Math Problem: Fractions And Operations

Hey math enthusiasts! Let's dive into a classic fraction problem. We're going to break down the question, go through each option step by step, and figure out the correct answer. This isn't just about finding the right choice; it's about understanding how fractions work and how to perform basic operations with them. Ready to get started?

Decoding the Question: Fractions at Play

So, the question is straightforward: "Which of the following is incorrect?" We're given four different equations involving fractions, and our job is to pinpoint the one that doesn't hold true. The core mathematical concepts here are addition, subtraction, multiplication, and division of fractions. It's like a quick review of the basics, so let's refresh our knowledge and get our thinking caps on. Remember, fractions represent parts of a whole, and knowing how to manipulate them is a fundamental skill in math. This skill goes beyond just solving this one problem; it's crucial for various other mathematical concepts you'll encounter.

Fractions can be intimidating at first, but once you grasp the underlying principles, they become much easier to manage. The key is to remember the rules for each operation: finding a common denominator for addition and subtraction, multiplying numerators and denominators for multiplication, and inverting and multiplying for division. We'll examine each option and apply these rules methodically to ensure we arrive at the correct answer. The goal here isn't just to get the right answer but also to reinforce our understanding of fractions through practice. Let’s face it, practice makes perfect, right? The more you work with fractions, the more comfortable and confident you will become. Let's make sure we review all the steps to fully grasp the concepts.

Let's get started, shall we?

Option A: Addition of Fractions

Let's start with option A: 3/2 + 2/3 = 13/6. This involves adding two fractions together. To add fractions, you need to have a common denominator. The least common denominator (LCD) of 2 and 3 is 6. So, we'll convert both fractions to have a denominator of 6. To do this, multiply the numerator and denominator of 3/2 by 3, which gives us 9/6. Then, multiply the numerator and denominator of 2/3 by 2, giving us 4/6. Now, add the two fractions together: 9/6 + 4/6 = 13/6. This matches the result given in the option. So, option A is correct.

This step highlights the importance of finding a common denominator in fraction addition. It reinforces the concept that you can't simply add the numerators if the denominators are different. You must ensure that you are adding equivalent parts of a whole, making the comparison accurate and the operation valid. This process also shows how important the concept of equivalent fractions is. Changing the fractions to the same denominator doesn't change their value, but it changes their representation. It's like seeing the same thing from a different angle to make it easier to understand.

Make sure that you're comfortable with this process. It forms the basis of many other mathematical operations and concepts. Keep practicing this fundamental step until it becomes second nature. Fractions are not as scary as they initially seem! When you break them down into smaller steps, they become much easier to manage, so don't be afraid to take your time and review each step until you fully understand it. Now let's move on to the next option.

Option B: Subtraction of Fractions

Next up is option B: 3/2 - 2/3 = 7/6. This requires subtracting fractions, which also requires a common denominator. As we discussed earlier, the least common denominator of 2 and 3 is 6. We already know the conversion from option A: 3/2 becomes 9/6 and 2/3 becomes 4/6. Now, we subtract: 9/6 - 4/6 = 5/6. However, the option states the result is 7/6. This does not match. So, we have found a potential incorrect option.

This part emphasizes the subtraction process with fractions. Just like addition, subtraction relies on equivalent fractions with a common denominator. It's important to remember the order of operations and ensure the subtraction is performed correctly. Mistakes can easily happen if you're not careful. Paying attention to detail and double-checking your work is very important. Recognizing that 9/6 - 4/6 should give us 5/6 and not 7/6, as provided, is key to identifying the incorrect option. The common denominator is key in subtraction, just like in addition, and should not be overlooked.

Now, let’s go through the rest of the options to make absolutely sure our answer is correct. Although we have already found a potential incorrect answer, it's always a good practice to examine all choices to be absolutely sure that we've found the correct one.

Option C: Multiplication of Fractions

Now, let's look at option C: 3/2 x 2/3 = 1. Multiplying fractions is much simpler than adding or subtracting because we don't need a common denominator. You just multiply the numerators together and the denominators together. So, 3 x 2 = 6 and 2 x 3 = 6. Thus, the result is 6/6, which simplifies to 1. The option is correct.

This step makes the process of multiplying fractions easier, and it’s important to reinforce the contrast with addition and subtraction. Multiplying fractions is straightforward and doesn't require finding a common denominator. This part highlights a fundamental rule: to multiply fractions, multiply the numerators and the denominators. This step is a good example of how to check your work; sometimes, you might find easier ways to solve the problem by simplifying the fractions. For example, before multiplying, you can cancel out the 2s and the 3s, which would immediately give you the answer of 1. Knowing both methods can increase the speed at which you can solve problems. This also helps improve your overall efficiency when working on more complex problems.

As you become more comfortable with multiplying fractions, you'll start to see these shortcuts and will be able to solve problems much more quickly. Knowing the shortcuts is not required, but it does help. Let's look at the last option and wrap this up!

Option D: Division of Fractions

Finally, let's examine option D: 3/2 / 2/3 = 9/4. Dividing fractions involves inverting the second fraction (the divisor) and then multiplying. Inverting 2/3 gives us 3/2. So, we now have 3/2 x 3/2. Multiplying these two fractions, we get (3 x 3) / (2 x 2) = 9/4. This matches the result in option D, making it correct.

This part highlights the steps of dividing fractions. Division of fractions follows a simple rule: invert the second fraction and multiply. This simple step can sometimes cause confusion if you forget to invert the fraction. Inverting a fraction means swapping its numerator and denominator. This concept is a core element in fraction division. So, the process of division is simplified when you understand this step. This step provides an opportunity to reinforce the importance of accurate steps when performing calculations. Many mistakes come from not following the steps exactly, so it’s key to work carefully and methodically.

Now that we have reviewed all the options, we can be certain of our answer.

The Verdict: Identifying the Incorrect Option

So, after carefully going through each option, we've identified the incorrect one. Option B, which states 3/2 - 2/3 = 7/6, is incorrect. The correct result of the subtraction is 5/6. Therefore, option B is the answer to the question.

This step is where we consolidate all the work we did. It brings together all the calculations and steps we took and confirms the answer. It is a good time to review how you arrived at the answer. Double-check your calculations to ensure there were no errors. This review solidifies your understanding of the concepts. This step gives you the satisfaction of reaching the solution. Remember that the goal is not just finding the answer but also understanding the process and reinforcing your knowledge of fraction operations.

Conclusion: Mastering Fractions

Great job, everyone! We successfully worked through a fraction problem. We not only identified the incorrect option but also reviewed all the fundamental operations with fractions. Remember, practice is key. The more you work with fractions, the more comfortable and confident you'll become. Keep up the good work, and you'll master fractions in no time. Keep practicing; keep learning. Math is just a bunch of puzzles that are solved step by step. Congratulations on working through the problem!