Mastering Math: Which Equation Is Calculated Correctly?

Hey math enthusiasts! Today, we're diving into the fascinating world of arithmetic and tackling a common challenge: ensuring our calculations are spot-on. In this article, we'll dissect a multiple-choice question that tests your understanding of basic operations involving negative numbers. Get ready to sharpen your skills as we explore each option and reveal the correct answer, along with a clear explanation. This isn't just about finding the right answer; it's about building a solid foundation in mathematical principles that will serve you well in all your academic and practical endeavors. We’ll break down the rules of multiplying and dividing negative numbers, a concept that often trips up learners. By the end of this discussion, you'll feel more confident in your ability to handle these types of problems and understand why the correct answer is indeed correct. So, grab your calculators (or your trusty mental math skills!) and let's get started on this mathematical adventure.

Decoding the Rules of Negative Numbers in Multiplication and Division

Before we jump into the specific equations, let's refresh our memory on the fundamental rules that govern operations with negative numbers. These rules are the bedrock upon which correct calculations are built. When multiplying or dividing two numbers, if both numbers have the same sign (both positive or both negative), the result will always be positive. Think of it this way: two negatives working together cancel each other out, resulting in a positive outcome. Conversely, if the two numbers have different signs (one positive and one negative), the result will always be negative. This is because the differing signs don't cancel each other out; one 'negative influence' remains, making the final product or quotient negative. Understanding this simple, yet crucial, principle is key to solving the problem at hand. We’ll be applying these rules directly to each option presented, so keep them firmly in mind as we progress. It’s like having a secret code that unlocks the mystery of negative number calculations, making what might seem complex suddenly clear and manageable. Mastering these rules is not just about passing a test; it's about building mathematical fluency and confidence, empowering you to tackle more complex problems down the line. Remember, consistency in applying these rules is paramount; a single slip can lead to an incorrect result, so always double-check your signs!

Analyzing Option A: $(-2) \times (-5) = -10$?

Let's begin our detailed analysis with Option A: $(-2) \times (-5) = -10$. Here, we are multiplying two negative numbers: -2 and -5. According to the rules we just discussed, when we multiply two negative numbers, the result should be positive. Therefore, $(-2) \times (-5)$ should equal +10, not -10. The equation states the result is -10, which is incorrect. This option fails to adhere to the fundamental rule of multiplying signed numbers. It's a common mistake to simply multiply the absolute values of the numbers and forget to apply the correct sign rule. In this case, the absolute values are 2 and 5, and their product is indeed 10. However, because both original numbers were negative, their product must be positive. This mismatch between the expected positive result and the given negative result clearly indicates that Option A is not the correctly calculated equation. It serves as a good reminder that paying close attention to the signs is just as important as performing the arithmetic correctly. The product of two negatives is a positive, a principle that is non-negotiable in the realm of arithmetic. So, while the magnitude of the answer (10) is correct, the sign is not, making this option definitively wrong.

Examining Option B: $(-6) \div 6 = 1$?

Next up is Option B: $(-6) \div 6 = 1$. In this scenario, we are dividing a negative number (-6) by a positive number (6). Applying our rules for division of signed numbers, when a negative number is divided by a positive number, the result is negative. So, $(-6) \div 6$ should yield -1. The equation presented claims the answer is 1, which is a positive number. This again contradicts the established rules. The absolute values of the numbers are 6 and 6, and their division is indeed 1. However, because we are dividing a negative by a positive, the outcome must be negative. Therefore, the correct calculation should be $(-6) \div 6 = -1$. Option B incorrectly suggests a positive result. This highlights the importance of remembering that signs matter in every operation. Whether it’s multiplication or division, the rule of differing signs resulting in a negative outcome is consistent. This equation, like Option A, demonstrates a misunderstanding or misapplication of these essential sign rules, making it an incorrect statement. It’s crucial to visualize these operations: imagine sharing -6 apples among 6 friends; each friend would get -1 apple, representing a deficit or debt, hence the negative sign.

Evaluating Option C: $(-20) \div (-5) = -4$?

Now, let's scrutinize Option C: $(-20) \div (-5) = -4$. Here, we have the division of two negative numbers: -20 and -5. Following our established rule for division, when two negative numbers are divided, the result is positive. Therefore, $(-20) \div (-5)$ should result in a positive number. The absolute values of 20 and 5, when divided, give us 4. Since both original numbers were negative, their quotient should be $+4$, not -4. The equation states the result is -4, which is a negative number. This is incorrect. This option, much like the previous ones, makes the error of assigning the wrong sign to the result. The division of -20 by -5 logically leads to a positive outcome because the two negative signs effectively cancel each other out. It's essential to reinforce this: a negative divided by a negative is always positive. The magnitude is correct (4), but the sign is incorrect. This makes Option C also an invalid calculation. It’s a common pitfall, and recognizing this pattern of error is part of mastering arithmetic with signed numbers.

Assessing Option D: $(-6) \times (-6) = 36$?

Finally, let’s turn our attention to Option D: $(-6) \times (-6) = 36$. We are multiplying two negative numbers here: -6 and -6. Recall our rule for multiplication: when two negative numbers are multiplied, the result is positive. The absolute value of -6 is 6. Multiplying 6 by 6 gives us 36. Since we are multiplying two negative numbers, the resulting sign must be positive. Therefore, $(-6) \times (-6)$ correctly equals $+36$. The equation states the result is 36, which is indeed positive 36. This option accurately follows the rules of multiplying signed numbers. Both numbers being multiplied are negative, and their product is positive, as expected. This is the one equation that has been calculated correctly according to the principles of arithmetic with negative numbers. It serves as a perfect example of the rule in action: a negative times a negative equals a positive. This confirmation brings us to the conclusion of our analysis, having identified the single correct calculation among the options.

The Correct Calculation Revealed!

After meticulously examining each option based on the established rules of arithmetic with negative numbers, we can definitively state which equation is correctly calculated. The key principles we've relied upon are: (1) Multiplying or dividing two numbers with the same sign (both positive or both negative) results in a positive number. (2) Multiplying or dividing two numbers with different signs (one positive and one negative) results in a negative number.

Let’s recap:

• Option A: $(-2) \times (-5)$. Two negatives multiplied should result in a positive. $(-2) \times (-5) = +10$, not -10. Incorrect.
• Option B: $(-6) \div 6$. A negative divided by a positive should result in a negative. $(-6) \div 6 = -1$, not 1. Incorrect.
• Option C: $(-20) \div (-5)$. Two negatives divided should result in a positive. $(-20) \div (-5) = +4$, not -4. Incorrect.
• Option D: $(-6) \times (-6)$. Two negatives multiplied should result in a positive. $(-6) \times (-6) = +36$. Correct.

Therefore, the only equation that has been correctly calculated is Option D. This exercise underscores the importance of understanding and consistently applying the rules of signs in mathematical operations. By mastering these foundational concepts, you build confidence and accuracy in your problem-solving abilities.

For further exploration into the rules of arithmetic and number properties, you can visit educational resources like Khan Academy or Math is Fun. These sites offer comprehensive explanations and practice exercises that can help solidify your understanding. Happy calculating!