Solving Math: Value Of The Expression

by Andrew McMorgan 38 views

Hey Plastik Magazine readers! Let's dive into a fun math problem. Today, we're going to break down how to find the value of a particular expression. This isn't just about getting an answer; it's about understanding the steps and the why behind them. So, grab your pencils, and let's get started! The main keyword here is value of the expression, and we'll learn to calculate it step-by-step. Get ready to flex those brain muscles! Understanding expressions is crucial in math because they form the building blocks for more complex equations and problem-solving. It's like learning the alphabet before you write a novel—you gotta get the basics down first. This problem looks like a mixed operation problem involving addition, subtraction and fractions. We'll start by making the mixed numbers into improper fractions. Then we'll group the addition, and subtraction together. Finally, we'll perform the operations and simplify as much as we can. This approach not only helps solve the problem but also builds a solid foundation for future math concepts. It also keeps us from getting confused with the different operations and signs involved. It's all about precision and clarity, guys!

Step 1: Convert Mixed Numbers to Improper Fractions

Alright, let's start with the expression: (10 1/4 + 6 1/4) - (3 3/5 - 3 1/2). The first thing we need to do is convert the mixed numbers into improper fractions. Why? Because it makes the arithmetic easier and less prone to errors. When working with fractions, improper fractions simplify the process of adding and subtracting. So, how do we do this, you ask? Easy peasy! To convert a mixed number (a whole number and a fraction) to an improper fraction, multiply the whole number by the denominator of the fraction, and add the numerator. Then, keep the same denominator. Let's break it down for each term:

  • 10 1/4 becomes (10 * 4 + 1) / 4 = 41/4.
  • 6 1/4 becomes (6 * 4 + 1) / 4 = 25/4.
  • 3 3/5 becomes (3 * 5 + 3) / 5 = 18/5.
  • 3 1/2 becomes (3 * 2 + 1) / 2 = 7/2.

See? It's not rocket science! Now our expression looks like this: (41/4 + 25/4) - (18/5 - 7/2). We've successfully transformed our mixed numbers into improper fractions. This step is super crucial because it sets us up for smooth sailing through the rest of the problem. Remember, always double-check your calculations to ensure accuracy. Small mistakes here can lead to big errors later on. We want to be accurate and confident! It's all about the details, guys, and making sure we get every step right. We're building a solid foundation here, and precision is our best friend. This stage is all about converting the fractions so that we can easily operate and solve the expression. Let's move on!

Step 2: Perform Operations Inside Parentheses

Now that we have all the fractions in improper form, the next step is to simplify the expressions within the parentheses. Following the order of operations (PEMDAS/BODMAS), we address what's inside the parentheses first. So, let's start with the first set: (41/4 + 25/4). These fractions have the same denominator (4), which means we can directly add the numerators. Keep the denominator the same. Therefore, 41/4 + 25/4 = (41 + 25) / 4 = 66/4.

Next, let's tackle the second set of parentheses: (18/5 - 7/2). Here, we have different denominators (5 and 2). To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 5 and 2 is 10. So, we'll convert both fractions to have a denominator of 10. For 18/5, multiply both the numerator and the denominator by 2: (18 * 2) / (5 * 2) = 36/10. For 7/2, multiply both the numerator and the denominator by 5: (7 * 5) / (2 * 5) = 35/10. Now, we can subtract: 36/10 - 35/10 = (36 - 35) / 10 = 1/10. So, we've simplified both sets of parentheses: (41/4 + 25/4) = 66/4 and (18/5 - 7/2) = 1/10. The expression now looks like this: 66/4 - 1/10. Remember, keeping the order of operations straight is key here! You don’t want to mess up the calculations in the parentheses. It's like following a recipe – you have to do things in the right order to get the desired result. The ability to manipulate and simplify fractions is a fundamental skill in mathematics, so kudos to you for practicing this process!

Step 3: Subtract the Fractions and Simplify

Alright, we're in the home stretch now, guys! We've simplified the parentheses, and now we're left with 66/4 - 1/10. To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 4 and 10 is 20. So, we'll convert both fractions to have a denominator of 20.

  • For 66/4, multiply both the numerator and the denominator by 5: (66 * 5) / (4 * 5) = 330/20.
  • For 1/10, multiply both the numerator and the denominator by 2: (1 * 2) / (10 * 2) = 2/20.

Now, we can subtract: 330/20 - 2/20 = (330 - 2) / 20 = 328/20. We've got our answer, but we're not quite done. Let's simplify the fraction 328/20. Both the numerator and denominator are divisible by 4. So, we divide both by 4: 328 / 4 = 82 and 20 / 4 = 5. Therefore, 328/20 simplifies to 82/5. This is an improper fraction, but we can convert it to a mixed number for a cleaner representation. To convert 82/5 to a mixed number, we divide 82 by 5. 5 goes into 82 sixteen times with a remainder of 2. So, 82/5 = 16 2/5.

There you have it! The value of the expression (10 1/4 + 6 1/4) - (3 3/5 - 3 1/2) is 16 2/5. We have successfully navigated through all the steps, from converting mixed numbers to improper fractions, performing operations inside parentheses, subtracting fractions, and simplifying our answer. Awesome work, guys! Remember, practice makes perfect. The more you work through these problems, the more comfortable and confident you'll become. Keep up the great work and keep exploring the amazing world of mathematics! It is very important to simplify at the end. It allows you to represent the answer in its simplest form. This final stage involves all the prior steps and requires careful calculation.

Conclusion: The Final Answer

So, the value of the expression (10 1/4 + 6 1/4) - (3 3/5 - 3 1/2) is 16 2/5. We broke down a math problem, step by step, and found the final answer. This involved converting mixed numbers, performing operations, and simplifying our result. You guys did amazing! Keep practicing, and don't be afraid to tackle challenging problems. Mathematics is all about logical thinking and gradual learning. Every step builds your skills and confidence. You now know how to solve this kind of expression, and you're building a strong mathematical foundation. Pat yourselves on the back, you all deserve it. See you in the next math adventure, guys!