Hey everyone! Today, we're diving into a classic math problem: 95 divided by one-fifth plus 35. Sounds simple enough, right? Well, let's break it down step-by-step and see how to solve it. This isn't just about getting the right answer; it's about understanding the process and why the order of operations matters. We'll explore the fractions, the division, the addition, and every step in between. Ready to crunch some numbers? Let's go!

The Breakdown: Understanding the Components of the Math Problem

First off, let's make sure we're all on the same page. The core of our problem is 95 / (1/5) + 35. What does this mean, exactly? Well, we've got three main components: a whole number (95), a fraction (one-fifth or 1/5), and another whole number (35). The mathematical operations at play are division and addition. But here's the kicker: the order in which we perform these operations is crucial. Mess it up, and you'll end up with the wrong answer! We'll use the order of operations, often remembered by the acronym PEMDAS or BODMAS, to make sure we do things in the right sequence.

So, what does PEMDAS/BODMAS stand for? It’s a handy way to remember the order: Parentheses / Brackets, Exponents / Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This means we tackle anything inside parentheses or brackets first, then exponents or orders, then multiplication and division, and finally, addition and subtraction. In our case, we have a fraction enclosed within the division operation, so that's the first thing we'll handle. Understanding these components and the order of operations sets the foundation for getting the correct solution.

Now, let's talk about why this problem is more than just numbers. It's about problem-solving. It's about taking a seemingly complex task and breaking it down into manageable steps. This skill isn't just useful in math class; it's something you can use in everyday life. Whether you're planning a project, organizing your day, or even just following a recipe, the ability to break things down and follow a logical sequence is invaluable. So, as we go through this, think about how the same principles can be applied to other challenges you face.

Diving into Division: 95 Divided by One-Fifth

Alright, let's get down to the nitty-gritty. Our first step is to tackle the division part of the problem: 95 divided by one-fifth (95 / (1/5)). Dividing by a fraction might seem a little tricky at first, but here’s a neat trick: dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is simply flipping the numerator and the denominator. So, the reciprocal of 1/5 is 5/1, which is just 5. Therefore, 95 divided by 1/5 becomes 95 multiplied by 5.

So, what's 95 multiplied by 5? You can do this in a couple of ways. You can either multiply it out directly (95 * 5), or you can break it down a bit. For instance, you could think of 95 as (90 + 5). Then, multiply each part by 5: (90 * 5) + (5 * 5). That gives you 450 + 25. Adding those together, you get 475. Easy, right?

So, the result of 95 divided by one-fifth is 475. We've simplified the division, which was the trickiest part of the problem. This step is a perfect example of how understanding the rules of fractions and division can simplify complex calculations. It’s also a good reminder that there's often more than one way to solve a math problem. Choose the method that makes the most sense to you and allows you to understand the steps clearly. Keep this in mind, and you'll find that many seemingly complicated problems become much more manageable.

Adding It Up: Completing the Calculation

Okay, we've done the division, and now we're ready for the final step: adding 35 to the result we got from the division. Remember, we found that 95 divided by one-fifth is 475. So, our final calculation is 475 + 35.

This is a simple addition problem. You can do it in your head, on paper, or with a calculator – whatever works best for you. Adding 475 and 35, you get a total of 510. And that’s it! We’ve solved the problem: 95 divided by one-fifth plus 35 equals 510.

So, what's the big takeaway here? Besides the answer itself, the key is the methodical process. We broke down the problem into smaller, more manageable steps, and we followed the order of operations to ensure we got the right answer. This approach isn't just about math; it's about problem-solving in general. Breaking complex tasks into smaller, more digestible parts is a skill that can be applied in all sorts of situations. Whether you're planning a trip, organizing your budget, or even just figuring out what to cook for dinner, the principles of breaking down a problem and tackling it step-by-step will serve you well. Congratulations, you've not only solved the math problem but also honed your problem-solving skills along the way!

Why the Order of Operations Matters: Avoiding Common Mistakes

Let's talk about why the order of operations is so important. Without it, you could easily end up with a completely different (and wrong) answer. Imagine if you just went through the problem from left to right without thinking about PEMDAS/BODMAS. You might add 35 to the 95 first, getting 130, and then try to divide that by one-fifth. That would lead you way off track! This is a classic example of why sticking to the rules is essential.

Another common mistake is misinterpreting the fraction. Some people might get confused and try to divide 95 by 1 and then divide the result by 5. That's not how it works! Remember, dividing by a fraction is the same as multiplying by its reciprocal. So, by understanding and correctly applying the order of operations, you avoid these common pitfalls and arrive at the right answer every time.

Think about it this way: the order of operations is like the grammar rules of mathematics. They provide a structure and ensure clarity. Without these rules, mathematical expressions would be ambiguous, and everyone would interpret them differently, leading to chaos! So, by using the order of operations, you're essentially speaking the same mathematical language as everyone else, which is crucial for clear communication and getting the right results. Make sure to always remember PEMDAS/BODMAS to solve mathematical problems correctly.

Putting It All Together: Recap and Conclusion

So, let’s wrap things up! We started with the problem: 95 divided by one-fifth plus 35. We broke it down step-by-step, first tackling the division, remembering that dividing by a fraction means multiplying by its reciprocal. Then, we multiplied 95 by 5, getting 475. Finally, we added 35 to 475, arriving at the answer: 510.

What did we learn beyond the answer? We reinforced the importance of the order of operations (PEMDAS/BODMAS) and how it guides us through complex calculations. We practiced working with fractions and understood how to simplify them through reciprocals. Most importantly, we demonstrated the value of breaking down a problem into smaller, manageable steps. This approach is fundamental to problem-solving, not just in math but in everyday life.

Hopefully, this breakdown has helped you understand the process and given you some useful tools for approaching similar problems. Remember, practice makes perfect! The more you work through problems like this, the more comfortable and confident you'll become. Keep practicing, keep exploring, and most importantly, keep enjoying the process of learning. And that, my friends, is how you solve 95 divided by one-fifth plus 35!