Solving $\frac{7 X^2}{12 W}-\frac{3 X^2}{12 W}$ A Step-by-Step Guide

Hey guys! Today, we're diving into a common yet crucial type of problem in algebra: performing indicated operations with fractions. Specifically, we're going to tackle the expression $\frac{7 x^2}{12 w}-\frac{3 x^2}{12 w}$. This kind of problem often pops up in various mathematical contexts, so mastering it is super important. We will break down this problem step by step, ensuring you understand not just the solution, but also the underlying principles. By the end of this article, you'll not only be able to solve this particular problem but also similar ones with confidence. Let's get started and make math a little less intimidating and a lot more fun!

Understanding the Basics of Fraction Subtraction

Before we jump into the specifics of our problem, let's quickly review the fundamentals of subtracting fractions. This will give us a solid foundation to work from. The most important thing to remember when dealing with fraction subtraction (or addition, for that matter) is that you need a common denominator. Think of it like this: you can't easily subtract apples from oranges, right? You need to have the same 'unit' to work with. Similarly, fractions need to have the same denominator before you can subtract their numerators.

Why Common Denominators Matter

The denominator of a fraction tells us how many parts a whole is divided into. If you're subtracting fractions with different denominators, you're essentially subtracting pieces of different sizes. Getting a common denominator is like resizing the pieces so they're all the same size, making subtraction straightforward. For instance, if you want to subtract $\frac{1}{2}$ from $\frac{3}{4}$, you can't directly subtract the numerators because the denominators are different. However, if you convert $\frac{1}{2}$ to $\frac{2}{4}$, now both fractions have the same denominator, and you can easily subtract: $\frac{3}{4} - \frac{2}{4} = \frac{1}{4}$.

Steps to Find a Common Denominator

1. Identify the Denominators: Look at the fractions you're working with and note their denominators.
2. Find the Least Common Multiple (LCM): The LCM of the denominators will be your common denominator. The LCM is the smallest multiple that both denominators share. There are several ways to find the LCM, including listing multiples or using prime factorization.
3. Adjust the Fractions: Once you have the common denominator, you need to adjust each fraction so that it has this new denominator. To do this, multiply both the numerator and the denominator of each fraction by the same number. This ensures that the value of the fraction remains unchanged.

Applying This to Our Problem

Now that we've covered the basics, let's bring it back to our original problem: $\frac{7 x^2}{12 w}-\frac{3 x^2}{12 w}$. Notice anything special about the denominators? They're the same! This makes our job a whole lot easier. We don't need to find a common denominator because we already have one. This is a crucial observation, and it simplifies the problem significantly. In the next section, we'll dive into how to actually perform the subtraction when the denominators are already aligned. Stay tuned, and let's keep this math train rolling!

Step-by-Step Solution of $\frac{7 x^2}{12 w}-\frac{3 x^2}{12 w}$

Okay, guys, let's get down to business and solve the expression $\frac{7 x^2}{12 w}-\frac{3 x^2}{12 w}$ step by step. We've already established that we have a common denominator, which is fantastic news! This means we can skip the sometimes tricky process of finding the least common multiple. When the denominators are the same, subtracting fractions becomes super straightforward. We simply subtract the numerators and keep the denominator the same. Think of it like this: if you have 7 slices of a pizza and you eat 3, you're left with 4 slices, and the size of the slices (the denominator) hasn't changed.

Step 1: Subtract the Numerators

In our case, the numerators are $7x^2$ and $3x^2$. We need to subtract $3x^2$ from $7x^2$. This is a simple algebraic subtraction. Since both terms have the same variable part ($x^2$), we can treat them as like terms and subtract their coefficients (the numbers in front of the variable part). So, we have:

$7x^2 - 3x^2$

To perform the subtraction, we subtract the coefficients: $7 - 3 = 4$. Therefore, $7x^2 - 3x^2 = 4x^2$. This is our new numerator.

Step 2: Keep the Denominator

The denominator in our expression is $12w$. Since we're subtracting fractions with a common denominator, we keep this denominator the same. So, our denominator remains $12w$.

Step 3: Write the Resulting Fraction

Now that we have our new numerator ($4x^2$) and our denominator ($12w$), we can write the resulting fraction:

$\frac{4x^2}{12w}$

Step 4: Simplify the Fraction (if possible)

This is a crucial step! Always check if your fraction can be simplified. In this case, both the numerator and the denominator have common factors. We can see that both 4 and 12 are divisible by 4. So, we can simplify the fraction by dividing both the numerator and the denominator by their greatest common factor, which is 4.

