Distance Calculation: Gopi, Kohli & Town Travel Speeds

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Let's break down this classic distance, speed, and time problem involving Gopi and Kohli's journey between town A and town B. We'll explore how to calculate the distance between the two towns, considering their different speeds and the time difference in their arrivals. This is a common type of question you might encounter in quantitative aptitude tests, so understanding the approach is super helpful, guys!

Understanding the Problem Setup

So, in this problem, we've got Gopi and Kohli trekking from Town A to Town B. Gopi is moving at a steady 81 km/hr, while Kohli is zooming along at 90 km/hr. The key detail here is that Kohli beats Gopi to Town B by 42 minutes. Our mission? Figure out the distance separating Town A and Town B.

To nail this, we've got to remember the golden rule: Distance = Speed × Time. The trick lies in using the time difference to link Gopi and Kohli's journeys. We will use this formula to derive a relationship between the distances, speeds, and times of their travel. The difference in their travel times is the key to unlocking the solution, so let's dive deeper into how we can express this mathematically.

Setting Up the Equations

Let's use 'D' to represent the distance between Town A and Town B (in kilometers). This is the magic number we're hunting for! Now, let's bring in 'T' to signify the time (in hours) Gopi takes to make the journey. Since Kohli is faster, he'll take less time, right? He arrives 42 minutes earlier, which translates to 42/60 hours or 7/10 hours. So, Kohli's travel time would be (T - 7/10) hours.

Now, let's translate this into equations using our trusty Distance = Speed × Time formula:

  • For Gopi: D = 81T
  • For Kohli: D = 90(T - 7/10)

See what we've done? We've expressed the distance 'D' in two different ways, once for each traveler. Since the distance is the same for both, we can equate these expressions. This is where the algebra fun begins, guys!

Solving for Time (T)

Now comes the crucial step where we equate Gopi's and Kohli's distance equations. Remember, both covered the same distance, just at different speeds and times. So, we can set their distance equations equal to each other:

81T = 90(T - 7/10)

Let's break down the equation. We have 81 times Gopi's travel time equals 90 times (Gopi's travel time minus 7/10 hours). To solve for 'T', we'll first distribute the 90 on the right side of the equation:

81T = 90T - 90 * (7/10) 81T = 90T - 63

Now, let's gather the 'T' terms on one side. Subtract 81T from both sides:

0 = 9T - 63

Next, let's isolate the 'T' term by adding 63 to both sides:

63 = 9T

Finally, to find 'T', divide both sides by 9:

T = 63 / 9 T = 7 hours

So, Gopi took 7 hours to travel from Town A to Town B. We are one giant leap closer to finding the distance now!

Calculating the Distance (D)

We've cracked the code for 'T'! We know Gopi took 7 hours for the trip. Now, to find the distance 'D', we can simply plug this value back into either Gopi's or Kohli's distance equation. Let's use Gopi's equation – it looks a bit simpler:

D = 81T

Substitute T = 7 hours:

D = 81 * 7 D = 567 kilometers

Boom! We've got our answer. The distance between Town A and Town B is a whopping 567 kilometers. See how breaking the problem down step-by-step makes it super manageable, guys?

Alternative Approach: Using Time Difference Directly

There's another cool way to solve this problem that's a bit more direct. It involves focusing on the time difference right from the start. This method can be quicker once you get the hang of it.

Let's call the time Gopi takes 'T' (in hours) and the time Kohli takes 'T - 7/10' (since he's 42 minutes, or 7/10 of an hour, faster). We know their speeds are 81 km/hr and 90 km/hr, respectively. Since they both travel the same distance, we can say:

Distance = Speed × Time

So,

81T = 90(T - 7/10)

Notice anything familiar? This is the same equation we arrived at earlier! The rest of the steps are identical – solving for T and then plugging it back in to find the distance. This approach highlights that you can often tackle these problems from different angles, and it's awesome to have multiple strategies in your toolkit.

Key Takeaways and Strategies

This problem perfectly illustrates how to tackle distance, speed, and time questions. The core idea is to relate the given information (speeds, time difference) using the fundamental formula: Distance = Speed × Time. Here are some key takeaways, guys:

  • Identify the unknowns: What are you trying to find? In this case, it was the distance.
  • Define variables: Assign variables to the unknowns (like 'D' for distance and 'T' for time).
  • Formulate equations: Translate the problem's information into mathematical equations. This is often the trickiest part!
  • Solve the equations: Use algebra to solve for the unknowns.
  • Check your answer: Does your answer make sense in the context of the problem?

The beauty of these problems is that there's often more than one way to crack them. The alternative approach we discussed, focusing directly on the time difference, is a prime example. Practice spotting these different approaches – it'll make you a much more versatile problem-solver.

Practice Problems to Sharpen Your Skills

Alright, guys, time to put those problem-solving muscles to work! Here are a couple of practice problems, that are similar to the one we just tackled. Try applying the same strategies and see how you do:

  1. Two trains leave a station at the same time, traveling in opposite directions. One train travels at 80 km/hr, and the other travels at 100 km/hr. How long will it take for them to be 900 km apart?
  2. A cyclist travels from Town P to Town Q at a speed of 15 km/hr. On the return trip, the cyclist travels at 10 km/hr. If the entire journey takes 5 hours, what is the distance between Town P and Town Q?

These problems will help you solidify your understanding of distance, speed, and time relationships. Remember to break them down step-by-step, define your variables, and formulate those equations! You've got this!

By mastering these types of problems, you'll be well-prepared for any quantitative challenges that come your way. Remember, practice makes perfect, so keep at it, guys! And don't be afraid to explore different approaches and strategies. Happy problem-solving!