Step By Step Solution $2 \times(-21) \times 7$

When tackling mathematical problems involving the multiplication of integers, a systematic approach ensures accuracy and clarity. This article delves into the step-by-step solution of the expression $2 \times (-21) \times 7$, providing a comprehensive understanding of the underlying principles. Whether you're a student grappling with integer multiplication or simply seeking to refresh your mathematical skills, this guide offers valuable insights and practical techniques to master this fundamental concept.

Breaking Down the Problem

To solve the expression $2 \times (-21) \times 7$, we need to understand the rules of integer multiplication and apply them sequentially. Integer multiplication involves multiplying numbers that can be positive, negative, or zero. The key rule to remember is that the product of two numbers with the same sign (both positive or both negative) is positive, while the product of two numbers with different signs (one positive and one negative) is negative. Let's break down the calculation step by step.

Step 1: Multiplying the First Two Numbers

The first step is to multiply the first two numbers in the expression: $2$ and $-21$. Here, we have a positive number ($2$) multiplied by a negative number ($-21$). According to the rules of integer multiplication, the result will be negative. The product of the absolute values of the numbers is $2 \times 21 = 42$. Therefore, the result of the first step is $-42$. This initial step is crucial as it sets the stage for the subsequent calculations. A clear understanding of this step helps in avoiding common mistakes and ensures a solid foundation for the rest of the problem. The negative sign is a critical component, and handling it correctly is paramount for arriving at the correct final answer.

Step 2: Multiplying the Result by the Third Number

Now that we have the result of the first multiplication, $-42$, we need to multiply it by the third number in the expression, which is $7$. Again, we are multiplying a negative number ($-42$) by a positive number ($7$). The rule remains the same: the product of two numbers with different signs is negative. The product of the absolute values of the numbers is $42 \times 7$. To calculate this, we can break it down further: $40 \times 7 = 280$ and $2 \times 7 = 14$. Adding these together, we get $280 + 14 = 294$. Since we are multiplying a negative number by a positive number, the result is negative, so the final result of this step is $-294$. This step reinforces the importance of maintaining accuracy in calculations and adhering to the rules of sign conventions in multiplication.

Step 3: The Final Answer

After completing the two multiplication steps, we arrive at the final answer. Multiplying $2 \times (-21) \times 7$ gives us $-294$. This result encapsulates the entire process, from understanding the initial expression to applying the rules of integer multiplication. The journey to the final answer underscores the necessity of methodical calculation and a keen eye for detail. The negative sign, determined by the interaction of positive and negative numbers in the multiplication process, is a crucial component of the final result. The correct handling of this sign is pivotal in achieving the right solution.

Why the Other Options Are Incorrect

Understanding why the correct answer is $-294$ also involves recognizing why the other options are incorrect. Let's examine each option:

• B) -273: This answer is incorrect because it likely results from a miscalculation in the multiplication process. It could be due to an arithmetic error or a misunderstanding of how to combine the numbers. The correct multiplication should yield $-294$, not $-273$. A thorough review of the steps would highlight the discrepancy and reinforce the need for precise calculations.

• C) 294: This answer has the correct numerical value but the wrong sign. It represents the product of the absolute values of the numbers but fails to account for the negative sign resulting from multiplying numbers with different signs. This error emphasizes the critical importance of sign conventions in integer multiplication. The absence of the negative sign transforms the answer from correct to incorrect, underscoring the significance of this detail.

• D) -7: This answer is significantly smaller in magnitude and does not reflect the cumulative effect of multiplying $2$, $-21$, and $7$. It indicates a fundamental misunderstanding of the multiplication process or a confusion of the operations involved. This option serves as a stark contrast to the correct answer, highlighting the magnitude of the error and the necessity of a comprehensive approach to solving the problem.

By analyzing these incorrect options, we gain a deeper appreciation for the nuances of integer multiplication and the importance of accuracy in each step. The process of elimination not only reinforces the correct solution but also enhances our understanding of the potential pitfalls in mathematical problem-solving.

Key Concepts in Integer Multiplication

To master integer multiplication, several key concepts must be understood. These concepts not only facilitate accurate calculations but also provide a solid foundation for more advanced mathematical topics. Let's explore these essential principles:

Sign Conventions

The cornerstone of integer multiplication is understanding sign conventions. As mentioned earlier, the product of two numbers with the same sign is positive, while the product of two numbers with different signs is negative. This rule is fundamental and must be applied consistently throughout the calculation. Ignoring or misapplying this rule is a common source of errors in integer multiplication. The sign conventions serve as a navigational guide, ensuring that the answer reflects the correct direction and magnitude.

Order of Operations

In expressions involving multiple operations, the order of operations is crucial. While this problem primarily involves multiplication, it's important to remember the general order (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). In this case, we perform the multiplications from left to right. Adhering to the order of operations prevents ambiguity and ensures that the expression is evaluated correctly. This systematic approach is applicable across various mathematical contexts, making it a valuable skill to develop.

