Calculating 19 × 6 + 19 × 4 Using The Distributive Law

Introduction to the Distributive Law

In the realm of mathematics, the distributive law stands as a fundamental principle that simplifies complex arithmetic expressions. It's a cornerstone concept for anyone looking to enhance their mathematical skills, whether they're students tackling algebraic equations or professionals applying mathematical principles in their fields. The distributive law essentially allows us to break down multiplication problems involving sums or differences into simpler, more manageable parts. It posits that multiplying a single number by the sum (or difference) of two numbers is the same as multiplying the single number by each of the two numbers individually and then adding (or subtracting) the results. This seemingly simple concept is incredibly powerful and versatile, enabling us to perform calculations mentally, simplify algebraic expressions, and solve equations more efficiently.

The essence of the distributive law can be encapsulated in the formula: a × (b + c) = (a × b) + (a × c). This formula illustrates that when we multiply ‘a’ by the sum of ‘b’ and ‘c’, it's equivalent to multiplying ‘a’ by ‘b’ and ‘a’ by ‘c’ separately, and then adding the products. This principle holds true for subtraction as well, where a × (b – c) = (a × b) – (a × c). Understanding this fundamental property is crucial for mastering various mathematical operations, particularly when dealing with larger numbers or algebraic variables. For instance, consider the expression 7 × (10 + 2). Applying the distributive law, we can break this down into (7 × 10) + (7 × 2), which equals 70 + 14, resulting in 84. This breakdown not only simplifies the calculation but also provides a clear pathway to understanding the underlying mathematical principle. The distributive law is not just a computational tool; it's a lens through which we can view the structure of mathematical expressions, making them more accessible and less daunting. By mastering this law, individuals can approach complex mathematical problems with greater confidence and clarity.

Problem Statement: 19 × 6 + 19 × 4

To effectively demonstrate the application of the distributive law, let's consider the specific mathematical problem at hand: 19 × 6 + 19 × 4. This expression presents a perfect opportunity to showcase how the distributive law can streamline calculations and make arithmetic operations more manageable. At first glance, one might approach this problem by performing the two multiplications separately (19 × 6 and 19 × 4) and then adding the results. While this method is certainly valid, it can be time-consuming and may involve larger numbers that are more prone to calculation errors. The distributive law offers a more elegant and efficient solution by allowing us to factor out the common term, which in this case is 19. By recognizing that 19 is being multiplied by both 6 and 4, we can rewrite the expression in a way that simplifies the arithmetic. This process involves identifying the common factor, which is the number that is being multiplied across multiple terms, and then using the distributive law in reverse to consolidate the expression. This technique not only reduces the computational burden but also enhances our understanding of the underlying mathematical relationships within the problem. By applying the distributive law, we can transform the original expression into a more concise and easier-to-solve format, highlighting the law's practical utility in simplifying complex calculations. This approach not only saves time but also minimizes the risk of errors, making it a valuable tool in any mathematical toolkit. The problem 19 × 6 + 19 × 4 thus serves as an excellent example to illustrate the power and efficiency of the distributive law in action.

Applying the Distributive Law: Step-by-Step

Now, let's delve into the step-by-step application of the distributive law to solve the problem 19 × 6 + 19 × 4. This process will not only provide a clear solution but also reinforce the understanding of how the distributive law works in practice. The first crucial step is to identify the common factor within the expression. In this case, the number 19 appears in both terms, making it the common factor. Recognizing this shared element is the key to unlocking the simplification potential of the distributive law. Once the common factor is identified, the next step involves factoring it out. This is essentially the reverse of the distributive property, where we rewrite the expression by placing the common factor outside a set of parentheses. Inside the parentheses, we include the remaining terms, which in this case are 6 and 4, connected by the addition operation that was present in the original expression. This transformation results in the expression 19 × (6 + 4). By factoring out the 19, we've effectively consolidated the multiplication into a single operation, which will significantly simplify the calculation process. The expression 19 × (6 + 4) is now in a form that is much easier to manage, as we can first perform the addition within the parentheses and then multiply the result by 19. This step-by-step approach not only makes the problem more approachable but also highlights the elegance and efficiency of the distributive law in simplifying arithmetic expressions. By understanding and applying these steps, anyone can confidently tackle similar problems and appreciate the power of this fundamental mathematical principle.

