Find The Smaller Number: Sum 3408, Difference 1904
Hey guys! Let's tackle this math problem together. We've got a classic scenario where we know the sum and the difference of two numbers, and our mission is to pinpoint the smaller one. It might sound tricky at first, but trust me, we'll break it down into easy-to-understand steps. So, if you're ready to sharpen those math skills, let's dive right in and figure out how to solve this! We'll explore the concepts and methods needed to crack this kind of problem, making it super clear and straightforward for everyone.
Understanding the Problem
So, the core of our problem states that the sum of two distinct numbers is 3408, and their difference clocks in at 1904. Our mission, should we choose to accept it, is to identify the smaller of these two elusive numbers. To make sure we're all on the same page, let's clarify what sum and difference mean in the context of mathematics. The sum, as you probably already know, is the result you get when you add two or more numbers together. For instance, the sum of 5 and 3 is 8 because 5 + 3 = 8. The difference, on the other hand, is what you get when you subtract one number from another. For example, the difference between 10 and 4 is 6, since 10 - 4 = 6. Understanding these basic concepts is crucial because they form the foundation upon which we'll build our solution. We need to be crystal clear about what we're trying to find, which is the smaller number in a pair where we know both their combined total and the gap between them. With this solid understanding, we're well-equipped to move forward and start thinking about strategies to solve the problem.
Setting Up Equations
Alright, let's translate this word problem into the language of math – equations! This is where we take the information we have and turn it into something we can actually work with. To start, we're going to assign variables to our unknown numbers. This is a standard practice in algebra because it allows us to represent quantities we don't know with symbols, making it easier to manipulate them. Let's call the larger number "x" and the smaller number "y." This simple step is super important because it gives us something concrete to refer to when we're setting up our equations. Now, let's use the information given in the problem to create our equations. We know that the sum of the two numbers is 3408. In equation form, that looks like this: x + y = 3408. This equation tells us that if we add our larger number (x) to our smaller number (y), we get 3408. We also know that the difference between the two numbers is 1904. That translates to another equation: x - y = 1904. This equation tells us that if we subtract the smaller number (y) from the larger number (x), we're left with 1904. So now we have a system of two equations with two variables:
- x + y = 3408
- x - y = 1904
This is a classic setup for solving using methods like substitution or elimination, which we'll dive into next. Creating these equations is a crucial step because it transforms the word problem into a mathematical puzzle that we can solve systematically. Stick with me, guys, we're getting closer to cracking this!
Solving the System of Equations
Now comes the fun part – actually solving for our unknowns! We've got our system of equations:
- x + y = 3408
- x - y = 1904
There are a couple of ways we could tackle this, but I think the easiest method here is the elimination method. The elimination method works best when you can easily add or subtract the equations to eliminate one of the variables. Looking at our equations, we can see that the 'y' terms have opposite signs (+y and -y). This is perfect because if we add the two equations together, the 'y' terms will cancel each other out, leaving us with just 'x' to solve for. So, let's do it! We'll add the left sides of the equations together and the right sides together:
(x + y) + (x - y) = 3408 + 1904
When we simplify, the 'y' terms disappear:
2x = 5312
Now we have a simple equation with just one variable. To solve for 'x', we just need to divide both sides of the equation by 2:
x = 5312 / 2 x = 2656
Great! We've found the value of 'x', which is the larger number. But remember, we're trying to find the smaller number, 'y'. No sweat, though! Now that we know 'x', we can plug it into either of our original equations to solve for 'y'. Let's use the first equation, x + y = 3408, because it looks a little simpler:
2656 + y = 3408
To isolate 'y', we subtract 2656 from both sides:
y = 3408 - 2656 y = 752
Fantastic! We've done it. We've solved for both 'x' and 'y'. We found that the larger number (x) is 2656 and the smaller number (y) is 752. But let's not forget what the question asked: we needed to find the smaller number. So, our final answer is 752. This elimination method is a powerful tool for solving systems of equations, and it really shone in this problem because of the opposite signs on the 'y' terms. Remember, the key is to look for ways to simplify the equations and eliminate variables until you can solve for the unknowns. You guys are doing awesome!
Identifying the Smaller Number
Okay, we've crunched the numbers and arrived at our two solutions: 2656 and 752. But remember, the original question specifically asked for the smaller number. This is a crucial step – always go back to the question and make sure you're answering exactly what's being asked! It's easy to get caught up in the calculations and forget the ultimate goal. So, a quick glance at our two numbers tells us that 752 is indeed smaller than 2656. Therefore, the smaller number is 752. This might seem like a really obvious step, but it's a vital habit to develop when you're solving math problems. It's like double-checking your work – you're making sure you haven't made a simple mistake or overlooked a key detail. In fact, before we move on, let's do a quick check to make sure our answer makes sense in the context of the original problem. We know the sum of the two numbers should be 3408, so let's add our numbers together: 2656 + 752 = 3408. Check! And we know the difference should be 1904, so let's subtract: 2656 - 752 = 1904. Double check! Our answers fit the conditions of the problem, which gives us a nice boost of confidence that we've done everything correctly. Identifying the smaller number and double-checking our work are essential steps in the problem-solving process. They ensure accuracy and help us avoid those silly mistakes that can sometimes trip us up. You guys are doing great at paying attention to these details! Now, let's wrap things up and summarize what we've learned.
Conclusion
Alright guys, we did it! We successfully navigated this math problem and found the smaller number. Let's take a moment to recap the steps we took to get there, because understanding the process is just as important as getting the right answer. First, we carefully read the problem and made sure we understood what it was asking. This is always the crucial first step – you can't solve a problem if you don't know what you're trying to find! Then, we translated the words into mathematical equations. We assigned variables to the unknown numbers (x and y) and used the given information (the sum and the difference) to create two equations. This step transformed the word problem into a more manageable algebraic problem. Next, we solved the system of equations. We chose the elimination method because it was a particularly good fit for this problem, but we could have also used substitution. By adding the equations together, we eliminated one variable and were able to solve for the other. Once we found one variable, we plugged it back into one of the original equations to solve for the second variable. Finally, we identified the smaller number and double-checked our work to make sure our answer made sense in the context of the problem. We added the two numbers together to make sure they summed to 3408, and we subtracted them to make sure the difference was 1904. This step gave us confidence in our solution. So, what have we learned? We've reinforced our understanding of how to translate word problems into equations, how to solve systems of equations using elimination, and the importance of carefully checking our work. These are valuable skills that will serve you well in all sorts of math problems. You guys rocked this! Keep practicing, and you'll become even more confident and skilled at problem-solving. Now, go forth and conquer those math challenges!