Evaluating -1/5 - (-1/8) Step-by-Step Simplest Form Solution

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Introduction

In this article, we will delve into the process of evaluating the expression -\frac{1}{5}-\left(-\frac{1}{8} ight) and expressing the result in its simplest form. This involves understanding the rules of subtracting negative fractions and finding a common denominator to perform the operation. Fraction arithmetic is a fundamental concept in mathematics, crucial for various applications in algebra, calculus, and real-world problem-solving. By meticulously following each step, we aim to provide a comprehensive guide that not only answers the given question but also enhances your understanding of fraction manipulation.

Understanding the Basics of Fraction Arithmetic

Before diving into the specific problem, it's essential to grasp the fundamentals of fraction arithmetic. Fractions represent parts of a whole, and operations involving them require careful consideration of their numerators and denominators. The numerator is the number above the fraction bar, indicating the number of parts we have, while the denominator is the number below the fraction bar, indicating the total number of equal parts the whole is divided into.

When adding or subtracting fractions, a crucial step is to ensure they have a common denominator. This means that the denominators of the fractions must be the same. The reason for this is that we can only directly add or subtract fractions that represent parts of the same whole. If the denominators are different, we need to find a common multiple of the denominators, which will become the new common denominator. The least common multiple (LCM) is often the most convenient choice as it keeps the numbers smaller and the calculations simpler.

Once the fractions have a common denominator, we can add or subtract the numerators while keeping the denominator the same. For example, if we have $\frac{a}{c}$ and $\frac{b}{c}$, then $\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$ and $\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}$.

Dealing with negative fractions involves understanding the rules of signs. Subtracting a negative number is the same as adding its positive counterpart. This concept is crucial in simplifying expressions like the one we are about to evaluate. The expression -\frac{1}{5}-\left(-\frac{1}{8} ight) involves subtracting a negative fraction, which will be transformed into an addition.

Furthermore, simplifying fractions to their simplest form means reducing the fraction to its lowest terms. This is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD). A fraction is in simplest form when the numerator and the denominator have no common factors other than 1. For instance, if we end up with a fraction like $\frac{4}{16}$, we can simplify it by dividing both the numerator and the denominator by their GCD, which is 4, resulting in $\frac{1}{4}$.

In summary, mastering fraction arithmetic involves understanding common denominators, the rules of signs, and simplification techniques. These skills are not only essential for solving mathematical problems but also for everyday situations involving proportions and ratios. With a solid grasp of these basics, evaluating complex expressions involving fractions becomes straightforward and manageable. As we proceed with our specific problem, we will apply these principles to arrive at the solution in its simplest form.

Step-by-Step Evaluation of $-\frac{1}{5}-\left(-\frac{1}{8}

ight)$

To evaluate the expression -\frac{1}{5}-\left(-\frac{1}{8} ight), we'll proceed step-by-step, ensuring each operation is clearly explained. This methodical approach will help in understanding the underlying concepts and avoiding common mistakes. The key steps involve dealing with the subtraction of a negative number, finding a common denominator, performing the addition, and simplifying the final result.

Step 1: Dealing with the Subtraction of a Negative Number

The first part of our expression is -\frac{1}{5}-\left(-\frac{1}{8} ight). The crucial observation here is the subtraction of a negative fraction. In mathematics, subtracting a negative number is equivalent to adding its positive counterpart. Therefore, we can rewrite the expression as:

$-\frac{1}{5} + \frac{1}{8}$

This transformation simplifies the problem significantly. Instead of dealing with subtraction, we now have an addition problem, which is often more straightforward to handle. The negative sign in front of $\frac{1}{5}$ remains, indicating that we are dealing with a negative fraction. The subtraction of the negative $\frac{1}{8}$ has been correctly transformed into the addition of positive $\frac{1}{8}$. This step is fundamental in ensuring the correct sign in the final answer.

Step 2: Finding a Common Denominator

To add two fractions, they must have a common denominator. In our expression, we have $-\frac{1}{5}$ and $\frac{1}{8}$. The denominators are 5 and 8, which are different. To find a common denominator, we need to find the least common multiple (LCM) of 5 and 8.

The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, ... The multiples of 8 are: 8, 16, 24, 32, 40, 48, ...

The least common multiple of 5 and 8 is 40. Therefore, our common denominator will be 40. Now, we need to convert both fractions to equivalent fractions with a denominator of 40.

For $-\frac{1}{5}$, we multiply both the numerator and the denominator by 8:

$-\frac{1}{5} \times \frac{8}{8} = -\frac{8}{40}$

For $\frac{1}{8}$, we multiply both the numerator and the denominator by 5:

$\frac{1}{8} \times \frac{5}{5} = \frac{5}{40}$

Now our expression becomes:

$-\frac{8}{40} + \frac{5}{40}$

Step 3: Performing the Addition

Now that we have a common denominator, we can add the fractions. This involves adding the numerators while keeping the denominator the same:

$-\frac{8}{40} + \frac{5}{40} = \frac{-8 + 5}{40}$

Adding the numerators, we have:

$-8 + 5 = -3$

So the expression simplifies to:

$\frac{-3}{40}$

Which is usually written as:

$-\frac{3}{40}$

Step 4: Simplifying the Result

The final step is to simplify the fraction to its simplest form. To do this, we look for common factors between the numerator and the denominator. In our case, the numerator is -3 and the denominator is 40.

