Calculating 3 Times 7 Using A Custom Operation

Introduction

Hey guys! Today, we're diving into a fun math problem where we're not using the regular multiplication symbol. Instead, we've got a special operation represented by the asterisk *. This operation is defined as a * b = (a * b) ÷ 2. Our mission, should we choose to accept it, is to calculate 3 * 7 using this unique operation. Sounds intriguing, right? Let's break it down step by step and make sure we understand the logic behind it. We'll explore the definition, apply it to our specific numbers, and walk through the calculation to get our final answer. This kind of problem is super cool because it shows us that math isn't just about memorizing rules, but also about understanding how operations and definitions work. So, let's put on our thinking caps and get started!

Understanding the Operation

Okay, so first things first, let's really get what this operation a * b = (a * b) ÷ 2 is all about. What this means is that whenever we see two numbers with an asterisk between them, we're not just multiplying them in the usual way. Instead, we're multiplying them first, and then we're dividing the result by 2. It's like a mini-recipe for how to combine these two numbers. Think of it as a special mathematical function that takes two inputs (a and b), performs a calculation (multiplying a and b, then dividing by 2), and gives us a unique output. This is a crucial concept in mathematics because it allows us to define our own operations and explore their properties. This operation isn't your everyday multiplication, it's a modified version! It's super important to understand this definition before we jump into any calculations. Imagine if we tried to use regular multiplication – we'd end up with a totally different answer, and we wouldn't be solving the problem correctly. So, let's make sure we've got this down. Whenever we see a * b in this context, we know we need to multiply a and b together, and then, the most important part, we divide that result by 2. Got it? Great! Now we're ready to move on to applying this to our specific numbers.

Applying the Operation to 3 * 7

Alright, now that we've got a solid grip on what our special operation means, let's put it to work with the numbers we've got: 3 * 7. Remember, 3 * 7 isn't just regular multiplication here; it's the operation we just talked about. So, following our definition a * b = (a * b) ÷ 2, we can substitute a with 3 and b with 7. This gives us 3 * 7 = (3 * 7) ÷ 2. See how we're plugging in the numbers into our formula? It's like we're filling in the blanks. This step is super important because it helps us translate the abstract definition into a concrete calculation. We're not just staring at symbols anymore; we're actually setting up the math we need to do. Think of it like this: the operation is the recipe, and we're gathering the ingredients (our numbers 3 and 7) and getting ready to cook up the answer. The next thing we need to do is actually perform the calculation. We've set it up, we know what to do, and now it's time to crunch those numbers. So, let's move on to the next step, where we'll actually do the multiplication and division to find out what 3 * 7 equals in this special operation.

Performing the Calculation

Okay, let's get down to the actual calculation now. We've got 3 * 7 = (3 * 7) ÷ 2. The first thing we need to do, according to our order of operations (and the definition of our special operation), is the multiplication inside the parentheses. So, what's 3 multiplied by 7? That's right, it's 21. So now we can rewrite our equation as 3 * 7 = 21 ÷ 2. We've taken the first step and simplified things a bit. Now we're down to just one operation: division. We need to divide 21 by 2. If you're comfortable with division, you probably already know the answer. But if not, let's think about it. How many times does 2 go into 21? Well, it goes in 10 whole times (because 2 times 10 is 20), and we've got 1 left over. So, 21 divided by 2 is 10 with a remainder of 1. We can also write this as a decimal: 21 ÷ 2 = 10.5. And there we have it! We've performed the calculation. We multiplied 3 and 7, got 21, and then divided by 2 to get 10.5. This means that, according to our special operation, 3 * 7 = 10.5. Awesome! We've solved the problem. But before we celebrate too much, let's make sure we understand what we've done and why.

Final Answer: 10.5

So, after all that calculation, we've arrived at our final answer: 3 * 7 = 10.5 according to the given operation a * b = (a * b) ÷ 2. Isn't that neat? We took a simple multiplication problem and twisted it with a new rule, and we were able to find the answer by carefully following the definition. This really highlights the importance of understanding definitions in mathematics. If we hadn't understood that * meant something different than regular multiplication, we would have gotten a totally wrong answer. Remember, in math (and in life!), it's crucial to pay attention to the details and understand the rules of the game. We've seen how a seemingly small change – like dividing by 2 after multiplying – can completely change the outcome. This kind of problem is also a great example of how math can be creative and playful. We're not just memorizing formulas; we're actually using our brains to apply rules in new and interesting ways. So, the next time you see a strange symbol or a new operation, don't be intimidated! Just take a deep breath, break it down step by step, and remember the fundamental principles. You might just surprise yourself with what you can figure out. And who knows, maybe you'll even invent your own cool mathematical operation someday!

Conclusion

Alright guys, we've reached the end of our mathematical adventure for today! We successfully calculated 3 * 7 using the operation a * b = (a * b) ÷ 2, and we found that the answer is 10.5. We started by making sure we really understood the definition of the operation, which was super important. Then, we applied that definition to our specific numbers, 3 and 7. We carefully performed the multiplication and division, step by step, and arrived at our final answer. We also talked about how important it is to pay attention to definitions and rules in math, and how even small changes can make a big difference. This kind of problem shows us that math isn't just about memorizing formulas, it's about thinking critically and applying what we know in creative ways. I hope you had fun working through this problem with me! Remember, math is like a puzzle, and it's so satisfying when you finally put all the pieces together. Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!