Solve $478 \longdiv{27 + 36}$: A Step-by-Step Guide

Hey guys! Today, we're diving deep into a mathematical problem that might seem a bit daunting at first glance: $478 \longdiv{27 + 36}$. Don't worry, we're going to break it down step by step, making sure everyone understands the process. This isn't just about getting the right answer; it's about understanding the underlying principles of arithmetic and how to apply them confidently. So, grab your calculators (or your mental math muscles), and let's get started!

Understanding the Order of Operations

Before we even think about dividing, we need to tackle the order of operations. Remember PEMDAS/BODMAS? It's our golden rule for solving mathematical expressions. It stands for:

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

In our problem, $478 \longdiv{27 + 36}$, we have addition within what the division bar acts as a bracket (or grouping symbol). Therefore, according to the order of operations, we must perform the addition before we perform the division. Understanding this fundamental rule is crucial. Without it, we'd end up with the wrong answer, and more importantly, we wouldn't be building a solid foundation for more complex math problems. This isn't just a one-time thing; the order of operations is a concept we'll use throughout our mathematical journey. So, let's make sure we've got it down pat. Think of it as the grammar of mathematics – just as grammar helps us structure sentences, the order of operations helps us structure mathematical expressions. We need to know which operations take precedence over others to ensure we arrive at the correct solution. For instance, if we were to ignore the order of operations and divide 478 by 27 first, then add 36, we'd get a completely different result. This highlights the importance of following the rules. It's not just about getting the right answer; it's about understanding the logical sequence of mathematical operations. So, always remember PEMDAS/BODMAS, and you'll be well on your way to mastering mathematical expressions!

Step-by-Step Solution: $27 + 36$

The first part of our equation, $27 + 36$, is a straightforward addition problem. We're simply adding two numbers together. Now, you might be able to do this in your head, and that's awesome! But let's break it down for those who prefer a more visual approach. We can think of this as combining two groups of objects. Imagine you have 27 apples, and someone gives you 36 more. How many apples do you have in total? That's what we're trying to figure out. There are a couple of ways we can tackle this. One way is to add the tens digits first, then the ones digits. So, we have 2 tens in 27 and 3 tens in 36. Adding those together gives us 5 tens, or 50. Then, we add the ones digits: 7 and 6. That gives us 13. Now, we just add 50 and 13 together, which gives us 63. Another way to think about it is to use the column method. We write the numbers on top of each other, aligning the ones and tens columns. Then, we add the ones column first: 7 + 6 = 13. We write down the 3 and carry over the 1 to the tens column. Then, we add the tens column: 1 (carried over) + 2 + 3 = 6. So, we get 63 again. No matter which method you choose, the answer is the same: $27 + 36 = 63$. This simple addition is the first crucial step in solving our larger problem. We've reduced the expression inside the division to a single number, which makes the next step much easier. So, we've conquered the addition, and we're ready to move on to the division. Let's keep this momentum going!

Dividing 478 by the Result

Now that we've solved the addition inside the parentheses, we know that $27 + 36 = 63$. Our problem now looks much simpler: $478 \div 63$. This is where the division magic happens! We need to figure out how many times 63 fits into 478. Long division can seem intimidating, but it's really just a systematic way of breaking down a division problem into smaller, more manageable steps. We're essentially asking,