Solve: A Number's Quarter Plus Four-Thirds Equals Itself

Hey guys! Ever get those math problems that seem like they're written in another language? Well, let's break down one of those today. We're going to tackle a problem where we need to find a number that fits a specific condition. Think of it like a puzzle – we have all the pieces; we just need to put them together in the right way. So, let's get started and make math a little less mysterious!

Understanding the Problem

So, let's dive into the problem we've got here. The main keyword is understanding the core question: "What number, when one-fourth of it is increased by four-thirds, equals the number itself?" It might sound a bit complicated at first, but don't worry, we'll break it down step by step. The trick to solving these kinds of problems is to translate the words into mathematical expressions. Think of it like learning a new language – we're just translating English into Math! We'll identify the unknowns, the operations, and the relationships between them. This is where the magic happens – when we turn words into equations. Trust me, once you get the hang of this, these problems become way less intimidating.

Translating Words into Math

Okay, let's get down to the nitty-gritty of translating this word problem into a mathematical equation. This is where we put on our detective hats and decode the language of math. The most important part here is to identify the unknown. What is it that we're actually trying to find? In this case, it's "the number." Since we don't know what it is yet, we'll call it something like "x." This is a classic move in algebra – using letters to represent unknown values. Now, let's look at the other parts of the problem. "One-fourth of a number" translates to (1/4) * x, or x/4. "Increased by four-thirds" means we're adding 4/3 to something. And "equals the number itself" means it all adds up to x. See how we're piecing it together? It's like building a sentence, but with math symbols. So, let's put it all together and write out the equation. This is the key step in solving the problem.

Setting Up the Equation

Alright, guys, this is where we get to the heart of the matter: setting up the equation. We've already translated the words into mathematical bits and pieces, and now it's time to assemble them. Remember, our unknown number is "x." "One-fourth of the number" is x/4. "Increased by four-thirds" means we add 4/3. And the whole thing "equals the number itself," which is just x. So, when we put it all together, we get the equation: x/4 + 4/3 = x. This might look a little intimidating if you're not used to it, but trust me, it's just a mathematical sentence. We're saying that if you take one-fourth of a number and add four-thirds to it, you get the original number back. Pretty neat, huh? Now, the next step is to actually solve this equation, which means finding out what value of "x" makes this statement true. We'll use some algebraic techniques to isolate "x" and figure out its value. So, let's move on to the solving part!

Solving the Equation

Okay, now for the fun part: solving the equation! This is where we put on our algebraic hats and manipulate the equation to find out what "x" actually is. Remember, our equation is x/4 + 4/3 = x. The goal here is to get "x" all by itself on one side of the equation. To do that, we'll need to use some basic algebraic principles, like adding and subtracting the same thing from both sides, and multiplying or dividing both sides by the same thing. It's like balancing a scale – whatever you do to one side, you have to do to the other to keep it even. We'll start by getting rid of the fractions, because let's be honest, fractions can be a bit of a pain to work with. Then, we'll move all the "x" terms to one side and the constants to the other. Step by step, we'll whittle down the equation until we have "x = something." That "something" is our answer! So, let's dive in and get our hands dirty with some algebra.

Isolating the Variable

Alright, let's talk about isolating the variable. Isolating the variable is a crucial step in solving equations. This basically means getting "x" (our unknown number) all by itself on one side of the equals sign. Think of it like giving "x" its own VIP section in the equation. To do this, we need to undo all the operations that are happening to "x." If something is being added to "x," we subtract it from both sides. If "x" is being multiplied by something, we divide both sides by that something. It's all about doing the opposite operation to get "x" alone. This might involve a few steps, but each step brings us closer to the solution. We're like mathematical detectives, carefully following the clues until we find our answer. So, let's put on our detective hats and start isolating that "x"!

Step-by-Step Solution

Alright, guys, let's get into the step-by-step solution of our equation: x/4 + 4/3 = x. This is where we put our algebra skills to the test. First things first, let's get rid of those pesky fractions. The easiest way to do this is to multiply both sides of the equation by the least common multiple (LCM) of the denominators, which in this case is 12 (the LCM of 4 and 3). So, we multiply both sides by 12: 12 * (x/4 + 4/3) = 12 * x. Now, we distribute the 12 on the left side: (12 * x/4) + (12 * 4/3) = 12x. This simplifies to 3x + 16 = 12x. See how the fractions are gone? Much cleaner, right? Next, we want to get all the "x" terms on one side. So, let's subtract 3x from both sides: 3x + 16 - 3x = 12x - 3x. This gives us 16 = 9x. Now, we're almost there! To isolate "x," we need to divide both sides by 9: 16/9 = 9x/9. This simplifies to x = 16/9. And there you have it! We've solved for "x." The number we're looking for is 16/9. High five! We conquered the equation. Now, let's move on to verifying our solution to make sure we got it right.

Verifying the Solution

Okay, so we've found a solution, but how do we know if it's the right solution? That's where verifying comes in. Verifying the solution is like double-checking your work – it's a crucial step in problem-solving. We're going to take the value we found for "x" (which is 16/9) and plug it back into the original equation: x/4 + 4/3 = x. If both sides of the equation are equal after we substitute "x," then we know we've got the correct answer. It's like a mathematical truth serum – if our solution makes the equation true, then we know it's legit. This step not only confirms our answer but also helps us catch any mistakes we might have made along the way. So, let's plug in 16/9 and see if it works!

Plugging the Solution Back In

Alright, let's plug our solution back into the original equation and see if it holds up. Remember, our equation is x/4 + 4/3 = x, and our solution is x = 16/9. So, we're going to substitute 16/9 for "x" in the equation: (16/9)/4 + 4/3 = 16/9. Now, let's simplify. (16/9)/4 is the same as (16/9) * (1/4), which equals 16/36. We can simplify this fraction by dividing both the numerator and denominator by 4, which gives us 4/9. So, now we have 4/9 + 4/3 = 16/9. To add these fractions, we need a common denominator. The least common denominator of 9 and 3 is 9. So, we rewrite 4/3 as 12/9. Now our equation looks like this: 4/9 + 12/9 = 16/9. Adding the fractions on the left side, we get 16/9 = 16/9. Hooray! The equation balances. Both sides are equal. This means our solution, x = 16/9, is correct. We did it! We found the number that, when one-fourth of it is increased by four-thirds, equals the number itself. Feels good, right? Now, let's wrap things up with a final summary of our solution.

Conclusion

So, guys, we've reached the end of our mathematical journey! We started with a word problem that seemed a bit tricky, but we broke it down step by step and conquered it. We translated the words into an equation, solved the equation using algebraic techniques, and then verified our solution to make sure it was correct. Our problem was: "What number, when one-fourth of it is increased by four-thirds, equals the number itself?" And after all our hard work, we found the answer: the number is 16/9. Isn't it satisfying to solve a math puzzle like this? It's like unlocking a secret code. Remember, the key to solving these kinds of problems is to take them one step at a time, translate the words carefully, and don't be afraid to get your hands dirty with some algebra. Math can be challenging, but it can also be really rewarding. So, keep practicing, keep exploring, and keep those mathematical muscles strong! You've got this! Now go out there and conquer some more math problems!