Simplifying Fractions A Step-by-Step Guide To Solve -5/6 - 7/-8 + 11/12 - (-1/6) + 0 - (-7/-16)

by Kenji Nakamura 96 views

Hey guys! Today, we're diving into the world of fractions and tackling a problem that might seem a bit daunting at first glance. But don't worry, we'll break it down step-by-step so you can conquer it with confidence. Our mission? To simplify the expression: -5/6 - 7/-8 + 11/12 - (-1/6) + 0 - (-7/-16). So, grab your pencils, and let's get started!

Understanding the Basics of Fraction Arithmetic

Before we jump into the main problem, it's crucial to understand the fundamental principles of fraction arithmetic. Fraction arithmetic involves performing operations such as addition, subtraction, multiplication, and division with fractions. Fractions are essentially parts of a whole, represented by a numerator (the top number) and a denominator (the bottom number). The denominator tells us how many equal parts the whole is divided into, and the numerator tells us how many of those parts we have.

When it comes to adding or subtracting fractions, the golden rule is that they must have a common denominator. This means the denominators of all the fractions involved must be the same. Why is this important? Because we can only directly add or subtract quantities that are expressed in the same units. Think of it like trying to add apples and oranges – you need a common unit, like “fruit,” to combine them. Similarly, with fractions, the common denominator provides that common unit.

To find a common denominator, we typically look for the least common multiple (LCM) of the denominators. The LCM is the smallest number that is a multiple of all the denominators. Once we have the LCM, we convert each fraction to an equivalent fraction with the LCM as the denominator. This involves multiplying both the numerator and the denominator of each fraction by a suitable factor. Remember, multiplying both the top and bottom by the same number doesn't change the value of the fraction, it just changes how it looks.

After converting the fractions to have a common denominator, we can then add or subtract the numerators while keeping the denominator the same. For example, if we have 1/4 + 2/4, we simply add the numerators (1 + 2) and keep the denominator (4), resulting in 3/4. Subtraction works the same way – we subtract the numerators and keep the denominator.

Simplifying fractions is another essential skill. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. To simplify a fraction, we find the greatest common divisor (GCD) of the numerator and denominator and divide both by it. For example, the fraction 4/8 can be simplified by dividing both 4 and 8 by their GCD, which is 4. This gives us 1/2, which is the simplified form.

Understanding these basic concepts is the key to confidently tackling more complex fraction problems. So, with these tools in our arsenal, let's dive into our main challenge!

Step-by-Step Solution: Simplifying -5/6 - 7/-8 + 11/12 - (-1/6) + 0 - (-7/-16)

Okay, let's break down this beast of an expression: -5/6 - 7/-8 + 11/12 - (-1/6) + 0 - (-7/-16). It looks intimidating, but we'll conquer it step-by-step. Here’s how we're going to do it:

  1. Simplify the Signs: The first thing we need to do is tackle those pesky negative signs. Remember, subtracting a negative is the same as adding a positive, and a negative divided by a negative is a positive. So, let's rewrite the expression with simplified signs:

    -5/6 + 7/8 + 11/12 + 1/6 + 0 - 7/16

    Notice how -7/-8 became +7/8, -(-1/6) became +1/6, and -(-7/-16) became -7/16 (since a negative divided by a negative is positive, and then we subtract it).

  2. Find the Least Common Denominator (LCD): To add or subtract fractions, we need a common denominator. Let’s find the LCD of 6, 8, 12, and 16. We can do this by listing the multiples of each number or by prime factorization. Let's go with prime factorization:

    • 6 = 2 x 3
    • 8 = 2 x 2 x 2 = 2^3
    • 12 = 2 x 2 x 3 = 2^2 x 3
    • 16 = 2 x 2 x 2 x 2 = 2^4

    The LCD is the product of the highest powers of all prime factors present. In this case, it's 2^4 x 3 = 16 x 3 = 48. So, our LCD is 48.

  3. Convert Fractions to Equivalent Fractions with the LCD: Now we need to convert each fraction to an equivalent fraction with a denominator of 48. To do this, we'll multiply the numerator and denominator of each fraction by the factor that makes the denominator 48:

    • -5/6 = (-5 x 8) / (6 x 8) = -40/48
    • 7/8 = (7 x 6) / (8 x 6) = 42/48
    • 11/12 = (11 x 4) / (12 x 4) = 44/48
    • 1/6 = (1 x 8) / (6 x 8) = 8/48
    • -7/16 = (-7 x 3) / (16 x 3) = -21/48

  4. Rewrite the Expression with the Common Denominator: Now our expression looks like this:

    -40/48 + 42/48 + 44/48 + 8/48 - 21/48

  5. Add and Subtract the Numerators: Now we can simply add and subtract the numerators, keeping the denominator the same:

    (-40 + 42 + 44 + 8 - 21) / 48

    Let's simplify the numerator: -40 + 42 = 2; 2 + 44 = 46; 46 + 8 = 54; 54 - 21 = 33.

    So, our expression simplifies to 33/48.

