Solve Math Problems Find Values And Products Using Properties
In this comprehensive guide, we will dive deep into solving mathematical expressions and finding products using suitable properties. This article aims to provide clear, step-by-step solutions to the given problems, ensuring a thorough understanding of the concepts involved. We'll tackle expressions that require careful application of the order of operations and the distributive property, as well as explore how to efficiently calculate products using these properties. Whether you're a student looking to improve your math skills or someone who enjoys problem-solving, this guide will offer valuable insights and techniques to enhance your mathematical prowess.
i. (368 × 12) + (18 × 368)
To find the value of the expression (368 × 12) + (18 × 368), we can leverage the distributive property of multiplication over addition. This property states that a(b + c) = ab + ac. In this case, we can factor out the common factor, which is 368. This strategic approach not only simplifies the calculation but also underscores the power of algebraic manipulation in problem-solving.
Step 1: Identify the Common Factor
Observe that 368 appears in both terms of the expression. Recognizing common factors is a crucial step in simplifying mathematical expressions. It allows us to rewrite the expression in a more manageable form, making subsequent calculations easier and less prone to errors. The common factor here is the cornerstone of applying the distributive property effectively.
Step 2: Apply the Distributive Property
By factoring out 368, we rewrite the expression as: 368 × (12 + 18). This transformation is a direct application of the distributive property in reverse. Instead of expanding a product, we are condensing a sum of products into a single product involving a sum. This technique is widely used in algebra and arithmetic to simplify calculations and solve equations.
Step 3: Simplify the Parentheses
Next, we simplify the expression inside the parentheses: 12 + 18 = 30. This step involves basic addition, but it is essential to follow the order of operations (PEMDAS/BODMAS), which prioritizes operations within parentheses. Simplifying the parentheses first ensures that we proceed with the rest of the calculation accurately.
Step 4: Perform the Multiplication
Now, we multiply 368 by 30: 368 × 30 = 11040. This multiplication step gives us the final numerical value of the expression. It’s crucial to perform this step accurately, as it directly determines the solution to the problem. The result, 11040, is the value of the original expression.
Therefore, the value of (368 × 12) + (18 × 368) is 11040. This solution demonstrates the elegance and efficiency of the distributive property in simplifying complex expressions. By recognizing and utilizing common factors, we can transform seemingly complicated calculations into manageable steps.
ii. (79 × 4319) + (4319 × 11)
To determine the value of the expression (79 × 4319) + (4319 × 11), we again employ the distributive property. This approach is particularly effective when dealing with expressions that share a common factor. By identifying and factoring out this common element, we can significantly simplify the calculation process, making it more efficient and less prone to errors. The distributive property is a fundamental concept in algebra and arithmetic, and its skillful application is key to solving a wide range of mathematical problems.
Step 1: Identify the Common Factor
In this expression, the common factor is 4319, which appears in both terms. Recognizing common factors is the first crucial step in applying the distributive property. It allows us to rewrite the expression in a form that is easier to work with, transforming a complex sum of products into a simpler product of a sum.
Step 2: Apply the Distributive Property
Factoring out 4319, we rewrite the expression as: 4319 × (79 + 11). This step is a direct application of the distributive property in reverse, effectively condensing two terms into one. This technique is a powerful tool in mathematical simplification, allowing us to manipulate expressions to make them more manageable.
Step 3: Simplify the Parentheses
Next, we simplify the expression inside the parentheses: 79 + 11 = 90. This addition is a straightforward arithmetic operation, but it is vital for maintaining the accuracy of the calculation. Simplifying within parentheses is a key aspect of following the order of operations, ensuring that we proceed with the rest of the problem correctly.
Step 4: Perform the Multiplication
Now, we multiply 4319 by 90: 4319 × 90 = 388710. This final multiplication step yields the value of the original expression. Accuracy in this step is paramount, as it determines the final answer. The result, 388710, is the solution to the problem.
Therefore, the value of (79 × 4319) + (4319 × 11) is 388710. This solution demonstrates how the distributive property can be used to efficiently solve seemingly complex calculations. By recognizing the common factor and applying the property, we transformed the problem into a series of simpler steps, leading to an accurate result.
