Solving And Verifying (4/5) × (3/8) = (12/40) ÷ 2 × (6/20) ÷ 2 A Step-by-Step Guide
Breaking Down the Mathematical Expression
In this article, we will delve into the mathematical expression (4/5) × (3/8) = (12/40) ÷ 2 × (6/20) ÷ 2 to understand the steps involved in solving it. This equation involves fractions, multiplication, and division, providing a comprehensive look at basic arithmetic operations. Understanding such calculations is fundamental in mathematics, and this detailed explanation aims to make the process clear and straightforward. Whether you are a student learning these concepts for the first time or someone looking to refresh your knowledge, this guide will break down each step, ensuring a thorough grasp of the material. The main goal is to clarify how to solve this particular equation and also to reinforce the principles of fraction manipulation. This includes simplifying fractions, performing multiplication and division, and ensuring the equality holds true. So, let’s begin by dissecting each part of the equation, ensuring that we understand the logic behind every step and how it contributes to the final result. This in-depth analysis will not only help in solving this particular problem but also in building a stronger foundation for more complex mathematical problems in the future. Remember, mathematics is a sequential subject, and mastering the basics is crucial for progressing further. Therefore, focusing on these foundational concepts is an investment in your mathematical journey.
Step 1: Multiplying the Fractions (4/5) and (3/8)
The initial part of the equation involves multiplying two fractions: (4/5) × (3/8). To multiply fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers) separately. This is a straightforward process that forms the basis for more complex fraction manipulations. When multiplying fractions, it's essential to ensure that you are correctly multiplying the numerators together and the denominators together. A common mistake is to add the numerators or denominators, which would lead to an incorrect result. So, let’s take a closer look at how this multiplication works. First, we multiply the numerators: 4 multiplied by 3 equals 12. Next, we multiply the denominators: 5 multiplied by 8 equals 40. Therefore, the result of (4/5) × (3/8) is 12/40. This fraction represents the product of the two original fractions and is a crucial intermediate step in solving the overall equation. Understanding this basic rule of fraction multiplication is fundamental for various mathematical applications, including algebra, calculus, and even real-world problem-solving. The resulting fraction, 12/40, can be further simplified, which we will discuss in the next step. Remember, simplifying fractions makes them easier to work with and understand. This step is not just about getting the right answer; it’s about developing a deep understanding of how fractions work and interact with each other. By mastering this step, you are setting a solid foundation for more advanced mathematical concepts.
Step 2: Simplifying the Fraction 12/40
The result from the multiplication step is the fraction 12/40. Simplifying fractions is a critical step in mathematics as it reduces the fraction to its simplest form, making it easier to understand and work with. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator and then divide both by this GCD. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. In this case, we need to find the GCD of 12 and 40. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. The greatest common factor between 12 and 40 is 4. So, to simplify the fraction 12/40, we divide both the numerator and the denominator by 4. Dividing 12 by 4 gives us 3, and dividing 40 by 4 gives us 10. Therefore, the simplified form of 12/40 is 3/10. This simplification process not only makes the fraction easier to handle but also provides a clearer understanding of the fraction's value. A simplified fraction is in its lowest terms, meaning there are no common factors other than 1 between the numerator and denominator. This step is essential in various mathematical operations, such as comparing fractions, adding and subtracting fractions, and solving equations. By simplifying fractions, we can avoid working with larger numbers and reduce the chances of making errors. The process of finding the GCD and simplifying fractions is a valuable skill that is used extensively in mathematics and everyday life.
