Dividing -162 By -9 Explained Step-by-Step Solution

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#h1 Divide: $-162 \div (-9)$

In this article, we will delve into the intricacies of dividing negative numbers, specifically focusing on the mathematical expression $-162 \div (-9)$. This is a fundamental concept in mathematics, crucial for understanding more complex arithmetic and algebraic problems. We will explore the rules governing division with negative numbers, step-by-step solutions, and practical applications of this principle. Whether you're a student grappling with basic arithmetic or someone looking to brush up on your math skills, this guide aims to provide a clear and comprehensive understanding of the topic.

Understanding the Basics of Division

Before we tackle the problem at hand, it’s important to have a solid grasp of what division entails. Division is one of the four basic arithmetic operations, the others being addition, subtraction, and multiplication. At its core, division is the process of splitting a quantity into equal parts. It's the inverse operation of multiplication, which means that if you multiply two numbers, dividing the product by one of the original numbers will give you the other number. For example, if $5 \times 3 = 15$, then $15 \div 5 = 3$ and $15 \div 3 = 5$.

In the expression $-162 \div (-9)$, $-162$ is the dividend, which is the number being divided, and $-9$ is the divisor, the number by which the dividend is being divided. The result of the division is called the quotient. Understanding these terms is essential for comprehending the process and the result.

The Role of Negative Numbers

The introduction of negative numbers adds another layer to the concept of division. Negative numbers are numbers less than zero and play a significant role in mathematics and real-world applications. When dealing with negative numbers in division, the rules governing the signs are crucial. A fundamental rule to remember is that dividing a negative number by a negative number yields a positive result. This is because the negative signs essentially cancel each other out. Conversely, dividing a negative number by a positive number, or a positive number by a negative number, results in a negative quotient.

To illustrate this, consider the following examples:

1. $-10 \div (-2) = 5$ (Negative divided by negative equals positive)
2. $10 \div (-2) = -5$ (Positive divided by negative equals negative)
3. $-10 \div 2 = -5$ (Negative divided by positive equals negative)
4. $10 \div 2 = 5$ (Positive divided by positive equals positive)

These rules are derived from the properties of multiplication and division as inverse operations. The same sign rules apply to both operations, which simplifies the process of calculating expressions involving both positive and negative numbers. In our case, $-162 \div (-9)$, we are dividing a negative number by another negative number, which means our result will be positive.

Step-by-Step Solution for $-162 \div (-9)$

Now, let's apply these principles to solve the problem $-162 \div (-9)$. We'll break down the process step by step to ensure clarity.

1. Identify the Dividend and Divisor: In this expression, $-162$ is the dividend, and $-9$ is the divisor.

2. Determine the Sign of the Quotient: Since we are dividing a negative number by a negative number, the quotient will be positive. This is a crucial step as it helps prevent errors in the final answer.

3. Perform the Division: Next, we perform the division without considering the signs initially. We divide the absolute values of the numbers: $162 \div 9$. To do this, we can use long division or recognize that 9 multiplied by 18 equals 162. So, $162 \div 9 = 18$.

4. Apply the Sign: Since we determined in step 2 that the quotient would be positive, we apply the positive sign to our result. Therefore, $-162 \div (-9) = 18$.

So, the answer to the expression $-162 \div (-9)$ is 18. This step-by-step approach is crucial for solving similar problems and building confidence in handling arithmetic operations with negative numbers.

Detailed Explanation of the Solution

To ensure a thorough understanding, let's delve deeper into each step of the solution. The key to solving division problems involving negative numbers is to separate the process into two parts: determining the sign of the result and performing the numerical division. This approach simplifies the problem and reduces the chances of making errors.

Step 1: Identifying the Dividend and Divisor

The first step in any division problem is to correctly identify the dividend and the divisor. The dividend is the number being divided, and the divisor is the number by which the dividend is divided. In the expression $-162 \div (-9)$, the dividend is $-162$, and the divisor is $-9$. This might seem straightforward, but correctly identifying these components is crucial for setting up the problem correctly and proceeding with the calculation.

Step 2: Determining the Sign of the Quotient

The next step is to determine the sign of the quotient. This is where the rules for dividing negative numbers come into play. As we discussed earlier, when dividing a negative number by a negative number, the result is always positive. This rule is a cornerstone of arithmetic operations with negative numbers and is essential for obtaining the correct answer. In our case, since we are dividing $-162$ by $-9$, the quotient will be positive. This means that our final answer will be a positive number.

Understanding this rule not only helps in solving the problem but also provides a foundation for more advanced mathematical concepts. For instance, in algebra, understanding how signs interact in division is crucial for simplifying expressions and solving equations.

Step 3: Performing the Division

Once we have determined the sign of the quotient, the next step is to perform the division itself. This involves dividing the absolute values of the numbers. The absolute value of a number is its distance from zero on the number line, regardless of direction. In other words, it is the non-negative value of the number. The absolute value of $-162$ is 162, and the absolute value of $-9$ is 9. So, we need to calculate $162 \div 9$.