The factors of 3 are: 1, 3 The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40

The only common factor between 3 and 40 is 1. Therefore, the fraction $-\frac{3}{40}$ is already in its simplest form.

Final Answer

In conclusion, the evaluation of the expression -\frac{1}{5}-\left(-\frac{1}{8} ight) in simplest form is $-\frac{3}{40}$. This result was obtained by first transforming the subtraction of a negative fraction into addition, finding a common denominator, performing the addition, and verifying that the resulting fraction is in its simplest form. The process highlights the importance of understanding fraction arithmetic and the step-by-step approach required to solve such problems accurately. By mastering these techniques, you can confidently tackle more complex mathematical challenges involving fractions.

Common Mistakes to Avoid When Evaluating Fraction Expressions

Evaluating expressions involving fractions can be tricky, and several common mistakes can lead to incorrect answers. Being aware of these pitfalls and understanding how to avoid them is crucial for mastering fraction arithmetic. Let's discuss some of the most frequent errors and how to navigate them effectively.

Mistake 1: Forgetting to Find a Common Denominator

One of the most common mistakes is attempting to add or subtract fractions without first finding a common denominator. As mentioned earlier, fractions can only be added or subtracted directly if they represent parts of the same whole, which is indicated by having the same denominator. For example, trying to add $\frac{1}{2}$ and $\frac{1}{3}$ without finding a common denominator would lead to an incorrect result. The fractions need to be converted to equivalent fractions with a common denominator, such as $\frac{3}{6}$ and $\frac{2}{6}$, before they can be added.

How to Avoid It: Always check if the fractions have the same denominator before adding or subtracting. If they don't, find the least common multiple (LCM) of the denominators and convert each fraction to an equivalent fraction with the LCM as the new denominator. This ensures that you are adding or subtracting comparable parts.

Mistake 2: Incorrectly Applying the Rules of Signs

The rules of signs, particularly when dealing with negative fractions, can be a source of confusion. For instance, subtracting a negative fraction is the same as adding a positive fraction, and vice versa. A common error is misinterpreting the operation and applying the wrong sign, which can drastically alter the outcome. For example, incorrectly simplifying -\frac{1}{5}-\left(-\frac{1}{8} ight) as $-\frac{1}{5} - \frac{1}{8}$ instead of $-\frac{1}{5} + \frac{1}{8}$ is a frequent mistake.

How to Avoid It: Always double-check the signs before performing any operation. Remember that subtracting a negative number is equivalent to addition, and adding a negative number is equivalent to subtraction. Writing out the steps clearly can help prevent sign errors.

Mistake 3: Not Simplifying the Final Answer

Simplifying fractions to their simplest form is a crucial step that is sometimes overlooked. A fraction is in simplest form when the numerator and the denominator have no common factors other than 1. Failing to simplify the fraction can lead to an incomplete answer, even if the initial calculations are correct. For example, if you arrive at an answer of $\frac{4}{16}$, it is not in simplest form and should be reduced to $\frac{1}{4}$.

How to Avoid It: After performing the addition or subtraction, always check if the resulting fraction can be simplified. Find the greatest common divisor (GCD) of the numerator and the denominator, and divide both by the GCD. This will ensure that your answer is in its simplest form.

Mistake 4: Errors in Finding the Least Common Multiple (LCM)

Finding the correct LCM is essential for determining the common denominator. An incorrect LCM will lead to incorrect equivalent fractions and, consequently, an incorrect final answer. Common errors include choosing a common multiple that is not the least, or simply multiplying the denominators without considering if a smaller multiple exists. For example, using 80 as the common denominator for $\frac{1}{5}$ and $\frac{1}{8}$ instead of 40 will still lead to the correct answer if simplified, but it involves larger numbers and more complex calculations.

How to Avoid It: Practice finding the LCM of different sets of numbers. Use methods such as listing multiples or prime factorization to ensure accuracy. Always aim for the least common multiple to keep the calculations manageable.

Mistake 5: Arithmetic Errors in Numerator or Denominator Operations

Simple arithmetic errors, such as incorrect addition, subtraction, multiplication, or division, can derail the entire calculation. These errors can occur when converting fractions to equivalent fractions, adding numerators, or simplifying the final result. For example, a small mistake in multiplying the numerator and denominator by the same number when finding equivalent fractions can lead to a significantly different answer.

How to Avoid It: Double-check each step of your calculations. Write out each step clearly and systematically to minimize the chances of making arithmetic errors. If possible, use a calculator to verify your calculations, especially for more complex numbers.

By being mindful of these common mistakes and actively working to avoid them, you can significantly improve your accuracy and confidence in evaluating expressions involving fractions. Remember to take your time, double-check your work, and practice regularly to reinforce your understanding.

