Solving 10 + 8 - (-5) × 6 A Step-by-Step Math Guide

Hey guys! Ever stumbled upon a math problem that looks like a jumbled mess of numbers and symbols? Don't worry, we've all been there. One of the most common challenges in math comes from not knowing where to start. That's where the order of operations comes in, our trusty guide through the mathematical wilderness. In this article, we're going to break down a specific problem – 10 + 8 - (-5) × 6 – using the order of operations. By the end, you'll not only know the answer but also understand the why behind each step. So, grab your pencils and let's dive into the world of mathematical problem-solving!

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we even think about touching the problem, we need to understand the golden rule of math: the order of operations. You might have heard of it as PEMDAS or BODMAS, depending on where you learned it. Either way, it's the same concept! This acronym helps us remember the correct sequence in which to solve mathematical expressions. PEMDAS stands for:

• Parentheses (or Brackets)
• Exponents (or Orders)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Following this order ensures that everyone arrives at the same correct answer. Think of it as the grammar of mathematics – it provides structure and clarity. Imagine if we just solved problems in the order they appear; we'd get wildly different answers, and math class would be even more confusing! So, let's keep PEMDAS (or BODMAS) in our mental toolkit as we tackle the problem at hand.

Why is this order so crucial, you ask? Well, math isn't just a random collection of symbols; it's a carefully constructed language. Operations have different levels of priority. For instance, multiplication and division are stronger operations than addition and subtraction. They tell us how many groups we have or how many times we're scaling something. Ignoring this hierarchy would be like ignoring punctuation in a sentence – the meaning gets completely lost! So, mastering PEMDAS/BODMAS is like learning the grammar of math, and it's the first step to solving complex problems with confidence. Without it, we'd be swimming in a sea of numbers without a life raft. Trust me, adhering to this order saves you from a lot of mathematical headaches.

Step 1: Identifying the Operations

Okay, now that we've got PEMDAS/BODMAS in our heads, let's look at our problem: 10 + 8 - (-5) × 6. The first step is to identify all the mathematical operations present. What do we see? We've got addition (+), subtraction (-), and multiplication (×). Notice that we also have a negative number inside parentheses, (-5). While the parentheses here aren't grouping terms in the same way they might in a more complex expression, they do indicate that the negative sign belongs to the 5. It's like saying, “This is a negative five, not just a five that we're going to subtract later.”

Now, let's prioritize these operations according to PEMDAS/BODMAS. Are there any parentheses that group an expression? Not really in this case, the parentheses are just marking the negative sign. Are there any exponents? Nope. So, what comes next? That's right, multiplication and division. We see a multiplication operation: (-5) × 6. This is the first thing we need to tackle. We can almost think of this as the most important part of the mathematical sentence we're trying to decipher. It's setting the stage for the rest of the operations.

Why do we pinpoint the operations first? Because it gives us a roadmap. It prevents us from getting lost in the numbers and symbols and makes the whole process feel much less overwhelming. It's like planning a road trip – you wouldn't just start driving without knowing your destination, would you? In the same way, identifying the operations is our first step towards solving the problem correctly. It's about breaking down the big, scary problem into smaller, manageable chunks. By clearly seeing what operations are at play, we're setting ourselves up for success. Trust me, this simple step can make a world of difference.

Step 2: Performing the Multiplication

Alright, we've identified that the multiplication operation, (-5) × 6, is our next target. Let's get to it! Remember, when multiplying a negative number by a positive number, the result is always negative. So, we know our answer is going to be a negative number. Now, we just need to multiply the absolute values: 5 × 6. Most of us probably know this one off the top of our heads: 5 times 6 equals 30. But if you're not sure, you can always break it down – 5 groups of 6, or 6 groups of 5. Either way, you'll get 30.

Since we're multiplying a negative number by a positive number, our result is -30. So, (-5) × 6 = -30. We've conquered the multiplication! Now, let's rewrite our original problem with this result: 10 + 8 - (-30). Notice how we've replaced the (-5) × 6 with -30. This is a crucial step because it simplifies the expression and makes the remaining operations clearer.

Why is it so important to get the sign right in multiplication (and division)? Because a single wrong sign can throw off the entire calculation. It's like misplacing a decimal point – the result can be wildly inaccurate. Think of the negative sign as a direction indicator. It tells us we're moving in the opposite direction from positive. In the context of our problem, the -30 represents a quantity that's 30 units less than zero. Getting this right is non-negotiable in math! We're one step closer to the final answer, and by carefully following the order of operations, we're building a solid foundation for future problem-solving. Go us!

