Multiplying And Dividing Fractions Simplifying To Lowest Terms

In the realm of mathematics, mastering the art of fraction manipulation is paramount. Fractions, the building blocks of rational numbers, play a crucial role in various mathematical disciplines and real-world applications. Among the fundamental operations involving fractions, multiplication and division stand out as essential skills. This comprehensive guide delves into the intricacies of multiplying and dividing fractions, emphasizing the importance of reducing answers to their lowest terms. We will explore the underlying principles, step-by-step procedures, and practical examples to solidify your understanding of these concepts.

Fraction multiplication might seem daunting at first, but it's a straightforward process that involves multiplying the numerators and denominators separately. The numerator is the top number in a fraction, representing the number of parts we have, while the denominator is the bottom number, indicating the total number of equal parts the whole is divided into. To illustrate, consider the fractions ${\frac{20}{9}}$ and ${\frac{-7}{12}}$. Multiplying these fractions involves multiplying the numerators (20 and -7) and the denominators (9 and 12).

Before we dive into the step-by-step procedure, let's establish the fundamental rule of multiplying fractions:

${\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}}$

where 'a' and 'c' represent the numerators, and 'b' and 'd' represent the denominators.

Now, let's break down the multiplication process into manageable steps:

1. Multiply the Numerators: Begin by multiplying the numerators of the fractions. In our example, we have 20 multiplied by -7, which yields -140.

2. Multiply the Denominators: Next, multiply the denominators of the fractions. In this case, 9 multiplied by 12 equals 108.

3. Form the New Fraction: The product of the numerators becomes the new numerator, and the product of the denominators becomes the new denominator. Thus, we obtain the fraction ${\frac{-140}{108}}$.

Fraction division introduces an extra layer of complexity, but it's easily conquered by understanding the concept of reciprocals. The reciprocal of a fraction is obtained by swapping its numerator and denominator. For instance, the reciprocal of ${\frac{2}{3}}$ is ${\frac{3}{2}}$. Dividing by a fraction is equivalent to multiplying by its reciprocal. This crucial principle transforms division into a multiplication problem, making it much easier to solve.

The general rule for dividing fractions is:

${\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}}$

where ${\frac{d}{c}}$ is the reciprocal of ${\frac{c}{d}}$.

Let's outline the steps involved in dividing fractions:

1. Find the Reciprocal of the Second Fraction: Identify the fraction you're dividing by and determine its reciprocal. Swap the numerator and denominator. For example, if you're dividing by ${\frac{1}{2}}$, its reciprocal is ${\frac{2}{1}}$.

2. Change Division to Multiplication: Replace the division sign with a multiplication sign. Now, you're essentially multiplying by the reciprocal.

3. Multiply the Fractions: Follow the same procedure as multiplying fractions. Multiply the numerators and denominators separately.

4. Simplify the Result: Reduce the resulting fraction to its lowest terms, if possible.

Reducing fractions to their lowest terms, also known as simplifying fractions, is an essential step in fraction manipulation. A fraction is in its lowest terms when the numerator and denominator have no common factors other than 1. In other words, the fraction cannot be further simplified.

The process of reducing fractions involves finding the greatest common factor (GCF) of the numerator and denominator. The GCF is the largest number that divides both the numerator and denominator without leaving a remainder. Once you've found the GCF, divide both the numerator and denominator by it to obtain the simplified fraction.

Let's illustrate this with an example. Consider the fraction ${\frac{24}{36}}$. To reduce this fraction to its lowest terms, we need to find the GCF of 24 and 36.

The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.

The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

The greatest common factor of 24 and 36 is 12. Now, we divide both the numerator and denominator by 12:

${\frac{24 \div 12}{36 \div 12} = \frac{2}{3}}$

Therefore, the fraction ${\frac{24}{36}}$ reduced to its lowest terms is ${\frac{2}{3}}$.

Now, let's apply the concepts we've learned to solve the original problem: multiply or divide as indicated and reduce the answer to lowest terms for ${\frac{20}{9} \cdot \frac{-7}{12}}$.

Since the operation is multiplication, we follow the steps outlined earlier:

1. Multiply the Numerators: 20 multiplied by -7 equals -140.

2. Multiply the Denominators: 9 multiplied by 12 equals 108.

3. Form the New Fraction: The resulting fraction is ${\frac{-140}{108}}$.

Now, we need to reduce this fraction to its lowest terms. To do this, we find the GCF of 140 and 108.

The factors of 140 are: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140.

The factors of 108 are: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.

The greatest common factor of 140 and 108 is 4. We divide both the numerator and denominator by 4:

${\frac{-140 \div 4}{108 \div 4} = \frac{-35}{27}}$

Therefore, the product of ${\frac{20}{9}}$ and ${\frac{-7}{12}}$, reduced to its lowest terms, is ${\frac{-35}{27}}$.

Multiplying and dividing fractions are fundamental skills in mathematics. By understanding the underlying principles and following the step-by-step procedures outlined in this guide, you can confidently tackle these operations. Remember to reduce your answers to their lowest terms for a complete and accurate solution. With practice and dedication, you'll master the art of fraction manipulation and unlock its vast applications in mathematics and beyond. Mastering these concepts will build a strong foundation for more advanced mathematical topics. So, embrace the challenge, practice diligently, and watch your mathematical prowess soar.

This comprehensive guide has equipped you with the knowledge and skills to confidently multiply and divide fractions, ensuring you can simplify and reduce answers to their lowest terms. As you continue your mathematical journey, remember that fractions are more than just numbers; they are essential tools for understanding and solving problems in various fields. Embrace the challenge, practice consistently, and watch your mathematical abilities flourish.