Difference Of Squares And Ratios Explained Solve Math Problems Easily
Hey there, math enthusiasts! Ever stumbled upon a math problem that seems like a tangled web of numbers and relationships? Well, today, we're going to dissect one such intriguing puzzle. Let's dive into a problem that involves the difference of squares and ratios, and together, we'll unravel the mystery behind it. Our mission, should we choose to accept it, is to find two numbers given a couple of clues: their squares have a difference of 80, and their ratio is 3 to 2. Sounds like a challenge? Absolutely! But fear not, because with a sprinkle of algebraic magic and a dash of logical deduction, we'll crack this code in no time. So, buckle up, sharpen your pencils, and let's embark on this mathematical journey together!
Setting the Stage: Decoding the Problem
Before we jump into calculations, let's make sure we truly understand the challenge that lies before us. Understanding the problem is often half the battle in mathematics, guys. We've got two key pieces of information here, like clues in a detective novel. First up, we know that the difference between the squares of our two mystery numbers is 80. In mathematical lingo, if we call our numbers 'x' and 'y', this translates to the equation x² - y² = 80. Got it? Great! That's our first clue locked in.
Now, for the second clue: the ratio between these very same numbers is 3 to 2. Ah, ratios! They tell us how our numbers compare to each other. In equation form, this clue whispers to us that x/y = 3/2. Think of it like this: for every 3 units of 'x', there are 2 units of 'y'. This relationship is crucial, and it's going to help us connect 'x' and 'y' in a way that lets us solve for their actual values. So, with these two clues in hand – the difference of squares and the ratio – we're ready to transform this word problem into a solvable mathematical puzzle. We've laid the groundwork; now, let's get our hands dirty with some algebra!
The Algebraic Adventure Begins: Equations and Manipulations
Alright, with our clues decoded, it's time to roll up our sleeves and dive into the heart of the problem: the algebra. Don't worry, it's not as scary as it sounds! We're going to use our two equations – x² - y² = 80 and x/y = 3/2 – like tools in a toolbox, each helping us build our solution step by step. Our main goal here is to somehow combine these equations so that we can solve for 'x' and 'y' individually. It's like solving a jigsaw puzzle, where each equation is a piece that fits together to reveal the bigger picture.
Let's start by working on that ratio equation, x/y = 3/2. To make it a bit more user-friendly, we can rearrange it. Imagine we want to express 'x' in terms of 'y'. A simple multiplication trick does the charm: multiply both sides of the equation by 'y'. Voila! We get x = (3/2)y. This is a fantastic breakthrough! Why? Because now we have a direct relationship between 'x' and 'y'. We know that 'x' is exactly 3/2 times 'y'. This is going to be super handy when we tackle our other equation, the difference of squares.
Now, let's bring in the big guns: the equation x² - y² = 80. This might look intimidating, but remember, we've got a secret weapon – our newfound relationship between 'x' and 'y'. What if we substituted our expression for 'x' – that (3/2)y – into this equation? It's like plugging one piece of the puzzle into another. When we do this, we replace 'x' with (3/2)y, and our equation transforms into ((3/2)y)² - y² = 80. See what we did there? We've now got an equation that involves only 'y'. This is a huge step forward because it means we're closer to solving for 'y' and, consequently, for 'x'. So, let's keep going, guys! We're on the right track.
Cracking the Code: Solving for 'y'
Okay, so we've reached a pivotal moment in our mathematical quest. We've got the equation ((3/2)y)² - y² = 80, and it's our golden ticket to finding the value of 'y'. Don't be intimidated by the fractions and squares; we'll break it down step by step, just like seasoned problem-solvers do. Remember, the key is to take things one operation at a time.
First things first, let's tackle that squared term, ((3/2)y)². To square a fraction, we simply square both the numerator and the denominator. So, (3/2)² becomes 9/4. And, of course, y² remains y². So, our equation now looks like (9/4)y² - y² = 80. We're making progress! The equation is becoming clearer, like a picture slowly coming into focus.
Now, we've got two terms with y² in them. Think of this as having 9/4 of something and taking away 1 of that same something. To combine these terms, we need a common denominator. Remember those fraction rules from math class? We can rewrite y² as (4/4)y². So, our equation morphs into (9/4)y² - (4/4)y² = 80. Now we're talking! We can easily subtract the fractions: (9/4) - (4/4) equals 5/4. So, our equation simplifies beautifully to (5/4)y² = 80. Feels good, doesn't it, guys? We're really honing in on the solution now.
We're almost there! Our next goal is to isolate y². To do this, we need to get rid of that 5/4 that's hanging out in front. Remember, we can undo multiplication by dividing. So, we'll divide both sides of the equation by 5/4. But hold on! Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 5/4 is 4/5. So, we multiply both sides of our equation by 4/5. This gives us y² = 80 * (4/5). A little multiplication and simplification, and we find that y² = 64. Wow! We're so close we can taste it.
One final step to unveil the value of 'y'. We have y² = 64, and we want 'y' itself. What operation undoes squaring? That's right, the square root! So, we take the square root of both sides of the equation. The square root of y² is simply 'y', and the square root of 64 is... well, it's 8! So, we've cracked the code: y = 8. High fives all around! We've solved for one of our mystery numbers. But remember, we've got two numbers to find. So, our adventure isn't quite over yet.
The Grand Finale: Discovering 'x' and Checking Our Work
With the value of 'y' triumphantly in our grasp, finding 'x' is going to be a piece of cake. Remember that handy relationship we unearthed earlier, x = (3/2)y? This is where it really shines. We know 'y' is 8, so we can simply substitute that value into our equation. This gives us x = (3/2) * 8. A little bit of multiplication, and we discover that x = 12. Boom! We've found our other number. Give yourselves a pat on the back, guys – we're on a roll!
But hold on a second. Before we declare victory and bask in the glory of our mathematical prowess, there's one crucial step we absolutely must take: checking our work. It's like the final sweep of a detective at a crime scene, making sure we haven't missed any clues or made any mistakes. We need to make sure that our values for 'x' and 'y' – 12 and 8, respectively – actually satisfy the conditions of our original problem.
First, let's check the difference of squares. Is 12² - 8² really equal to 80? Let's calculate: 12² is 144, and 8² is 64. Subtracting, we get 144 - 64 = 80. Bingo! Our numbers pass the first test with flying colors. Now, let's tackle the ratio. Is the ratio of 12 to 8 really 3/2? We can simplify the fraction 12/8 by dividing both the numerator and denominator by their greatest common divisor, which is 4. This gives us 3/2. Double bingo! Our numbers nail the ratio test too.
So, there you have it, folks! We've not only found the two mystery numbers, but we've also rigorously verified that they fit all the clues. Our algebraic adventure has come to a satisfying conclusion. We can confidently say that the two numbers are indeed 12 and 8. What a journey! We've flexed our algebraic muscles, conquered fractions and squares, and emerged victorious. Remember, math problems like these aren't just about finding the right answer; they're about the thrill of the chase, the satisfaction of solving a puzzle, and the joy of learning something new. So, keep exploring, keep questioning, and keep unraveling those mathematical mysteries!
Difference of squares and ratio problem: Find two numbers where the difference of their squares is 80 and their ratio is 3/2.
Difference of Squares and Ratios Explained Solve Math Problems Easily
