Solving The Expression 4 11/18 * 6/7 - 1 4/9 Step By Step
Hey guys! In this article, we're going to dive deep into solving the expression 4 11/18 * 6/7 - 1 4/9. This might seem a bit daunting at first, but trust me, we'll break it down step by step so you can tackle it with confidence. Our main goal here is not just to get the right answer, but to understand the underlying principles of fraction manipulation and order of operations. So, grab your pencils, and let's get started!
Understanding the Expression
Before we jump into calculations, let's take a closer look at the expression: 4 11/18 * 6/7 - 1 4/9. The expression involves mixed numbers, multiplication, and subtraction, so we'll need to be mindful of the order of operations (PEMDAS/BODMAS), which dictates that we perform multiplication before subtraction. The mixed numbers need to be converted into improper fractions to make the multiplication process smoother. So, the first key step in finding the value of this expression is to convert the mixed numbers into improper fractions. Understanding how to do this conversion is crucial for simplifying the expression. A mixed number combines a whole number and a fraction, like 4 11/18. To convert it, we multiply the whole number by the denominator of the fraction and add the numerator. This result becomes the new numerator, and we keep the same denominator. For example, to convert 4 11/18, we calculate (4 * 18) + 11, which equals 83. So, 4 11/18 becomes 83/18. Similarly, we can convert 1 4/9 into an improper fraction. We multiply 1 by 9 and add 4, which gives us 13. So, 1 4/9 becomes 13/9. Now, let's rewrite the expression with the improper fractions: 83/18 * 6/7 - 13/9. This form is much easier to work with because we can now perform the multiplication of fractions directly. Remember, when multiplying fractions, we multiply the numerators together and the denominators together. So, the next step is to perform the multiplication operation. Once we've done that, we'll be left with a subtraction problem between two fractions, which we'll need to simplify further. This initial conversion is a foundational step, and mastering it will make solving complex expressions like this much easier. Remember, practice makes perfect, so the more you work with mixed numbers and improper fractions, the more comfortable you'll become with the conversion process.
Step-by-Step Solution
Now, let's break down the solution step-by-step. As we discussed, the first thing we need to do is convert the mixed numbers into improper fractions. We already did this in the previous section, but let's reiterate for clarity. 4 11/18 becomes (4 * 18) + 11 / 18 = 83/18. Similarly, 1 4/9 becomes (1 * 9) + 4 / 9 = 13/9. So, the expression now looks like this: 83/18 * 6/7 - 13/9. Next, we need to perform the multiplication operation. Remember, we multiply the numerators together and the denominators together: (83/18) * (6/7) = (83 * 6) / (18 * 7) = 498/126. Before we move on, let's simplify this fraction. Both 498 and 126 are divisible by 6, so we can simplify the fraction to 83/21. Now, our expression looks like this: 83/21 - 13/9. To subtract fractions, we need a common denominator. The least common multiple (LCM) of 21 and 9 is 63. So, we need to convert both fractions to have a denominator of 63. To convert 83/21, we multiply both the numerator and the denominator by 3: (83 * 3) / (21 * 3) = 249/63. To convert 13/9, we multiply both the numerator and the denominator by 7: (13 * 7) / (9 * 7) = 91/63. Now, our expression looks like this: 249/63 - 91/63. Finally, we can subtract the fractions: 249/63 - 91/63 = (249 - 91) / 63 = 158/63. This is an improper fraction, so let's convert it back to a mixed number. We divide 158 by 63, which gives us 2 with a remainder of 32. So, 158/63 is equal to 2 32/63. Therefore, the final answer to the expression 4 11/18 * 6/7 - 1 4/9 is 2 32/63. This step-by-step approach helps break down the problem into manageable parts. Each step, from converting mixed numbers to finding a common denominator, is essential for arriving at the correct solution. By understanding each of these steps, you'll be well-equipped to tackle similar problems in the future.
Common Mistakes to Avoid
When solving expressions like this, there are a few common mistakes that students often make. Let's talk about these so you can avoid them. One of the biggest mistakes is forgetting the order of operations. Remember PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In our expression, we had to perform the multiplication before the subtraction. If you subtract before multiplying, you'll get the wrong answer. Another common mistake is not converting mixed numbers into improper fractions before multiplying or dividing. It's much easier to work with improper fractions when performing these operations. Trying to multiply or divide mixed numbers directly can lead to errors. A third mistake is making errors when finding a common denominator. When adding or subtracting fractions, you need to have a common denominator. Make sure you find the least common multiple (LCM) correctly and adjust the numerators accordingly. A simple mistake in finding the LCM can throw off your entire calculation. Another mistake to watch out for is incorrect simplification of fractions. Always try to simplify fractions to their lowest terms. This makes the numbers smaller and easier to work with. If you don't simplify, you might end up with larger numbers that are harder to manage. Finally, double-check your calculations! Math errors can easily happen, especially when dealing with multiple steps. Take a few extra moments to review your work and make sure you haven't made any mistakes in arithmetic. Paying attention to these common mistakes and actively working to avoid them can significantly improve your accuracy and confidence in solving math problems. Remember, practice and careful attention to detail are key!
Practice Problems
To really master solving these types of expressions, it's crucial to practice. Here are a few practice problems for you guys to try out. These problems are similar to the one we just solved, so you can use the same techniques and strategies. Remember to break each problem down step by step and pay close attention to the order of operations. 1. Solve: 3 1/4 * 2/5 - 1 1/2 2. Evaluate: 5 2/3 + 1/4 * 8/9 3. Simplify: 2 5/8 - 1/3 * 3/4 4. Calculate: 4/5 * 1 2/7 + 2/3 For each of these problems, start by converting any mixed numbers to improper fractions. Then, perform the multiplication and division operations before moving on to addition and subtraction. Don't forget to find a common denominator when adding or subtracting fractions. Once you've solved each problem, double-check your work to make sure you haven't made any calculation errors. If you're unsure about your answer, try working through the problem again or asking a friend or teacher for help. The key to improving your math skills is consistent practice. The more problems you solve, the more comfortable you'll become with the concepts and techniques. So, grab a pencil and paper, and give these practice problems a try. Good luck, and happy solving! Remember, math is like learning a new language; the more you practice, the more fluent you become.
Conclusion
So, guys, we've successfully navigated the process of finding the value of the expression 4 11/18 * 6/7 - 1 4/9. We've covered converting mixed numbers to improper fractions, the importance of the order of operations, finding common denominators, and simplifying fractions. The final answer, as we found, is 2 32/63. But more than just getting the answer, we've focused on understanding the process. The ability to break down complex problems into smaller, manageable steps is a valuable skill, not just in math, but in many areas of life. Remember the common mistakes we discussed, like overlooking the order of operations or miscalculating common denominators. Avoiding these pitfalls will help you approach math problems with greater confidence and accuracy. And don't forget the importance of practice! The more you work with these concepts, the more natural they will become. Math isn't about memorizing formulas; it's about understanding the underlying logic and developing a problem-solving mindset. So, keep practicing, keep asking questions, and keep exploring the world of mathematics. You've got this! If you ever encounter a similar problem, take a deep breath, remember the steps we've outlined, and tackle it one step at a time. And remember, there are plenty of resources available to help you along the way, from online tutorials to textbooks to teachers and classmates. Embrace the challenge, and enjoy the journey of learning. Math can be fun, and it's a skill that will serve you well throughout your life. Keep up the great work, and happy calculating!
