Simplifying The Expression (2/15 + 2/5) + (9/10 × 4/3) A Step-by-Step Guide
This article provides a detailed walkthrough of simplifying the mathematical expression (2/15 + 2/5) + (9/10 × 4/3). We will break down each step, ensuring clarity and understanding for those looking to improve their arithmetic skills. This involves the fundamental principles of fraction addition and multiplication, crucial concepts in mathematics. By understanding these operations, you'll be better equipped to tackle more complex mathematical problems.
Understanding the Order of Operations
Before we dive into the simplification, it's important to understand the order of operations, often remembered by the acronym PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order dictates the sequence in which we perform mathematical operations to arrive at the correct answer. Ignoring this order can lead to incorrect results. Therefore, adhering to PEMDAS/BODMAS is crucial for accurate mathematical calculations. For instance, in our expression, we first address the operations within the parentheses and then proceed with addition.
The core of this mathematical journey lies in understanding the order of operations, often symbolized by the acronym PEMDAS or BODMAS. This guiding principle dictates the precise sequence in which mathematical operations should be executed to achieve an accurate outcome. Standing for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (performed from left to right), and Addition and Subtraction (also performed from left to right), PEMDAS/BODMAS serves as a roadmap for navigating complex expressions. Neglecting this established order can lead to a cascade of errors and an incorrect final answer. Therefore, it's imperative to diligently adhere to PEMDAS/BODMAS as the cornerstone of mathematical problem-solving. In the specific expression we're tackling, this means prioritizing the operations enclosed within the parentheses, setting the stage for subsequent addition. Mastering this order isn't merely a procedural exercise; it's the bedrock of mathematical accuracy and efficiency. Consider the implications of misapplying PEMDAS/BODMAS in more intricate equations – the deviations can quickly compound, leading to significant inaccuracies. By internalizing this principle, we not only solve individual problems correctly but also cultivate a systematic approach to mathematical challenges, fostering a deeper comprehension of numerical relationships. The order of operations isn't a suggestion; it's a fundamental rule that ensures consistency and reliability in mathematical endeavors. Think of it as the grammar of mathematics, providing structure and clarity to our calculations. When applied meticulously, PEMDAS/BODMAS empowers us to navigate the complexities of mathematical expressions with confidence and precision.
Step 1: Simplifying Inside the Parentheses
Our expression has two sets of parentheses. Let's start with the first one: (2/15 + 2/5). To add fractions, they need a common denominator. The least common multiple (LCM) of 15 and 5 is 15. So, we need to convert 2/5 to an equivalent fraction with a denominator of 15. We multiply both the numerator and denominator of 2/5 by 3, resulting in 6/15. Now we can add the fractions: 2/15 + 6/15 = 8/15. Therefore, the first set of parentheses simplifies to 8/15.
Delving into the initial set of parentheses, (2/15 + 2/5), we encounter the foundational principle of fraction addition. A critical prerequisite for adding fractions is the attainment of a common denominator, a shared numerical base upon which the numerators can be combined. In this instance, we identify the least common multiple (LCM) of 15 and 5 as the ideal common denominator, which is 15. To achieve this common denominator, we focus on transforming the fraction 2/5 into an equivalent form with a denominator of 15. This transformation is accomplished by multiplying both the numerator and the denominator of 2/5 by 3, a strategic maneuver that maintains the fraction's inherent value while aligning it with our desired denominator. The result of this multiplication is the equivalent fraction 6/15. Now, with both fractions sharing a common denominator, the addition process becomes straightforward: 2/15 + 6/15. We simply add the numerators, keeping the denominator constant, leading us to the sum 8/15. This outcome represents the simplified value of the expression within the first set of parentheses. Therefore, we can confidently state that the first set of parentheses, (2/15 + 2/5), has been successfully simplified to 8/15. This seemingly simple step underscores the fundamental importance of understanding fraction manipulation in the broader context of mathematical problem-solving. The ability to find common denominators and create equivalent fractions is a cornerstone skill that underpins more advanced algebraic and calculus concepts. Moreover, this process highlights the elegance and precision inherent in mathematical operations, where each step is logically connected and contributes to the final solution. Mastering these basic skills not only empowers us to solve specific problems but also cultivates a deeper appreciation for the interconnectedness of mathematical ideas.
Next, we focus on the second set of parentheses, (9/10 × 4/3), which involves fraction multiplication. Unlike addition, fraction multiplication doesn't require a common denominator. We can multiply the numerators together and the denominators together: (9 × 4) / (10 × 3) = 36/30. Therefore, the result of the multiplication is 36/30, which can be simplified further. This initial multiplication sets the stage for simplification, a crucial step in expressing the answer in its most concise form. Understanding fraction multiplication is essential, as it frequently appears in various mathematical contexts, from solving equations to understanding proportions.
