Elena's Walk: Total Distance Calculation

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Hey guys! Let's break down this math problem step by step. We're trying to figure out how far Elena walked in total. She started by walking 12 miles, and then she walked an additional distance equal to 1/4 of her initial 12-mile walk. So, how do we calculate the total distance? Let's dive in!

Understanding the Problem

The core of this problem lies in understanding the two parts of Elena's journey. First, she walked a solid 12 miles. That's our baseline. Then, she walked an additional 1/4 of that 12-mile distance. The keyword here is "of," which in math often means multiplication. So, to find out how far she walked in the second part of her journey, we need to calculate 1/4 of 12 miles. This is a crucial step in solving the problem, and recognizing this relationship helps us set up the correct equation.

To make it even clearer, imagine dividing the initial 12 miles into four equal parts. Elena walked one of those parts in the second leg of her journey. Now, we need to figure out how many miles are in that one part. To find 1/4 of 12, we multiply these two numbers together. This gives us the distance Elena walked in the second part, which we then add to the initial 12 miles to get the total distance. Thinking about it this way can help you visualize the problem and avoid common mistakes. So, we’re not just adding 1/4 to 12; we're adding 1/4 of 12. This is a common trick in word problems, so keep an eye out for it!

Another way to think about it is to consider what 1/4 represents. It’s a fraction, a part of a whole. In this case, the "whole" is the initial 12 miles. We're not adding a quarter of a mile (0.25 miles); we're adding a quarter of the entire 12-mile distance. This is why multiplication is the correct operation here. By multiplying 1/4 by 12, we're effectively finding out what one-fourth of 12 is. This distinction is important for setting up the correct equation and solving the problem accurately. So remember, when you see "of" in a word problem, think multiplication! It's your key to unlocking the solution. Now that we understand the core concept, let's look at the answer choices and see which ones correctly represent this calculation.

Analyzing the Options

Let's break down the answer options to see which ones correctly represent the total distance Elena walked. Remember, we're looking for the options that show both the initial 12 miles and the additional 1/4 of 12 miles.

  • Option A: 12 + 1/4

    This option seems simple, but it's incorrect. It adds 1/4 to 12, which means it's adding a quarter of a mile (0.25 miles) to the initial 12 miles. However, the problem states that Elena walked 1/4 of the 12-mile distance, not just an additional quarter of a mile. This option misses the crucial multiplication step. It doesn't account for the fact that the second part of her walk is a fraction of the initial distance, not just a fraction itself. So, while addition is involved in finding the total distance, this option doesn't correctly represent what we need to add. It's a common mistake to simply add the fraction without considering what it's a fraction of. Therefore, this option is not the correct representation of the total distance Elena walked.

  • Option B: 12 + (1/4) * 12

    This is one of the correct options! It perfectly represents the problem. It shows the initial 12 miles plus 1/4 of 12 miles. The (1/4) * 12 part calculates the distance Elena walked in the second part of her journey, and then we add it to the initial 12 miles. This option follows the correct order of operations (multiplication before addition) and accurately reflects the problem's conditions. The multiplication shows that we're finding a fraction of a distance, and the addition combines the two parts of her walk to give us the total distance. This option clearly demonstrates a solid understanding of the problem and the mathematical operations required to solve it. So, give yourself a pat on the back if you identified this one!

  • Option C: 12 * (3/4)

    This option is incorrect. While it involves multiplication, it doesn't represent the total distance Elena walked. This calculation would be useful if we were trying to find out what 3/4 of 12 is, but that's not what the problem asks. It's tempting to see a fraction and multiplication and think it might be related, but it's crucial to understand what the calculation represents in the context of the problem. In this case, multiplying by 3/4 doesn't help us find the total distance Elena walked. It's a distraction, and understanding why it's incorrect is just as important as understanding why the correct options are correct. So, we can confidently eliminate this option.

  • Option D: 12 * (1/4)

    This option is partially correct, but it's not the complete answer. It correctly calculates the distance Elena walked in the second part of her journey (1/4 of 12 miles). However, it only gives us part of the total distance. We still need to add the initial 12 miles to this result. This option is a good start, but it's not the final answer. It highlights the importance of reading the problem carefully and making sure you're answering the entire question, not just a piece of it. So, while this calculation is necessary, it's not sufficient on its own. We need to remember the first 12 miles as well.

  • Option E: 12 * (5/4)

    This is another correct option! This might look a little different, but it’s actually another way to calculate the total distance. Think of it this way: Elena walked 12 miles, which is the whole distance (1 or 4/4 of the distance), and then an additional 1/4 of that distance. So, in total, she walked 1 + 1/4 = 5/4 of the initial 12 miles. Multiplying 12 by 5/4 gives us the same total distance as 12 + (1/4) * 12. This option demonstrates a deeper understanding of fractions and how they can represent different ways of calculating the same thing. If you chose this option, you’re thinking like a math pro! It's a more concise way of expressing the total distance, and it shows a good grasp of fraction operations.

Final Answer

Alright guys, after analyzing each option, we've nailed down the correct answers. The options that accurately represent the total distance Elena walked are:

  • B. 12 + (1/4) * 12
  • E. 12 * (5/4)

These options both correctly calculate the total distance, just in slightly different ways. Option B breaks it down into the initial 12 miles plus the additional 1/4 of 12 miles. Option E combines those steps by recognizing that the total distance is 5/4 of the initial 12 miles. Both are valid, and understanding why they both work is key to mastering these types of problems. So, great job working through this with me! Remember to always read the problem carefully, identify the key information, and break it down step by step. You got this! Now go conquer some more math challenges!