Adante's Evaluation Of 3 1/3 × 5 1/4 A Step-by-Step Analysis
#h1 Adante's Approach to Evaluating 3 1/3 × 5 1/4
This article delves into Adante's method for evaluating the expression 3 1/3 × 5 1/4, providing a comprehensive, step-by-step analysis of his approach. We'll break down each stage of his calculation, ensuring a clear understanding of the mathematical principles involved. This exploration will not only illuminate Adante's technique but also serve as a valuable resource for anyone seeking to master mixed number multiplication.
Understanding the Initial Problem: 3 1/3 × 5 1/4
Before we dissect Adante's solution, let's first understand the problem he's tackling. The expression 3 1/3 × 5 1/4 involves the multiplication of two mixed numbers. A mixed number, as the name suggests, is a combination of a whole number and a fraction. In this case, we have 3 1/3 (three and one-third) multiplied by 5 1/4 (five and one-quarter). To solve this, Adante employs a specific strategy that leverages the distributive property of multiplication. The goal here is to transform these mixed numbers into a more manageable form for calculation. One common approach is to convert mixed numbers into improper fractions, which simplifies the multiplication process. However, Adante takes a slightly different route, which we will explore in detail. Understanding the properties of numbers and operations, especially the distributive property, is crucial for solving this type of problem efficiently. This property allows us to break down complex multiplications into simpler, more manageable steps. For example, multiplying a mixed number by another number can be simplified by multiplying the whole and fractional parts separately and then adding the results. This strategy is precisely what Adante uses in his solution, making his method an excellent example of applying fundamental mathematical principles to solve a seemingly complex problem. Furthermore, recognizing that mixed numbers represent a combination of whole and fractional parts is essential for accurate calculations. Failing to account for both components can lead to significant errors. By carefully considering each part and applying the appropriate operations, Adante ensures a precise and reliable solution. This methodical approach is a hallmark of effective mathematical problem-solving.
Step 1: Distributing 3 1/3 across (5 + 1/4)
Adante's first step involves applying the distributive property. He recognizes that 5 1/4 can be expressed as the sum of its whole number part (5) and its fractional part (1/4), essentially rewriting the expression as 3 1/3 × (5 + 1/4). This is a crucial step because it sets the stage for applying the distributive property, which states that a(b + c) = ab + ac. By breaking down 5 1/4 into its components, Adante makes it easier to multiply it by 3 1/3. The distributive property is a fundamental concept in mathematics, particularly in algebra, and understanding its application is key to solving a wide range of problems. In this case, it allows Adante to avoid directly multiplying mixed numbers, which can be cumbersome. Instead, he can multiply 3 1/3 by 5 and 3 1/3 by 1/4 separately, and then add the results. This strategy simplifies the calculation and reduces the chance of errors. Recognizing and applying the distributive property is a powerful tool in mathematical problem-solving, enabling us to break down complex expressions into simpler components. It's also a great example of how understanding mathematical principles can lead to more efficient and accurate solutions. This initial step by Adante demonstrates a strong grasp of fundamental mathematical concepts and sets the foundation for a clear and logical solution to the problem. By carefully applying the distributive property, he transforms the original expression into a form that is much easier to work with.
Step 2: Applying the Distributive Property: (3 1/3 × 5) + (3 1/3 × 1/4)
Following the distributive property, Adante expands the expression to (3 1/3 × 5) + (3 1/3 × 1/4). This step is a direct application of the distributive property, where he multiplies 3 1/3 by both 5 and 1/4. Now, he has two separate multiplication problems, each of which is simpler than the original. This transformation is a key step in making the overall calculation more manageable. By separating the multiplication into two parts, Adante can focus on each part individually, reducing the risk of errors. This approach also highlights the power of the distributive property in simplifying complex calculations. It allows us to break down a single multiplication involving a mixed number and a sum into two separate multiplications, each of which can be solved more easily. The expression (3 1/3 × 5) + (3 1/3 × 1/4) clearly shows how the original problem has been transformed into a sum of two products. This form is much easier to work with, as each product can be calculated independently. Understanding the order of operations is also crucial at this stage. The multiplication operations within the parentheses must be performed before the addition. By carefully following the correct order of operations, Adante ensures that the solution is accurate and logically sound. This step is a clear demonstration of how mathematical principles can be applied strategically to simplify complex problems.
