Dividing Negative Numbers A Step-by-Step Guide To Solving -8 ÷ -2 6/7

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In mathematics, dividing negative numbers can sometimes seem tricky, but with a clear understanding of the rules, it becomes straightforward. This article delves into the specifics of dividing -8 by -2 6/7, offering a step-by-step explanation to ensure clarity. We will explore the underlying principles of dividing negative numbers, convert mixed fractions to improper fractions, and perform the division to arrive at the final answer. Understanding these concepts is crucial for anyone studying basic arithmetic or algebra. The division of negative numbers follows a fundamental rule: when you divide a negative number by another negative number, the result is positive. This is because division is the inverse operation of multiplication, and a negative times a negative results in a positive. Keeping this rule in mind will help us as we proceed with the calculation.

Converting Mixed Fractions to Improper Fractions

Before we can divide -8 by -2 6/7, we need to convert the mixed fraction -2 6/7 into an improper fraction. A mixed fraction is a combination of a whole number and a proper fraction (where the numerator is less than the denominator). An improper fraction, on the other hand, has a numerator that is greater than or equal to the denominator. To convert a mixed fraction to an improper fraction, we follow these steps:

  1. Multiply the whole number part by the denominator of the fractional part.
  2. Add the result to the numerator of the fractional part.
  3. Place the sum over the original denominator.

For -2 6/7, we multiply the whole number 2 by the denominator 7, which gives us 14. Then, we add this result to the numerator 6, yielding 20. Finally, we place this sum over the original denominator 7. So, -2 6/7 converts to -20/7. This conversion is crucial because it allows us to perform division more easily, especially when dealing with fractions. By converting to an improper fraction, we transform the mixed number into a single fraction, simplifying the subsequent division process. The ability to convert mixed fractions to improper fractions is a fundamental skill in arithmetic and algebra, essential for performing various mathematical operations accurately.

Performing the Division

Now that we have -8 and -20/7, we can proceed with the division. Remember, dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. So, the reciprocal of -20/7 is -7/20.

Therefore, -8 ÷ (-20/7) becomes -8 × (-7/20). When multiplying fractions, we multiply the numerators together and the denominators together. In this case, we have:

(-8/1) × (-7/20) = (8 × 7) / (1 × 20) = 56/20

Now, we simplify the fraction 56/20 by finding the greatest common divisor (GCD) of 56 and 20. The GCD is the largest number that divides both 56 and 20 without leaving a remainder. The GCD of 56 and 20 is 4. We divide both the numerator and the denominator by 4:

56 ÷ 4 = 14 20 ÷ 4 = 5

So, the simplified fraction is 14/5. This is an improper fraction, and we can convert it back to a mixed fraction to make it more understandable. To do this, we divide 14 by 5. The quotient is 2, and the remainder is 4. Therefore, 14/5 as a mixed fraction is 2 4/5. This final result represents the answer to our original division problem, -8 ÷ -2 6/7.

To recap, let's go through the solution step-by-step to ensure a clear understanding:

  1. Convert the mixed fraction -2 6/7 to an improper fraction: -2 6/7 = -20/7.
  2. Rewrite the division problem as multiplication by the reciprocal: -8 ÷ (-20/7) = -8 × (-7/20).
  3. Multiply the fractions: (-8/1) × (-7/20) = 56/20.
  4. Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 4: 56/20 = 14/5.
  5. Convert the improper fraction 14/5 back to a mixed fraction: 14/5 = 2 4/5.

Therefore, -8 ÷ -2 6/7 = 2 4/5. This step-by-step approach breaks down the problem into manageable parts, making it easier to follow and understand each operation. The conversion of mixed fractions to improper fractions, the multiplication by the reciprocal, and the simplification of the resulting fraction are all critical steps in solving this problem. Mastering these steps will improve your ability to tackle similar mathematical challenges.

Importance of Understanding Fraction Division

Understanding fraction division is a fundamental concept in mathematics with wide-ranging applications. It is not only essential for basic arithmetic but also crucial for more advanced topics like algebra, calculus, and various real-world applications. Whether you are calculating measurements, splitting resources, or solving complex equations, the ability to divide fractions accurately is indispensable. In algebra, for instance, fraction division is often used when simplifying expressions and solving equations involving rational numbers. It is also a key component in understanding ratios and proportions, which are used in various fields, including science, engineering, and finance. Moreover, in everyday life, we encounter situations where fraction division is necessary, such as when scaling recipes, calculating time, or determining quantities.

For example, consider a recipe that calls for 2 1/2 cups of flour and you only want to make half the recipe. You would need to divide 2 1/2 by 2 to find the new amount of flour needed. Similarly, in construction, you might need to divide a length of wood into equal parts, which requires fraction division. The ability to confidently divide fractions allows for precise calculations and better decision-making in various contexts. Furthermore, mastering fraction division builds a strong foundation for understanding other mathematical concepts. It enhances problem-solving skills and logical reasoning, which are valuable in any academic or professional pursuit. Therefore, a thorough understanding of fraction division is not just about memorizing steps but about grasping the underlying principles and applying them effectively.

Common Mistakes to Avoid

When dividing fractions, there are several common mistakes that students often make. Being aware of these pitfalls can help in avoiding them and ensuring accurate calculations. One of the most common mistakes is forgetting to take the reciprocal of the second fraction before multiplying. Remember, division is equivalent to multiplying by the reciprocal, so this step is crucial. Another frequent error is incorrectly converting mixed fractions to improper fractions. Ensure that you multiply the whole number by the denominator and add the numerator, placing the result over the original denominator. Careless mistakes in this conversion can lead to significant errors in the final answer.

Additionally, students sometimes forget to simplify the resulting fraction after performing the multiplication. Simplifying fractions by dividing both the numerator and the denominator by their greatest common divisor is essential for expressing the answer in its simplest form. Failing to simplify can result in an answer that is technically correct but not fully reduced. Another mistake is not paying attention to the signs. Remember the rules for dividing negative numbers: a negative divided by a negative is a positive, and a negative divided by a positive (or vice versa) is a negative. Neglecting the signs can lead to errors in the sign of the final answer. Finally, some students may attempt to divide the fractions directly without converting to multiplication by the reciprocal. This method is incorrect and will lead to wrong results. To avoid these common mistakes, practice is key. Working through a variety of problems and paying close attention to each step will help solidify your understanding and improve your accuracy.

In conclusion, dividing -8 by -2 6/7 involves several steps, including converting mixed fractions to improper fractions, multiplying by the reciprocal, simplifying the fraction, and converting back to a mixed fraction if necessary. By following these steps carefully and understanding the underlying principles, you can confidently solve similar problems. Remember the rule for dividing negative numbers, and always double-check your work to avoid common mistakes. Fraction division is a foundational skill in mathematics, and mastering it will benefit you in various areas of study and in everyday life. Through practice and a clear understanding of the process, you can develop proficiency in this essential mathematical operation. The final answer to -8 ÷ -2 6/7 is 2 4/5, illustrating the importance of each step in arriving at the correct solution. Continue to practice and apply these concepts to build a strong mathematical foundation.