Dividing Negative Numbers Solving -12 -12 Step By Step

Dividing negative numbers can seem tricky at first, but with a solid understanding of the basic principles, it becomes quite straightforward. The key concept to remember is that dividing two negative numbers results in a positive number, while dividing a negative number by a positive number (or vice versa) results in a negative number. This rule is fundamental and applies universally across all division problems involving negative numbers. To truly grasp this, let's break down the process step by step and explore why these rules hold true. First, consider the relationship between multiplication and division. Division is essentially the inverse operation of multiplication. For example, if we know that 3 x 4 = 12, then we also know that 12 / 3 = 4 and 12 / 4 = 3. This relationship helps us understand why the rules for dividing negative numbers work the way they do. When dividing negative numbers, think about what number you need to multiply the divisor by to get the dividend. For instance, if we have -12 / -4, we are asking ourselves, "What number multiplied by -4 equals -12?" The answer is 3, because -4 x 3 = -12. This illustrates the principle that a negative divided by a negative yields a positive result. Conversely, when dividing a negative number by a positive number, the result will be negative. For example, -12 / 4 means, "What number multiplied by 4 equals -12?" The answer is -3, since 4 x -3 = -12. This demonstrates the rule that a negative divided by a positive is negative. Similarly, a positive divided by a negative also results in a negative answer. For example, 12 / -4 asks, "What number multiplied by -4 equals 12?" The answer is -3, as -4 x -3 = 12. These rules are not arbitrary; they are rooted in the fundamental properties of arithmetic and ensure that mathematical operations remain consistent and logical. Understanding these rules is crucial not only for solving simple division problems but also for tackling more complex algebraic equations and mathematical concepts. It's also essential to practice these concepts with various examples to reinforce your understanding. This practice will help you build confidence and accuracy when dealing with negative numbers in division.

Negative divided by negative yields a positive result. This might seem counterintuitive at first, but it's a crucial rule in mathematics. Let's delve deeper into why this rule exists and how it works in practice. The foundation of this rule lies in the relationship between multiplication and division. As mentioned earlier, division is the inverse operation of multiplication. To understand why a negative divided by a negative is positive, consider the following: We know that a negative number multiplied by a negative number gives a positive number. For example, -3 x -4 = 12. Now, if we reverse this operation using division, we have 12 / -4 = -3 and 12 / -3 = -4. However, what if we start with -12 and divide it by -4? We are essentially asking, "What number multiplied by -4 gives -12?" The answer is 3, because -4 x 3 = -12. This illustrates why -12 / -4 = 3, a positive number. Another way to think about this is through the concept of opposites. When you divide a number by its opposite, you get -1. For example, 5 / -5 = -1. However, when you divide a negative number by another negative number, you are essentially canceling out the negativity. Think of it like this: the first negative sign indicates a quantity below zero, and the second negative sign indicates a reversal of direction. Combining these two negations results in a positive direction, thus a positive result. This concept can be visualized on a number line. Imagine you are facing the negative direction, and then you make a 180-degree turn (another negative action). You are now facing the positive direction. This analogy helps to understand the logic behind the rule. Let's look at some examples to solidify this concept. Consider -20 / -5. We are asking, "What number multiplied by -5 equals -20?" The answer is 4, because -5 x 4 = -20. Therefore, -20 / -5 = 4, a positive number. Similarly, -36 / -9 asks, "What number multiplied by -9 equals -36?" The answer is 4, so -36 / -9 = 4. It's important to remember this rule as it applies not only to simple division problems but also to more complex mathematical equations and expressions. When you encounter a problem involving dividing two negative numbers, confidently apply this rule and remember that the result will always be positive. Practice with various examples to build your proficiency and accuracy. The more you practice, the more natural this rule will become, and you'll be able to apply it effortlessly in different mathematical contexts.

