Simplifying Fractions A Step-by-Step Solution For (50 + 5100) / (150 + 5100)

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Understanding the Problem

At the heart of this mathematical problem lies a fraction: (50 + 5100) / (150 + 5100). Our primary goal is to simplify this fraction and arrive at its most reduced form. This seemingly simple problem provides an opportunity to explore fundamental arithmetic operations, order of operations, and the crucial concept of fraction simplification. Before diving into the step-by-step solution, it's beneficial to grasp the underlying principles that govern these operations. The numerator, which is the top part of the fraction (50 + 5100), and the denominator, which is the bottom part of the fraction (150 + 5100), are both expressions that need to be evaluated before we can simplify the overall fraction. Remember the order of operations (often remembered by the acronym PEMDAS or BODMAS) which dictates that we perform any operations within parentheses first, followed by exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). In this case, we have addition operations within both the numerator and the denominator. Fraction simplification involves finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by the GCD. This process reduces the fraction to its simplest form, where the numerator and denominator have no common factors other than 1. Understanding these core concepts will make solving the problem not just a mechanical exercise, but a meaningful application of mathematical principles. We aim to not just get the correct answer but also to comprehend why each step is taken, building a solid foundation for tackling more complex mathematical problems in the future. By the end of this exploration, you should be comfortable with simplifying fractions and understanding the reasoning behind each step.

Step-by-Step Solution

To effectively solve the fraction (50 + 5100) / (150 + 5100), we'll break it down into a series of straightforward steps. This methodical approach ensures clarity and minimizes the risk of errors.

  1. Simplify the Numerator: Our first step involves simplifying the numerator, which is the expression 50 + 5100. This is a simple addition operation. Adding 50 to 5100, we get 5150. So, the numerator simplifies to 5150. This step is crucial because it reduces the complexity of the fraction, making it easier to work with in subsequent steps. We're essentially consolidating the addition operation into a single numerical value. This is a direct application of the order of operations, where we address the operations within the parentheses (in this case, the implicit parentheses around the numerator) before dealing with the fraction as a whole. It's important to perform this addition accurately, as any error here will propagate through the rest of the solution. Double-checking this step can save time and prevent mistakes. In essence, we're converting the expression in the numerator into its simplest form, preparing it for the next stage of the simplification process.

  2. Simplify the Denominator: Next, we tackle the denominator, which is 150 + 5100. Similar to the numerator, this involves a straightforward addition. Adding 150 to 5100, we arrive at 5250. Therefore, the denominator simplifies to 5250. This step mirrors the simplification of the numerator and is equally crucial for accurately solving the problem. By simplifying the denominator, we're again reducing the complexity of the fraction and making it easier to manage. The same principles of the order of operations apply here as they did with the numerator. Accuracy in this step is paramount, as any mistake will affect the final result. Just as with the numerator, it's wise to double-check the addition to ensure that it's correct. With both the numerator and the denominator now simplified to single numerical values, we've transformed the original fraction into a more manageable form, setting the stage for the next key step: finding the greatest common divisor.

  3. The Fraction Now: After simplifying both the numerator and the denominator, our fraction now looks like this: 5150 / 5250. This simplified form is much easier to work with than the original expression. We've reduced the complexity by performing the additions, and now we have a fraction with two clear numbers. This is a crucial intermediate step, as it allows us to focus on the core task of fraction simplification. By presenting the fraction in this form, we can more easily identify common factors between the numerator and the denominator. This is a visual representation of the progress we've made so far. We started with a complex fraction involving addition operations, and we've transformed it into a simpler fraction involving just two numbers. This step highlights the power of breaking down a problem into smaller, manageable parts. The fraction 5150/5250 is now our focus, and the next step will involve finding the greatest common divisor (GCD), which will allow us to reduce the fraction to its simplest form.

  4. Find the Greatest Common Divisor (GCD): To simplify the fraction 5150 / 5250, we need to find the greatest common divisor (GCD) of 5150 and 5250. The GCD is the largest number that divides both 5150 and 5250 without leaving a remainder. There are several methods to find the GCD, but one common method is the Euclidean algorithm. However, for smaller numbers, we can often find the GCD by inspection or by prime factorization. Let's start by looking for common factors. Both numbers end in 0, so they are both divisible by 10. Dividing both by 10 gives us 515 and 525. Both these numbers are divisible by 5. Dividing both by 5 gives us 103 and 105. Now, we need to check if 103 and 105 have any common factors. 103 is a prime number, meaning its only factors are 1 and itself. So, we only need to check if 105 is divisible by 103. It is not. Therefore, the GCD of 5150 and 5250 is 10 * 5 = 50. Finding the GCD is a crucial step in simplifying fractions. It allows us to reduce the fraction to its lowest terms, making it easier to understand and compare. The Euclidean algorithm is a more systematic approach for larger numbers, but for this problem, we were able to find the GCD relatively easily by identifying common factors. This step demonstrates the importance of number sense and the ability to recognize divisibility rules.

