Nidia & Naomi's Race: Find The Distance Difference!
Let's break down this problem involving Nidia and Naomi's race! This is a classic physics problem that involves understanding ratios and distances. Basically, we're trying to figure out how much farther one person ran than the other, given some information about their speeds and positions. It might sound a bit tricky at first, but don't worry, we'll go through it step by step. So, buckle up and get ready to solve this fun little puzzle! We'll use some basic math and logical reasoning to get to the bottom of it. Let's dive in!
Understanding the Problem
Okay, guys, let's make sure we really get what the problem is asking. We know Nidia and Naomi are running at different speeds, and those speeds have a ratio of 12:9. That means for every 12 meters Nidia runs, Naomi runs 9 meters. Now, here's the key part: When Nidia (the faster one) gets to the spot where Naomi started, Naomi still has 352 meters left to run to reach Nidia's starting point. Our mission is to find the difference between the total distance Nidia ran and the total distance Naomi ran. Think of it like this: Nidia gets a head start, and we need to figure out how much further she went overall. We need to find the exact difference in distance covered by Nidia and Naomi. To solve this, we will look into the ratios and the remaining distance. This should be fun!
Setting up the Equations
Alright, let's get a little mathematical here. We can represent the speeds of Nidia and Naomi as 12x and 9x, respectively. Here, 'x' is just a common multiplier that keeps the ratio correct. Now, let's say the distance between their starting points is 'D' meters. When Nidia reaches Naomi's starting point, she has covered a distance of 'D' meters. In the same time, Naomi has covered a distance of 'D - 352' meters. Since time is the same for both when Nidia reaches Naomi's starting point, we can set up an equation using the formula: time = distance / speed. So, we have: D / 12x = (D - 352) / 9x. This equation relates the distance 'D' with the known speeds and the remaining distance of 352 meters. Solving this equation will give us the value of 'D', which is the distance between their starting points. This will help us later find the difference in the distances they traveled.
Solving for the Distance
Now, let's solve that equation we set up! We had D / 12x = (D - 352) / 9x. First, we can cancel out the 'x' from both sides, which simplifies things to D / 12 = (D - 352) / 9. Next, cross-multiply to get rid of the fractions: 9D = 12(D - 352). Expanding that gives us 9D = 12D - 4224. Now, let's isolate 'D' by subtracting 9D from both sides: 0 = 3D - 4224. Add 4224 to both sides: 4224 = 3D. Finally, divide by 3: D = 1408. So, the distance 'D' between their starting points is 1408 meters. That's a key piece of information! Now we know exactly how far apart Nidia and Naomi were at the beginning of the race. This will lead us to calculating the difference in total distances traveled.
Calculating the Distances Traveled
Okay, we know that the distance between their starting points, 'D', is 1408 meters. Nidia ran this full distance, so Nidia's distance is 1408 meters. Naomi, on the other hand, ran 'D - 352' meters, which is 1408 - 352 = 1056 meters. Now we need to find the difference in the distances they traveled. To do this, we simply subtract Naomi's distance from Nidia's distance: 1408 - 1056 = 352 meters. So, the difference in the distances they traveled when Nidia reaches Naomi's starting point is 352 meters. It may seem counterintuitive that the difference is the same as the remaining distance, but it is true!
Finding the Difference in Total Distances
But hold on! We're not quite done yet. The problem asks for the difference in the total distances they traveled. When Nidia reaches Naomi's starting point, Naomi still has 352 meters to go. That means that by the time Nidia has run distance D (1408 meters) Naomi has run 1408-352 = 1056 meters. The question asks for the difference in the distances covered, and we know Nidia covered more distance than Naomi. We can easily calculate this by subtracting Naomi's distance from Nidia's distance: 1408 meters (Nidia) - 1056 meters (Naomi) = 352 meters.
However, the key here is that Nidia is faster. The ratio of their speeds is 12:9, which simplifies to 4:3. This means that for every 4 meters Nidia runs, Naomi runs 3 meters. The problem states that when Nidia reaches Naomi's starting point, Naomi still has 352 meters to go. Let's denote the distance Nidia runs as 4x, and the distance Naomi runs as 3x. When Nidia reaches Naomi's starting point, Nidia has run 'D' meters and Naomi has run 'D - 352' meters. So, we can say that 4x = D and 3x = D - 352. Subtracting the second equation from the first gives us x = 352. Now we can find the actual distances Nidia and Naomi ran.
Nidia's distance = 4x = 4 * 352 = 1408 meters Naomi's distance = 3x = 3 * 352 = 1056 meters
The difference in their distances is 1408 - 1056 = 352 meters. But this is not the final answer. We need to consider what happens after Nidia reaches Naomi's start point. The problem isn't clear on what happens after Nidia reaches Naomi's start. Let's assume the problem is only asking for the distance difference when Nidia reaches Naomi's start point.
The Final Answer
So, after all that calculating, here's our answer: The difference in the distances traveled by Nidia and Naomi when Nidia reaches Naomi's starting point is 352 meters. We found this by first determining the distance between their starting points, and then calculating how far each of them ran. Finally, we subtracted Naomi's distance from Nidia's distance to find the difference. Hope you enjoyed solving this problem with us! Remember, breaking down the problem into smaller steps and using the right formulas can make even the trickiest physics problems manageable. Keep practicing, and you'll become a pro at solving these types of questions in no time! This problem showcased the use of ratios and their significance, as well as the use of speeds and distances in the real world.