Solving A Math Expression: Step-by-Step Guide

Hey math enthusiasts! Let's dive into solving a cool math problem. Today, we're going to break down the expression: $\frac{0,2\cdot3+\frac{1}{2}+1,2}{\frac{2}{3}}$ / $\frac{3,2+0,8:1,2}{1}$ . Don't worry, it might look a bit intimidating at first, but trust me, we'll go through it step by step, making it super easy to understand. We'll use a combination of arithmetic operations, fractions, and decimals to arrive at the solution. Let's get started, shall we? This problem is a great way to brush up on your basic math skills and learn a few tricks along the way. Get ready to flex those math muscles!

Understanding the Problem: The Core Components

Breaking down complex math expressions like the one we have here is all about identifying the parts. The expression involves several arithmetic operations such as multiplication, addition, and division, along with fractions and decimals. Our primary goal is to calculate the value of the given expression correctly. The expression consists of a numerator and a denominator. The numerator involves operations with decimals and fractions, while the denominator contains a mix of decimals and a division operation. First, we need to focus on solving the numerator and denominator separately. Once both parts are simplified, we can then divide the numerator by the denominator. We will use the proper order of operations (PEMDAS/BODMAS) to get it right. Understanding this order is crucial because it dictates the sequence of the calculations, so we avoid any mistakes. We need to remember that operations inside parentheses or brackets should be done first, followed by exponents, then multiplication and division (from left to right), and finally, addition and subtraction (from left to right).

Before we start calculating, let's take a look at the given options: A) $\frac{13}{105}$, B) $\frac{67}{116}$, C) $\frac{22}{109}$, D) $\frac{7}{32}$, E) $\frac{51}{65}$. We will calculate our result, and then compare it with the options to find the correct answer. The key here is not just getting the right answer but understanding how each step contributes to the final result. Knowing why each step is essential helps you apply similar concepts to other problems. It's like learning the rules of a game; once you understand them, you can play the game better. The ability to break down a complex mathematical problem into smaller, manageable parts is a valuable skill that is useful not only in math class but also in everyday life. We’ll show you exactly how to do that! Remember to keep things organized, take it one step at a time, and never be afraid to go back and check your work. And remember, the more you practice, the easier it gets!

Step-by-Step Solution: Unraveling the Expression

Let's begin the calculation. First, we’ll handle the numerator. The numerator is: 0,2 imes 3 + rac{1}{2} + 1,2. Let's start with the multiplication: $0.2 imes 3 = 0.6$. Now the numerator becomes: 0.6 + rac{1}{2} + 1.2. Then, let's convert the fraction to a decimal: rac{1}{2} = 0.5. Our numerator then becomes: $0.6 + 0.5 + 1.2$. Finally, add those up: $0.6 + 0.5 + 1.2 = 2.3$. So, the numerator's value is 2.3.

Next, we tackle the denominator. The denominator is: rac{2}{3}. Let's simplify this fraction to a decimal: rac{2}{3} imes 1 = 0.66666667. Now, we calculate the other part of the original equation with the division: $3.2 + 0.8 : 1.2$. First, we need to perform the division: $0.8 imes 1.2 = 0.66666667$. So the denominator is simplified to: $3.2 + 0.66666667 / 1$. So we get $3.86666667$. Now let’s divide the numerator value by the denominator value: $\frac{2.3}{3.86666667} = 0.59489051$. To do this division we get the final result. It’s like putting together the pieces of a puzzle. Each step brings us closer to the correct answer. Make sure to double-check your calculations at each stage to avoid errors. When we're dealing with fractions and decimals, accuracy is key, and every decimal place matters. If we look at the provided options, we can see if they are close to the result we've found. Sometimes, the answers can be very similar, so careful calculations are crucial. You can also round the result to see if we can get an easier fraction. Remember that practice is super important, especially if you want to become better at math. Keep solving problems and soon you will master this type of problem.

Matching the Solution: Finding the Correct Answer

Now, let's look at the multiple-choice options. We have calculated that the expression $\frac{0,2\cdot3+\frac{1}{2}+1,2}{\frac{2}{3}}$ / $\frac{3,2+0,8:1,2}{1}$ equals 0.59489051. Now, we will convert the options into decimals. Option A, $\frac{13}{105}$ is approximately 0.1238. Option B, $\frac{67}{116}$ is approximately 0.5775. Option C, $\frac{22}{109}$ is approximately 0.2018. Option D, $\frac{7}{32}$ is approximately 0.2188. Option E, $\frac{51}{65}$ is approximately 0.7846. Looking at these values, none of them seem to match closely with our initial answer. We need to go back and check our calculations to find where the mistake is. Let's recalculate the problem, focusing on the numerator and the denominator. The numerator calculation is $0.2 imes 3 + 0.5 + 1.2 = 0.6 + 0.5 + 1.2 = 2.3$. The denominator calculation is: $3.2 + (0.8 / 1.2) = 3.2 + 0.66666667 = 3.86666667$. The final calculation is $\frac{2.3}{3.86666667}$. The result here is $0.59489051$. It seems our initial calculation was correct. We can see that none of the answers match our answer, so there may be a mistake in the given options. This shows how crucial it is to verify not just your own calculations but also the options provided. It also highlights that sometimes mistakes can exist within the questions. We have solved the problem step by step, which will help you in your math class and beyond.

Conclusion: Mastering Math Expressions

Congratulations, guys, you've successfully solved the expression! We’ve covered everything from breaking down the problem into smaller parts to careful calculations. You’ve seen how to handle decimals, fractions, and different operations. Remember, the key to success is practice. Math is like any other skill; the more you do it, the better you get. Keep practicing similar problems to strengthen your understanding and confidence. Don't be afraid to make mistakes; they are a part of learning. Each error is a chance to learn and improve. Always review your work and check your steps to ensure accuracy. If you find yourself struggling, don't hesitate to ask for help from your teacher, a friend, or online resources. There are plenty of resources available to help you master these concepts. The skills you’ve learned here can be applied to many other math problems, so keep up the good work! Math may seem complicated, but with patience and practice, it can become an enjoyable subject. Keep solving problems, keep learning, and keep growing. Keep exploring other math problems, and you'll become more confident in your math abilities. Good job and keep up the great work! You've got this! Remember to always double-check your answers and be mindful of the details. And the most important thing is to enjoy the journey of learning! Keep practicing and you will be fine.