When dividing a negative number by a positive number or a positive number by a negative number, the result is always negative. This is another fundamental rule in mathematics that is essential for understanding how to work with negative numbers. Let's explore why this rule holds true and how it applies in different scenarios. This rule is also based on the relationship between multiplication and division. If a negative number multiplied by a positive number results in a negative number, then dividing a negative number by a positive number (or vice versa) must yield a negative result. For example, we know that -3 x 4 = -12. If we reverse this using division, we get -12 / 4 = -3 and -12 / -3 = 4. This shows that when a negative number is divided by a positive number, the result is negative. Similarly, if we have 3 x -4 = -12, then -12 / -4 = 3 and -12 / 3 = -4, which demonstrates that a negative number divided by a negative number yields a positive number. Now, let's focus on the cases where the result is negative. When we divide a negative number by a positive number, we are essentially splitting a negative quantity into positive groups. For instance, if we have -12 / 4, we are asking, "If we divide -12 into 4 equal groups, how much is in each group?" The answer is -3, meaning each group contains a negative quantity. Conversely, when we divide a positive number by a negative number, we are asking, "How many negative groups are needed to make a positive quantity?" For example, if we have 12 / -4, we are asking, "How many groups of -4 are needed to make 12?" The answer is -3, because three groups of -4 equal -12. This concept can also be understood using the number line. When you divide a positive number by a negative number, you are essentially moving in the opposite direction on the number line, resulting in a negative value. Similarly, when you divide a negative number by a positive number, you are still moving towards the negative side of the number line. Let's look at some examples to further illustrate this rule. Consider -20 / 5. We are asking, "What number multiplied by 5 equals -20?" The answer is -4, because 5 x -4 = -20. Therefore, -20 / 5 = -4, a negative number. Similarly, if we have 20 / -5, we are asking, "What number multiplied by -5 equals 20?" The answer is -4, so 20 / -5 = -4. It's crucial to internalize this rule as it is fundamental to solving a wide range of mathematical problems. Whether you are working with simple arithmetic or complex algebraic equations, understanding that a negative divided by a positive or a positive divided by a negative always results in a negative number is essential. To reinforce your understanding, practice with various examples and scenarios. The more you practice, the more comfortable and confident you will become in applying this rule.

To solve the expression $\frac{-12}{-12}$ step-by-step, we apply the rules of dividing negative numbers. This particular problem involves dividing a negative number by another negative number, so we know the result will be positive. Here’s a detailed breakdown of the solution: First, identify the problem: The problem is a fraction, $\frac{-12}{-12}$, which represents the division of -12 by -12. This can also be written as -12 ÷ -12. Next, recall the rule for dividing negative numbers: As we discussed earlier, a negative number divided by a negative number results in a positive number. This is because the two negative signs essentially “cancel” each other out. Now, apply the rule: We have -12 divided by -12. Both numbers are negative, so the result will be positive. We can rewrite the problem as: $\frac{-12}{-12}$ = $\frac{12}{12}$ Now, perform the division: Divide the numbers without the negative signs. We are dividing 12 by 12, which is a basic division operation. 12 divided by 12 equals 1. So, $\frac{12}{12}$ = 1 Finally, state the result: Since we determined that the result would be positive, the answer is 1. Therefore, $\frac{-12}{-12}$ = 1 Let's reinforce this with another perspective: Think of division as the inverse of multiplication. We are asking, “What number multiplied by -12 equals -12?” The answer is 1, because -12 x 1 = -12. This confirms our solution. This step-by-step approach ensures clarity and accuracy when dealing with division problems involving negative numbers. By breaking down the problem into smaller steps, we can easily apply the appropriate rules and arrive at the correct solution. Understanding the underlying principles, such as the relationship between multiplication and division, helps to solidify our grasp of the concept. To further enhance your understanding, try solving similar problems. For example, try solving $\frac{-25}{-5}$ or $\frac{-36}{-4}$. These problems follow the same principle of dividing a negative number by a negative number, resulting in a positive quotient. Practice is key to mastering these mathematical operations and building confidence in your abilities. Remember to always consider the rules for negative numbers when performing division and apply them methodically to ensure accurate results. With consistent practice, you’ll become proficient in solving such problems effortlessly.