  6. Simplify the Resulting Fraction: Finally, let's simplify 33/48. The greatest common divisor (GCD) of 33 and 48 is 3. So, we'll divide both the numerator and the denominator by 3:

    33/48 = (33 ÷ 3) / (48 ÷ 3) = 11/16

    And there you have it! Our final simplified answer is 11/16.

Key Takeaways and Common Mistakes to Avoid

Alright, we've successfully simplified a pretty complex fraction expression. Let's recap some key takeaways and highlight common mistakes to watch out for.

Key Takeaways:

  • Simplify Signs First: Always start by simplifying the signs. Subtracting a negative is the same as adding, and a negative divided by a negative is a positive. Getting this right from the start prevents errors down the line.
  • Find the LCD: The least common denominator (LCD) is crucial for adding and subtracting fractions. Make sure you find the correct LCD by using methods like listing multiples or prime factorization.
  • Convert Carefully: When converting fractions to equivalent fractions with the LCD, ensure you multiply both the numerator and the denominator by the same factor. This keeps the value of the fraction the same.
  • Simplify at the End: Always simplify your final answer. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1.

Common Mistakes to Avoid:

  • Forgetting to Simplify Signs: This is a big one! Misinterpreting the signs can throw off your entire calculation.
  • Incorrect LCD: Choosing the wrong LCD will lead to incorrect equivalent fractions and ultimately a wrong answer. Double-check your LCD calculation.
  • Adding/Subtracting Numerators Without a Common Denominator: This is a fundamental mistake. You can only add or subtract fractions once they have the same denominator.
  • Forgetting to Simplify the Final Answer: Leaving your answer in an unsimplified form, while technically correct, isn't the best practice. Always reduce to the simplest form.
  • Arithmetic Errors: Simple addition or subtraction mistakes can happen, especially with larger numbers. Take your time and double-check your calculations.

By keeping these key takeaways and common mistakes in mind, you'll be well-equipped to tackle any fraction problem that comes your way. Practice makes perfect, so don't hesitate to try out more examples and build your confidence.

Practice Problems to Boost Your Fraction Skills

Now that we've walked through the solution and discussed key takeaways, it's time to put your skills to the test! Practice is the key to mastering fraction arithmetic. Here are a few problems similar to the one we just solved. Try tackling them on your own, and feel free to refer back to the steps and tips we discussed earlier.

Practice Problems:

  1. Simplify: 3/4 - (-1/2) + 5/8 - 1/4
  2. Simplify: -2/3 + 5/6 - 1/2 + (-1/3)
  3. Simplify: 7/10 - 2/5 + 3/4 - (-1/20)
  4. Simplify: -5/9 + 1/3 - 2/6 + 4/18
  5. Simplify: 1/2 - (-3/4) + 5/6 - 7/12

Remember to follow the steps we outlined: simplify signs, find the LCD, convert fractions, add/subtract numerators, and simplify the final answer. Don't rush, and double-check your work along the way.

If you get stuck, try breaking the problem down into smaller steps. Can you find the LCD for the first two fractions? Can you simplify any signs? Sometimes, just focusing on one small part of the problem can help you make progress.

You can also check your answers using online fraction calculators or ask a friend or teacher to review your work. The important thing is to keep practicing and learning from your mistakes. Each problem you solve will help solidify your understanding and build your confidence.

So, grab a pen and paper, and get to work! You've got this!

Conclusion: Mastering Fraction Simplification

Great job, guys! We've covered a lot in this comprehensive guide to simplifying fractions. From understanding the basics of fraction arithmetic to tackling a complex problem step-by-step, you've gained valuable skills and insights. Remember, mastering fractions is a fundamental skill in mathematics, and it opens the door to more advanced concepts down the road.

We started by emphasizing the importance of a solid foundation in fraction arithmetic. We discussed the need for a common denominator when adding or subtracting fractions, the process of finding the least common multiple (LCM), and the technique of simplifying fractions to their lowest terms. These building blocks are essential for tackling more challenging problems.

Then, we dove into the step-by-step solution of our main problem: -5/6 - 7/-8 + 11/12 - (-1/6) + 0 - (-7/-16). We meticulously worked through each step, from simplifying signs to finding the LCD, converting fractions, adding/subtracting numerators, and simplifying the final result. By breaking down the problem into manageable chunks, we demonstrated how even complex expressions can be conquered with a systematic approach.

We also highlighted key takeaways and common mistakes to avoid. Simplifying signs first, finding the correct LCD, converting fractions carefully, and always simplifying the final answer are crucial for accuracy. Avoiding mistakes like forgetting to simplify signs or adding numerators without a common denominator will save you from frustrating errors.

Finally, we provided practice problems to help you solidify your skills. Remember, practice is the key to mastery. The more problems you solve, the more confident and proficient you'll become in working with fractions.

So, keep practicing, keep learning, and keep simplifying! You've got the tools and the knowledge to conquer any fraction challenge that comes your way. And remember, even if you stumble along the way, don't give up. Every mistake is an opportunity to learn and grow. Now go out there and rock those fractions!