Solving Fraction Subtraction (6a)/5 - (3a)/4 A Step-by-Step Guide

Hey guys! Today, let's dive into a common algebraic problem: fraction addition and subtraction. Specifically, we're going to tackle the expression (6a)/5 - (3a)/4. This might seem a bit daunting at first, but don't worry, we'll break it down step-by-step so you can master these types of problems. So let's get started and unlock the secrets of fraction manipulation!

The Core Concept: Common Denominators

The key to both adding and subtracting fractions lies in finding a common denominator. Think of it like this: you can't directly compare or combine slices from two pizzas if they're cut into different numbers of slices. You need to make sure they're cut into the same number of slices first. Similarly, with fractions, we need a common denominator, which is a shared multiple of the original denominators. The least common denominator (LCD) is usually the easiest to work with, as it keeps the numbers smaller.

For our problem, (6a)/5 - (3a)/4, we need to find the LCD of 5 and 4. What's the smallest number that both 5 and 4 divide into evenly? That's right, it's 20. So, 20 is our LCD. Now, we need to rewrite each fraction with a denominator of 20. To do this, we'll multiply both the numerator and the denominator of each fraction by a suitable factor.

For the first fraction, (6a)/5, we need to multiply the denominator 5 by 4 to get 20. So, we also multiply the numerator 6a by 4. This gives us (6a * 4) / (5 * 4) = (24a) / 20. For the second fraction, (3a)/4, we need to multiply the denominator 4 by 5 to get 20. So, we multiply the numerator 3a by 5 as well. This results in (3a * 5) / (4 * 5) = (15a) / 20. Now we have equivalent fractions with the same denominator: (24a)/20 and (15a)/20. This is a crucial step, so make sure you understand how we arrived here. Mastering this skill will make fraction operations much smoother for you. Remember, the goal is to transform the fractions without changing their value, only their appearance. This allows us to perform the subtraction accurately and confidently.

Performing the Subtraction

Now that we have our fractions with a common denominator, (24a)/20 and (15a)/20, the subtraction becomes much simpler. We can now subtract the numerators directly, keeping the denominator the same. Think of it like this: if you have 24 slices of a pizza that's cut into 20 slices and you take away 15 slices of the same pizza, you're left with 9 slices. So, we have (24a - 15a) / 20. Subtracting the numerators, 24a - 15a, we get 9a. Remember, we're combining like terms here. The 'a' is just a variable, so we treat it like any other unit when we subtract. It’s like saying 24 apples minus 15 apples equals 9 apples.

Therefore, our result is (9a) / 20. This is the fraction representing the difference between the two original fractions. It’s essential to understand that the denominator stays the same because it represents the size of the slices (or parts) we're dealing with. Only the number of slices (the numerator) changes when we add or subtract fractions with a common denominator. This principle is fundamental to working with fractions, and it’s something you’ll use time and again in algebra and beyond. So, always keep in mind that once you have a common denominator, the focus shifts to the numerators for addition or subtraction, and the denominator acts as the constant frame of reference.

Simplifying the Result (If Possible)

After performing the subtraction, it's always a good idea to check if the resulting fraction can be simplified. Simplification means reducing the fraction to its lowest terms, where the numerator and denominator have no common factors other than 1. This makes the fraction easier to understand and work with in future calculations. In our case, we have (9a) / 20. To simplify, we look for the greatest common divisor (GCD) of 9 and 20. The GCD is the largest number that divides both 9 and 20 without leaving a remainder.

The factors of 9 are 1, 3, and 9. The factors of 20 are 1, 2, 4, 5, 10, and 20. Looking at these factors, we can see that the only common factor between 9 and 20 is 1. This means that the fraction (9a) / 20 is already in its simplest form. There's no further simplification possible because the numerator and denominator are relatively prime – they share no common factors other than 1. This is important to recognize because sometimes fractions can be simplified, and simplifying them is a crucial step to presenting the answer in its most concise and understandable form. So, while in this particular case the fraction is already simplified, always make it a habit to check for simplification opportunities after any fraction operation.

