Solve It! $5- rac{1}{6}+2 rac{1}{3}$ Explained
Hey math enthusiasts! Today, we're diving into a classic arithmetic problem: 5- rac{1}{6}+2 rac{1}{3}. It might look a little intimidating at first glance with the fractions and mixed numbers, but trust me, it's totally manageable. We'll break it down step-by-step, making sure everyone understands how to conquer this calculation. This guide is designed to be super clear, even if you're not a math whiz. So, grab your pencils and let's get started. We'll go through each part of the problem and explain the rationale behind every step. By the end of this, you will not only know the answer, but also you will be able to apply the same concept in similar problems. This is about building the foundation and knowledge of math problem-solving. This problem is fundamental in the realm of arithmetic, which provides a basis for higher-level mathematics. Understanding how to handle fractions, negative numbers, and mixed numbers is crucial. Without a strong grasp of these core concepts, more advanced topics can become difficult. With some practice, you will become comfortable and confident in solving such problems. Also, you will understand how important it is to work with fractions.
Understanding the Problem: 5- rac{1}{6}+2 rac{1}{3}
Alright, let's unpack this problem. We have a whole number (5), a fraction subtracted from it ( rac{1}{6}), and a mixed number added to it (2 rac{1}{3}). The key here is to handle the fractions correctly. Remember, the order of operations (PEMDAS/BODMAS) tells us we do addition and subtraction from left to right. Before we start solving, let's clarify some crucial points. The problem involves a whole number, a fraction, and a mixed number, each requiring a specific method for operations. The whole number (5) can be thought of as . The fraction rac{1}{6} represents one part of six equal parts. The mixed number 2 rac{1}{3} is a combination of a whole number (2) and a fraction ( rac{1}{3}). Before we do anything else, let's transform the mixed number into an improper fraction. This involves multiplying the whole number by the denominator of the fraction and adding the numerator. Keep the same denominator. This will simplify our calculations later. It is super important to ensure we are comfortable with these. Because it is the foundation. Now, since we have only addition and subtraction operations, we will go through the problem from left to right. Now that we have a solid understanding of the problem and the components, we are ready to take it to the next level. Let's start transforming and calculating. You will see how simple it is.
Converting Mixed Numbers to Improper Fractions
First things first: let's deal with that mixed number, 2 rac{1}{3}. We need to convert it into an improper fraction. To do this, we multiply the whole number (2) by the denominator (3) and add the numerator (1). This gives us (2 * 3) + 1 = 7. We keep the same denominator, so 2 rac{1}{3} becomes rac{7}{3}. Now our problem looks like this: 5 - rac{1}{6} + rac{7}{3}. Transforming mixed numbers into improper fractions simplifies calculations. It also facilitates easier addition and subtraction of fractions. You will see that everything aligns in terms of calculations after transforming everything into fractions. The main idea here is to convert everything into fractions so we can perform the calculations. This approach also makes it easier to find a common denominator, which is essential for adding and subtracting fractions. By converting the mixed number, we're ensuring that we're working with a single type of mathematical entity which streamlines the entire process. Remember, understanding the basic concept and operations will make you feel confident in dealing with similar problems. Let's go to the next level.
Solving Step-by-Step
Okay, now we've got 5 - rac{1}{6} + rac{7}{3}. Next, we'll want to address the subtraction and addition from left to right. To do this, we need a common denominator for all our fractions. The number 6 works perfectly since both 6 and 3 can be divided by it. First, let's rewrite 5 as rac{30}{6} (because 5 * 6 = 30). This gives us rac{30}{6} - rac{1}{6} + rac{7}{3}. Next, we need to convert rac{7}{3} to have a denominator of 6. We do this by multiplying both the numerator and the denominator by 2. rac{7}{3} becomes rac{14}{6}. Now our equation is rac{30}{6} - rac{1}{6} + rac{14}{6}.
Subtraction: rac{30}{6} - rac{1}{6}
Now, let's subtract. rac{30}{6} - rac{1}{6} = rac{29}{6}. This is pretty straightforward since we have the same denominator. We simply subtract the numerators (30 - 1 = 29) and keep the denominator. So far, we're doing great! This step involves a simple subtraction of fractions that share a common denominator. The process involves subtracting the numerators while keeping the same denominator. It’s important to understand this step as it demonstrates the basic operation of subtracting fractions. After converting our problem into proper fractions, we easily performed the subtraction because both fractions shared a common denominator. This step highlights the importance of the common denominator, as it’s crucial for performing addition and subtraction of fractions. Now, let's go on to the next step and see what we get.
Addition: rac{29}{6} + rac{14}{6}
Finally, we add the last two fractions: rac{29}{6} + rac{14}{6}. Again, we have a common denominator, so this is simple. We add the numerators (29 + 14 = 43) and keep the denominator. So, rac{29}{6} + rac{14}{6} = rac{43}{6}. This step involves the addition of fractions with a common denominator. Adding fractions with a common denominator is a fundamental arithmetic operation, which is essential for various mathematical concepts. Performing the addition is straightforward since the denominators are the same. Add the numerators (29+14 = 43). And keep the same denominator, which is 6. We get rac{43}{6}. It is also important to note that the result can be converted into a mixed number. This would involve dividing 43 by 6. This is where it gets fun.
Simplifying the Answer
We have our answer as rac{43}{6}. However, it's often preferred to express the answer as a mixed number. To do this, we divide 43 by 6. 6 goes into 43 seven times (7 * 6 = 42), with a remainder of 1. So, rac{43}{6} can be written as 7 rac{1}{6}. And there you have it! The final answer is 7 rac{1}{6}. The final step involves expressing the result as a simplified mixed number. By dividing the numerator by the denominator, we convert the improper fraction to a mixed number. This allows for a more intuitive understanding of the value. The mixed number representation helps in visualizing the magnitude of the answer. It emphasizes the importance of converting the answer to the simplest form. By dividing the numerator by the denominator, we get the whole number part (7) and the remainder becomes the new numerator (1), and the denominator remains the same. Finally, we arrived at our answer in a simplified format.
Conclusion
Awesome work, everyone! We successfully solved the problem 5- rac{1}{6}+2 rac{1}{3}. We started by converting the mixed number to an improper fraction, then found a common denominator to add and subtract the fractions. Finally, we simplified the answer. Math problems like this are all about understanding the steps and practicing. The most important lesson is not the answer itself, but the journey of getting there. Keep practicing, and you'll find these problems become easier and easier. If you keep practicing the concepts and understanding the steps, you will be able to do this with your eyes closed! You will develop a solid foundation in arithmetic skills. Keep up the good work and keep practicing! If you have any questions or want to try another problem, feel free to ask. Practice makes perfect. So, keep at it, and you'll become a fraction master in no time. This problem highlighted several key mathematical concepts. Working with fractions, understanding the order of operations, and simplifying answers are fundamental to math. These principles apply to more advanced topics. Remember, every step counts, and with practice, you'll feel more confident in solving similar problems.