Step 3: Handling Subtraction of a Negative Number

Now our problem looks like this: 10 + 8 - (-30). Notice something interesting? We're subtracting a negative number. This is a situation that often trips people up, but it's actually quite straightforward once you understand the rule. Subtracting a negative number is the same as adding its positive counterpart. Think of it like this: if you're taking away a debt, you're actually increasing your wealth. The two negatives essentially cancel each other out.

So, 10 + 8 - (-30) becomes 10 + 8 + 30. We've transformed the subtraction of a negative into addition. This makes the problem much easier to solve. We've eliminated a potential source of confusion and simplified the expression further. Now we're left with a series of additions, which we can tackle from left to right.

Why does subtracting a negative number equal addition? It might seem counterintuitive at first, but there's a logical explanation. Imagine a number line. Subtracting a positive number moves us to the left. Subtracting a negative number is the opposite – it moves us to the right. This movement to the right is exactly what addition does. So, subtracting a negative is like doing the opposite of subtracting, which is adding! It's a fundamental concept in math, and understanding it will save you from countless errors. Mastering this rule is like unlocking a secret level in math – you'll be able to solve problems with greater confidence and accuracy.

Step 4: Performing Addition from Left to Right

We've reached the final stage! Our problem is now simplified to 10 + 8 + 30. According to PEMDAS/BODMAS, we perform addition and subtraction from left to right. So, let's start with the first addition: 10 + 8. This is a pretty straightforward one. 10 plus 8 equals 18. We've taken the first step in our addition sequence.

Now, let's rewrite the problem with this result: 18 + 30. We've reduced the problem to a single addition. This is the home stretch! We're almost there.

Why do we add from left to right? Because addition and subtraction have equal priority in the order of operations. Just like we read a sentence from left to right, we perform these operations in the same order. It's a convention that ensures consistency and prevents ambiguity. Imagine if we added 8 + 30 first – we'd get a different result, and math would be a chaotic mess! Sticking to the left-to-right rule keeps things organized and ensures we arrive at the correct answer. Think of it as following the flow of the mathematical sentence – we're reading it in the proper order.

Step 5: The Final Calculation

We're at the finish line! Our problem is now 18 + 30. This is the last calculation we need to make. 18 plus 30. Let's break it down if we need to. We can think of it as 10 + 30 = 40, then add the remaining 8: 40 + 8 = 48. Or, we can stack the numbers vertically and add the ones place first (8 + 0 = 8) and then the tens place (1 + 3 = 4). Either way, we arrive at the same answer:

18 + 30 = 48

So, the final answer to our problem, 10 + 8 - (-5) × 6, is 48! We did it! We started with a seemingly complex expression and, by carefully following the order of operations, we arrived at a clear and correct answer. Give yourself a pat on the back – you've earned it!

Why is it so satisfying to reach the final answer in a math problem? Because it's a testament to our problem-solving skills. It shows that we can take a complex challenge, break it down into smaller steps, and methodically work our way to the solution. It's like climbing a mountain – the view from the top is all the more rewarding because of the effort we put in. This feeling of accomplishment is what makes math so engaging, and it's something we can carry with us in all areas of life.

Conclusion: Mastering Math One Step at a Time

Guys, we've successfully navigated the mathematical terrain and solved the problem 10 + 8 - (-5) × 6. We learned that the key to success lies in understanding and applying the order of operations (PEMDAS/BODMAS). We broke down the problem into manageable steps: identifying the operations, performing multiplication, handling subtraction of a negative number, and finally, adding from left to right. Each step built upon the previous one, leading us to the correct answer: 48.

Remember, math isn't about memorizing formulas; it's about understanding the underlying principles. The order of operations is one such principle, a fundamental rule that governs how we solve mathematical expressions. By mastering this rule, you've equipped yourself with a powerful tool for tackling a wide range of problems. So, the next time you encounter a mathematical challenge, don't shy away from it. Embrace it! Break it down, follow the order of operations, and you'll be amazed at what you can achieve. And now you know step by step how to solve it.

What's the biggest takeaway from this exercise? It's that even the most complex problems can be solved if we approach them methodically. We didn't just jump to the answer; we took our time, understood each step, and carefully executed the operations in the correct order. This approach – breaking down challenges into smaller, manageable steps – is a valuable skill that extends far beyond the realm of mathematics. It's a life skill! So, keep practicing, keep learning, and keep mastering math one step at a time. You've got this!