Dividing Fractions Step-by-Step A Comprehensive Guide

In the realm of mathematics, fractions play a crucial role, and understanding how to manipulate them is essential for various calculations and problem-solving scenarios. One fundamental operation involving fractions is division. Dividing fractions might seem daunting at first, but with a clear understanding of the underlying principles and a step-by-step approach, it becomes a straightforward process. This comprehensive guide will walk you through the process of dividing fractions, focusing on the specific example of ${\frac{15}{6} \div \frac{27}{28}}$. We will break down each step, ensuring clarity and comprehension, and ultimately demonstrate how to reduce the resulting quotient to its lowest terms.

Understanding the Basics of Fraction Division

Before diving into the specifics of our example, let's establish a solid foundation by revisiting the core concept of fraction division. Dividing by a fraction is conceptually equivalent to multiplying by its reciprocal. The reciprocal of a fraction is obtained by simply swapping the numerator and the denominator. For instance, the reciprocal of ${\frac{2}{3}}$ is ${\frac{3}{2}}$. This fundamental principle forms the bedrock of fraction division. In essence, when you divide one fraction by another, you are essentially asking, "How many times does the second fraction fit into the first fraction?" This understanding provides a concrete context for the mathematical operation.

To further solidify this concept, consider a real-world analogy. Imagine you have a pizza cut into 6 slices, representing the fraction ${\frac{6}{6}}$ or a whole pizza. If you want to divide this pizza among friends, and each friend gets ${\frac{1}{3}}$ of the pizza, you are essentially dividing 1 (the whole pizza) by ${\frac{1}{3}}$. To find out how many friends can be served, you would multiply 1 by the reciprocal of ${\frac{1}{3}}$, which is 3. This means you can serve 3 friends. This simple example illustrates the practical application of dividing fractions and highlights the connection between division and multiplication by the reciprocal.

Understanding this concept is crucial because it transforms the potentially complex operation of division into a more familiar operation: multiplication. By grasping this fundamental principle, you can approach fraction division problems with confidence and clarity.

Step 1: Rewriting the Division Problem as Multiplication

Now, let's apply this principle to our specific problem: ${\frac{15}{6} \div \frac{27}{28}}$. The first step in solving this division problem is to rewrite it as a multiplication problem. To do this, we keep the first fraction, ${\frac{15}{6}}$, as it is, and change the division sign (${\div}$) to a multiplication sign (${\times}$). Then, we replace the second fraction, ${\frac{27}{28}}$, with its reciprocal. As we discussed earlier, the reciprocal of a fraction is obtained by swapping the numerator and the denominator. Therefore, the reciprocal of ${\frac{27}{28}}$ is ${\frac{28}{27}}$.

By applying this transformation, we rewrite the original division problem as a multiplication problem: ${\frac{15}{6} \times \frac{28}{27}}$. This seemingly simple step is the key to unlocking the solution. By converting the division into multiplication, we can leverage the well-established rules of fraction multiplication to arrive at the answer.

This transformation not only simplifies the calculation but also provides a clearer understanding of the operation we are performing. Instead of trying to visualize how many times ${\frac{27}{28}}$ fits into ${\frac{15}{6}}$, we are now dealing with a straightforward multiplication problem. This shift in perspective can make the problem less intimidating and more accessible.

Step 2: Multiplying the Fractions

With the division problem transformed into a multiplication problem, we can now proceed with the multiplication process. Multiplying fractions is a relatively straightforward operation. To multiply two fractions, you multiply the numerators (the top numbers) together to get the new numerator, and you multiply the denominators (the bottom numbers) together to get the new denominator.

In our case, we have ${\frac{15}{6} \times \frac{28}{27}}$. To multiply these fractions, we multiply the numerators: 15 multiplied by 28. This gives us 420. Next, we multiply the denominators: 6 multiplied by 27. This gives us 162. Therefore, the result of the multiplication is ${\frac{420}{162}}$.

So, ${\frac{15}{6} \times \frac{28}{27} = \frac{15 \times 28}{6 \times 27} = \frac{420}{162}}$. This fraction, ${\frac{420}{162}}$, represents the quotient of the original division problem. However, it is not yet in its simplest form. The next step involves reducing this fraction to its lowest terms.

The process of multiplying fractions is a fundamental skill in mathematics, and mastering it is crucial for various applications. This step-by-step approach ensures accuracy and clarity in the calculation. By breaking down the process into smaller steps, we can avoid errors and gain a deeper understanding of the underlying principles.

Step 3: Reducing the Fraction to Lowest Terms

The fraction ${\frac{420}{162}}$ we obtained in the previous step is a valid answer, but it is not in its simplest form. Reducing a fraction to its lowest terms means finding the simplest equivalent fraction, where the numerator and the denominator have no common factors other than 1. In other words, we want to express the fraction using the smallest possible whole numbers.

To reduce ${\frac{420}{162}}$ to its lowest terms, we need to find the greatest common divisor (GCD) of 420 and 162. The GCD is the largest number that divides both 420 and 162 without leaving a remainder. There are several methods to find the GCD, such as listing the factors of each number or using the Euclidean algorithm. For this example, we will use the method of listing factors.

Factors of 420: 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420 Factors of 162: 1, 2, 3, 6, 9, 18, 27, 54, 81, 162

By comparing the lists of factors, we can identify the greatest common factor (GCD) of 420 and 162, which is 6.

Now that we have found the GCD, we can reduce the fraction by dividing both the numerator and the denominator by the GCD. So, we divide 420 by 6, which gives us 70, and we divide 162 by 6, which gives us 27. Therefore, the fraction ${\frac{420}{162}}$ reduced to its lowest terms is ${\frac{70}{27}}$.

This process of reducing fractions to their simplest form is essential in mathematics. It not only presents the answer in its most concise form but also makes it easier to compare and work with fractions in subsequent calculations. By finding the GCD and dividing both the numerator and denominator, we ensure that the resulting fraction is in its lowest terms.

Final Answer: The Quotient in Lowest Terms

After following the steps outlined above, we have successfully found the quotient of ${\frac{15}{6} \div \frac{27}{28}}$ and reduced it to its lowest terms. We started by rewriting the division problem as a multiplication problem, multiplying the fractions, and then reducing the resulting fraction to its simplest form.

The final answer, the quotient of ${\frac{15}{6} \div \frac{27}{28}}$ in lowest terms, is ${\frac{70}{27}}$. This fraction represents the result of the division in its most simplified form, where the numerator and denominator have no common factors other than 1. We can also express this improper fraction (where the numerator is greater than the denominator) as a mixed number: 2 ${\frac{16}{27}}$.

This comprehensive guide has demonstrated the step-by-step process of dividing fractions and reducing them to their lowest terms. By understanding the underlying principles and following these steps, you can confidently tackle any fraction division problem. Remember, the key is to rewrite the division as multiplication by the reciprocal, multiply the fractions, and then reduce the resulting fraction to its simplest form.

Mastering fraction division is a crucial skill in mathematics, and this guide provides a solid foundation for further exploration and application of fractions in various mathematical contexts. With practice and a clear understanding of the concepts, you can confidently navigate the world of fractions and their operations.