The correct answer to the question $\frac{-12}{-12}$ is D. 1. This is because, as we've thoroughly discussed, dividing a negative number by another negative number yields a positive result. Additionally, any number (except zero) divided by itself equals 1. Let’s break down why each of the other options is incorrect and why option D is the definitive answer. First, let's consider the problem again: $\frac{-12}{-12}$ The expression is asking us to divide -12 by -12. Understanding the rules for dividing negative numbers is crucial here. Option A: 12 This option is incorrect because it fails to account for the negative signs. While it correctly identifies the numerical result of dividing 12 by 12, it overlooks the impact of the negative signs on both the numerator and the denominator. Option B: -1 This option is incorrect because it correctly applies the rule that a negative number divided by a positive number results in a negative number, but it mistakenly applies this rule to the division of two negative numbers. When dividing -12 by -12, both numbers are negative, so the result should be positive, not negative. Option C: -12 This option is incorrect because it misunderstands the operation entirely. It seems to treat the problem as if it were a subtraction problem (i.e., -12 - 0) or an attempt to simply state the numerator. This does not align with the rules of division. Option D: 1 This is the correct answer. When we divide -12 by -12, we are essentially asking, “What number multiplied by -12 gives -12?” The answer is 1, because -12 x 1 = -12. Furthermore, we know that a negative divided by a negative results in a positive. So, the result is positive 1. Option E: None of these This option is incorrect because we have already established that option D is the correct answer. The correct answer is present among the choices. To summarize, option D is correct because it accurately applies the rule of dividing negative numbers and the fundamental principle that any non-zero number divided by itself equals 1. Let's further reinforce why this is the case. When you divide a number by itself, you are essentially asking how many times that number fits into itself. In this case, -12 fits into -12 exactly once. This is a fundamental property of division. Additionally, the negative signs “cancel” each other out, resulting in a positive result. Therefore, understanding the rules of dividing negative numbers and the basic principles of division is essential to correctly answer this type of problem. By recognizing these concepts, we can confidently determine that the correct answer is indeed 1.

To further solidify your understanding of dividing negative numbers, working through additional examples and practice problems is highly beneficial. This hands-on approach will not only reinforce the rules but also help you apply them confidently in various scenarios. Let's explore some additional examples and practice problems to help you master this concept. Example 1: Simplify $\frac{-24}{-6}$ First, we recognize that we are dividing a negative number by a negative number, so the result will be positive. Next, we divide the absolute values: 24 ÷ 6 = 4. Therefore, $\frac{-24}{-6}$ = 4. Example 2: Simplify $\frac{30}{-5}$ In this case, we are dividing a positive number by a negative number, so the result will be negative. Then, we divide the absolute values: 30 ÷ 5 = 6. Therefore, $\frac{30}{-5}$ = -6. Example 3: Simplify $\frac{-45}{9}$ Here, we are dividing a negative number by a positive number, so the result will be negative. We divide the absolute values: 45 ÷ 9 = 5. Therefore, $\frac{-45}{9}$ = -5. Example 4: Simplify $\frac{-100}{-10}$ This is another case of dividing a negative number by a negative number, so the result will be positive. Divide the absolute values: 100 ÷ 10 = 10. Therefore, $\frac{-100}{-10}$ = 10. Now, let's try some practice problems. Try solving these on your own, applying the rules we've discussed: Practice Problem 1: $\frac{-18}{-3}$ Practice Problem 2: $\frac{56}{-8}$ Practice Problem 3: $\frac{-72}{12}$ Practice Problem 4: $\frac{-150}{-15}$ Practice Problem 5: $\frac{90}{-10}$ Answers: Practice Problem 1: $\frac{-18}{-3}$ = 6 (negative divided by negative is positive) Practice Problem 2: $\frac{56}{-8}$ = -7 (positive divided by negative is negative) Practice Problem 3: $\frac{-72}{12}$ = -6 (negative divided by positive is negative) Practice Problem 4: $\frac{-150}{-15}$ = 10 (negative divided by negative is positive) Practice Problem 5: $\frac{90}{-10}$ = -9 (positive divided by negative is negative) By working through these examples and practice problems, you are actively engaging with the material and reinforcing your understanding of the rules. This practice will help you develop the skills and confidence needed to tackle more complex mathematical problems involving negative numbers. Remember, the key is to focus on the signs of the numbers first and then perform the division operation. This methodical approach will help you avoid common errors and ensure accurate results. Continue practicing with various problems to further enhance your proficiency and master the concept of dividing negative numbers. The more you practice, the easier it will become, and you’ll be able to apply these rules effortlessly in different mathematical contexts.