There are several ways to perform this division. One common method is long division, which provides a structured way to divide larger numbers. However, for smaller numbers like 9, it may be possible to perform the division mentally or through simple multiplication knowledge. In this case, we can recall that $9 \times 10 = 90$, and $9 \times 8 = 72$. Adding these together, $90 + 72 = 162$. Thus, $9 \times (10 + 8) = 9 \times 18 = 162$. Therefore, $162 \div 9 = 18$.

Another way to think about this is to break down 162 into multiples of 9. We know that 9 goes into 90 ten times, leaving 72. Then, 9 goes into 72 eight times. Adding these, we get $10 + 8 = 18$. This approach demonstrates the flexibility and interconnectedness of mathematical operations.

Step 4: Applying the Sign

The final step is to apply the sign we determined in Step 2 to the result we obtained in Step 3. We established that the quotient would be positive, so we apply a positive sign to the result of the division, which is 18. Therefore, $-162 \div (-9) = 18$.

This final step is crucial because it ensures that we present the correct answer, considering the rules of signs in division. Neglecting this step can lead to an incorrect answer, even if the numerical division was performed correctly. By following this structured approach, we can confidently solve division problems involving negative numbers and ensure accuracy.

Real-world Applications of Dividing Negative Numbers

Understanding how to divide negative numbers isn't just an academic exercise; it has numerous real-world applications. From finance to physics, the ability to work with negative numbers is essential for solving practical problems. Let's explore some specific examples.

Finance

In finance, negative numbers are frequently used to represent debt, losses, or decreases in value. Division involving negative numbers can help calculate average losses, rates of return, or the impact of financial decisions. For example:

• Average Loss Calculation: If a company has a total loss of $1,620 over 9 months, you can use division to find the average monthly loss. The calculation would be $-1620 \div 9 = -180$, indicating an average loss of $180 per month.

• Investment Analysis: Suppose an investment fund lost $162 in value, and this loss was distributed equally among 9 investors. Dividing the total loss by the number of investors gives the loss per investor: $-162 \div 9 = -18$. Each investor lost $18.

These examples highlight how dividing negative numbers provides insights into financial performance and helps in making informed decisions.

Temperature Measurement

Negative numbers are also commonly used to represent temperatures below zero degrees. Division can be used to calculate average temperature changes or temperature variations over time. For example:

• Average Temperature Drop: If the temperature dropped by 162 degrees Fahrenheit over 9 hours, the average hourly temperature drop can be calculated by dividing the total drop by the number of hours: $-162 \div 9 = -18$ degrees per hour.

This type of calculation is crucial in meteorology and other fields where temperature fluctuations are significant.

Physics

In physics, negative numbers are used to represent quantities like negative charge, direction of force, or potential energy. Division involving negative numbers can be used to calculate quantities such as acceleration or electric potential. For example:

• Calculating Acceleration: If an object's velocity changes by -162 meters per second over 9 seconds, the acceleration can be found by dividing the change in velocity by the time: $-162 \div 9 = -18$ meters per second squared. The negative sign indicates deceleration.

These examples illustrate the broad applicability of dividing negative numbers in various scientific and practical contexts.

Common Mistakes and How to Avoid Them

When dividing negative numbers, there are several common mistakes that students often make. Recognizing these mistakes and understanding how to avoid them is crucial for ensuring accuracy and building confidence in mathematical problem-solving. Let's discuss some of these common errors and strategies to prevent them.

Mistake 1: Incorrectly Applying the Sign Rule

The most common mistake is misapplying the rules for dividing numbers with the same or different signs. Forgetting that a negative divided by a negative yields a positive result, or incorrectly applying this rule, can lead to errors. For instance, a student might incorrectly calculate $-162 \div (-9)$ as $-18$ instead of 18.

How to Avoid: The key is to memorize and consistently apply the sign rules: negative divided by negative is positive, positive divided by positive is positive, and negative divided by positive (or vice versa) is negative. Practicing with various examples and double-checking the sign of the result before calculating the numerical value can also help.

Mistake 2: Dividing the Divisor by the Dividend

Another frequent error is confusing the roles of the dividend and the divisor, leading to the division being performed in the wrong order. This can happen when the problem is presented in a format that isn't explicitly written as a division problem, such as a fraction or a ratio. For example, in $-162 \div (-9)$, a student might mistakenly divide $-9$ by $-162$.

How to Avoid: Always identify the dividend (the number being divided) and the divisor (the number by which you are dividing) before performing the operation. Writing the division problem in a standard format, such as dividend ${\div}$ divisor, can help clarify the order of operations.

Mistake 3: Arithmetic Errors in Division

Arithmetic errors during the division process itself are also common. These can include mistakes in long division, incorrect recall of multiplication facts, or simple calculation errors. These errors can occur regardless of whether negative numbers are involved, but they are just as critical to avoid.