Step 1: Identify the Common Factor

The first crucial step in applying the distributive law to the expression 19 × 6 + 19 × 4 is to identify the common factor. This is the number that is being multiplied in both terms of the expression. In this particular case, it's quite evident that the number 19 appears in both 19 × 6 and 19 × 4. Recognizing this commonality is the cornerstone of simplifying the expression using the distributive law. The common factor acts as a bridge connecting the two terms, allowing us to rewrite the expression in a more manageable form. Without identifying this common element, applying the distributive law would not be possible. It's like finding the keystone in an arch; once you've located it, the structure begins to make sense and the rest of the process falls into place. Therefore, the ability to quickly and accurately identify common factors is a crucial skill in mathematical problem-solving. It not only simplifies calculations but also deepens our understanding of the underlying mathematical relationships. In this instance, pinpointing 19 as the common factor sets the stage for the next steps in applying the distributive law, paving the way for a more efficient and elegant solution. This initial identification is not just a procedural step; it's a fundamental insight that transforms the problem from a potentially cumbersome calculation into a straightforward application of a mathematical principle. By focusing on this critical first step, we lay a solid foundation for the subsequent steps in the simplification process.

Step 2: Factor out the Common Factor

Having successfully identified the common factor as 19 in the expression 19 × 6 + 19 × 4, the next pivotal step is to factor out this common factor. Factoring out is essentially the reverse process of distribution, where we rewrite the expression by extracting the common element and placing it outside a set of parentheses. This maneuver is crucial in simplifying the expression and making it easier to calculate. When we factor out 19, we are essentially saying that we can rewrite the expression as 19 multiplied by the sum of the remaining factors. This transformation is based on the distributive law, which allows us to distribute a factor across a sum or difference. In this case, we are applying the law in reverse to consolidate the multiplication into a single operation. The process of factoring out 19 involves rewriting the expression as 19 × (6 + 4). This new form clearly shows that 19 is being multiplied by the sum of 6 and 4. The parentheses act as a grouping symbol, indicating that the addition within should be performed before the multiplication. This step not only simplifies the expression visually but also reduces the computational complexity. By factoring out the common factor, we have transformed the original expression, which involved two separate multiplications and an addition, into a single multiplication operation. This significantly reduces the mental effort required to solve the problem and minimizes the risk of errors. Factoring out the common factor is a powerful technique in mathematics, allowing us to streamline calculations and gain a deeper understanding of the relationships between numbers and operations. It's a skill that is widely applicable in various mathematical contexts, from basic arithmetic to advanced algebra.

Step 3: Simplify Inside the Parentheses

With the expression now in the form 19 × (6 + 4), the next logical step is to simplify inside the parentheses. This involves performing the addition operation within the parentheses, which is a fundamental aspect of following the order of operations in mathematics. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. Parentheses take precedence, meaning that any operations contained within them should be carried out before any operations outside of them. In this case, we have the addition of 6 and 4 within the parentheses. Performing this addition is a straightforward process: 6 + 4 equals 10. This simplification transforms the expression into 19 × 10, which is a much simpler multiplication problem than the original expression. By simplifying inside the parentheses, we have reduced the complexity of the calculation and made it more manageable. This step is not just about performing the arithmetic; it's about strategically reducing the expression to its simplest form before proceeding with further calculations. It demonstrates the importance of following the order of operations and utilizing parentheses to group operations in a way that facilitates simplification. This strategic approach is a key aspect of mathematical problem-solving, allowing us to tackle complex problems by breaking them down into smaller, more manageable steps. By simplifying inside the parentheses, we have paved the way for the final step in solving the problem, which is a simple multiplication operation.

Step 4: Perform the Final Multiplication

Having simplified the expression to 19 × 10, the final step in solving the problem is to perform the final multiplication. This is a straightforward arithmetic operation that will yield the solution to the original expression, 19 × 6 + 19 × 4. Multiplying 19 by 10 is a relatively simple task, especially since multiplying by 10 involves simply adding a zero to the end of the number. Therefore, 19 × 10 equals 190. This result represents the final answer to the problem, demonstrating the effectiveness of applying the distributive law to simplify complex calculations. By breaking down the original expression into smaller, more manageable steps, we were able to arrive at the solution with ease and efficiency. This final multiplication step is not just the culmination of the calculation process; it's a validation of the entire approach. It demonstrates that by strategically applying mathematical principles, such as the distributive law, we can transform complex problems into simple ones. The result of 190 is not just a number; it's a testament to the power of mathematical reasoning and problem-solving techniques. This step-by-step process, from identifying the common factor to performing the final multiplication, highlights the elegance and efficiency of the distributive law. It's a valuable tool in any mathematical toolkit, allowing us to tackle a wide range of problems with confidence and clarity. The final multiplication step is the payoff for all the previous efforts, providing a clear and concise solution to the problem.

