Finding Two Numbers Two Thirds Of Sum Is 92 And Three Eighths Of Difference Is 3
Hey guys! Ever get those math problems that seem like they're speaking another language? Well, let's break one down together. We're going to tackle a problem where we need to find two numbers, but we're given some clues about their sum and difference. It sounds tricky, but trust me, we'll get through it! This is like a fun puzzle, and who doesn’t love puzzles? So, grab your thinking caps, and let’s dive into the world of algebra! We’ll turn this word problem into something super understandable. Math can be super fun when you approach it the right way, and we’re gonna make sure it is! So, let’s get started and unlock the secrets hidden in these numbers. Ready to become math wizards? Let’s go!
Understanding the Problem
So, the problem goes like this: We need to find two numbers. We're told that two-thirds of their sum is 92, and three-eighths of their difference is 3. Sounds like a mouthful, right? But don't worry, we’ll take it step by step. The first key to solving any math problem is really understanding what it’s asking. Let's break down each piece of information. “Two-thirds of their sum is 92” – what does that mean? It means if you add the two numbers together and then take two-thirds of that total, you'll get 92. Okay, got it! And then, “three-eighths of their difference is 3.” This means if you subtract the smaller number from the larger one and then take three-eighths of that difference, you’ll end up with 3. See? We’re already making progress just by understanding the words. Now, let’s think about how we can turn these words into math equations. Equations are like the secret language of math, and once we translate our problem into equations, solving it becomes so much easier. We’re basically becoming codebreakers, but with numbers instead of secret messages. The more comfortable you get with turning word problems into equations, the more math problems you'll be able to conquer. It’s like having a superpower! So, let’s keep going and see how we can do just that.
Setting Up the Equations
Alright, let's get those math equations set up! This is where we turn our words into symbols and numbers. Let’s call our two mystery numbers x and y. It’s super common to use x and y in algebra, so you’ll see these a lot. Now, let’s tackle the first part of the problem: “Two-thirds of their sum is 92.” How do we write that as an equation? Well, the sum of x and y is simply x + y. Two-thirds of that sum is (2/3)(x + y). And we know this equals 92. So, our first equation is (2/3)(x + y) = 92. Boom! We’ve got one equation down. Feels good, doesn’t it? Now, let’s move on to the second part: “Three-eighths of their difference is 3.” The difference between x and y is x - y (we'll assume x is the larger number). Three-eighths of that difference is (3/8)(x - y). And this equals 3. So, our second equation is (3/8)(x - y) = 3. Awesome! We now have two equations, and with two equations, we can solve for our two unknowns, x and y. Think of it like having two pieces of a puzzle – we can now fit them together to see the whole picture. Setting up equations is a crucial step in solving word problems. It’s like building the foundation of a house – if your foundation is solid, the rest of the house will stand strong. So, let’s make sure our foundation is rock solid, and then we can move on to solving the equations.
Solving the Equations
Okay, we've got our equations set up, now comes the fun part – solving them! We have two equations:
- (2/3)(x + y) = 92
- (3/8)(x - y) = 3
Let’s start with equation 1. To get rid of that fraction, we can multiply both sides of the equation by 3/2 (the reciprocal of 2/3). This gives us x + y = 92 * (3/2). Calculating that, we get x + y = 138. Great! We’ve simplified our first equation. Now, let’s tackle equation 2. Similarly, to get rid of the fraction, we multiply both sides by 8/3 (the reciprocal of 3/8). This gives us x - y = 3 * (8/3). Calculating that, we get x - y = 8. Fantastic! We’ve also simplified our second equation. Now we have a much simpler system of equations:
- x + y = 138
- x - y = 8
We can use a method called elimination to solve this. Notice that we have a +y in the first equation and a -y in the second equation. If we add these two equations together, the y terms will cancel out! So, let’s do it: (x + y) + (x - y) = 138 + 8. This simplifies to 2x = 146. To find x, we divide both sides by 2: x = 73. Yay! We’ve found one of our numbers! Now that we know x, we can plug it into either equation to find y. Let’s use the first equation, x + y = 138. Substituting x = 73, we get 73 + y = 138. Subtracting 73 from both sides, we find y = 65. We did it! We’ve solved for both x and y. Solving equations can feel like a bit of a workout for your brain, but it’s so rewarding when you finally crack the code. And the more you practice, the easier it gets. So, let’s celebrate this victory, and then we’ll make sure our answer makes sense.
Checking the Solution
Alright, we’ve found our numbers: x = 73 and y = 65. But before we do a victory dance, we need to make sure our solution actually works. It’s like double-checking your work on a test – you want to be absolutely sure you’ve got the right answer. So, let’s plug our values back into the original equations and see if they hold true. Remember our first equation? It was (2/3)(x + y) = 92. Let’s substitute x = 73 and y = 65: (2/3)(73 + 65) = (2/3)(138) = 92. Bingo! It checks out. Our first equation is happy. Now, let’s check the second equation: (3/8)(x - y) = 3. Substituting again, we get (3/8)(73 - 65) = (3/8)(8) = 3. Woohoo! That one checks out too. Both of our original conditions are satisfied, so we can confidently say that our solution is correct. x = 73 and y = 65 are indeed the numbers we were looking for. Checking your solution is a super important habit to get into. It’s like having a built-in safety net – it catches any mistakes you might have made along the way. And it gives you that extra boost of confidence knowing that you’ve got the right answer. So, always, always, always check your solution! Now that we’ve done that, we can finally celebrate our math victory. High fives all around!
Final Answer
Okay, guys, after all that brainpower, we’ve finally arrived at the final answer! We set up our equations, we solved them, and we even checked our work to make sure we were spot on. So, what are the two numbers we were searching for? Drumroll, please… The two numbers are 73 and 65! Isn’t it satisfying when you finally solve a tricky problem? It’s like completing a challenging level in a video game or finishing a really good book. You get that sense of accomplishment and the satisfaction of knowing you’ve conquered something tough. This problem might have seemed intimidating at first, but we broke it down step by step, and now we’ve got our answer. Remember, math is all about taking things one piece at a time. Don’t get overwhelmed by the whole problem – just focus on understanding each step, setting up your equations correctly, and solving them carefully. And always, always check your work! With practice, you’ll become a math-solving superstar. So, keep challenging yourself, keep learning, and keep having fun with math. And now, let’s celebrate our success! We found those numbers, and we did it together. You guys rock!
