Finding Positive Numbers Ratio 7 5 Difference Of Squares 96

Hey guys! Let's dive into a fun math problem today. We're going to tackle how to find two positive numbers that have a specific ratio and a difference of squares. It might sound a bit intimidating at first, but trust me, we'll break it down step by step and make it super easy to understand. This kind of problem often pops up in algebra, and mastering it will definitely boost your problem-solving skills. So, grab your thinking caps, and let's get started!

Understanding the Problem

Okay, first things first, let's make sure we're all on the same page about what the problem is asking. We're looking for two positive numbers. Remember, positive numbers are just numbers greater than zero – no negatives allowed here! These numbers have a special relationship with each other: their ratio is 7:5. What does that mean? Well, it means that if you divide the larger number by the smaller number, you'll get 7/5. Another way to think about it is that the larger number is 7 parts of some value, and the smaller number is 5 parts of the same value. This concept of ratios is super important in many areas of math and even in real-life situations like scaling recipes or understanding proportions in art and design. We can represent these numbers algebraically, which is a fancy way of saying we'll use letters to stand for them. Let's call the larger number 7x and the smaller number 5x. See how the 7:5 ratio is built right into our variables? This is a crucial step in setting up the problem correctly. Now, here's the other key piece of information: the difference of their squares is 96. Whoa, what does that mean? It simply means that if you take the square of the larger number (that's the number multiplied by itself) and subtract the square of the smaller number, you'll get 96. This involves a bit of algebra, specifically dealing with squares and differences. So, we have our ratio represented algebraically (7x and 5x) and we understand the difference of squares concept. We're well on our way to cracking this problem! Remember, the key to solving any math problem is to understand what it's asking and then break it down into smaller, manageable parts. And that's exactly what we're doing here. By understanding the ratio and the difference of squares, we've laid the groundwork for setting up an equation and finding our mystery numbers. Keep this clear understanding in mind as we move forward, and you'll see how everything starts to fit together like a puzzle.

Setting Up the Equation

Alright, guys, now that we've wrapped our heads around the problem, it's time to translate that understanding into an equation. This is where the algebra magic happens! Remember, we've already defined our numbers as 7x and 5x, which nicely captures the 7:5 ratio. And we know that the difference of their squares is 96. So, how do we write that mathematically? Well, the square of the larger number (7x) is (7x)^2, which simplifies to 49x^2. Similarly, the square of the smaller number (5x) is (5x)^2, which simplifies to 25x^2. Now, the difference of these squares is simply 49x^2 minus 25x^2. And we know this difference equals 96. Boom! We've got our equation: 49x^2 - 25x^2 = 96. See how we took the words and turned them into a powerful algebraic statement? This is a key skill in problem-solving – being able to bridge the gap between a written description and a mathematical representation. Now, before we jump into solving this equation, let's take a moment to appreciate what we've done. We've taken a word problem, identified the important information (the ratio and the difference of squares), represented the unknowns with variables (7x and 5x), and translated the problem's condition into a concrete equation. This process of setting up the equation is often the most challenging part of these types of problems, but it's also the most crucial. A well-set-up equation is half the battle won! So, take a deep breath and give yourself a pat on the back. You've successfully transformed a word problem into an algebraic equation. Now, the fun part begins – actually solving for x! We're on the home stretch, guys. With our equation in hand, we're just a few steps away from uncovering the two positive numbers we're seeking. Remember, math is like a puzzle, and we've just found a key piece. Let's keep going and fit all the pieces together to reveal the solution.