Solution: 19 × 6 + 19 × 4 = 190

In conclusion, by applying the distributive law, we have efficiently calculated the value of the expression 19 × 6 + 19 × 4. The step-by-step process involved identifying the common factor (19), factoring it out to rewrite the expression as 19 × (6 + 4), simplifying inside the parentheses to obtain 19 × 10, and finally, performing the multiplication to arrive at the solution: 190. This result demonstrates the power and utility of the distributive law in simplifying arithmetic expressions. The distributive law not only makes calculations easier but also provides a deeper understanding of the relationships between numbers and operations. By mastering this fundamental principle, individuals can approach complex mathematical problems with greater confidence and clarity. The solution of 190 is not just a numerical answer; it's a testament to the effectiveness of strategic problem-solving in mathematics. The process of applying the distributive law showcases how breaking down a problem into smaller, more manageable steps can lead to a simpler and more elegant solution. This approach is applicable not only to arithmetic problems but also to more advanced mathematical concepts, such as algebra and calculus. The distributive law is a cornerstone of mathematical reasoning, and its mastery is essential for anyone seeking to excel in mathematics. The final answer of 19 × 6 + 19 × 4 = 190 serves as a clear illustration of the law's practical application and its value in simplifying calculations. This example reinforces the importance of understanding and utilizing mathematical principles to solve problems efficiently and effectively.

Real-World Applications of the Distributive Law

The distributive law, while a fundamental concept in mathematics, extends far beyond the classroom and finds numerous real-world applications in various fields. Understanding how to apply this law can significantly enhance problem-solving skills in everyday life and professional settings. One common application is in calculating costs and discounts. For instance, imagine you're buying 5 items that each cost $9.99. Instead of multiplying 5 by 9.99 directly, you can use the distributive law to simplify the calculation. You can think of $9.99 as $10 - $0.01, so the total cost can be calculated as 5 × ($10 - $0.01), which is equal to (5 × $10) - (5 × $0.01), resulting in $50 - $0.05 = $49.95. This method not only makes the calculation easier but also reduces the chances of making errors. In business and finance, the distributive law is frequently used in budgeting and financial analysis. For example, if a company needs to calculate the total cost of producing 200 units of a product, where each unit requires 3 hours of labor at $15 per hour and $5 in materials, the distributive law can be applied. The total cost can be calculated as 200 × (3 × $15 + $5), which simplifies to 200 × ($45 + $5) = 200 × $50 = $10,000. This approach helps in breaking down complex calculations into simpler steps, making it easier to manage and analyze financial data. In engineering and design, the distributive law is crucial for scaling up or down designs and calculations. For example, if an architect is designing a building and needs to calculate the total area of multiple rooms, the distributive law can be used to simplify the process. If there are 10 rooms, each with an area of 250 square feet, the total area can be calculated as 10 × 250 = 2500 square feet. This simple calculation, based on the principles of distribution, is essential for planning and executing construction projects effectively. These examples highlight just a few of the many ways the distributive law is used in the real world. Its versatility and applicability make it an invaluable tool for problem-solving in various contexts, demonstrating that mathematics is not just an abstract subject but a practical and essential skill for everyday life.

Conclusion: The Power of the Distributive Law

In conclusion, the distributive law stands as a powerful and versatile tool in the realm of mathematics, offering a streamlined approach to solving complex arithmetic problems. As we've demonstrated with the example 19 × 6 + 19 × 4, this law allows us to simplify expressions by factoring out common terms, thereby reducing the computational burden and minimizing the risk of errors. The ability to recognize and apply the distributive law is not just a mathematical skill; it's a strategic problem-solving technique that can be applied in various real-world scenarios. From calculating costs and discounts to managing budgets and designing structures, the distributive law provides a framework for breaking down complex problems into smaller, more manageable steps. Its applications extend beyond the classroom, making it an essential tool for professionals in fields such as finance, engineering, and architecture. The step-by-step process of applying the distributive law, as outlined in this discussion, highlights the importance of understanding fundamental mathematical principles. By identifying common factors, factoring them out, simplifying expressions within parentheses, and performing final calculations, we can efficiently arrive at solutions with clarity and confidence. The solution to the problem 19 × 6 + 19 × 4, which is 190, serves as a clear testament to the power and elegance of the distributive law. This example not only reinforces the law's practical utility but also underscores the value of strategic thinking in mathematics. Mastering the distributive law is a crucial step in developing mathematical proficiency, enabling individuals to tackle a wide range of problems with greater ease and effectiveness. It's a fundamental concept that empowers us to see the underlying structure of mathematical expressions, making them more accessible and less daunting. In essence, the distributive law is a key that unlocks the potential for simplified problem-solving, both in academic pursuits and in everyday life.