In this section, we will explore how to find products efficiently by employing suitable properties of multiplication. These properties, such as the distributive property and the associative property, allow us to simplify complex calculations and arrive at the solution more easily. Understanding and applying these properties is crucial for developing mathematical fluency and problem-solving skills. We will delve into specific examples, demonstrating how these properties can be used to break down multiplication problems into more manageable steps.
i. 205 × 1989
To find the product of 205 and 1989, we can employ the distributive property to simplify the calculation. The distributive property, which states that a × (b + c) = (a × b) + (a × c), allows us to break down one of the numbers into more manageable parts. This approach is particularly useful when dealing with large numbers, as it reduces the complexity of the multiplication process. By strategically applying the distributive property, we can transform a single multiplication problem into a series of simpler multiplications and additions.
Step 1: Break Down One Number
We can break down 205 as 200 + 5. This decomposition allows us to apply the distributive property effectively. By expressing 205 as a sum of two numbers, we can distribute the multiplication of 1989 over these two parts, making the calculation more manageable. This step is a key element in simplifying the problem and paving the way for an easier solution.
Step 2: Apply the Distributive Property
Now, we can rewrite the expression as (200 + 5) × 1989 = (200 × 1989) + (5 × 1989). This transformation is a direct application of the distributive property. We have effectively distributed the multiplication of 1989 over the sum 200 + 5, resulting in two separate multiplication problems that are easier to handle individually.
Step 3: Perform the Multiplications
Next, we perform the two multiplications:
- 200 × 1989 = 397800
- 5 × 1989 = 9945
These multiplications can be done using standard multiplication techniques or with the aid of a calculator. The key is to ensure accuracy in these calculations, as they directly contribute to the final answer. By breaking down the original problem into these smaller multiplications, we have made the overall calculation more manageable and less prone to errors.
Step 4: Add the Results
Finally, we add the results: 397800 + 9945 = 407745. This addition combines the products obtained in the previous step to give us the final answer. It is a crucial step in completing the calculation and arriving at the solution. Accuracy in this final addition is essential to ensure the correctness of the result.
Therefore, the product of 205 and 1989 is 407745. This solution demonstrates the power of the distributive property in simplifying multiplication problems. By breaking down one of the numbers into smaller parts, we were able to transform a complex calculation into a series of simpler steps, leading to an accurate and efficient solution.
ii. 1991 × 1005
To efficiently find the product of 1991 and 1005, we can utilize the distributive property. This property is a powerful tool in mathematics that allows us to break down complex multiplication problems into simpler, more manageable calculations. By strategically applying the distributive property, we can transform a single multiplication into a series of multiplications and additions, making the overall process easier and less prone to errors. This approach is particularly effective when dealing with numbers that are close to multiples of 10, 100, or 1000.
Step 1: Break Down One Number
We can break down 1005 as 1000 + 5. This decomposition is key to applying the distributive property effectively. By expressing 1005 as a sum of two numbers, we can distribute the multiplication of 1991 over these two parts. This step simplifies the problem and sets the stage for an easier calculation.
Step 2: Apply the Distributive Property
Now, we rewrite the expression as (1991 × (1000 + 5) = (1991 × 1000) + (1991 × 5). This transformation is a direct application of the distributive property, which states that a × (b + c) = (a × b) + (a × c). We have effectively distributed the multiplication of 1991 over the sum 1000 + 5, resulting in two separate multiplication problems that are easier to handle individually.
Step 3: Perform the Multiplications
Next, we perform the two multiplications:
- 1991 × 1000 = 1991000
- 1991 × 5 = 9955
These multiplications are relatively straightforward. Multiplying by 1000 simply involves adding three zeros to the end of the number, while multiplying by 5 can be done using standard multiplication techniques or a calculator. Accuracy in these calculations is crucial, as they directly contribute to the final answer.
Step 4: Add the Results
Finally, we add the results: 1991000 + 9955 = 2000955. This addition combines the products obtained in the previous step to give us the final answer. It is the crucial step in completing the calculation and arriving at the solution. Accuracy in this final addition is essential to ensure the correctness of the result.
Therefore, the product of 1991 and 1005 is 2000955. This solution highlights the efficiency of the distributive property in simplifying multiplication problems. By breaking down one of the numbers into smaller parts, we were able to transform a complex calculation into a series of simpler steps, leading to an accurate and efficient solution.
In conclusion, mastering the distributive property and other mathematical properties is essential for simplifying calculations and solving complex problems. By understanding and applying these techniques, we can enhance our mathematical skills and approach problem-solving with greater confidence and efficiency. Whether you're a student or simply someone who enjoys math, the ability to manipulate expressions and apply mathematical properties is a valuable asset.