Solving the Equation

Okay, let's get down to business and solve this equation! We've got 49x^2 - 25x^2 = 96. The first thing we can do is simplify the left side of the equation. We have two terms with x^2, so we can combine them. 49x^2 minus 25x^2 is 24x^2. So, our equation now looks like this: 24x^2 = 96. Much simpler, right? Now, our goal is to isolate x^2. To do that, we need to get rid of the 24 that's multiplying it. How do we do that? We divide both sides of the equation by 24. This is a fundamental principle of algebra: whatever you do to one side of the equation, you must do to the other to keep it balanced. So, dividing both sides by 24, we get x^2 = 96 / 24. And 96 divided by 24 is 4. So, now we have x^2 = 4. We're getting closer! But we're not quite there yet. We need to find x, not x^2. How do we undo squaring? We take the square root! Remember, the square root of a number is a value that, when multiplied by itself, gives you the original number. So, we take the square root of both sides of the equation. The square root of x^2 is simply x. And the square root of 4 is 2 (since 2 * 2 = 4). But hold on a second! There's a little wrinkle here. Mathematically, the square root of 4 could be either 2 or -2, because (-2) * (-2) also equals 4. However, remember the original problem? We were looking for positive numbers. So, we can discard the negative solution. This is an important point to always keep in mind: pay attention to the context of the problem! In this case, the word "positive" is a crucial clue that helps us narrow down our answer. So, we have x = 2. We've found x! Give yourselves a round of applause! But wait, we're not quite done yet. Finding x is just one piece of the puzzle. We still need to find the two actual numbers we were looking for. And that's what we'll do in the next section. We're on the verge of solving this problem completely, so let's keep that momentum going!

Finding the Numbers

Alright, we've done the hard work and found that x = 2. Awesome! But remember, x is just a tool we used to help us find the actual numbers. The numbers themselves are 7x and 5x. So, how do we find them? It's simple: we just substitute the value of x (which is 2) back into these expressions. Let's start with the larger number, which is 7x. If x = 2, then 7x is 7 * 2, which equals 14. So, our larger number is 14. Easy peasy! Now, let's find the smaller number, which is 5x. Again, we substitute x = 2. So, 5x is 5 * 2, which equals 10. Our smaller number is 10. And there you have it! We've found the two positive numbers: 14 and 10. But before we declare victory, let's just double-check that these numbers actually satisfy the conditions of the problem. This is a crucial step in problem-solving: always verify your solution! First, let's check the ratio. Is the ratio of 14 to 10 equal to 7:5? Well, we can simplify the fraction 14/10 by dividing both the numerator and denominator by 2. This gives us 7/5. So, the ratio checks out! Next, let's check the difference of squares. Is 14^2 - 10^2 equal to 96? Let's calculate it. 14 squared (14 * 14) is 196. 10 squared (10 * 10) is 100. And 196 minus 100 is indeed 96! So, the difference of squares also checks out! We've done it! We've found two positive numbers that have a ratio of 7:5 and a difference of squares of 96. They are 14 and 10. Give yourselves a huge pat on the back! You've successfully navigated a multi-step problem, using algebra and logical reasoning. This is the kind of problem-solving skill that will serve you well in math and in life. Remember, math isn't just about getting the right answer; it's about the process of thinking, analyzing, and strategizing. And you've demonstrated those skills beautifully today.

Conclusion

So, guys, we've reached the end of our mathematical journey for today. We started with a seemingly complex problem – finding two positive numbers with a given ratio and a difference of squares – and we broke it down step by step until we arrived at a clear and confident solution. We saw how understanding the problem, setting up the equation, solving the equation, and verifying the solution are all essential parts of the problem-solving process. This wasn't just about finding the numbers 14 and 10; it was about learning a method that you can apply to a wide range of problems. And that's what makes math so powerful! The ability to think logically, break down complex issues, and find solutions is a skill that will benefit you in countless ways, whether you're building a bridge, designing a computer program, or making important decisions in your daily life. Remember, the key to mastering math is practice. The more problems you solve, the more comfortable you'll become with the techniques and the more confident you'll feel in your abilities. So, don't be afraid to tackle challenging problems. Embrace the struggle, learn from your mistakes, and celebrate your successes. You've shown today that you have the potential to be a fantastic problem solver. Keep that spirit alive, and who knows what mathematical heights you'll reach! And remember, if you ever get stuck, don't hesitate to ask for help. There are always people who are willing to guide you and support you on your mathematical journey. Keep exploring, keep learning, and most importantly, keep enjoying the beauty and power of mathematics! You guys rock!