Evaluating (1/3)x - (3/4) When X = (1/4) A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little math problem. We need to figure out the value of the expression (1/3)x - 3/4 when x is equal to 1/4. It might seem a bit daunting at first, but trust me, it's super straightforward once we break it down step by step. Think of it like following a recipe – we just need to plug in the right ingredients (or in this case, numbers) and follow the instructions. So, grab your thinking caps, and let’s get started!

Understanding the Expression

Before we jump into plugging in the value, let's really understand what the expression (1/3)x - 3/4 is telling us. In math, an expression is a combination of numbers, variables, and operations. Here, we have a variable x, which is a placeholder for a number. We're told that x is equal to 1/4, so we're going to replace x with 1/4 in our expression. The expression involves two operations: multiplication and subtraction. The term (1/3)x means we're multiplying 1/3 by the value of x, which is 1/4. Then, we're subtracting 3/4 from the result of that multiplication. It’s like saying, “Take one-third of a number, then subtract three-fourths.” Understanding this basic structure is crucial because it guides us on how to solve the problem correctly. We need to remember the order of operations (PEMDAS/BODMAS), which tells us to do multiplication before subtraction. So, first, we'll handle the (1/3) * (1/4) part, and then we'll subtract 3/4 from the result. This foundational understanding makes the rest of the problem a breeze. We're not just blindly plugging in numbers; we're understanding the relationship between the numbers and the operations. This approach will help you tackle more complex problems later on. Remember, math is like building blocks – each concept builds upon the previous one, so mastering the basics is key. Now that we've got a solid grasp of the expression, let's move on to the next step: substituting the value of x.

Substituting x = 1/4

Okay, guys, now comes the fun part – substituting! We know that x = 1/4, so we're going to replace every instance of x in our expression with 1/4. Our expression (1/3)x - 3/4 becomes (1/3)(1/4) - 3/4. See? It’s like swapping out a player in a game. We’re taking x out and putting 1/4 in its place. This substitution is a crucial step because it allows us to move from an expression with a variable to one with only numbers, which we can then simplify. Now, let’s talk about why this step is so important. In algebra, variables represent unknown quantities. By giving x a specific value, we’re turning the unknown into a known. This is the foundation of solving equations and evaluating expressions. We’re essentially saying, “If x is this, then what is the value of the whole thing?” This process of substitution is used everywhere in math, from simple algebraic expressions to complex calculus problems. So, mastering it here will really pay off down the road. Imagine trying to bake a cake without knowing how much flour to use – you'd end up with a mess! Similarly, in math, you can't solve an expression without knowing the values of the variables. Substitution is how we get those values into the mix. Now that we've successfully substituted, we have a purely numerical expression. We’re ready to roll up our sleeves and simplify it, which means performing the operations in the correct order. Remember PEMDAS/BODMAS? It's about to come in super handy! Let's move on to the next part where we'll simplify this expression and find our answer.

Simplifying the Expression

Alright, let’s simplify! We've got (1/3)(1/4) - 3/4. Remember our order of operations (PEMDAS/BODMAS)? We need to handle the multiplication before we tackle the subtraction. So, first things first, let’s multiply (1/3) by (1/4). When multiplying fractions, it's super simple – we just multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, (1/3) * (1/4) = (1 * 1) / (3 * 4) = 1/12. Easy peasy, right? Now our expression looks like this: 1/12 - 3/4. We're one step closer to the finish line! But before we can subtract these fractions, there's a little catch. We need to make sure they have the same denominator. Why? Because we can only directly add or subtract fractions that are “cut into the same number of pieces.” Think of it like trying to compare apples and oranges – they’re different until you put them in the same context (like saying they're both fruits). In our case, 1/12 is cut into 12 pieces, and 3/4 is cut into 4 pieces. We need to find a common denominator, which is a number that both 12 and 4 divide into evenly. The smallest such number is 12 (which is awesome because one of our fractions already has a denominator of 12!). So, we need to convert 3/4 into an equivalent fraction with a denominator of 12. To do this, we ask ourselves, “What do we multiply 4 by to get 12?” The answer is 3. So, we multiply both the numerator and the denominator of 3/4 by 3: (3/4) * (3/3) = 9/12. Now our expression is 1/12 - 9/12. We’ve got common denominators, and we’re ready for the final subtraction step. We are almost there, guys!

Final Calculation and Answer

Okay, guys, we're in the home stretch! We've simplified our expression down to 1/12 - 9/12. Now, to subtract fractions with the same denominator, we simply subtract the numerators and keep the denominator the same. So, 1/12 - 9/12 = (1 - 9) / 12. What is 1 minus 9? It’s -8! So, we have -8/12. We’re not quite done yet, though. We should always try to simplify our fractions to their simplest form. This means finding the greatest common factor (GCF) of the numerator and the denominator and dividing both by it. In this case, the GCF of 8 and 12 is 4. So, we divide both -8 and 12 by 4: (-8 / 4) / (12 / 4) = -2/3. And there you have it! The value of the expression (1/3)x - 3/4 when x = 1/4 is -2/3. Woohoo! We did it! This final simplification step is important because it presents our answer in the most concise and understandable way. Think of it like giving someone directions – you want to give them the simplest, most direct route, not a roundabout, confusing one. Similarly, in math, we want to express our answers in their simplest form. Simplifying fractions not only makes them easier to understand but also makes them easier to work with in future calculations. It's a fundamental skill that will serve you well in all your math adventures. So, always remember to check if your fraction can be simplified before you declare victory! Now, let’s recap what we did and celebrate our success!

Recap and Conclusion

Let's do a quick recap of what we've accomplished today, guys! We started with the expression (1/3)x - 3/4 and the value x = 1/4. Our mission? To find the value of the expression when we substitute x with 1/4. We broke the problem down into manageable steps: First, we understood the expression, recognizing the multiplication and subtraction operations. Then, we substituted x with 1/4, turning our expression into (1/3)(1/4) - 3/4. Next, we simplified. We started by multiplying 1/3 and 1/4 to get 1/12. Then, we dealt with the subtraction. We found a common denominator for 1/12 and 3/4, converting 3/4 to 9/12. This allowed us to subtract the fractions: 1/12 - 9/12 = -8/12. Finally, we simplified our fraction -8/12 by dividing both the numerator and denominator by their greatest common factor, 4, resulting in our final answer: -2/3. See? When we break down a problem into smaller, more manageable steps, it becomes much less intimidating. This is a great strategy to use in all areas of life, not just math! By understanding each step and why we're doing it, we gain a deeper understanding of the concepts involved. Math isn't just about memorizing formulas; it's about understanding the logic and reasoning behind them. And you guys totally rocked it today! You’ve tackled a problem involving fractions, substitution, and the order of operations. These are crucial skills that will help you in more advanced math topics. So, give yourselves a pat on the back! Remember, practice makes perfect. The more you work with expressions and equations, the more comfortable you'll become. Keep challenging yourselves, and don't be afraid to ask questions. Math is a journey, and we're all in it together. Keep up the awesome work, everyone!