Further Steps Required to Complete the Evaluation
While Adante has made significant progress, the evaluation is not yet complete. The expression (3 1/3 × 5) + (3 1/3 × 1/4) still needs to be simplified further. To proceed, Adante needs to address each multiplication separately. This involves converting the mixed number 3 1/3 into an improper fraction. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. Converting mixed numbers to improper fractions is a common technique in multiplication and division, as it simplifies these operations. To convert 3 1/3 to an improper fraction, we multiply the whole number (3) by the denominator (3) and add the numerator (1), placing the result over the original denominator. This gives us (3 * 3 + 1) / 3 = 10/3. Once the mixed number is converted to an improper fraction, the multiplication becomes more straightforward. The expression then transforms into (10/3 × 5) + (10/3 × 1/4). Now, we have two multiplications involving fractions. To multiply fractions, we multiply the numerators and the denominators separately. For the first multiplication, 10/3 × 5, we can rewrite 5 as 5/1, giving us (10/3) × (5/1) = 50/3. For the second multiplication, 10/3 × 1/4, we multiply the numerators (10 * 1 = 10) and the denominators (3 * 4 = 12), resulting in 10/12. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This simplifies 10/12 to 5/6. Now the expression becomes 50/3 + 5/6. To add these fractions, we need a common denominator. The least common multiple of 3 and 6 is 6. So, we convert 50/3 to an equivalent fraction with a denominator of 6 by multiplying both the numerator and the denominator by 2, resulting in 100/6. The expression now is 100/6 + 5/6. Adding the numerators gives us 105/6. Finally, we can simplify this improper fraction and convert it back to a mixed number. Dividing 105 by 6 gives us 17 with a remainder of 3, so the simplified mixed number is 17 3/6. The fraction 3/6 can be further simplified to 1/2, so the final answer is 17 1/2.
Key Takeaways from Adante's Method
Adante's approach highlights several important mathematical concepts. First and foremost, it demonstrates the power of the distributive property in simplifying complex multiplication problems. By breaking down the mixed number 5 1/4 into its components (5 + 1/4), Adante was able to apply the distributive property effectively, transforming the original problem into a sum of two simpler products. This strategy is a valuable tool in mathematical problem-solving, allowing us to tackle seemingly daunting calculations with greater ease. Secondly, Adante's method underscores the importance of understanding the structure of mixed numbers. Recognizing that a mixed number is a combination of a whole number and a fraction is crucial for applying the distributive property correctly. It also lays the groundwork for converting mixed numbers to improper fractions, a technique that is often necessary for performing multiplication and division operations. Furthermore, Adante's approach emphasizes the need for a step-by-step, methodical approach to problem-solving. Each step in his solution is clearly defined and logically connected to the previous step. This systematic approach not only reduces the risk of errors but also makes the solution process easier to follow and understand. By breaking down the problem into smaller, more manageable steps, Adante makes the solution process less intimidating and more accessible. Finally, Adante's method serves as a reminder that there are often multiple ways to solve a mathematical problem. While converting mixed numbers to improper fractions is a common strategy for multiplication, Adante's approach offers an alternative that can be particularly useful in certain situations. By exploring different methods, we can deepen our understanding of mathematical concepts and develop a more flexible and adaptable approach to problem-solving. Overall, Adante's method provides valuable insights into effective mathematical problem-solving strategies. It demonstrates the importance of understanding fundamental principles, applying them strategically, and adopting a methodical approach to complex calculations.
In conclusion, Adante's initial steps provide a solid foundation for evaluating the expression. By applying the distributive property, he simplifies the problem into manageable parts. Further calculations, including converting mixed numbers to improper fractions, performing the multiplications, finding a common denominator, and simplifying the result, are necessary to arrive at the final answer. This detailed breakdown showcases the importance of understanding and applying fundamental mathematical principles to solve complex problems.
