Clara's Multiplication: Is Her Answer Right?

Hey math enthusiasts! Today, we're diving into a fun little problem involving Clara and her multiplication skills. The core question: Did Clara get the correct answer when she multiplied $(-6)(-7)(-1)$ and got 42? Let's break it down, shall we? We'll explore the rules of multiplying integers, paying special attention to how those pesky negative signs can trip us up, and then we'll figure out if Clara's answer makes sense. Trust me, it's not as scary as it sounds, and it's a great opportunity to refresh some fundamental math concepts! So, buckle up, grab your calculators (or your brains!), and let's get started. We'll approach this with a bit of a detective's mindset, examining each step to see where the potential pitfalls might be. It's all about precision and understanding the basics. Whether you're a math whiz or just trying to brush up on your skills, this is a great chance to review integer multiplication rules and practice applying them. Plus, we'll get to analyze a real-world (or at least, a problem-set-world) scenario and determine if someone's math is on point or needs a little adjustment. Let's make sure we understand the question, the concepts involved, and the implications of Clara's answer. The goal is to make sure we truly understand why or why not, so we can be confident in our response. Ready to unravel this multiplication mystery? Let's go!

Understanding Integer Multiplication: The Basics

Alright, before we jump into Clara's problem, let's refresh our memories on the rules of integer multiplication. These rules are absolutely crucial for getting the right answer. Basically, when you're multiplying integers, the signs of the numbers determine the sign of the result. Here's a quick rundown:

• Positive times Positive: If you multiply two positive numbers, the answer is always positive. For example, $2 * 3 = 6$.
• Negative times Negative: If you multiply two negative numbers, the answer is also positive. Think of it like this: two wrongs make a right! For example, $(-2) * (-3) = 6$.
• Positive times Negative or Negative times Positive: If you multiply a positive number by a negative number (or vice versa), the answer is always negative. For example, $2 * (-3) = -6$ and $(-2) * 3 = -6$.

These simple rules are the key to unlocking this type of problem, and any problem like it. Keep these in mind! The core idea is that the number of negative signs involved matters. If there's an even number of negative signs, the answer is positive. If there's an odd number of negative signs, the answer is negative. Let’s try an example. If we had $(-2) * (-3) * (-4)$, we have three negative signs (an odd number), which means the answer will be negative. The actual calculation would be $(-2) * (-3) = 6$, and then $6 * (-4) = -24$. So, the answer is -24. Got it? These rules might seem like simple memorization, but they're incredibly important when dealing with more complex equations. Once you get the hang of it, you'll be able to quickly determine the correct sign of your answer without much thought. Keep these principles in mind as we approach Clara's problem. Understanding these basic principles will provide a solid foundation for evaluating Clara's work. The key takeaway? The sign of your answer is directly determined by the number of negative numbers you are multiplying. Let's apply these rules to solve this problem.

Analyzing Clara's Calculation

Now, let's take a closer look at Clara's problem: $(-6)(-7)(-1) = 42$. We need to figure out if her answer is correct. First, we need to apply the rules of integer multiplication to the given expression. Let's break it down step by step: The initial expression involves the product of three numbers: -6, -7, and -1. First, we'll multiply the first two numbers, (-6) and (-7). Applying our rules, we know that a negative times a negative equals a positive. Therefore, $(-6) * (-7) = 42$. Now we have $42 * (-1)$. When multiplying a positive number by a negative number, the result is negative. This means that $42 * (-1) = -42$. So, the correct answer should be -42, and Clara’s answer was 42. By applying the rules of integer multiplication step-by-step, we were able to determine the right answer. This comparison helps us understand where Clara might have made an error. If she had properly understood the rules of multiplying integers, then she would have gotten the correct answer. The critical thing here is that the final result should be negative, because there is an odd number of negative signs in the original equation. Let's see if we can identify why Clara got it wrong.

Identifying Clara's Error

So, where did Clara go wrong? Let's think about it. The issue arises from a misunderstanding of how the negative signs interact during multiplication. Let's walk through the steps again to see if we can find the mistake. When multiplying $(-6) * (-7) * (-1)$, she might have overlooked the effect of the negative sign on the final result. In the initial stage, $(-6) * (-7)$ should result in a positive number (42). However, in the second stage, Clara might have overlooked the last negative number. This is where the error typically occurs. Because there is a negative number at the end, she likely thought the answer would be positive, like the result from the beginning. She didn't account for how the negative one would change the sign of the answer. This is a common mistake when dealing with multiple negative numbers. It’s easy to focus on pairs of negatives and forget about the implications of the final negative sign. The likely error is that she calculated $(-6) * (-7) = 42$ correctly, but then incorrectly multiplied $42 * (-1)$ and somehow got a positive result (42). Perhaps she forgot the rule that a positive number multiplied by a negative number gives a negative result. This might be a simple oversight, or maybe a deeper misunderstanding of the rules. The key takeaway is to carefully track the signs throughout each step of the calculation. Make sure each step makes sense in the context of the rules! This is a great opportunity to remind ourselves that the correct application of the multiplication rules is essential for solving these types of problems correctly. By understanding exactly where Clara went wrong, we can avoid making the same mistakes ourselves in the future. Now, let’s choose the correct answer for Clara's multiplication problem. We'll be able to decide which of the given options correctly explains why Clara's answer is not reasonable.

Is Clara's Answer Reasonable? - The Final Verdict

Based on our analysis, we know that the correct answer to $(-6)(-7)(-1)$ is -42. Clara, however, arrived at 42. Since the solution should be negative (because we have an odd number of negative signs), her answer is incorrect. Looking at the options, we can choose the one that explains why Clara's answer isn't reasonable. The correct answer is:

A. No, because the solution should have a negative answer.

This option directly addresses the error in Clara's calculation. It points out that the correct answer should have been negative, which clearly indicates that Clara's answer is unreasonable. She did not properly account for all the negative signs. Therefore, option A accurately reflects the math and the error. By choosing this answer, we demonstrate our understanding of the rules of integer multiplication. We also demonstrate the importance of paying attention to detail when solving such problems. In summary, Clara's answer is not reasonable because she failed to correctly apply the rules of multiplying integers. This is a common mistake and serves as a good reminder to always double-check our work and be mindful of the signs. Now that we've found the answer, we can be confident in our evaluation of Clara's multiplication. This entire exercise helps reinforce the key concepts of integer multiplication, making it easier for us to apply these rules to other problems we may encounter.