Hey everyone! Let's dive into this mathematical expression and break it down together. We're going to explore the solution to the problem: What is the result of (4x / 2y) * (3x + 2) / (5y - 2)? Don't worry, we'll take it one step at a time, so it's super clear for everyone. Math can be fun, and we're here to make it easy to understand.
Understanding the Problem
Before we jump into solving, let's make sure we understand exactly what we're dealing with. The expression we're tackling is (4x / 2y) * (3x + 2) / (5y - 2). This looks a bit complicated, but it's really just a multiplication of two fractions. Remember, when we multiply fractions, we multiply the numerators (the top parts) together and the denominators (the bottom parts) together. Our goal is to simplify this expression as much as possible. So, understanding the problem is half the battle won, right? We've got two fractions that are being multiplied. The first fraction is 4x divided by 2y, and the second fraction is (3x + 2) divided by (5y - 2). We need to combine these two fractions into one, and then we'll simplify it to get our final answer. It’s like putting together pieces of a puzzle; each part has its place, and together they form a complete picture. We'll be focusing on how to properly multiply these fractions and then how to make the resulting fraction as simple as it can be. Think of it as cooking – we have the ingredients, and now we're following the recipe to get the best dish possible. Each step is crucial, and we’ll make sure to explain every bit so you can follow along easily. So, let's get started and break down this mathematical puzzle piece by piece!
Step 1: Multiplying the Numerators
The first thing we need to do is multiply the numerators. In our expression, the numerators are 4x and (3x + 2). So, we need to calculate 4x * (3x + 2). This involves using the distributive property, which means we multiply 4x by both terms inside the parentheses. Let's break it down: 4x * 3x equals 12x², and 4x * 2 equals 8x. Therefore, when we multiply the numerators, we get 12x² + 8x. Now, multiplying the numerators might sound like a daunting task, but it's actually quite straightforward once you remember the rules of algebra. We're essentially combining the top parts of our fractions. Think of it as expanding a recipe – we're taking the ingredients from the top shelf and mixing them together. The distributive property is our best friend here. It ensures that we multiply each term inside the parentheses by the term outside. So, we take 4x and multiply it by both 3x and 2. This gives us two separate products, which we then add together. The result, 12x² + 8x, is the new numerator of our combined fraction. We've successfully combined the top parts, and we're one step closer to our final answer. This step is like laying the foundation for our mathematical solution. We need a strong numerator to build upon, and we've just created it by carefully multiplying the individual numerators together. Remember, each term is important, and paying attention to the details will help us avoid mistakes. So, let's move on to the next step with confidence!
Step 2: Multiplying the Denominators
Next up, we need to multiply the denominators. In our expression, the denominators are 2y and (5y - 2). Just like we did with the numerators, we'll use the distributive property here. We need to calculate 2y * (5y - 2). So, 2y * 5y equals 10y², and 2y * -2 equals -4y. Thus, when we multiply the denominators, we get 10y² - 4y. Multiplying the denominators is very similar to what we did with the numerators. It's like working on the bottom part of our fraction puzzle. We're taking the bottom pieces and fitting them together. Again, the distributive property is key. We're going to multiply 2y by both 5y and -2. This gives us two separate products, which we then combine. The result, 10y² - 4y, becomes the new denominator of our combined fraction. Think of this step as setting the stage for our final act. We've created a strong foundation by multiplying the numerators, and now we're building the base by multiplying the denominators. The denominator is just as important as the numerator, and we need to make sure we get it right. This step is like balancing an equation – we need to ensure that the bottom part is correct to keep everything in harmony. So, with the denominators multiplied, we're well on our way to simplifying the entire expression. Remember, accuracy is crucial, and paying close attention to the signs and terms will help us avoid errors. Now, let's put the numerators and denominators together and see what we've got!
Step 3: Combining the Results
Now that we've multiplied the numerators and the denominators, we can combine the results to form a single fraction. We found that the product of the numerators is 12x² + 8x, and the product of the denominators is 10y² - 4y. So, our combined fraction is (12x² + 8x) / (10y² - 4y). This is the result of the multiplication, but we're not done yet! We can simplify this fraction further. Combining the results is like putting the final touches on a masterpiece. We've done the hard work of multiplying the top and bottom parts, and now we're putting it all together. Our fraction, (12x² + 8x) / (10y² - 4y), is the result of our multiplication, but it's not in its simplest form. Think of this as having a rough draft – we've got all the ideas down, but now we need to polish it up. The next step is to look for ways to simplify this fraction. This is where our algebra skills come in handy. We're going to look for common factors in both the numerator and the denominator. Simplifying fractions is like editing a piece of writing – we're cutting out the unnecessary parts and making it as clear and concise as possible. This step is like assembling a machine – we've got all the components, and now we're fitting them together to make it work smoothly. We need to make sure everything is aligned and in its proper place. So, with our combined fraction in hand, we're ready to move on to the final step: simplifying. Remember, simplifying is key, and it will give us the most elegant and clear answer. Let's see how we can make this fraction even simpler!
