Simplifying (12/7 ÷ 5/4) + (1/7 ⋅ 4/5) A Step-by-Step Guide

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Hey there, math enthusiasts! Ever stumbled upon an expression that looks like a jumbled mess of fractions and operations? Don't worry, we've all been there. Today, we're going to break down and simplify the expression (12/7 ÷ 5/4) + (1/7 ⋅ 4/5) step-by-step. This isn't just about getting the right answer; it's about understanding the process, the why behind each step. So, grab your pencils, and let's dive in!

Understanding the Order of Operations

Before we even touch the fractions, it's crucial to remember the golden rule of math: order of operations. You might have heard of the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). Both essentially tell us the same thing: the order in which we should tackle different operations. In our expression, we have division, multiplication, and addition. According to PEMDAS/BODMAS, we perform division and multiplication before addition. This is a fundamental concept, guys, and getting it right ensures we're on the right track from the start. Think of it as the foundation upon which we build our solution. If the foundation is shaky, the whole thing might crumble! So, let's make sure we've got this down pat. Remember, division and multiplication have equal priority, so we work from left to right. Similarly, addition and subtraction have equal priority and are also worked from left to right. Keeping this hierarchy in mind is the key to unlocking the solution. It's like having a map that guides us through the mathematical terrain, preventing us from getting lost in the complexity of the expression. Understanding this order isn't just about solving this particular problem; it's a skill that will serve you well in countless mathematical challenges.

Step 1: Tackling the Division (12/7 ÷ 5/4)

Our first task is to handle the division: 12/7 ÷ 5/4. Now, dividing fractions might seem intimidating at first, but there's a neat trick to it: we can turn division into multiplication by flipping the second fraction (taking its reciprocal). So, 12/7 ÷ 5/4 becomes 12/7 multiplied by 4/5. Remember, the reciprocal of a fraction is simply flipping the numerator and the denominator. This transformation is a powerful tool in our arsenal. It simplifies the process and allows us to apply the much simpler rule of multiplying fractions. When we multiply fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, (12/7) * (4/5) equals (12 * 4) / (7 * 5). Now, let's do the math: 12 multiplied by 4 is 48, and 7 multiplied by 5 is 35. Therefore, 12/7 ÷ 5/4 simplifies to 48/35. We've successfully conquered the division part! This is a significant milestone in our journey. By converting division to multiplication, we've made the problem much more manageable. We're not just solving a problem here; we're learning strategies that can be applied to a wide range of mathematical scenarios. Remember, math is not just about memorizing rules; it's about understanding the why behind them. And in this case, the why is that dividing by a fraction is the same as multiplying by its reciprocal. This is a fundamental concept that will help you navigate the world of fractions with confidence.

Step 2: Multiplying the Fractions (1/7 ⋅ 4/5)

Next up, we have the multiplication part: 1/7 ⋅ 4/5. As we discussed earlier, multiplying fractions is straightforward: we multiply the numerators together and the denominators together. So, (1/7) * (4/5) equals (1 * 4) / (7 * 5). This is where the simplicity of fraction multiplication shines. Unlike division, there's no need to flip fractions or perform any transformations. It's a direct and concise process. Let's do the math: 1 multiplied by 4 is 4, and 7 multiplied by 5 is 35. Thus, 1/7 ⋅ 4/5 simplifies to 4/35. We've successfully handled the multiplication part! This step highlights the elegance of fraction operations. Multiplication, in this context, becomes a seamless process of combining numerators and denominators. This simple yet powerful rule forms the backbone of many mathematical calculations involving fractions. By mastering this technique, you're equipping yourself with a valuable tool for tackling more complex problems down the line. Remember, guys, each step we take is building upon the previous one. We're not just solving this particular expression; we're developing a skillset that will empower us to tackle a wide range of mathematical challenges. The beauty of math lies in its consistency and its logical structure. Once you grasp the fundamental principles, you can apply them to various scenarios and unlock the solutions.