Step 3: Evaluating (12/40) ÷ 2 and (6/20) ÷ 2
Moving on to the next part of the equation, we have (12/40) ÷ 2 and (6/20) ÷ 2. Dividing a fraction by a whole number is the same as multiplying the fraction by the reciprocal of that number. The reciprocal of a number is simply 1 divided by that number. So, dividing by 2 is the same as multiplying by 1/2. Let’s first consider (12/40) ÷ 2. To perform this division, we multiply the fraction 12/40 by 1/2. This gives us (12/40) × (1/2). Multiplying the numerators, we get 12 × 1 = 12. Multiplying the denominators, we get 40 × 2 = 80. So, (12/40) ÷ 2 equals 12/80. Now, let’s simplify the fraction 12/80. The GCD of 12 and 80 is 4. Dividing both the numerator and the denominator by 4, we get 12 ÷ 4 = 3 and 80 ÷ 4 = 20. Thus, 12/80 simplifies to 3/20. Next, we evaluate (6/20) ÷ 2. We multiply the fraction 6/20 by 1/2. This gives us (6/20) × (1/2). Multiplying the numerators, we get 6 × 1 = 6. Multiplying the denominators, we get 20 × 2 = 40. So, (6/20) ÷ 2 equals 6/40. Simplifying the fraction 6/40, we find the GCD of 6 and 40, which is 2. Dividing both the numerator and the denominator by 2, we get 6 ÷ 2 = 3 and 40 ÷ 2 = 20. Thus, 6/40 simplifies to 3/20. This step is crucial in understanding how division affects fractions and how simplification makes the results more manageable.
Step 4: Final Verification of the Equation
Now that we have simplified both sides of the equation, let's verify if they are equal. The left side of the original equation, (4/5) × (3/8), simplifies to 3/10, as we calculated in the earlier steps. The right side of the equation, (12/40) ÷ 2 × (6/20) ÷ 2, involves two division operations. We found that (12/40) ÷ 2 simplifies to 3/20 and (6/20) ÷ 2 also simplifies to 3/20. Therefore, the right side of the equation can be rewritten as 3/20 × 3/20. Multiplying these two fractions, we multiply the numerators: 3 × 3 = 9. We also multiply the denominators: 20 × 20 = 400. So, the result is 9/400. Comparing the simplified left side (3/10) and the simplified right side (9/400), we need to determine if they are equal. To do this, we can try to find a common denominator or convert both fractions to decimals. Converting 3/10 to a fraction with a denominator of 400, we multiply both the numerator and the denominator by 40: (3 × 40) / (10 × 40) = 120/400. Now, comparing 120/400 and 9/400, it is clear that they are not equal. This indicates that there may have been an error in the original equation or in our calculations. To ensure accuracy, let’s re-examine each step. We know that (4/5) × (3/8) = 12/40, which simplifies to 3/10. We also know that (12/40) ÷ 2 = 3/20 and (6/20) ÷ 2 = 3/20. The error lies in interpreting the right side of the equation as a series of multiplications. It should have been interpreted as separate terms. The correct interpretation is (12/40) ÷ 2 results 3/20 and (6/20) ÷ 2 results 3/20. Therefore, the final calculation should compare 3/10 with 3/20, since the multiplication operation has no presence in the right part of original equation. It is clear that 3/10 is not equal to 3/20. The right interpretation of the equation should be done in such way that mathematical rules of simplification are not compromised. This detailed verification step is essential to catch any mistakes and ensure the correctness of the solution.
Conclusion: Key Takeaways from the Calculation
In conclusion, the initial equation (4/5) × (3/8) = (12/40) ÷ 2 × (6/20) ÷ 2 presented a comprehensive problem involving fractions, multiplication, and division. Through a step-by-step breakdown, we were able to solve each part of the equation and identify a critical error in the interpretation of the right side of the equation. The correct interpretation of the equation is vital for arriving at the correct solution. We began by multiplying (4/5) and (3/8), which resulted in 12/40. This fraction was then simplified to 3/10. Next, we tackled the right side of the equation, dividing (12/40) by 2, which gave us 3/20, and dividing (6/20) by 2, which also gave us 3/20. We initially misinterpreted the right side as a series of multiplications, leading to an incorrect comparison. However, through careful re-evaluation, we recognized that the correct approach was to compare 3/10 with individual terms 3/20. This process highlighted the importance of accurately interpreting mathematical expressions and following the correct order of operations. One of the key takeaways from this exercise is the significance of simplifying fractions. Simplifying fractions not only makes them easier to work with but also helps in understanding their true value. Additionally, this problem underscored the importance of double-checking calculations and interpretations to avoid errors. Mathematical problem-solving is not just about arriving at an answer; it’s about understanding the process and ensuring accuracy in each step. This exercise provides a valuable lesson in mathematical rigor and critical thinking. By mastering these fundamental concepts and techniques, we can build a strong foundation for tackling more complex mathematical problems in the future. Remember, mathematics is a cumulative subject, and a thorough understanding of the basics is essential for success.