How to Avoid: Take your time and double-check each step of the division. Practice basic multiplication and division facts to improve recall and speed. Using a calculator can be helpful for complex calculations, but it’s important to understand the underlying process.

Mistake 4: Forgetting the Zero Rule

The rule that zero divided by any non-zero number is zero, but dividing by zero is undefined, can sometimes be forgotten or misapplied. This is particularly relevant when dealing with algebraic expressions that might involve division by zero under certain conditions.

How to Avoid: Remember the fundamental rule: $0 \div a = 0$ (where $a$ is any non-zero number), and $a \div 0$ is undefined. Always check for situations where the divisor could be zero, especially in algebraic expressions.

Mistake 5: Not Simplifying Fractions Correctly

When dealing with fractions involving negative numbers, incorrect simplification can lead to errors. This often involves mishandling the signs or not reducing the fraction to its simplest form.

How to Avoid: Ensure you apply the sign rules correctly when simplifying fractions (e.g., a negative fraction can be written with the negative sign in the numerator, denominator, or in front of the entire fraction). Always reduce fractions to their simplest form by dividing both the numerator and the denominator by their greatest common divisor.

By being aware of these common mistakes and employing the strategies outlined above, you can significantly reduce errors and improve your accuracy when dividing negative numbers. Consistent practice and a methodical approach to problem-solving are key to mastering these concepts.

Practice Problems

To reinforce your understanding of dividing negative numbers, let's work through some practice problems. These exercises will help you apply the concepts we've discussed and build your confidence in solving similar problems.

1. Calculate $-225 \div (-15)$.
2. Evaluate $336 \div (-14)$.
3. What is $-504 \div 21$?
4. Solve for $x$ in the equation $-6x = -198$.
5. A company's losses totaled $2,520 over 12 months. What was the average monthly loss?

Solutions and Explanations

1. Calculate $-225 \div (-15)$

   • Step 1: Identify the Dividend and Divisor: The dividend is $-225$, and the divisor is $-15$.
   • Step 2: Determine the Sign of the Quotient: A negative divided by a negative is positive.
   • Step 3: Perform the Division: $225 \div 15 = 15$ (since $15 \times 15 = 225$)
   • Step 4: Apply the Sign: The quotient is positive, so the answer is 15.

   Solution: $-225 \div (-15) = 15$

2. Evaluate $336 \div (-14)$

   • Step 1: Identify the Dividend and Divisor: The dividend is 336, and the divisor is $-14$.
   • Step 2: Determine the Sign of the Quotient: A positive divided by a negative is negative.
   • Step 3: Perform the Division: $336 \div 14 = 24$ (using long division or knowledge of multiples of 14)
   • Step 4: Apply the Sign: The quotient is negative, so the answer is $-24$.

   Solution: $336 \div (-14) = -24$

3. What is $-504 \div 21$?

   • Step 1: Identify the Dividend and Divisor: The dividend is $-504$, and the divisor is 21.
   • Step 2: Determine the Sign of the Quotient: A negative divided by a positive is negative.
   • Step 3: Perform the Division: $504 \div 21 = 24$ (using long division or knowledge of multiples of 21)
   • Step 4: Apply the Sign: The quotient is negative, so the answer is $-24$.

   Solution: $-504 \div 21 = -24$

4. Solve for $x$ in the equation $-6x = -198$

   • Step 1: Isolate $x$: Divide both sides of the equation by $-6$.
   • Step 2: Perform the Division: $x = -198 \div (-6)$
   • Step 3: Determine the Sign of the Quotient: A negative divided by a negative is positive.
   • Step 4: Perform the Numerical Division: $198 \div 6 = 33$ (since $6 \times 30 = 180$ and $6 \times 3 = 18$, so $180 + 18 = 198$)
   • Step 5: Apply the Sign: The quotient is positive, so $x = 33$.

   Solution: $x = 33$

5. A company's losses totaled $2,520 over 12 months. What was the average monthly loss?

   • Step 1: Set up the Division: Divide the total loss by the number of months: $-2520 \div 12$
   • Step 2: Determine the Sign of the Quotient: A negative divided by a positive is negative.
   • Step 3: Perform the Division: $2520 \div 12 = 210$ (using long division or knowledge of multiples of 12)
   • Step 4: Apply the Sign: The quotient is negative, so the average monthly loss is $-210$.

   Solution: The average monthly loss was $210.

By working through these practice problems and understanding the solutions, you can enhance your skills in dividing negative numbers and prepare for more complex mathematical challenges.

Conclusion

In conclusion, understanding how to divide negative numbers is a crucial skill in mathematics with practical applications across various fields. By following a step-by-step approach—identifying the dividend and divisor, determining the sign of the quotient, performing the division, and applying the sign—you can confidently solve these problems. Remember to pay close attention to the sign rules and practice regularly to reinforce your knowledge. Mastering these concepts will not only improve your math skills but also enhance your ability to tackle real-world problems involving negative numbers. Whether you're calculating financial losses, analyzing temperature changes, or solving physics equations, the principles of dividing negative numbers are fundamental to success.