Simplifying Fractions Solve {2/5 + [(13/5 + 12/3) - (2/30 + 4/5)]} - 1/5

by Kenji Nakamura 73 views

Hey everyone! Today, we're diving into the world of fractions with a detailed walkthrough on how to solve and simplify the expression: {2/5 + [(13/5 + 12/3) - (2/30 + 4/5)]} - 1/5. Don't worry if it looks a bit intimidating at first. We'll break it down step by step, making it super easy to understand. Let's get started!

Understanding the Order of Operations

Before we even touch the fractions, it’s crucial to remember our good old friend, the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order tells us the sequence in which we should perform the operations to get the correct answer. So, we'll tackle the parentheses first, then move on to the rest.

Tackling the Innermost Parentheses: (13/5 + 12/3)

Our initial focus is on simplifying the innermost parentheses: (13/5 + 12/3). To add these fractions, we need a common denominator. The least common multiple (LCM) of 5 and 3 is 15. So, let's convert each fraction to have this denominator:

  • 13/5 becomes (13 * 3) / (5 * 3) = 39/15
  • 12/3 becomes (12 * 5) / (3 * 5) = 60/15

Now, we can easily add them:

39/15 + 60/15 = (39 + 60) / 15 = 99/15

So, the sum inside the first set of parentheses simplifies to 99/15. But wait, we can simplify this fraction further! Both 99 and 15 are divisible by 3:

99/15 = (99 Ă· 3) / (15 Ă· 3) = 33/5

So, (13/5 + 12/3) simplifies to 33/5. Keep this result handy; we'll need it for the next step.

Moving to the Second Set of Parentheses: (2/30 + 4/5)

Now, let's conquer the second set of parentheses: (2/30 + 4/5). Again, we need a common denominator to add these fractions. The LCM of 30 and 5 is 30. Let's convert the fractions:

  • 2/30 remains as it is.
  • 4/5 becomes (4 * 6) / (5 * 6) = 24/30

Adding these fractions together:

2/30 + 24/30 = (2 + 24) / 30 = 26/30

We can simplify this fraction as well. Both 26 and 30 are divisible by 2:

26/30 = (26 Ă· 2) / (30 Ă· 2) = 13/15

Thus, (2/30 + 4/5) simplifies to 13/15. Great job! We're making excellent progress.

Combining the Results Inside the Brackets

Now that we've simplified the expressions within the parentheses, let's tackle the brackets: [(13/5 + 12/3) - (2/30 + 4/5)]. We've already found that:

  • (13/5 + 12/3) = 33/5
  • (2/30 + 4/5) = 13/15

So, our expression within the brackets now looks like this:

[33/5 - 13/15]

To subtract these fractions, we need a common denominator. The LCM of 5 and 15 is 15. Let’s convert 33/5:

33/5 becomes (33 * 3) / (5 * 3) = 99/15

Now we can subtract:

99/15 - 13/15 = (99 - 13) / 15 = 86/15

So, the expression within the brackets simplifies to 86/15. We’re on the home stretch now!

Completing the Expression: {2/5 + 86/15} - 1/5

Our original expression now looks like this: {2/5 + 86/15} - 1/5. First, let’s deal with the curly braces. We need to add 2/5 and 86/15. The LCM of 5 and 15 is 15. Converting 2/5:

2/5 becomes (2 * 3) / (5 * 3) = 6/15

Now we add:

6/15 + 86/15 = (6 + 86) / 15 = 92/15

So, the expression within the curly braces simplifies to 92/15. Finally, we subtract 1/5 from this result:

92/15 - 1/5

We need a common denominator, which is 15. Convert 1/5:

1/5 becomes (1 * 3) / (5 * 3) = 3/15

Now subtract:

92/15 - 3/15 = (92 - 3) / 15 = 89/15

The Final Simplified Answer

Therefore, the final simplified answer to the expression {2/5 + [(13/5 + 12/3) - (2/30 + 4/5)]} - 1/5 is 89/15. You did it!

Converting to a Mixed Number (Optional)

For those who prefer mixed numbers, we can convert 89/15. Divide 89 by 15:

89 Ă· 15 = 5 with a remainder of 14

So, 89/15 as a mixed number is 5 14/15. Both 89/15 and 5 14/15 are correct answers, just in different forms.

Tips for Mastering Fraction Simplification

Working with fractions can be tricky, but here are some handy tips to help you become a pro:

  • Always use PEMDAS: Remember the order of operations to avoid mistakes.
  • Find the LCM: The least common multiple is your best friend when adding or subtracting fractions.
  • Simplify, simplify, simplify: Always reduce your fractions to the simplest form.
  • Practice makes perfect: The more you work with fractions, the easier it becomes.

