69 km/h. After 1.75 hours, the first train would have traveled about 169.03 * 1.75 = 295.8 km, and the second train would have traveled about 112.69 * 1.75 = 197.2 km. Adding these distances gives us 295.8 + 197.2 = 493 km, which matches the total distance we calculated earlier (124 km initial separation + 369 km final separation).

Also, let's check the ratio of the speeds: 169.03 / 112.69 is approximately 1.5, which is the same as 3/2. So, our answers seem to be consistent with the information given in the problem. This gives us confidence that we've solved it correctly.

Key Takeaways

This problem illustrates several important problem-solving strategies:

1. Read Carefully: Make sure you understand all the information given in the problem.
2. Define Variables: Assign variables to the unknowns you need to find.
3. Translate to Equations: Convert the word problem into mathematical equations.
4. Solve the Equations: Use algebraic techniques to solve for the unknowns.
5. Check Your Work: Make sure your answers make sense in the context of the problem.

By following these steps, you can tackle a wide variety of math problems with confidence. Remember, the key is to break down complex problems into smaller, manageable steps.

Conclusion

So there you have it! We've successfully navigated this train problem and figured out the speeds of the two trains. This kind of problem shows how math can be used to model real-world situations. Keep practicing, and you'll become a pro at solving these types of problems. Remember, guys, math is like a muscle – the more you use it, the stronger it gets! Good luck, and happy problem-solving!