  5. Divide by the GCD: Now that we've determined the GCD of 5150 and 5250 to be 50, the next step is to divide both the numerator and the denominator by this GCD. This process will reduce the fraction to its simplest form. Dividing the numerator, 5150, by 50, we get 103. Dividing the denominator, 5250, by 50, we get 105. This step is the core of fraction simplification. By dividing both the numerator and denominator by their GCD, we are essentially removing the common factors that they share. This results in a fraction where the numerator and denominator are relatively prime, meaning they have no common factors other than 1. This is the simplest possible representation of the fraction. The arithmetic in this step is straightforward, but it's crucial to perform the divisions accurately. Double-checking the results can prevent errors and ensure that the final answer is correct. This step demonstrates the practical application of the GCD and its role in simplifying fractions. It also highlights the importance of performing operations consistently on both the numerator and the denominator to maintain the fraction's value.

  6. The Simplified Fraction: After dividing both the numerator and the denominator by the GCD of 50, we arrive at the simplified fraction: 103 / 105. This fraction is in its simplest form, as 103 and 105 have no common factors other than 1. This is our final answer. We have successfully simplified the original fraction (50 + 5100) / (150 + 5100) to its most reduced form. This step represents the culmination of all the previous steps. We started with a seemingly complex fraction, and through a series of logical steps, we have simplified it to its most basic form. This demonstrates the power of mathematical simplification and the importance of following a systematic approach. The fraction 103/105 is easier to understand and work with than the original fraction. It represents the same value but in a more concise and manageable way. This is the goal of fraction simplification: to express a fraction in its simplest terms. We can be confident that this is the final answer because we have divided both the numerator and denominator by their GCD, ensuring that there are no further common factors.

Final Answer

Therefore, the simplified form of the fraction (50 + 5100) / (150 + 5100) is 103 / 105. This answer is the culmination of our step-by-step solution, where we first simplified the numerator and denominator separately, then found their greatest common divisor (GCD), and finally, divided both by the GCD to arrive at the simplest form of the fraction. This final answer is not just a number; it represents a complete understanding of the problem and the application of appropriate mathematical principles. We have successfully reduced the original complex fraction to its most basic and easily understandable form. The journey to this answer involved several key steps, each building upon the previous one. From simplifying the numerator and denominator to finding the GCD and performing the division, each step was crucial to arriving at the correct solution. The final answer of 103/105 demonstrates the power of simplification and the elegance of mathematical problem-solving. It is a clear and concise representation of the value expressed by the original fraction.

Key Concepts Revisited

Throughout the process of solving the fraction (50 + 5100) / (150 + 5100), we've employed several key mathematical concepts that are fundamental to understanding and simplifying fractions. Let's revisit these concepts to solidify our understanding. Firstly, the order of operations (PEMDAS/BODMAS) played a crucial role in correctly simplifying the numerator and denominator. We addressed the addition operations within the parentheses (or implied parentheses) before moving on to other operations. This principle ensures that we perform operations in the correct sequence, leading to an accurate result. Secondly, the concept of the greatest common divisor (GCD) was essential for simplifying the fraction. The GCD is the largest number that divides two or more numbers without leaving a remainder. Finding the GCD allowed us to identify common factors between the numerator and the denominator, which we could then divide out to reduce the fraction to its simplest form. Thirdly, fraction simplification itself is a core concept. It involves reducing a fraction to its lowest terms by dividing both the numerator and the denominator by their GCD. This process results in a fraction that is equivalent to the original but is expressed in its simplest form, making it easier to understand and work with. These three concepts – order of operations, GCD, and fraction simplification – are interconnected and essential for effectively solving problems involving fractions. By understanding and applying these concepts, we can confidently tackle a wide range of mathematical problems involving fractions and other numerical expressions. The ability to simplify fractions is a valuable skill in mathematics and has applications in various fields, including science, engineering, and finance.