Associative Property

The associative property of multiplication states that the grouping of numbers does not affect the final product. In other words, $(a \times b) \times c = a \times (b \times c)$. This property allows us to multiply the numbers in any order, as long as we maintain the correct signs. While we solved $2 \times (-21) \times 7$ by multiplying from left to right, we could also multiply $-21$ and $7$ first, then multiply the result by $2$. Understanding and applying the associative property provides flexibility in problem-solving and can sometimes simplify calculations.

Importance of Precision

In mathematical calculations, precision is paramount. Even a small error in multiplication or sign handling can lead to a significantly different final answer. Double-checking each step and ensuring accuracy in calculations is essential. This practice not only guarantees the correct solution but also fosters a disciplined approach to problem-solving. Precision extends beyond mere numerical accuracy; it encompasses a thorough understanding of the concepts and their application.

Real-World Applications of Integer Multiplication

Integer multiplication is not just a theoretical concept; it has numerous real-world applications across various fields. Understanding these applications can enhance our appreciation for the practical significance of integer multiplication. Let's explore some examples:

Finance

In finance, integer multiplication is used to calculate profits, losses, and compound interest. For instance, if an investor loses $250 per month for 3 months, the total loss can be calculated as $-250 \times 3 = -750$. The negative sign indicates a loss. This simple calculation highlights the direct application of integer multiplication in financial analysis. The ability to accurately calculate financial outcomes is crucial for informed decision-making and effective management of resources.

Temperature Changes

Integer multiplication is useful in scenarios involving temperature changes. If the temperature drops by $3$ degrees Celsius per hour for $5$ hours, the total temperature change is $-3 \times 5 = -15$ degrees Celsius. The negative sign indicates a decrease in temperature. This application underscores the relevance of integer multiplication in understanding and quantifying real-world phenomena. Predicting and interpreting temperature variations are essential in various fields, from meteorology to engineering.

Distance and Displacement

In physics, integer multiplication can be used to calculate distance and displacement. If a car travels at a speed of $-60$ miles per hour (indicating a westward direction) for $2$ hours, the total displacement is $-60 \times 2 = -120$ miles. The negative sign indicates the direction of displacement. This application illustrates the role of integer multiplication in representing and analyzing motion. The concept of displacement is fundamental in physics and engineering, and integer multiplication provides a valuable tool for its calculation.

Inventory Management

In business, integer multiplication is used in inventory management. If a store sells $15$ items per day for $7$ days, the total number of items sold is $15 \times 7 = 105$. This calculation helps businesses track sales and manage their inventory effectively. Accurate inventory management is crucial for optimizing costs and ensuring customer satisfaction. Integer multiplication provides a simple yet powerful method for analyzing sales data and making informed decisions.

Tips and Tricks for Mastering Integer Multiplication

Mastering integer multiplication requires practice and a strategic approach. Here are some tips and tricks to help you excel in this area:

Practice Regularly

Consistent practice is key to mastering any mathematical concept. Solve a variety of problems involving integer multiplication to reinforce your understanding and improve your speed and accuracy. Regular practice also helps in identifying areas of weakness and addressing them effectively. The more you practice, the more confident you will become in your ability to tackle complex problems.

Use Visual Aids

Visual aids, such as number lines, can be helpful in understanding integer multiplication, especially when dealing with negative numbers. Representing the multiplication process visually can make the concept more intuitive and easier to grasp. Visual aids are particularly beneficial for visual learners and can provide a concrete representation of abstract mathematical concepts.

Break Down Complex Problems

When faced with complex problems involving multiple operations, break them down into smaller, more manageable steps. This approach simplifies the problem and reduces the likelihood of errors. Each step can be tackled individually, ensuring accuracy and clarity. This methodical approach is applicable across various problem-solving contexts and fosters a structured and efficient approach to complex tasks.

Double-Check Your Work

Always double-check your work to ensure accuracy. Review each step of the calculation and verify that you have applied the rules of integer multiplication correctly. This practice can prevent careless errors and significantly improve your overall performance. Double-checking is a hallmark of meticulous work and reflects a commitment to accuracy and excellence.

Seek Help When Needed

If you encounter difficulties or have questions, don't hesitate to seek help from teachers, tutors, or online resources. Understanding the underlying concepts is crucial for long-term success in mathematics. Seeking clarification when needed is a sign of proactive learning and a commitment to mastering the material. Collaboration and discussion with others can provide valuable insights and perspectives.

Conclusion

In conclusion, solving $2 \times (-21) \times 7$ involves understanding the rules of integer multiplication and applying them systematically. The correct answer is $-294$. By mastering the concepts of sign conventions, order of operations, and the associative property, you can confidently tackle similar problems. Integer multiplication is not only a fundamental mathematical skill but also a valuable tool in various real-world applications. Consistent practice and a strategic approach will pave the way for success in this area and beyond. The journey of mastering integer multiplication is a testament to the power of methodical calculation and the pursuit of accuracy and understanding.