Step 3: Adding the Fractions (48/35 + 4/35)

Now comes the final step: adding the results from our division and multiplication. We have 48/35 + 4/35. Adding fractions is a breeze when they have the same denominator. Lucky for us, both fractions have a denominator of 35! When the denominators are the same, we simply add the numerators and keep the denominator. So, 48/35 + 4/35 equals (48 + 4) / 35. This is where the common denominator truly shines. It allows us to combine the fractions into a single, unified expression. The common denominator acts as a common unit, enabling us to add the numerators as if they were whole numbers. Let's do the addition: 48 plus 4 is 52. Therefore, 48/35 + 4/35 equals 52/35. We've added the fractions! But hold on, we're not quite done yet. It's always good practice to check if our final fraction can be simplified. This is the final polish that transforms a good answer into a great answer. Simplifying fractions is all about finding common factors between the numerator and the denominator and dividing them out. This process reduces the fraction to its simplest form, making it easier to understand and work with. In this case, 52 and 35 don't share any common factors other than 1, so the fraction 52/35 is already in its simplest form. We've reached the finish line! By following the order of operations and applying the rules of fraction arithmetic, we've successfully simplified the expression. This is a testament to the power of systematic problem-solving. Remember, math is not just about arriving at the correct answer; it's about the journey you take to get there. Each step is a learning opportunity, and the final answer is the reward for your diligent efforts.

Final Result

Therefore, the simplified form of (12/7 ÷ 5/4) + (1/7 ⋅ 4/5) is 52/35. We did it!

Key Takeaways

  • Order of Operations (PEMDAS/BODMAS): Always remember the order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This is the bedrock of mathematical calculations. Mastering this order is like having a compass that guides you through the complexities of mathematical expressions. It ensures that you're tackling the operations in the correct sequence, leading you to the accurate solution. Remember, consistency in applying the order of operations is key to avoiding errors and building a strong foundation in math. So, make it a habit to always double-check the order before diving into any calculation.
  • Dividing Fractions: Dividing by a fraction is the same as multiplying by its reciprocal. This is a powerful transformation that simplifies the division process. This trick allows us to convert a division problem into a multiplication problem, which is often easier to handle. By flipping the second fraction and changing the operation from division to multiplication, we unlock a more straightforward path to the solution. This technique is not just a shortcut; it's a fundamental concept that reveals the interconnectedness of mathematical operations. Understanding this relationship empowers you to manipulate expressions with greater confidence and flexibility.
  • Multiplying Fractions: Multiply the numerators and multiply the denominators. It's that simple! This is one of the most straightforward operations in fraction arithmetic. The simplicity of this rule makes it a powerful tool for combining fractions. By multiplying the top numbers and the bottom numbers separately, we efficiently arrive at the product of the fractions. This process highlights the beauty of mathematical patterns. The consistent application of this rule reinforces the underlying structure of fraction multiplication, making it easier to remember and apply in various contexts.
  • Adding Fractions: Fractions need a common denominator before they can be added. This is a crucial step in fraction addition. The common denominator acts as a unifying unit, allowing us to combine the fractions seamlessly. Think of it like adding apples and oranges. You need to convert them to a common unit, like pieces of fruit, before you can add them together. Similarly, fractions need a common denominator to represent the same underlying unit. Once the denominators are the same, adding fractions becomes as simple as adding the numerators. This principle underscores the importance of consistency in mathematical operations. By ensuring that fractions have a common denominator, we establish a level playing field for addition, leading to accurate and meaningful results.
  • Simplifying Fractions: Always check if your final answer can be simplified. This is the final polish that ensures your answer is in its most concise form. Simplifying fractions is like editing a piece of writing. You're removing unnecessary elements and ensuring that the core message is clear and impactful. By finding the greatest common factor between the numerator and the denominator and dividing both by it, we reduce the fraction to its simplest form. This process not only makes the fraction easier to understand but also demonstrates a commitment to precision and clarity in mathematical communication. So, always take that extra step to simplify your fractions and present your answer in its most elegant form.

By mastering these key concepts, you'll be well-equipped to tackle a wide range of fraction problems. Keep practicing, guys, and you'll become fraction ninjas in no time!