Calculating 3A + 4B: A Step-by-Step Solution

Hey guys! Today, we're diving into a simple yet important math problem. We've got A = (-4)/2 and B = (-5)/2, and our mission is to figure out the value of 3A + 4B. Don't worry; we'll break it down step by step so it's super easy to follow. Math can seem intimidating sometimes, but with a clear approach, it becomes much more manageable. We'll use basic arithmetic operations and the order of operations to solve this. Let's get started and make math a little less mysterious and a lot more fun!

Understanding the Problem

Before we jump into solving, let's make sure we really understand what the problem is asking. We're given two values: A equals (-4)/2, and B equals (-5)/2. Our ultimate goal? To find out what 3A + 4B equals. This means we need to multiply A by 3, multiply B by 4, and then add the results together. It's like following a recipe – each step is important, and the order matters! Understanding the problem is the first key to solving it accurately. If we misinterpret what's being asked, we might end up going down the wrong path. So, let’s take a deep breath, read the problem again, and make sure we've got it crystal clear. It's also helpful to think about the operations we'll be using. We'll need to remember our multiplication rules for negative numbers and how to add fractions or integers. A solid grasp of these basics will make the rest of the solution flow much smoother. Think of this step as laying the foundation for a strong building – if the foundation is solid, the rest will stand tall!

Step 1: Simplify A and B

Alright, let's kick things off by simplifying the values of A and B. Remember, we have A = (-4)/2 and B = (-5)/2. Simplifying these fractions is crucial because it makes the rest of the calculation much easier. For A, we have -4 divided by 2. This is a straightforward division: -4 ÷ 2 = -2. So, A simplifies to -2. Now, let's tackle B. We have -5 divided by 2, which can be written as -5/2. This is an improper fraction, but we can leave it as is for now or convert it to a decimal if that feels more comfortable for you. In decimal form, -5/2 is -2.5. Simplifying A and B is like decluttering your workspace before starting a project – it removes unnecessary complexity and lets you focus on the main task. By reducing these fractions to their simplest forms, we've already made our next steps a whole lot easier. It's a small step, but it makes a big difference in the overall process. Plus, working with smaller numbers generally reduces the chance of making errors. So, pat yourself on the back – you've just completed a vital step in solving this problem!

Step 2: Calculate 3A

Now that we've simplified A and B, let's move on to calculating 3A. We know that A is -2, so 3A means 3 multiplied by -2. Remember the rules for multiplying positive and negative numbers: a positive number times a negative number results in a negative number. So, 3 * (-2) equals -6. That's it! We've found that 3A = -6. This step might seem simple, but it's an important piece of the puzzle. We're methodically working through each part of the expression 3A + 4B, making sure we get each step right before moving on. Calculating 3A is like assembling one part of a larger machine – it needs to be done correctly so that the whole machine functions properly. By breaking down the problem into smaller, manageable steps, we're making the entire process less daunting and more achievable. So, we've successfully calculated 3A, and we're one step closer to finding the final answer. Keep up the great work!

Step 3: Calculate 4B

Okay, guys, let's tackle the next part of our problem: calculating 4B. We know that B is -5/2 (or -2.5 if you prefer decimals). So, 4B means we need to multiply 4 by -5/2. There are a couple of ways we can approach this. One way is to think of 4 as the fraction 4/1 and then multiply the fractions: (4/1) * (-5/2). When we multiply fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, (4 * -5) / (1 * 2) equals -20/2. Now, we can simplify -20/2 by dividing -20 by 2, which gives us -10. Another way to solve this is to multiply 4 by the decimal form of B, which is -2.5. So, 4 * (-2.5) also equals -10. Either way, we arrive at the same answer: 4B = -10. Calculating 4B is another crucial step in our journey to solve the problem. We're taking each term in the expression 3A + 4B and finding its value individually. This methodical approach helps prevent errors and makes the overall solution clearer. Think of it like building a bridge – you need to construct each section carefully before you can connect them all together. So, we've successfully calculated 4B, and we're now ready for the final step: adding the results together!

Step 4: Add 3A and 4B

We're almost there! We've calculated 3A and 4B separately, and now it's time to add them together to find the final answer. We found that 3A equals -6, and 4B equals -10. So, the expression 3A + 4B becomes -6 + (-10). Adding two negative numbers is like adding debts – they combine to create a larger debt. In this case, -6 plus -10 equals -16. Therefore, the value of 3A + 4B is -16. Congratulations! You've successfully solved the problem. Adding 3A and 4B is the final piece of the puzzle. We've taken all the individual components, calculated their values, and now we're putting them together to get the complete picture. This step highlights the importance of accuracy in the previous steps – if we had made a mistake earlier, it would affect our final answer. But because we took our time and worked methodically, we've arrived at the correct solution. Think of it like cooking a meal – you need to prepare each ingredient properly, but it's the final combination that creates the delicious dish. So, we've added 3A and 4B, and we've found the answer: -16. You did it!

Final Answer

So, after all our calculations, we've arrived at the final answer! We set out to find the value of 3A + 4B, given that A = (-4)/2 and B = (-5)/2. We carefully simplified A and B, calculated 3A and 4B, and then added the results together. And what did we find? The value of 3A + 4B is -16. That's it! We've solved the problem from start to finish. The final answer is the culmination of all our hard work. It's the destination we were aiming for, and we reached it by following a clear and methodical path. This process highlights the importance of breaking down complex problems into smaller, manageable steps. By tackling each step individually, we made the overall problem much less daunting. Think of it like climbing a mountain – you don't try to climb it all in one leap, but rather you take it one step at a time. So, we've found our final answer, -16, and we've learned a valuable lesson about problem-solving along the way. Great job, everyone!