1. Calculate (-10/11) × (-6/7). 2. Calculate (9/10) ÷ (-6/7). 3. Calculate (-11/24) ÷ (-5/8). 4. Calculate (-70/7) + (66/77) And Verify If The Result Is Equal To (-736/77).

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In the realm of mathematics, fraction operations are fundamental building blocks. Understanding how to multiply, divide, add, and subtract fractions is crucial for various mathematical concepts and real-world applications. This comprehensive guide delves into the intricacies of fraction operations, providing step-by-step explanations and examples to solidify your understanding. Let's embark on this mathematical journey to master fraction operations.

1. Multiplying Fractions: Unveiling the Process

Multiplying fractions is a straightforward process that involves multiplying the numerators (the top numbers) and the denominators (the bottom numbers). This seemingly simple operation forms the basis for more complex fraction calculations. To effectively multiply fractions, it's essential to grasp the underlying principle: when you multiply fractions, you're essentially finding a fraction of a fraction. For instance, multiplying 1/2 by 1/4 means you're finding half of one-quarter. This fundamental concept helps visualize the multiplication process and makes it more intuitive.

The procedure for multiplying fractions can be summarized in three concise steps:

  1. Multiply the numerators: Begin by multiplying the numerators of the fractions. The result becomes the numerator of the product.
  2. Multiply the denominators: Next, multiply the denominators of the fractions. The outcome will be the denominator of the product.
  3. Simplify the result (if possible): After obtaining the product, simplify it to its simplest form by dividing both the numerator and denominator by their greatest common factor (GCF). This step ensures that the fraction is expressed in its most reduced form.

Let's illustrate this process with the first example:

1. -10/11 × -6/7

In this example, we're multiplying two negative fractions. Remember, a negative times a negative results in a positive. So, the product will be positive.

  1. Multiply the numerators: -10 × -6 = 60
  2. Multiply the denominators: 11 × 7 = 77
  3. Simplify the result (if possible): In this case, 60/77 cannot be simplified further as 60 and 77 share no common factors other than 1.

Therefore,

-10/11 × -6/7 = 60/77

This example demonstrates the step-by-step process of multiplying fractions, emphasizing the importance of multiplying numerators and denominators and simplifying the result when necessary. Understanding this process is fundamental to mastering fraction operations and solving more complex mathematical problems.

2. Dividing Fractions: Unlocking the Inversion Technique

Dividing fractions might seem daunting initially, but it becomes manageable with a simple technique: inverting the second fraction and multiplying. This concept stems from the fundamental relationship between division and multiplication, where division is essentially the inverse operation of multiplication. To divide fractions effectively, it's crucial to understand why this inversion technique works. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator. For example, the reciprocal of 2/3 is 3/2. Understanding this principle simplifies fraction division and makes it less intimidating.

The process of dividing fractions involves the following steps:

  1. Invert the second fraction: The first step is to find the reciprocal of the second fraction (the divisor). This involves swapping its numerator and denominator.
  2. Change the division to multiplication: Once you've inverted the second fraction, change the division operation to multiplication.
  3. Multiply the fractions: Now, multiply the fractions as you would in a regular multiplication problem (multiply numerators and denominators).
  4. Simplify the result (if possible): After obtaining the product, simplify it to its simplest form by dividing both the numerator and denominator by their greatest common factor (GCF).

Let's apply this technique to the second example:

2. 9/10 ÷ -6/7

Here, we're dividing a positive fraction by a negative fraction. The result will be negative.

  1. Invert the second fraction: The reciprocal of -6/7 is -7/6.
  2. Change the division to multiplication: 9/10 ÷ -6/7 becomes 9/10 × -7/6.
  3. Multiply the fractions:
    • 9 × -7 = -63
    • 10 × 6 = 60
    • So, 9/10 × -7/6 = -63/60
  4. Simplify the result (if possible): -63/60 can be simplified by dividing both numerator and denominator by their GCF, which is 3.
    • -63 ÷ 3 = -21
    • 60 ÷ 3 = 20
    • Therefore, -63/60 simplifies to -21/20

Thus,

9/10 ÷ -6/7 = -21/20

This example showcases the inversion technique in action, highlighting how dividing fractions can be transformed into a multiplication problem by inverting the second fraction. Mastering this technique is essential for confidently tackling division problems involving fractions.

