Solving $6 / 7 \div 36 / 56$ In Simplest Form

Understanding the Problem

At the heart of this problem lies the concept of dividing fractions. When we encounter a division problem involving fractions, we're essentially asking how many times the second fraction (the divisor) fits into the first fraction (the dividend). To solve this, we use a handy trick: we flip the second fraction and multiply. This transforms our division problem into a multiplication problem, which is much easier to handle. To provide a comprehensive solution, we'll walk through each step, ensuring clarity and understanding. The goal is to not only arrive at the correct answer but also to grasp the underlying principles of fraction division and simplification. This foundational knowledge is crucial for tackling more complex mathematical problems in the future. Remember, mathematics is a journey, and each problem solved is a step forward. In this detailed exploration, we will address not only the mathematical process but also the reasoning behind it, reinforcing your understanding of fractions and their operations. Furthermore, we'll emphasize the importance of simplifying fractions to their lowest terms, a skill that's vital in various mathematical contexts. Our detailed breakdown is designed to equip you with the skills and confidence to approach similar problems with ease and accuracy. Throughout this explanation, we'll use clear language and provide concrete examples, making the concepts accessible and memorable. So, let's dive into the world of fractions and unravel this problem together, ensuring you gain a solid grasp of the fundamental principles involved.

Step-by-Step Solution

Let's break down how to solve this division problem step-by-step. We begin with the expression $6 / 7 \div 36 / 56$. As mentioned earlier, dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. So, the reciprocal of $36 / 56$ is $56 / 36$. Therefore, our division problem transforms into a multiplication problem: $6 / 7 \times 56 / 36$. Now, we can multiply the numerators together and the denominators together. This gives us $(6 \times 56) / (7 \times 36)$. Before we perform the multiplication, it's often helpful to see if we can simplify any factors. We notice that 6 and 36 share a common factor of 6, and 7 and 56 share a common factor of 7. We can divide 6 by 6 to get 1 and 36 by 6 to get 6. Similarly, we can divide 7 by 7 to get 1 and 56 by 7 to get 8. Our expression now becomes $(1 \times 8) / (1 \times 6)$, which simplifies to $8 / 6$. However, we're not done yet, as we need to express our answer in the simplest form. Both 8 and 6 are divisible by 2. Dividing both the numerator and the denominator by 2, we get $4 / 3$. This is the simplest form of the fraction. This meticulous approach ensures we not only find the correct answer but also understand the underlying principles, solidifying our grasp of fraction operations. Remember, simplification is key to presenting your answer in the most concise and understandable manner. This process not only makes the fraction easier to work with but also highlights the essential relationship between the numerator and the denominator. By mastering these steps, you'll be well-equipped to handle a variety of fraction problems with confidence and precision.

Detailed Calculation

To demonstrate the calculation more clearly, let's revisit the steps with specific numbers. We have the initial problem: $6 / 7 \div 36 / 56$. First, we find the reciprocal of the second fraction, $36 / 56$, which is $56 / 36$. Next, we change the division to multiplication: $6 / 7 \times 56 / 36$. Now, we multiply the numerators: $6 \times 56 = 336$. Then, we multiply the denominators: $7 \times 36 = 252$. This gives us the fraction $336 / 252$. While this fraction is correct, it's not in its simplest form. To simplify, we look for common factors between the numerator and the denominator. We can see that both 336 and 252 are divisible by 2. Dividing both by 2, we get $168 / 126$. We can continue to simplify. Both 168 and 126 are divisible by 2 again, resulting in $84 / 63$. Now, we notice that both 84 and 63 are divisible by 7. Dividing both by 7, we get $12 / 9$. Finally, both 12 and 9 are divisible by 3. Dividing both by 3, we get $4 / 3$. This detailed step-by-step simplification showcases how we arrive at the simplest form of the fraction. By breaking down the process into smaller steps, we minimize the chances of error and gain a deeper understanding of the simplification process. This thorough approach not only ensures accuracy but also reinforces the importance of meticulous calculation in mathematics. Moreover, it demonstrates the beauty of simplifying complex fractions to their most elegant form, making them easier to interpret and utilize in further calculations.

Identifying the Correct Option

Now that we have the solution in its simplest form, which is $4 / 3$, we can compare it to the given options. The options are:

A) $6 / 8$ B) $8 / 6$ C) $2 / 8$ D) $4 / 3$

By comparing our solution, $4 / 3$, to the options, we can clearly see that option D matches our answer. Therefore, the correct answer is D) $4 / 3$. This final step of identifying the correct option reinforces the importance of careful comparison and attention to detail. After going through the entire process of solving the problem, it's crucial to ensure that the final answer aligns perfectly with one of the provided choices. This not only validates our solution but also demonstrates our ability to accurately interpret and apply our mathematical skills. Moreover, it underscores the significance of double-checking our work to avoid any errors in the final selection. This meticulous approach to identifying the correct answer is a hallmark of strong mathematical proficiency and ensures that our efforts culminate in the accurate solution to the problem.

Why Other Options are Incorrect

To strengthen our understanding, let's briefly discuss why the other options are incorrect:

• A) $6 / 8$: This fraction can be simplified to $3 / 4$, which is not equal to $4 / 3$.
• B) $8 / 6$: This fraction can be simplified to $4 / 3$, but it's not in the simplest form initially. While it represents the same value, it wasn't the final simplified answer until it was further reduced. However, the question asked the answer in simplest form, so it is still incorrect.
• C) $2 / 8$: This fraction simplifies to $1 / 4$, which is also not equal to $4 / 3$.

Understanding why incorrect options are wrong is just as important as understanding why the correct option is right. It helps solidify our understanding of the problem and the concepts involved. By analyzing the incorrect options, we can identify common mistakes or misconceptions that might arise while solving similar problems. This process not only reinforces our knowledge but also enhances our critical thinking skills, allowing us to approach future mathematical challenges with greater confidence and accuracy. Furthermore, it underscores the importance of thoroughness and attention to detail in problem-solving, ensuring that we not only arrive at the correct answer but also understand the reasoning behind it. This comprehensive approach to learning is essential for building a strong foundation in mathematics and fostering a deeper appreciation for the subject.

Final Answer

Therefore, the final answer, in the simplest form, is:

D) $4 / 3$