Evaluating 6/5 - (1/3 + 1/4) A Step-by-Step Guide

Introduction

In this article, guys, we're going to break down how to evaluate the mathematical expression 6/5 - (1/3 + 1/4). This involves a mix of fractions and parentheses, so we'll go step by step to make sure everything is crystal clear. Whether you're brushing up on your math skills or tackling homework, understanding how to work with fractions is super important. We'll cover the basic principles and provide a detailed solution so you can follow along easily. So, grab a pen and paper, and let's dive in!

Understanding the Basics of Fraction Arithmetic

Before we get to the main problem, let's quickly recap the basics of fraction arithmetic. Fractions, as you know, represent a part of a whole. They consist of two main parts: the numerator (the top number) and the denominator (the bottom number). To add or subtract fractions, they need to have the same denominator. This common denominator allows us to perform the operation on the numerators while keeping the denominator the same. Think of it like this: you can only add or subtract slices of pizza if they're cut into the same size pieces. If the pieces are different sizes, you need to find a common size to work with.

When adding fractions with the same denominator, you simply add the numerators and keep the denominator the same. For example, 2/7 + 3/7 = (2+3)/7 = 5/7. Similarly, when subtracting fractions with the same denominator, you subtract the numerators. For instance, 5/8 - 2/8 = (5-2)/8 = 3/8. Now, what happens when the denominators are different? That's when we need to find the least common multiple (LCM) of the denominators. The LCM is the smallest number that both denominators can divide into evenly. Once you find the LCM, you convert each fraction to an equivalent fraction with the LCM as the new denominator. This is done by multiplying both the numerator and the denominator of each fraction by the number that makes the original denominator equal to the LCM. It might sound a bit complicated, but it’s really just about making the fractions comparable so you can add or subtract them accurately.

Step-by-Step Solution

Step 1: Addressing the Parentheses

The expression we're tackling is 6/5 - (1/3 + 1/4). As with any mathematical expression, we follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This means we need to tackle what's inside the parentheses first: (1/3 + 1/4). To add these fractions, we need a common denominator. The least common multiple (LCM) of 3 and 4 is 12. So, we need to convert both fractions to have a denominator of 12.

To convert 1/3 to an equivalent fraction with a denominator of 12, we multiply both the numerator and the denominator by 4: (1 * 4) / (3 * 4) = 4/12. Similarly, to convert 1/4 to a fraction with a denominator of 12, we multiply both the numerator and the denominator by 3: (1 * 3) / (4 * 3) = 3/12. Now we can add these fractions: 4/12 + 3/12 = (4 + 3) / 12 = 7/12. So, the expression inside the parentheses simplifies to 7/12. This step is crucial because it simplifies the overall problem, making it easier to handle the next operation. By finding a common denominator and adding the fractions correctly, we've set ourselves up for the next stage of the calculation.

Step 2: Subtracting the Fractions

Now that we've simplified the expression inside the parentheses, our problem looks like this: 6/5 - 7/12. Again, we need a common denominator to subtract these fractions. The least common multiple (LCM) of 5 and 12 is 60. This means we need to convert both 6/5 and 7/12 into equivalent fractions with a denominator of 60. Let's start with 6/5. To get the denominator to 60, we multiply both the numerator and the denominator by 12: (6 * 12) / (5 * 12) = 72/60. Next, we'll convert 7/12 to a fraction with a denominator of 60. We multiply both the numerator and the denominator by 5: (7 * 5) / (12 * 5) = 35/60.

Now we can subtract the fractions: 72/60 - 35/60 = (72 - 35) / 60 = 37/60. This is the result of our subtraction. It's important to ensure that the fraction is in its simplest form. In this case, 37 is a prime number, and it does not divide evenly into 60, so the fraction 37/60 is already in its simplest form. This step demonstrates the importance of finding the correct common denominator and performing the subtraction accurately. We've now arrived at our final answer after carefully following the order of operations and fraction arithmetic rules.

Step 3: Final Answer

After performing all the steps, we've found that 6/5 - (1/3 + 1/4) = 37/60. This is our final answer. To recap, we first tackled the expression inside the parentheses by finding a common denominator for 1/3 and 1/4, which was 12. We added these fractions to get 7/12. Then, we subtracted 7/12 from 6/5. For this, we found the least common multiple of 5 and 12, which is 60. We converted both fractions to have this denominator, performed the subtraction, and simplified the result. It’s always a good idea to double-check your work to ensure accuracy, especially when dealing with multiple steps.

Why is This Important?

Understanding how to evaluate expressions like this is crucial for several reasons. First, it’s a fundamental skill in mathematics. Fractions are everywhere, from cooking recipes to financial calculations. Being able to confidently work with fractions is essential for everyday life. Second, these types of problems help develop your problem-solving skills. They require you to think step-by-step and apply the order of operations correctly. This logical thinking is valuable in many areas, not just math. Finally, mastering these basics sets you up for more advanced math topics. Algebra, calculus, and other higher-level subjects build upon these foundational concepts. If you have a solid understanding of fractions, you’ll find it much easier to tackle more complex math problems in the future. So, investing time in mastering these skills now will pay off in the long run.

Common Mistakes to Avoid

When working with fractions, there are a few common mistakes that students often make. One of the biggest is trying to add or subtract fractions without finding a common denominator first. Remember, you can’t directly add or subtract fractions unless they have the same denominator. Another mistake is incorrectly finding the least common multiple (LCM). Double-check your calculations and make sure you’ve found the smallest number that both denominators can divide into evenly. A third common error is forgetting to simplify the final answer. Always reduce your fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD). Lastly, watch out for errors in basic arithmetic, such as adding or subtracting numerators incorrectly. Taking your time and double-checking each step can help you avoid these mistakes. Practice makes perfect, so the more you work with fractions, the more confident you’ll become in avoiding these pitfalls.

Practice Problems

To really nail down these concepts, here are a few practice problems you can try:

1. Evaluate: 3/4 + (1/2 - 1/3)
2. Evaluate: 5/6 - (2/5 + 1/10)
3. Evaluate: 7/8 - (1/4 + 1/2)

Work through these problems step by step, following the same methods we used in the example. Remember to tackle the parentheses first, find common denominators, and simplify your final answers. Working through these exercises will reinforce your understanding and build your confidence in handling fractions. Don't be afraid to make mistakes – they're a part of the learning process. Just review your work, identify where you went wrong, and try again. With consistent practice, you’ll become a fraction whiz in no time!

Conclusion

In conclusion, evaluating the expression 6/5 - (1/3 + 1/4) involves a clear understanding of fraction arithmetic and the order of operations. By breaking the problem down into manageable steps – first addressing the parentheses and then performing the subtraction – we arrived at the final answer of 37/60. Remember, the key to mastering fractions is practice and attention to detail. Understanding these fundamental concepts not only helps in math class but also in numerous real-life situations. So, keep practicing, and you’ll find working with fractions becomes second nature. Keep up the great work, guys!