Solving Fraction Equations: A Step-by-Step Guide

Hey guys! Let's dive into the world of fractions and tackle the equation: (4/7 - 1/2) + (4/3 - 4/7) / 1 + 1/7. Don't worry, it might look a bit intimidating at first, but with a step-by-step approach, we'll break it down and make it super easy to understand. We'll be using the order of operations (PEMDAS/BODMAS) and some basic fraction arithmetic. This will ensure we solve it correctly and confidently. Ready to get started? Let's go!

Step 1: Solving Parentheses (or the expressions inside the parentheses)

First things first, according to the order of operations, we need to deal with what's inside the parentheses. We have two sets of parentheses in this equation: (4/7 - 1/2) and (4/3 - 4/7). Let's start with the first set: (4/7 - 1/2). To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 7 and 2 is 14. So, we'll convert both fractions to have a denominator of 14.

To convert 4/7, we multiply both the numerator and denominator by 2: (4 * 2) / (7 * 2) = 8/14. For 1/2, we multiply both the numerator and denominator by 7: (1 * 7) / (2 * 7) = 7/14. Now we can subtract: 8/14 - 7/14 = 1/14. Awesome, we've solved the first set of parentheses! That gives us 1/14.

Now, let's look at the second set of parentheses: (4/3 - 4/7). Again, we need a common denominator. The LCM of 3 and 7 is 21. Convert 4/3 by multiplying the numerator and denominator by 7: (4 * 7) / (3 * 7) = 28/21. Convert 4/7 by multiplying the numerator and denominator by 3: (4 * 3) / (7 * 3) = 12/21. Now, subtract: 28/21 - 12/21 = 16/21. Cool, we've solved the second set of parentheses! So, this gives us 16/21.

Now our equation looks much simpler. Replacing the content of the parentheses, the equation looks like this: 1/14 + 16/21 / 1 + 1/7. See? We're making progress. Let's move on to the next step and continue the journey of simplification and solution. Remember, patience and a methodical approach are key when working with fractions. We'll soon have the final answer. We'll keep going, and you'll find that it's much easier than you initially thought. Let's break down the fractions and combine them together to get the final answer. The journey from complex fractions to a simplified answer is a rewarding one.

Step 2: Division

Alright, moving on to the next operation in our order of operations: division. We have 16/21 divided by 1, or (16/21) / 1. Dividing any number by 1 doesn't change the value. So, 16/21 / 1 is simply 16/21. Now our equation is looking even simpler: 1/14 + 16/21 + 1/7.

See how we're breaking it down step by step? With each operation, the equation gets less complicated. Now we know, division by 1 doesn't change anything, so the division step was actually quite easy. With one less thing to worry about, we can concentrate on the next stage. It is really awesome to see how an equation simplifies. It's like watching a puzzle come together! We're nearly there. The next steps will involve adding and simplifying to get the final answer. The key takeaway here is to always follow the correct order of operations. You'll never go wrong. So, keep your eye on the prize, and let's get that final answer!

Step 3: Addition

Okay, time for the final step: addition! Our equation now looks like this: 1/14 + 16/21 + 1/7. To add these fractions, we need a common denominator. The LCM of 14, 21, and 7 is 42. So, let's convert each fraction to have a denominator of 42.

For 1/14, we multiply both numerator and denominator by 3: (1 * 3) / (14 * 3) = 3/42. For 16/21, we multiply both numerator and denominator by 2: (16 * 2) / (21 * 2) = 32/42. For 1/7, we multiply both numerator and denominator by 6: (1 * 6) / (7 * 6) = 6/42. Now we can add the numerators: 3/42 + 32/42 + 6/42 = (3 + 32 + 6) / 42 = 41/42. And that's it! We've solved the equation. The final answer is 41/42. Congratulations!

We did it, guys! We started with a complex-looking fraction equation, but through methodical steps and by remembering the rules, we reached our final answer, 41/42. See? Fractions aren't so scary once you break them down. It is really important to use these rules step-by-step.

Summary of Steps and Key Takeaways

Let's recap what we did:

1. Parentheses: Solved the expressions inside the parentheses first, finding common denominators where necessary.
2. Division: Performed the division operation.
3. Addition: Added the fractions together, finding a common denominator.

Key Takeaways:

• Order of Operations: Always follow the order of operations (PEMDAS/BODMAS).
• Common Denominators: To add or subtract fractions, you must have a common denominator.
• Step-by-Step Approach: Break down the problem into smaller, manageable steps.
• Practice: The more you practice, the easier it becomes! Every equation becomes simpler. Always believe in yourself, and keep at it. We got this, guys.

Further Practice and Resources

Want to get even better? Here are some ways you can practice more and learn:

• Online Calculators: Use online fraction calculators to check your work and learn.
• Workbooks: Get a math workbook dedicated to fractions.
• Khan Academy: A great resource for free math lessons and practice problems.
• Practice, Practice, Practice: The more you work on fractions, the more comfortable you'll become.

Now that you know how to solve this type of equation, go out there and conquer those fractions! You've got this, and remember, the more you practice, the easier it gets. It is a rewarding achievement to correctly solve fraction equations. Good luck, and keep practicing! Fractions are challenging, but with the right methods, you will be successful.