Step 4: Simplifying the Fraction
To simplify the fraction (12x² + 8x) / (10y² - 4y), we need to look for common factors in both the numerator and the denominator. In the numerator, both terms have a common factor of 4x. If we factor out 4x, we get 4x(3x + 2). In the denominator, both terms have a common factor of 2y. If we factor out 2y, we get 2y(5y - 2). So, our fraction becomes 4x(3x + 2) / 2y(5y - 2). Now, we can simplify further by dividing both the numerator and the denominator by their common factor of 2. This gives us 2x(3x + 2) / y(5y - 2). This is the simplified form of our fraction. Simplifying the fraction is the final step in our mathematical journey. It's like putting the finishing touches on a painting – we're making it as beautiful and clear as possible. We started with a complex expression, and now we're bringing it down to its simplest form. Think of this as refining a diamond – we're cutting away the rough edges to reveal the brilliance underneath. Factoring out common factors is our key tool here. We're looking for the biggest numbers and variables that divide evenly into both the top and bottom parts of the fraction. This is like finding the perfect fit – we're matching up the pieces that belong together and removing the ones that don't. Simplifying fractions is like writing a good summary – we're capturing the essence of the problem in the most concise way possible. This step is like tuning an instrument – we're making sure everything is in perfect harmony. We need to balance the top and bottom parts of the fraction to get the most elegant result. So, with our simplified fraction, 2x(3x + 2) / y(5y - 2), we've reached the end of our journey. Remember, practice makes perfect, and the more you simplify fractions, the easier it will become. Let’s expand this to get the final simplified answer, we have (6x^2 + 4x) / (5y^2 - 2y). Going back to the options presented, we need to compare our simplified answer with the choices given to us.
Comparing with the Options
Looking at the options provided, we need to see which one matches our simplified result. Our simplified form is (12x² + 8x) / (10y² - 4y). Let's look at the options:
a. (12x² + 8x) / (10y² - 4y) b. (10y² - 4y) / (12x² + 8x) c. (12x + 8) / (10y - 4)
By comparing, we can see that option a, (12x² + 8x) / (10y² - 4y), is the same as our result before the final simplification. So, option a is the correct answer. Comparing with the options is a crucial step in solving any multiple-choice problem. It's like checking your work – we need to make sure our answer lines up with the available choices. We've gone through the entire process of multiplying and simplifying, and now we're making sure we haven't made any mistakes along the way. Think of this as proofreading a document – we're looking for any typos or errors that might have slipped through. The options are like a set of clues, and we need to use our solution to find the correct one. This step is like matching a key to a lock – we're seeing which option fits our answer perfectly. We need to be careful and methodical in our comparison. A slight difference in the terms or signs can make an option incorrect. So, with our simplified result and the options in front of us, we can confidently identify the correct answer. Remember, accuracy is key, and taking the time to compare will ensure that we choose the right option. With option a matching our result, we've successfully solved the problem!
Final Answer
So, guys, we've made it to the end! The result of (4x / 2y) * (3x + 2) / (5y - 2) is (12x² + 8x) / (10y² - 4y), which corresponds to option a. Great job to everyone who followed along! Math might seem tricky at first, but with a little patience and step-by-step thinking, we can conquer any problem. Keep practicing, and you'll become a math whiz in no time! The final answer is the culmination of all our hard work. It's like reaching the summit of a mountain – we've overcome all the obstacles and now we can enjoy the view. We started with a complex expression, and now we've broken it down and simplified it to its core. Think of this as solving a mystery – we've gathered all the clues and now we've cracked the code. Our final answer, (12x² + 8x) / (10y² - 4y), is the solution we've been searching for. It's the result of our careful calculations and simplifications. This is the moment of truth – we're presenting our answer and standing behind it with confidence. We've checked our work, compared with the options, and made sure everything lines up. So, with our final answer in hand, we can celebrate our success. Remember, every problem is solvable, and with the right approach, we can find the solution. Let's keep exploring the world of math and tackling new challenges!