Fraction Problem: Finding The Difference Of Terms

Hey guys! Let's dive into an exciting math problem today that involves equivalent fractions. We're going to break down a problem where the sum of the terms of a fraction equivalent to 9/7 equals 192. Our mission, should we choose to accept it, is to calculate the difference between these terms. Sounds like fun, right? Math can be like a puzzle, and we're here to solve it together. We will explore how to find equivalent fractions and use simple algebra to solve this problem. So, grab your pencils and let's get started!

Understanding Equivalent Fractions

First things first, let's chat about equivalent fractions. Equivalent fractions are fractions that might look different but actually represent the same value. Think of it like this: 1/2 is the same as 2/4, 3/6, and so on. They're all just different ways of showing the same amount. To find an equivalent fraction, you multiply (or divide) both the numerator (the top number) and the denominator (the bottom number) by the same number. For example, if we have the fraction 9/7, we can multiply both 9 and 7 by the same number, say 2, to get 18/14, which is an equivalent fraction. The key here is that the value of the fraction doesn't change; we're just expressing it in different terms. So, when we're dealing with equivalent fractions, we are essentially scaling the fraction up or down while maintaining its original proportion. This concept is super important because it helps us simplify fractions, compare them, and, as we'll see in our problem, solve some cool equations. This principle of equivalent fractions is crucial in many areas of math, from basic arithmetic to more complex algebra and calculus. So, understanding this concept well sets a solid foundation for future math adventures.

Setting Up the Problem

Now, let's get back to our original problem. We have a fraction that's equivalent to 9/7, and we know that the sum of its terms is 192. Let's call the equivalent fraction 9x/7x. Why? Because we're multiplying both the numerator and the denominator of the original fraction (9/7) by the same number, x, to get an equivalent fraction. This is a common technique in algebra – using variables to represent unknown quantities. So, 9x represents the numerator of our equivalent fraction, and 7x represents the denominator. The problem tells us that if we add these terms together, we get 192. So, we can write this as an equation: 9x + 7x = 192. This is a simple algebraic equation, and setting it up correctly is half the battle. Once we have this equation, we're on the right track to finding the value of x, which will then help us find the actual terms of the equivalent fraction. Remember, the goal here is to translate the word problem into mathematical language, and this equation does just that. It captures the essence of the problem in a concise and solvable form. So, with our equation in hand, let's move on to solving it and unraveling the mystery of this fraction!

Solving for x

Alright, guys, we've got our equation: 9x + 7x = 192. Now it's time to put on our algebra hats and solve for x. The first thing we can do is combine like terms. 9x and 7x are like terms because they both have x. When we add them together, we get 16x. So our equation now looks like this: 16x = 192. Much simpler, right? Now, to isolate x, we need to get it all by itself on one side of the equation. To do that, we can divide both sides of the equation by 16. Why 16? Because that's the number that's currently multiplying x. Dividing both sides by 16 keeps the equation balanced (which is super important in algebra) and helps us get x alone. So, we have 16x / 16 = 192 / 16. On the left side, the 16s cancel out, leaving us with just x. On the right side, 192 divided by 16 is 12. So, we've found that x = 12. Woohoo! We're one step closer to solving the whole problem. Remember, x is the number we multiplied both the numerator and denominator of our original fraction (9/7) by to get our equivalent fraction. So, now that we know x, we can find the actual terms of that fraction.

Finding the Terms of the Equivalent Fraction

Now that we know x = 12, let's find the terms of our equivalent fraction. Remember, we called the equivalent fraction 9x/7x. So, to find the numerator, we multiply 9 by x, which is 12. 9 * 12 = 108. That's our numerator! To find the denominator, we multiply 7 by x, which is also 12. 7 * 12 = 84. That's our denominator! So, our equivalent fraction is 108/84. We've found it! But we're not done yet. The problem asks us to find the difference between these terms, not just the terms themselves. So, we're in the home stretch now. We've done the hard work of setting up the equation, solving for x, and finding the equivalent fraction. Now, we just need to do one more simple calculation to get to our final answer. This is a great example of how math problems often have multiple steps, and it's important to take them one at a time. We've broken down the problem into smaller, manageable parts, and now we're ready to wrap it up. So, let's move on to the final calculation and get this problem solved!

Calculating the Difference

Okay, guys, we're almost there! We've found that our equivalent fraction is 108/84. Now, the final step is to calculate the difference between these terms. That means we need to subtract the smaller number (84) from the larger number (108). So, we have 108 - 84. This is a simple subtraction problem. When we do the math, we find that 108 - 84 = 24. And that's it! We've found the difference between the terms of the equivalent fraction. It's like reaching the top of a mountain after a long climb – we can finally see the view! This final calculation is a great reminder that sometimes the last step is the simplest, but it's still important to get it right. We've used all our math skills – from understanding equivalent fractions to solving algebraic equations – to get to this point. And now we have our answer. So, let's celebrate our success and write down our final answer in a clear and concise way.

Final Answer

So, after all that awesome math work, we've arrived at our final answer. The difference between the terms of the fraction equivalent to 9/7, where the sum of the terms is 192, is 24. We did it! We took a problem that might have seemed a bit tricky at first and broke it down into manageable steps. We understood what equivalent fractions are, set up an equation, solved for x, found the terms of the equivalent fraction, and then calculated the difference. That's a lot of math in one problem! But we tackled it together, step by step. This kind of problem-solving is what math is all about – taking a challenge and using our skills to find the solution. And remember, every math problem is a chance to learn something new and get a little bit better. So, keep practicing, keep exploring, and keep having fun with math! You've got this!