The Final Answer

Alright guys, we've gone through the entire process step-by-step. We found the common denominator, performed the subtraction, and checked for simplification. So, the final answer to (6a)/5 - (3a)/4 is (9a)/20. See, it wasn't so scary after all! By breaking down the problem into smaller, manageable steps, we were able to solve it with confidence.

Remember, the key takeaway here is the importance of common denominators when adding or subtracting fractions. Once you master that, the rest is just arithmetic. And don't forget to always check if your final answer can be simplified. This is a crucial step to ensure you're presenting your answer in its most elegant and useful form. This skill is not just important in algebra but will serve you well in many areas of mathematics and even in everyday life when you're dealing with proportions and ratios. Keep practicing, and you'll become a pro at fraction operations in no time!

Practice Makes Perfect

To really solidify your understanding, try tackling some similar problems. The more you practice, the more comfortable you'll become with the process. You can find tons of practice problems online or in your textbook. Try changing the coefficients and variables to see how the solution changes. Experiment with different denominators to challenge yourself. And don't be afraid to make mistakes! Mistakes are part of the learning process. When you make a mistake, take the time to understand why you made it and how to avoid it in the future. This is how you truly learn and grow in mathematics.

Consider trying problems with more complex expressions in the numerators or denominators. This will help you develop your algebraic skills further. Also, explore situations where you have to combine addition and subtraction within the same problem. This will require you to apply the same principles but with a slightly higher level of complexity. Remember, the goal is not just to get the right answer but to understand the underlying concepts and be able to apply them in various situations. So, keep pushing yourself, keep exploring, and keep practicing, and you'll be amazed at how much you can achieve!

By diligently working through practice problems and reflecting on your solutions, you'll build a strong foundation in fraction operations that will serve you well in your mathematical journey. And remember, math is not just about formulas and procedures; it's about developing problem-solving skills and critical thinking abilities that are valuable in all aspects of life. So, embrace the challenge, enjoy the process, and celebrate your successes along the way.

Real-World Applications

You might be thinking, “Okay, this is cool, but when am I ever going to use this in real life?” Well, you'd be surprised! Fractions are everywhere! Think about cooking: recipes often call for fractional amounts of ingredients. If you need to double a recipe, you'll need to be able to multiply fractions. Or what about measuring things? Construction workers, engineers, and even artists use fractions all the time. Understanding how to work with fractions is a valuable life skill, not just a math skill.

Consider situations like dividing a pizza among friends or calculating discounts at the store. These everyday scenarios often involve fractional calculations. Even understanding financial concepts like interest rates and investments requires a solid grasp of fractions and percentages. The ability to confidently manipulate fractions also extends to more advanced fields like science and technology. Scientists use fractions to represent proportions and ratios in experiments, and engineers rely on fractional calculations when designing structures and machines. So, the skills you're developing in algebra are not just for the classroom; they're building blocks for success in a wide range of endeavors. By mastering fraction operations, you're equipping yourself with a powerful tool that will help you navigate the complexities of the world around you.

So, next time you encounter a situation involving fractions, take it as an opportunity to apply your knowledge and reinforce your skills. You'll be amazed at how quickly you can solve problems and how much more confident you feel in your mathematical abilities.

Conclusion

So, there you have it! We've successfully tackled the problem (6a)/5 - (3a)/4. Remember the key steps: find the common denominator, perform the operation on the numerators, and simplify if possible. Keep practicing, and you'll be a fraction master in no time! And don't forget, understanding these concepts is not just about getting good grades; it's about building a solid foundation for future mathematical success and real-world problem-solving. You've got this guys! Keep up the awesome work, and don't hesitate to explore more complex problems and challenges. The world of mathematics is vast and fascinating, and the more you learn, the more you'll appreciate its beauty and power. So, keep your curiosity alive, keep asking questions, and keep pushing your boundaries. You're on a journey of discovery, and every step you take brings you closer to a deeper understanding of the world around you.

Remember, the skills you're developing now will serve you well in countless ways throughout your life. So, embrace the challenge, celebrate your progress, and never stop learning!