When dividing negative numbers, several common mistakes can lead to incorrect answers. Being aware of these pitfalls and understanding how to avoid them is crucial for achieving accuracy. Let's discuss some of these common errors and how to ensure you don't fall into these traps. One of the most frequent mistakes is confusing the rules for multiplying negative numbers with the rules for dividing negative numbers. While it's true that a negative times a negative is a positive, the same rule applies to division: a negative divided by a negative is also a positive. However, a negative times a positive (or vice versa) is negative, and similarly, a negative divided by a positive (or vice versa) is negative. The key is to remember that the rules are consistent for both multiplication and division. Mistake 1: Incorrectly applying the sign rule A common mistake is to forget that a negative divided by a negative yields a positive result. For instance, when solving $\frac{-20}{-4}$, some might mistakenly think the answer is -5. The correct answer is 5, because a negative divided by a negative is positive. To avoid this, always focus on the signs first. Determine whether the result should be positive or negative before performing the division. Mistake 2: Forgetting the negative sign when dividing a negative by a positive or vice versa When dividing a negative number by a positive number, or a positive number by a negative number, it’s essential to remember that the result will be negative. For example, when solving $\frac{-36}{9}$, some may correctly divide 36 by 9 to get 4 but then forget to include the negative sign, resulting in an incorrect answer of 4 instead of -4. Mistake 3: Misunderstanding the relationship between multiplication and division To truly understand division, it’s crucial to recognize its relationship with multiplication. Division is the inverse operation of multiplication. If you're unsure about a division problem, try thinking about the related multiplication problem. For instance, if you're solving $\frac{-48}{-8}$, ask yourself, “What number multiplied by -8 equals -48?” This will help you determine the correct sign and value. Mistake 4: Not simplifying fractions correctly When dealing with fractions that involve negative numbers, it's important to simplify them correctly. This means dividing both the numerator and the denominator by their greatest common divisor while paying attention to the signs. For example, if you have $\frac{-24}{36}$, both numbers are divisible by 12. Dividing both by 12 gives $\frac{-2}{3}$, which is the simplified form. Mistake 5: Errors in basic division skills Sometimes, mistakes in dividing negative numbers stem from errors in basic division skills. For instance, if you're not confident in your multiplication tables, you might make mistakes when dividing larger numbers. Practicing basic division facts can help prevent these errors. Strategies to avoid these mistakes: Always check the signs first: Before performing any division, determine whether the result should be positive or negative based on the signs of the numbers. Think of the related multiplication problem: If you're unsure about a division problem, ask yourself what multiplication problem corresponds to it. Simplify fractions carefully: When simplifying fractions, pay attention to the signs and make sure you're dividing both the numerator and the denominator by the greatest common divisor. Practice basic division facts: Ensure you have a strong foundation in basic division skills to avoid errors when working with larger numbers or more complex problems. By being mindful of these common mistakes and implementing these strategies, you can significantly improve your accuracy when dividing negative numbers. Consistent practice and attention to detail are key to mastering this important mathematical concept.

In conclusion, mastering the division of negative numbers is a crucial skill in mathematics that builds upon fundamental arithmetic principles. Throughout this comprehensive guide, we have explored the essential rules, step-by-step solutions, and common pitfalls to avoid, ensuring a solid understanding of this topic. We began by establishing the basic rules: a negative number divided by a negative number yields a positive result, while a negative number divided by a positive number (or vice versa) results in a negative number. These rules are rooted in the relationship between multiplication and division, where division is the inverse operation of multiplication. Understanding this connection helps to solidify the logic behind the rules. We then delved into the specific case of dividing a negative number by another negative number, illustrating why the result is positive. By considering examples and thinking about the concept of opposites, we can see that dividing two negative numbers essentially “cancels out” the negativity, leading to a positive outcome. Conversely, we explored the rule that dividing a negative number by a positive number (or a positive number by a negative number) results in a negative number. This concept can be visualized using the number line, where division by a negative number indicates movement in the opposite direction, leading to a negative value. Next, we tackled the problem $\frac{-12}{-12}$ step-by-step, demonstrating how to apply the rules and arrive at the correct answer of 1. This involved recognizing that dividing a number by itself always results in 1, and that the two negative signs “cancel” each other out. We also discussed why the other answer choices were incorrect, reinforcing the importance of understanding the rules and avoiding common mistakes. To further enhance understanding, we provided additional examples and practice problems, covering various scenarios and sign combinations. These exercises offer hands-on practice and help to build confidence in applying the rules correctly. The answers to these problems were also provided, allowing for self-assessment and reinforcement of learning. Common mistakes to avoid were also highlighted, such as confusing the rules for multiplication and division, forgetting the negative sign when needed, misunderstanding the relationship between the two operations, not simplifying fractions correctly, and making errors in basic division skills. Strategies to prevent these mistakes included checking signs first, thinking of the related multiplication problem, simplifying fractions carefully, and practicing basic division facts. Mastering the division of negative numbers not only enhances your arithmetic skills but also lays a strong foundation for more advanced mathematical concepts, such as algebra and calculus. The ability to confidently and accurately work with negative numbers is essential for solving a wide range of mathematical problems. Therefore, continued practice and a thorough understanding of the rules are key to success. By following the guidelines and examples provided in this comprehensive guide, you can effectively master the division of negative numbers and confidently apply this knowledge in your future mathematical endeavors.