Calculating The Distance Between Two Cities A Step-by-Step Guide

Hey guys! Ever found yourself scratching your head over a tricky distance problem? Well, you're not alone! Let's break down this classic algebra question step by step. We've got a car journey with some fractions and distances thrown in, but don't worry, we'll make it super clear.

The Problem: Unraveling the Road Trip

Let's recap the challenge we are tackling. A car travels the first leg of its journey, covering 1/4 of the total distance between two cities plus an extra 10 kilometers. In the second hour, the car eats up 2/5 of the remaining distance and tacks on another 12 kilometers. Just when you think we're done, the car goes on for a third hour, devouring 2/3 of the new remainder, plus another 10 kilometers. Finally, after all that driving, there are still 20 kilometers left to reach the destination. The million-dollar question is: What's the total distance between these two cities?

This might sound like a wild goose chase, but trust me, we can crack this! The trick is to work backward, unraveling the problem piece by piece. We'll start from the end and work our way back to the beginning, like reverse engineering a road trip. First, we'll figure out how far the car traveled in the third hour, then the second, and finally, we'll reveal the total distance. So buckle up, and let's hit the road!

Breaking Down the Final Leg: The Third Hour's Journey

Okay, let's zoom in on that last stretch of the journey. We know that after the third hour, there were 20 kilometers remaining. Before the car embarked on this third hour, there was a certain distance left. During the third hour, the car covered 2/3 of this distance and an additional 10 kilometers. This left us with that final 20-kilometer stretch. Let's translate this into an equation to make things clearer. If we let 'x' represent the distance remaining before the third hour, we can say that the car covered (2/3) * x + 10 kilometers during that hour. The remaining 20 kilometers can be expressed as x - [(2/3) * x + 10] = 20. See? We're turning words into math!

Now, let's solve for x. First, we simplify the equation: x - (2/3)x - 10 = 20. Combine the 'x' terms, and we get (1/3)x - 10 = 20. Next, we add 10 to both sides, giving us (1/3)x = 30. Finally, to isolate 'x', we multiply both sides by 3, which reveals that x = 90 kilometers. So, before the third hour, there were 90 kilometers left. This is a crucial step! It means that in the third hour, the car covered 90 - 20 = 70 kilometers. We're piecing the puzzle together, one hour at a time.

Rewinding to the Second Hour: Unveiling More of the Distance

Now that we've conquered the third hour, let's rewind our mental odometer to the second hour. We know that before the third hour, there were 90 kilometers remaining. This means that the distance remaining after the second hour was 90 kilometers. During the second hour, the car traveled 2/5 of the remaining distance (at that time) plus 12 kilometers. Let's use 'y' to represent the distance remaining before the second hour. We can then express the journey in the second hour as (2/5) * y + 12 kilometers.

This journey, when subtracted from the distance 'y', leaves us with the 90 kilometers that were remaining before the third hour. So, the equation looks like this: y - [(2/5) * y + 12] = 90. Ready to crack another equation? Let's simplify! We get y - (2/5)y - 12 = 90. Combine the 'y' terms to get (3/5)y - 12 = 90. Add 12 to both sides, and we have (3/5)y = 102. To solve for 'y', we multiply both sides by 5/3. This gives us y = 170 kilometers. Woohoo! We now know that before the second hour, there were 170 kilometers remaining.

The Grand Finale: Calculating the Total Distance

We're almost there, guys! We've worked our way backward, unraveling the distances hour by hour. Now, let's zoom out to the big picture and calculate the total distance between the two cities. We know that before the second hour, there were 170 kilometers remaining. This means that after the first hour, the car had 170 kilometers left to travel. In the first hour, the car covered 1/4 of the total distance plus 10 kilometers. Let's use 'z' to represent the total distance between the two cities. We can then say that in the first hour, the car covered (1/4) * z + 10 kilometers.

This distance, when subtracted from the total distance 'z', leaves us with the 170 kilometers that were remaining before the second hour. Our equation now looks like this: z - [(1/4) * z + 10] = 170. Time for the final equation showdown! Let's simplify: z - (1/4)z - 10 = 170. Combining the 'z' terms, we get (3/4)z - 10 = 170. Add 10 to both sides, resulting in (3/4)z = 180. To solve for 'z', we multiply both sides by 4/3. This grand finale gives us z = 240 kilometers. Boom! We've done it! The total distance between the two cities is 240 kilometers. Give yourselves a pat on the back; you've conquered a tricky algebra problem!

Putting It All Together: The Journey Unveiled

Let's take a moment to appreciate the journey we've taken, both mathematically and metaphorically. We started with a seemingly complex problem, a car weaving its way through fractions and distances. By breaking it down into smaller, manageable chunks, we were able to solve for each unknown piece by piece. We worked backward, unraveled each hour's progress, and finally arrived at the total distance.

• In the first hour, the car traveled 1/4 of the total distance (240 km), which is 60 km, plus an additional 10 km, totaling 70 km. This left 170 km remaining.
• In the second hour, the car traveled 2/5 of the remaining 170 km, which is 68 km, plus another 12 km, totaling 80 km. This left 90 km remaining.
• In the third hour, the car covered 2/3 of the new remainder of 90 km, which is 60 km, plus an additional 10 km, totaling 70 km. This left 20 km, as stated in the problem.

And there you have it! A complete breakdown of the journey, from start to finish. This problem beautifully illustrates the power of algebra in solving real-world problems. By using variables, equations, and a bit of logical thinking, we were able to navigate through the complexities and arrive at a clear solution. So next time you encounter a challenging problem, remember this journey and break it down step by step. You've got this!

*   Distance problem
*   Algebra
*   Word problem
*   Equation solving
*   Fractions
*   Problem-solving
*   Step-by-step solution
*   Working backwards
*   Total distance
*   Remaining distance

Calculate the Distance Between Two Cities A Step-by-Step Solution