Step 2: Simplifying the Second Set of Parentheses
The second set of parentheses contains the expression (9/10 × 4/3). When multiplying fractions, we multiply the numerators together and the denominators together. So, 9/10 multiplied by 4/3 is (9 * 4) / (10 * 3) = 36/30. Now, we can simplify this fraction by finding the greatest common divisor (GCD) of 36 and 30, which is 6. Dividing both the numerator and the denominator by 6, we get 6/5. Therefore, the second set of parentheses simplifies to 6/5.
Turning our attention to the second set of parentheses, we encounter the expression (9/10 × 4/3), a classic example of fraction multiplication. Unlike fraction addition, which necessitates a common denominator, fraction multiplication operates on a more direct principle: we multiply the numerators together and the denominators together. Applying this principle, we find that 9/10 multiplied by 4/3 is equivalent to (9 * 4) / (10 * 3), which yields the fraction 36/30. However, our journey doesn't conclude here. A crucial step in expressing mathematical answers in their most refined form is simplification. To simplify the fraction 36/30, we embark on a quest to identify the greatest common divisor (GCD) of 36 and 30. The GCD, in essence, is the largest number that divides both the numerator and the denominator without leaving a remainder. In this case, the GCD of 36 and 30 is 6. Armed with this knowledge, we divide both the numerator and the denominator of 36/30 by 6. This division process unveils the simplified fraction 6/5. Therefore, we can definitively state that the second set of parentheses, (9/10 × 4/3), simplifies to 6/5. This process of simplification is not merely an aesthetic exercise; it's a fundamental aspect of mathematical communication. By expressing fractions in their simplest form, we ensure clarity and conciseness, making our results more readily understandable and comparable. Moreover, the ability to simplify fractions is a crucial skill for tackling more complex mathematical problems, where it can significantly reduce computational burden and enhance accuracy. The concept of GCD extends beyond fraction simplification; it's a cornerstone of number theory and finds applications in various fields, from cryptography to computer science. Mastering the art of simplification empowers us to navigate the world of fractions with confidence and precision.
Step 3: Adding the Simplified Fractions
Now we have 8/15 + 6/5. Again, we need a common denominator to add these fractions. The least common multiple of 15 and 5 is 15. So, we convert 6/5 to an equivalent fraction with a denominator of 15. We multiply both the numerator and denominator of 6/5 by 3, resulting in 18/15. Now we can add the fractions: 8/15 + 18/15 = 26/15. Therefore, the final simplified answer is 26/15.
With the expressions within the parentheses neatly simplified, we now arrive at the final stage of our mathematical journey: adding the simplified fractions. Our task is to evaluate 8/15 + 6/5. As we've established earlier, adding fractions necessitates a common denominator, a shared numerical foundation upon which the numerators can be combined. In this scenario, we identify the least common multiple (LCM) of 15 and 5 as our target common denominator, which, in this case, is 15. Therefore, our focus shifts to converting the fraction 6/5 into an equivalent form with a denominator of 15. This transformation is achieved by multiplying both the numerator and the denominator of 6/5 by 3, a strategic maneuver that preserves the fraction's intrinsic value while aligning it with our desired denominator. The outcome of this multiplication is the equivalent fraction 18/15. Now, with both fractions harmoniously sharing a common denominator, the addition process becomes a straightforward application of a fundamental rule: we add the numerators, maintaining the denominator constant. Thus, 8/15 + 18/15 yields the fraction 26/15. Therefore, we can confidently proclaim that the final simplified answer to our original expression is 26/15. This result, a culmination of a series of logical steps, exemplifies the elegance and precision inherent in mathematical operations. Each stage, from identifying the common denominator to performing the addition, underscores the interconnectedness of mathematical concepts. Moreover, the ability to navigate fraction addition with proficiency is a crucial skill that underpins more advanced mathematical endeavors, such as solving algebraic equations and tackling calculus problems. This final step not only provides a numerical answer but also reinforces the importance of a systematic approach to problem-solving, where each step builds upon the previous one, culminating in a clear and concise solution. The fraction 26/15, while seemingly simple, encapsulates a journey through the fundamental principles of arithmetic, highlighting the power of mathematical reasoning and the beauty of numerical relationships.
Final Answer
The simplified form of the expression (2/15 + 2/5) + (9/10 × 4/3) is 26/15.
In conclusion, we have successfully simplified the given expression by following the order of operations, finding common denominators, and simplifying fractions. This exercise demonstrates the importance of understanding basic arithmetic principles in solving mathematical problems.