Dividing the numerator by 4, we get:

$\frac{4x^2}{4} = x^2$

Dividing the denominator by 4, we get:

$\frac{12w}{4} = 3w$

So, our simplified fraction is:

$\frac{x^2}{3w}$

And that's it! We've successfully performed the indicated operation and simplified the result. The final answer is $\frac{x^2}{3w}$. In the next section, we'll discuss how this answer corresponds to the multiple-choice options given and why it's the correct one. Keep up the great work!

Identifying the Correct Answer Choice

Alright, let's circle back to the original question and pinpoint the correct answer from the multiple-choice options provided. We've worked through the problem step-by-step, and we've arrived at the simplified fraction $\frac{x^2}{3w}$. Now, it's time to match our answer with the options and make sure we're on the right track. This is a critical skill in test-taking – you need to be able to not only solve the problem but also confidently identify the correct answer among the distractors.

Reviewing the Multiple-Choice Options

The options given were:

• A. $\frac{5 x^2}{6 w}$
• B. $\frac{x^2}{3 w}$
• C. $\frac{-x^2}{3 w}$
• D. $\frac{-5 x^2}{6 w}$

Matching Our Solution

We can clearly see that our solution, $\frac{x^2}{3w}$, matches option B. This is fantastic! It confirms that our step-by-step process has led us to the correct result. However, it's always a good idea to quickly examine the other options to understand why they are incorrect. This helps reinforce your understanding and prevents you from making similar mistakes in the future.

Why Other Options Are Incorrect

• Option A: $\frac{5 x^2}{6 w}$ – This option is incorrect because it seems to arise from either incorrectly subtracting the numerators or incorrectly simplifying the fraction. The numerator should be $7x^2 - 3x^2 = 4x^2$, not $5x^2$, and the simplified form should have a denominator of $3w$, not $6w$.
• Option C: $\frac{-x^2}{3 w}$ – This option introduces a negative sign, which is not present in our original problem. We are subtracting a positive term from a positive term, so the result should also be positive. This is a common mistake – watch out for sign errors!
• Option D: $\frac{-5 x^2}{6 w}$ – This option combines both the incorrect numerator from Option A and the incorrect negative sign from Option C. It's a double whammy of errors! Recognizing why these options are wrong is just as important as knowing why our answer is correct. It shows a deeper understanding of the underlying math and helps prevent future errors. In the next section, we'll wrap up by summarizing the key concepts and offering some final tips for tackling similar problems. Let's keep sharpening those math skills!

Key Takeaways and Tips for Similar Problems

Alright, guys, we've reached the end of our journey through solving $\frac{7 x^2}{12 w}-\frac{3 x^2}{12 w}$. Let's take a moment to recap the key takeaways and equip you with some tips for tackling similar problems in the future. Mastering these concepts will not only help you ace your math tests but also build a solid foundation for more advanced topics. Remember, math is like building a house – each concept builds upon the previous one, so a strong foundation is essential.

Key Takeaways

1. Common Denominators are Crucial: The most important thing to remember when adding or subtracting fractions is the need for a common denominator. If the denominators are different, you must find the least common multiple (LCM) and adjust the fractions accordingly.
2. Subtract (or Add) Numerators: Once you have a common denominator, you can simply subtract (or add) the numerators. The denominator remains the same.
3. Simplify, Simplify, Simplify: Always check if your resulting fraction can be simplified. Divide both the numerator and the denominator by their greatest common factor (GCF) to get the simplest form.
4. Pay Attention to Signs: Be extra careful with negative signs. A small mistake with a sign can completely change your answer.

Tips for Similar Problems

• Double-Check Your Work: Math can be tricky, and it's easy to make small errors. Always double-check your steps, especially when dealing with signs and simplification.
• Practice Makes Perfect: The more you practice, the more comfortable you'll become with these types of problems. Try working through similar examples and challenging yourself with more complex variations.
• Break Down Complex Problems: If you encounter a problem that seems daunting, break it down into smaller, more manageable steps. This will make the problem less intimidating and easier to solve.
• Understand the Concepts: Don't just memorize the steps. Make sure you understand the underlying concepts. This will help you adapt your knowledge to different types of problems.
• Use Real-World Examples: Sometimes, relating math concepts to real-world examples can make them easier to understand. Think about how fractions are used in cooking, measuring, or sharing things.

Final Thoughts

We've successfully navigated this problem, and hopefully, you now have a clearer understanding of how to subtract fractions with common denominators. Remember, math is a journey, not a destination. Keep practicing, keep asking questions, and most importantly, keep believing in yourself. You've got this! And remember, every problem you solve is a step closer to mastering mathematics. Keep up the great work, and I'll catch you in the next math adventure!