Practice Problems for Evaluating Fraction Expressions

To solidify your understanding of evaluating fraction expressions, it's essential to practice with a variety of problems. Practice not only reinforces the concepts but also helps you identify areas where you may need further clarification. Here are some practice problems that cover different scenarios, including addition, subtraction, and negative fractions.

Problem 1: $\frac{2}{3} + \frac{1}{4}$

This problem requires you to add two fractions with different denominators. The first step is to find a common denominator. The least common multiple of 3 and 4 is 12. Convert both fractions to equivalent fractions with a denominator of 12:

$\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$

$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$

Now, add the fractions:

$\frac{8}{12} + \frac{3}{12} = \frac{8 + 3}{12} = \frac{11}{12}$

The fraction $\frac{11}{12}$ is already in its simplest form, so the final answer is $\frac{11}{12}$.

Problem 2: $\frac{5}{6} - \frac{1}{3}$

This problem involves subtracting fractions with different denominators. The least common multiple of 6 and 3 is 6. Convert the fractions to equivalent fractions with a denominator of 6. Note that $\frac{5}{6}$ already has the desired denominator:

$\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$

Now, subtract the fractions:

$\frac{5}{6} - \frac{2}{6} = \frac{5 - 2}{6} = \frac{3}{6}$

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

$\frac{3}{6} = \frac{3 \div 3}{6 \div 3} = \frac{1}{2}$

The final answer in simplest form is $\frac{1}{2}$.

Problem 3: $-\frac{3}{4} + \frac{1}{2}$

This problem involves adding a negative fraction to a positive fraction. The least common multiple of 4 and 2 is 4. Convert the fractions to equivalent fractions with a denominator of 4:

$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$

Now, add the fractions:

$-\frac{3}{4} + \frac{2}{4} = \frac{-3 + 2}{4} = \frac{-1}{4}$

The fraction is already in its simplest form, so the final answer is $-\frac{1}{4}$.

Problem 4: $-\frac{2}{5} - \left(-\frac{1}{10}\right)$

This problem involves subtracting a negative fraction, which is the same as adding its positive counterpart. Rewrite the expression:

$-\frac{2}{5} + \frac{1}{10}$

The least common multiple of 5 and 10 is 10. Convert the fractions to equivalent fractions with a denominator of 10:

$-\frac{2}{5} = -\frac{2 \times 2}{5 \times 2} = -\frac{4}{10}$

Now, add the fractions:

$-\frac{4}{10} + \frac{1}{10} = \frac{-4 + 1}{10} = \frac{-3}{10}$

The fraction is already in its simplest form, so the final answer is $-\frac{3}{10}$.

Problem 5: $\frac{7}{8} - \frac{1}{4} + \frac{1}{2}$

This problem involves multiple operations. The least common multiple of 8, 4, and 2 is 8. Convert all fractions to equivalent fractions with a denominator of 8:

$\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$

$\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$

Now, perform the operations from left to right:

$\frac{7}{8} - \frac{2}{8} + \frac{4}{8} = \frac{7 - 2 + 4}{8} = \frac{9}{8}$

The fraction $\frac{9}{8}$ is an improper fraction. It can be written as a mixed number: $1\frac{1}{8}$. However, if the question asks for the answer in simplest form as a fraction, $\frac{9}{8}$ is correct.

By working through these practice problems, you'll gain confidence in your ability to evaluate fraction expressions accurately. Remember to take your time, show your work, and double-check each step. Consistent practice is the key to mastering these skills.

Conclusion

In this comprehensive guide, we've explored the process of evaluating the expression -\frac{1}{5}-\left(-\frac{1}{8} ight) and expressing the result in its simplest form. We've covered the fundamental concepts of fraction arithmetic, including the importance of finding a common denominator, dealing with negative fractions, and simplifying the final answer. By following a step-by-step approach, we demonstrated how to transform the initial expression, perform the necessary operations, and arrive at the solution, which is $-\frac{3}{40}$.

Throughout the article, we emphasized the significance of understanding the underlying principles of fraction manipulation. This understanding not only helps in solving specific problems but also builds a strong foundation for more advanced mathematical concepts. We also highlighted common mistakes to avoid when evaluating fraction expressions, such as forgetting to find a common denominator, incorrectly applying the rules of signs, and failing to simplify the final answer. By being aware of these pitfalls, you can minimize errors and improve your accuracy.

To further solidify your understanding, we provided a set of practice problems that cover a range of scenarios, including addition, subtraction, and working with negative fractions. These problems offer an opportunity to apply the concepts learned and develop confidence in your problem-solving abilities. Consistent practice is key to mastering fraction arithmetic and becoming proficient in evaluating fraction expressions.

In conclusion, evaluating expressions involving fractions requires a systematic approach, a solid grasp of the fundamental concepts, and careful attention to detail. By following the steps outlined in this guide and practicing regularly, you can confidently tackle any fraction-related challenge. Whether you are a student learning these concepts for the first time or someone looking to refresh your skills, this article serves as a valuable resource for mastering the art of fraction evaluation.