3. Dividing Fractions: Another Example

To further solidify your understanding of fraction division, let's analyze another example, focusing on the same inversion technique. Repetition and diverse examples are key to mastering mathematical concepts. By working through multiple examples, you'll gain confidence in applying the inversion technique and simplifying the resulting fractions. Each example presents a unique scenario, allowing you to hone your skills and develop a deeper understanding of the underlying principles. This approach will empower you to solve a wider range of fraction division problems with ease and accuracy.

3. -11/24 ÷ -5/8

In this case, we're dividing a negative fraction by another negative fraction, resulting in a positive quotient.

  1. Invert the second fraction: The reciprocal of -5/8 is -8/5.
  2. Change the division to multiplication: -11/24 ÷ -5/8 becomes -11/24 × -8/5.
  3. Multiply the fractions:
    • -11 × -8 = 88
    • 24 × 5 = 120
    • So, -11/24 × -8/5 = 88/120
  4. Simplify the result (if possible): 88/120 can be simplified by dividing both numerator and denominator by their GCF, which is 8.
    • 88 ÷ 8 = 11
    • 120 ÷ 8 = 15
    • Therefore, 88/120 simplifies to 11/15

Thus,

-11/24 ÷ -5/8 = 11/15

This example reinforces the inversion technique and emphasizes the importance of simplifying the final result. By consistently applying these steps, you'll develop fluency in dividing fractions and be able to solve a variety of problems efficiently.

4. Adding Fractions: Addressing the Common Denominator

Adding fractions requires a bit more attention than multiplying or dividing, primarily because fractions must have a common denominator before they can be added. The common denominator serves as a unifying unit, allowing us to combine the numerators accurately. This concept is crucial for understanding why a common denominator is necessary: we can only add fractions that represent parts of the same whole. If the denominators are different, we're essentially trying to add apples and oranges – they need to be expressed in the same unit before we can combine them. Finding the least common denominator (LCD) is the most efficient way to ensure accurate addition, as it minimizes the need for simplification later on.

The process of adding fractions involves the following steps:

  1. Find the least common denominator (LCD): The LCD is the smallest common multiple of the denominators. It's the smallest number that both denominators divide into evenly.
  2. Convert the fractions to equivalent fractions with the LCD: To do this, multiply both the numerator and denominator of each fraction by the factor that makes the denominator equal to the LCD.
  3. Add the numerators: Once the fractions have the same denominator, add their numerators. The denominator remains the same.
  4. Simplify the result (if possible): Simplify the resulting fraction to its simplest form by dividing both the numerator and denominator by their greatest common factor (GCF).

Let's examine the fourth example, which involves adding fractions:

4. -70/7 + 66/77 = -736/77

This example presents an addition problem where the fractions have different denominators. We need to find the LCD before we can add them.

  1. Find the least common denominator (LCD): The denominators are 7 and 77. The LCD of 7 and 77 is 77 (since 77 is a multiple of 7).
  2. Convert the fractions to equivalent fractions with the LCD:
    • -70/7 needs to be converted to an equivalent fraction with a denominator of 77. To do this, multiply both the numerator and denominator by 11 (since 7 × 11 = 77).
      • -70 × 11 = -770
      • 7 × 11 = 77
      • So, -70/7 becomes -770/77
    • 66/77 already has the LCD as its denominator, so no conversion is needed.
  3. Add the numerators:
    • -770/77 + 66/77 = (-770 + 66) / 77
    • -770 + 66 = -704
    • So, -770/77 + 66/77 = -704/77
  4. Simplify the result (if possible): -704/77 can be simplified by dividing both numerator and denominator by their GCF, which is 11.
    • -704 ÷ 11 = -64
    • 77 ÷ 11 = 7
    • Therefore, -704/77 simplifies to -64/7

Thus, the correct solution is:

-70/7 + 66/77 = -64/7

However, the given solution is:

-70/7 + 66/77 = -736/77

This indicates an error in the original calculation. The correct sum, after finding the LCD and simplifying, is -64/7, not -736/77. This highlights the importance of carefully following the steps for adding fractions and verifying the result to ensure accuracy.

Conclusion: Mastering Fraction Operations for Mathematical Proficiency

In conclusion, mastering fraction operations is crucial for developing a strong foundation in mathematics. This comprehensive guide has elucidated the fundamental principles and step-by-step procedures for multiplying, dividing, and adding fractions. By understanding these concepts and practicing consistently, you can confidently tackle various mathematical problems involving fractions. Remember, multiplying fractions involves multiplying numerators and denominators, while dividing fractions requires inverting the second fraction and multiplying. Adding fractions, on the other hand, necessitates finding a common denominator before combining the numerators. Consistent practice and a thorough understanding of these principles will pave the way for mathematical proficiency and success.