Solve: Number Divided By 7, Multiplied By 6 Equals 48
Hey guys! Today, we're diving into a fun math problem that might seem a bit tricky at first, but don't worry, we'll break it down step by step. The question we're tackling is: What number, when divided by 7 and then multiplied by 6, results in 48? This type of problem involves working backward to find the original number, which is a common strategy in algebra and problem-solving. We will use some basic algebraic principles to solve this and, trust me, it is not as daunting as it sounds. So, grab your thinking caps, and let's get started!
Understanding the Problem
Before we jump into solving, let's make sure we understand what the problem is asking. We're looking for a specific number, let’s call it "x". This number has been through two operations: first, it was divided by 7, and then the result of that division was multiplied by 6. The final outcome of these two operations is 48. Understanding the order of operations is crucial here. We need to reverse these operations in the correct order to find our original number. Think of it like retracing your steps – if you walked forward and then turned left, you need to turn right and then walk backward to return to your starting point. In math, we use inverse operations to achieve this. Division is the inverse of multiplication, and multiplication is the inverse of division. Keeping this in mind will help us navigate the solution more effectively.
Now, let's break down the problem into smaller, manageable parts. We know that some number (x) divided by 7 can be represented as x/7. Then, this result is multiplied by 6, which gives us (x/7) * 6. And we know that the final result of this entire expression is 48. So, we can write this as an equation: (x/7) * 6 = 48. This equation is the key to unlocking our answer. By solving for x, we'll find the mystery number that fits all the conditions of the problem. So, the next step is to isolate x, and we'll do that by reversing the operations applied to it. This involves using inverse operations in the reverse order. Ready to dive into the solution? Let’s do it!
Setting up the Equation
Okay, so as we discussed, we can translate the problem into a mathematical equation. The core equation that represents our problem is (x/7) * 6 = 48. This equation is a concise way of expressing the series of operations performed on our unknown number, x. It tells us exactly what happened to x and what the final outcome was. Writing the problem as an equation is a crucial step because it transforms a word problem into a format that we can manipulate using mathematical rules and principles. Think of it like a secret code – once you have the code written down, you can start to decipher it. In this case, our code is the equation, and deciphering it means solving for x.
The left side of the equation, (x/7) * 6, represents the operations performed on x, while the right side, 48, represents the result of those operations. The equal sign (=) is the balance point, indicating that both sides of the equation are equal in value. Our goal now is to isolate x on one side of the equation so that we can determine its value. This involves using inverse operations, which are operations that undo each other. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. We'll use these inverse operations to peel away the layers around x until we have x by itself. This is like unwrapping a present – each step reveals a little more of what's inside. In our case, the present is the value of x, and we're about to unwrap it!
Now that we have our equation set up, the next step is to start solving it. Remember, the key is to reverse the order of operations and use inverse operations to isolate x. So, let's roll up our sleeves and get to the business of solving this equation. We're one step closer to finding our mystery number!
Solving the Equation: Step-by-Step
Alright, let's get down to the nitty-gritty and solve this equation step-by-step. Remember our equation: (x/7) * 6 = 48. The first thing we want to do is undo the multiplication by 6. To do this, we'll use the inverse operation, which is division. We'll divide both sides of the equation by 6. It’s super important to perform the same operation on both sides of the equation to keep the equation balanced – like keeping both sides of a scale level. If we divide only one side, the equation will no longer be true.
So, we divide both sides by 6: ((x/7) * 6) / 6 = 48 / 6. On the left side, the multiplication by 6 and the division by 6 cancel each other out, leaving us with x/7. On the right side, 48 divided by 6 is 8. So, our equation now looks like this: x/7 = 8. We've made progress! We've managed to isolate the term with x on one side, but we're not quite there yet. We still need to get x completely by itself. Now, we have x divided by 7 equals 8. To get x alone, we need to undo the division by 7. The inverse operation of division is multiplication, so we'll multiply both sides of the equation by 7. Again, we do this to both sides to maintain the balance of the equation.
Multiplying both sides by 7 gives us: (x/7) * 7 = 8 * 7. On the left side, the division by 7 and the multiplication by 7 cancel each other out, leaving us with just x. On the right side, 8 multiplied by 7 is 56. So, finally, we have our answer: x = 56! That's it! We've successfully solved the equation and found the value of x. This step-by-step approach is key to solving any algebraic equation. By breaking it down into smaller parts and using inverse operations, we can systematically isolate the variable and find its value. Now, let's make sure our answer makes sense in the context of the original problem.
Verifying the Solution
Awesome! We've found that x = 56, but before we declare victory, it's crucial to verify our solution. Verifying the solution is a critical step in problem-solving because it ensures that our answer is correct and that we haven't made any mistakes along the way. It's like double-checking your work before you submit it – you want to be sure you've got it right. So, how do we verify our solution? We plug the value we found for x back into the original equation and see if it holds true. Our original equation was (x/7) * 6 = 48. Now, let's substitute x with 56 and see what happens.
So, we have (56/7) * 6. First, we divide 56 by 7, which gives us 8. Then, we multiply 8 by 6, which gives us 48. So, our equation becomes 48 = 48. This is a true statement! This means that our solution, x = 56, is correct. We've successfully verified that when we divide 56 by 7 and then multiply the result by 6, we indeed get 48. It's always a good feeling when the numbers work out perfectly! Verifying our solution not only confirms our answer but also reinforces our understanding of the problem and the steps we took to solve it. It's a great way to build confidence in our problem-solving skills. Now that we've verified our solution, we can confidently say that 56 is the number we were looking for. But let’s summarize our journey from understanding the problem to finding the solution.
Conclusion: The Magic Number is...
Woo-hoo! We did it! We successfully solved the problem and found the number that, when divided by 7 and then multiplied by 6, gives us 48. The magic number is 56! It’s pretty satisfying when a math problem clicks into place, isn't it? We started with a question that might have seemed a bit perplexing, but by breaking it down into smaller steps, setting up an equation, and using inverse operations, we were able to find the answer. And, most importantly, we verified our solution to make sure we were spot on.
This problem is a great example of how mathematical problems can be approached systematically. We learned the importance of understanding the problem, translating it into an equation, solving the equation step-by-step, and verifying the solution. These are valuable skills that can be applied to many other types of problems, not just in math but in life in general. Remember, problem-solving is like building a puzzle – each step brings you closer to the complete picture. And when you finally fit that last piece in, the feeling of accomplishment is amazing!
So, next time you encounter a challenging problem, remember the steps we took today. Break it down, set up your tools (like equations), use the right techniques (like inverse operations), and always, always verify your solution. And most importantly, don't be afraid to dive in and give it a try. You might just surprise yourself with what you can achieve. Keep practicing, keep learning, and keep solving those problems. You've got this! Now go out there and conquer the world, one math problem at a time! Well done, guys!