Common Mistakes to Avoid

Fractions can be confusing, and it’s easy to make mistakes. Here are some common pitfalls to watch out for:

  • Forgetting PEMDAS: Performing operations in the wrong order can lead to incorrect answers.
  • Incorrectly finding the LCM: A wrong common denominator will mess up your calculations.
  • Not simplifying: Failing to simplify your final answer means you’re not quite done.
  • Adding/subtracting numerators without a common denominator: This is a big no-no!

Real-World Applications of Fraction Simplification

You might be wondering, “When will I ever use this in real life?” Well, fractions are everywhere! Here are a few examples:

  • Cooking: Recipes often use fractions (e.g., 1/2 cup of flour).
  • Construction: Measuring materials often involves fractions (e.g., cutting wood to 3/4 of an inch).
  • Finance: Calculating interest rates or dividing expenses can use fractions.
  • Time Management: Splitting your day into fractions (e.g., spending 1/4 of your day at work).

So, mastering fractions is not just an academic exercise; it’s a practical skill that you'll use in many areas of life.

Why This Matters: The Importance of Understanding Fractions

Fractions are a fundamental concept in mathematics, and understanding them is essential for several reasons. Firstly, they lay the groundwork for more advanced math topics such as algebra, calculus, and statistics. Without a solid grasp of fractions, you might struggle with these subjects later on. Imagine trying to understand algebraic equations without knowing how to manipulate fractions – it would be like trying to build a house without a foundation!

Secondly, fractions are incredibly practical in everyday life. Think about sharing a pizza with friends, measuring ingredients for a recipe, or figuring out discounts at the store. All of these situations involve fractions. Mastering fractions allows you to make informed decisions and solve problems efficiently. For example, if you're baking a cake and need to double a recipe that calls for 2/3 cup of sugar, knowing how to multiply fractions will help you get the measurements right.

Moreover, understanding fractions enhances your problem-solving skills. Working with fractions requires logical thinking, attention to detail, and the ability to break down complex problems into smaller, manageable steps. These are valuable skills that can be applied in various aspects of life, from managing your finances to planning a project at work. When you solve a fraction problem, you're not just finding an answer; you're training your brain to think critically and creatively.

In addition, fractions are a building block for understanding proportions and ratios, which are crucial in fields like science, engineering, and economics. For instance, scientists use ratios to express concentrations of solutions, engineers use proportions to design structures, and economists use fractions and percentages to analyze financial data. A strong foundation in fractions opens doors to a wide range of career opportunities and academic pursuits.

Finally, learning fractions boosts your confidence in your mathematical abilities. Math can be intimidating for many people, but mastering fractions is a significant accomplishment that can increase your self-assurance. Each time you successfully solve a fraction problem, you build a sense of competence and a positive attitude toward mathematics. This confidence can motivate you to tackle even more challenging math problems and pursue further learning.

Practice Problems: Test Your Fraction Skills

Now that we've covered the step-by-step solution and some helpful tips, it's time to put your skills to the test! Here are a few practice problems to help you solidify your understanding of fraction simplification.

  1. Solve and simplify: (3/4 + 1/8) - 5/16
  2. Evaluate: {1/3 + [(2/5 + 1/10) - 1/6]}
  3. Simplify: (7/9 - 1/3) + 2/6

Take your time, follow the steps we discussed, and remember to simplify your answers as much as possible. Working through these problems will not only improve your fraction skills but also boost your confidence in tackling similar challenges in the future. Don't be afraid to make mistakes – they are a natural part of the learning process. The key is to learn from your errors and keep practicing.

If you get stuck, revisit the explanations and examples we've covered in this guide. Pay close attention to finding common denominators, using the order of operations, and simplifying fractions. And remember, practice makes perfect! The more you work with fractions, the easier they will become.

Answers to the Practice Problems:

  1. 1/16
  2. 8/15
  3. 7/9

How did you do? If you got all the answers correct, congratulations! You're well on your way to mastering fraction simplification. If you missed a few, don't worry. Review the steps and try again. Keep practicing, and you'll get there!

Conclusion: You've Got This!

Fractions might seem daunting at first, but with a systematic approach and a bit of practice, they become much more manageable. By understanding the order of operations, finding common denominators, and simplifying your answers, you can confidently tackle even the most complex fraction problems. Remember, practice makes perfect, so keep honing your skills, and you'll become a fraction-solving pro in no time!

We hope this guide has helped you understand how to solve and simplify the expression {2/5 + [(13/5 + 12/3) - (2/30 + 4/5)]} - 1/5. Keep up the great work, and happy fraction solving! Remember, guys, you've got this!