Solving Negative Fraction Multiplication: (-1/6) * (-2/3)

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In the realm of mathematics, fractions play a crucial role in representing parts of a whole. When dealing with negative fractions, the rules of multiplication remain consistent, but special attention must be paid to the signs. This article aims to provide a comprehensive guide to solving the multiplication of negative fractions, using the example of -1/6 × -2/3. By breaking down the process step-by-step, we will explore the fundamental principles and techniques involved, ensuring a clear understanding of the concept.

The Fundamentals of Fraction Multiplication

To grasp the multiplication of negative fractions, it is essential to first understand the basics of fraction multiplication. When multiplying fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. This can be expressed as:

(a/b) × (c/d) = (a × c) / (b × d)

For instance, if we want to multiply 1/2 by 2/3, we would multiply the numerators (1 and 2) to get 2 and the denominators (2 and 3) to get 6, resulting in 2/6, which can be simplified to 1/3. This fundamental principle forms the basis for multiplying negative fractions as well.

Multiplying Negative Fractions: Key Principles

When multiplying negative fractions, we need to consider the signs of the fractions involved. The following rules govern the multiplication of negative numbers:

  • A negative number multiplied by a negative number yields a positive number.
  • A negative number multiplied by a positive number yields a negative number.
  • A positive number multiplied by a negative number yields a negative number.

These rules are crucial when dealing with negative fractions, as they determine the sign of the final product. In our example of -1/6 × -2/3, we are multiplying two negative fractions, which means the result will be a positive fraction. This is because a negative times a negative results in a positive. Understanding this principle is paramount for accurately solving such problems.

Step-by-Step Solution of -1/6 × -2/3

Now, let's apply these principles to solve the given problem: -1/6 × -2/3. We will proceed step-by-step to ensure clarity and understanding.

Step 1: Multiply the Numerators

The first step is to multiply the numerators of the two fractions. In this case, the numerators are -1 and -2. Multiplying these gives us:

(-1) × (-2) = 2

As discussed earlier, the product of two negative numbers is positive, so -1 multiplied by -2 equals 2.

Step 2: Multiply the Denominators

Next, we multiply the denominators of the fractions. The denominators are 6 and 3. Multiplying these gives us:

6 × 3 = 18

This step is straightforward and involves simple multiplication of the denominators.

Step 3: Form the Resulting Fraction

Now that we have the product of the numerators and the product of the denominators, we can form the resulting fraction. The product of the numerators (2) becomes the new numerator, and the product of the denominators (18) becomes the new denominator. Thus, our fraction is:

2/18

This fraction represents the result of multiplying -1/6 and -2/3.

Step 4: Simplify the Fraction (if possible)

The final step is to simplify the fraction if possible. Simplification involves reducing the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD). In the case of 2/18, the GCD of 2 and 18 is 2. Dividing both the numerator and the denominator by 2, we get:

(2 ÷ 2) / (18 ÷ 2) = 1/9

Therefore, the simplified fraction is 1/9. This is the final answer to the multiplication of -1/6 and -2/3.

Analyzing the Answer Choices

Now, let's analyze the given answer choices to determine which one matches our solution:

  • F. 1/4
  • G. -1/9
  • H. 1/9
  • J. -1/4
  • K. None of these

Comparing our solution, 1/9, with the answer choices, we can see that option H, 1/9, is the correct answer. The other options are either incorrect fractions or have the wrong sign. Understanding how to simplify fractions is essential to arriving at the correct answer.

Common Mistakes to Avoid

When multiplying negative fractions, several common mistakes can occur. Being aware of these pitfalls can help prevent errors and ensure accurate solutions. Here are some mistakes to watch out for:

  1. Forgetting the Sign Rule: One of the most common mistakes is forgetting the rules for multiplying negative numbers. Remember that a negative times a negative is a positive, and a negative times a positive is a negative. Failing to apply this rule correctly can lead to the wrong sign in the final answer.
  2. Incorrectly Multiplying Numerators or Denominators: Another common mistake is making errors in the multiplication of either the numerators or the denominators. Double-checking these calculations can help avoid such mistakes. Accuracy in multiplication is key to solving these problems.
  3. Failing to Simplify: Sometimes, students arrive at the correct fraction but fail to simplify it to its lowest terms. This can lead to selecting an incorrect answer choice if the simplified form is required. Always simplify the fraction if possible.
  4. Misunderstanding the Question: It is essential to fully understand the question being asked. Misinterpreting the problem can lead to applying the wrong operations or methods. Read the question carefully and identify what is being asked before attempting to solve it.

Practical Applications of Multiplying Negative Fractions

Understanding the multiplication of negative fractions is not just an academic exercise; it has practical applications in various real-world scenarios. Here are a few examples:

  1. Finance: In financial calculations, negative fractions can represent losses or debts. Multiplying these fractions can help determine the overall financial impact. For example, if a business loses 1/3 of its revenue in one quarter and 2/5 of its remaining revenue in the next quarter, multiplying these fractions can help calculate the total loss.
  2. Measurement and Conversions: Negative fractions can be used in measurements and conversions, particularly when dealing with temperatures below zero or distances in opposite directions. Multiplying these fractions can help in converting units or calculating total distances.
  3. Engineering and Construction: In engineering and construction, fractions are commonly used to represent dimensions and proportions. Negative fractions can represent cuts or subtractions. Multiplying these fractions can help in calculating the final dimensions or the amount of material needed.
  4. Everyday Scenarios: Even in everyday scenarios like cooking, negative fractions can come into play. For instance, if a recipe calls for reducing the quantity of ingredients by a fraction, understanding how to multiply fractions is essential for accurate adjustments. Real-world applications make this mathematical skill highly valuable.

Conclusion: Mastering Negative Fraction Multiplication

In conclusion, the multiplication of negative fractions is a fundamental concept in mathematics with practical applications across various fields. By understanding the basic principles, following the step-by-step process, and avoiding common mistakes, one can master this skill. In the specific example of -1/6 × -2/3, the correct answer is 1/9, which corresponds to option H. Remember to multiply the numerators, multiply the denominators, and simplify the resulting fraction. With practice and a clear understanding of the rules, multiplying negative fractions can become a straightforward and essential mathematical tool.

By mastering this concept, you will enhance your mathematical skills and be better equipped to tackle real-world problems involving fractions and negative numbers. This step-by-step guide aims to provide a solid foundation for further mathematical explorations and applications.