Calculating The Quotient Of (a-3)/7 Divided By (3-a)/21

Quotients in mathematics represent the result of a division operation. Understanding how to calculate quotients is a fundamental skill in algebra and beyond. This comprehensive guide will walk you through the process of solving the expression (a-3)/7 ÷ (3-a)/21, providing a clear, step-by-step explanation to ensure you grasp the underlying concepts. We'll cover the key principles of dividing fractions, factoring, and simplifying expressions, equipping you with the knowledge to tackle similar problems with confidence. This detailed exploration is designed to enhance your understanding of quotients and algebraic manipulations, making it easier to succeed in your math studies.

Decoding the Division: (a-3)/7 ÷ (3-a)/21

At its core, this problem involves the division of two algebraic fractions. To effectively solve this, we need to understand the fundamental principle of dividing fractions: dividing by a fraction is the same as multiplying by its reciprocal. This means we'll transform the division problem into a multiplication problem by inverting the second fraction. The expression (a-3)/7 ÷ (3-a)/21 can be rewritten as (a-3)/7 * 21/(3-a). This transformation is the first crucial step in simplifying the problem and making it more manageable. By understanding this basic principle, we set the stage for further simplification and arrive at the correct solution. This initial transformation is key to unlocking the solution and provides a solid foundation for the subsequent steps in the problem-solving process.

Step 1: Transforming Division into Multiplication

As mentioned earlier, the cornerstone of solving this problem is understanding that dividing by a fraction is equivalent to multiplying by its reciprocal. This is a fundamental rule in fraction arithmetic and is crucial for simplifying complex expressions. When we apply this to our problem, (a-3)/7 ÷ (3-a)/21 becomes (a-3)/7 * 21/(3-a). This transformation is not just a mechanical step; it reflects a deeper mathematical principle. By converting division into multiplication, we make the expression more amenable to simplification. This step sets the foundation for the rest of the solution, allowing us to apply other algebraic techniques to arrive at the final answer. This conversion is a powerful tool in mathematics, allowing us to manipulate expressions and make them easier to work with. It's a technique that extends beyond this specific problem and is applicable in a wide range of mathematical contexts.

Step 2: Unveiling the Hidden Factor: Factoring and Simplification

After transforming the division into multiplication, the next step involves a clever manipulation of the expression to reveal a common factor. Notice that (3-a) is the negative of (a-3). This is a critical observation. We can rewrite (3-a) as -(a-3). This seemingly small change is a game-changer because it allows us to identify a common factor in the numerator and the denominator. Replacing (3-a) with -(a-3) transforms our expression into (a-3)/7 * 21/(-(a-3)). Now, we can clearly see the (a-3) term in both the numerator and the denominator. This allows us to simplify the expression significantly. Factoring and simplification are essential skills in algebra, and this step highlights their importance. By recognizing the relationship between (3-a) and (a-3), we unlock the potential for further simplification and move closer to the final solution.

Step 3: The Art of Cancellation: Simplifying Fractions

With the common factor (a-3) now apparent, we can proceed with simplification. The expression (a-3)/7 * 21/(-(a-3)) has (a-3) in both the numerator and the denominator, which can be cancelled out. This is a direct application of the principle that any non-zero term divided by itself equals 1. Cancelling (a-3) leaves us with 1/7 * 21/(-1). Now, we have a much simpler expression to deal with. Next, we can simplify the numerical part of the expression. 21 divided by 7 is 3, so our expression further simplifies to 1 * 3/(-1), which is simply 3/(-1). This step showcases the power of cancellation in simplifying fractions. By identifying and cancelling common factors, we reduce the complexity of the expression and make it easier to evaluate. Cancellation is a fundamental technique in fraction manipulation and is a skill that is invaluable in various mathematical contexts.

Step 4: The Grand Finale: Calculating the Quotient

We've reached the final stage of our journey! After the simplification process, we have arrived at the expression 3/(-1). This is a straightforward division problem. Dividing 3 by -1 yields -3. Therefore, the quotient of the original expression (a-3)/7 ÷ (3-a)/21 is -3. This final step brings together all the previous steps, demonstrating how each manipulation contributed to the final answer. The solution -3 is a concise and clear representation of the quotient. This final calculation underscores the importance of careful simplification and the application of basic arithmetic principles. By meticulously following each step, we have successfully navigated the problem and arrived at the correct solution. This step-by-step approach not only provides the answer but also reinforces the underlying mathematical concepts and techniques.

The Final Verdict: -3

In conclusion, by systematically applying the principles of fraction division, factoring, and simplification, we have determined that the quotient of (a-3)/7 ÷ (3-a)/21 is -3. This solution highlights the importance of understanding the underlying concepts and applying them methodically. Each step, from transforming division to multiplication to cancelling common factors, played a crucial role in arriving at the correct answer. This comprehensive guide not only provides the solution but also illuminates the problem-solving process, empowering you to tackle similar challenges with confidence and clarity. The journey to finding the quotient has reinforced key mathematical principles and demonstrated the power of step-by-step problem-solving. This understanding will serve as a valuable asset in your continued mathematical endeavors.