Dividing Fractions Perform The Operation And Reduce To Lowest Terms

When it comes to mathematics, particularly dealing with fractions, performing operations like division might seem daunting at first. However, with a clear understanding of the underlying principles, dividing fractions becomes a straightforward process. This article aims to elucidate the concept of dividing fractions, focusing on the specific example of $5/6 \\div 1/2$, and guiding you through the steps to arrive at the solution in its simplest form. We'll explore the methodology, the importance of reducing fractions to their lowest terms, and how to confidently tackle similar problems. This foundational knowledge is crucial not only for academic success but also for practical applications in everyday life, such as cooking, measuring, and financial calculations. So, let's dive into the world of fractions and conquer the division operation with ease.

The Key: Multiplying by the Reciprocal

The cornerstone of dividing fractions lies in the understanding that dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of a fraction is obtained by simply swapping the numerator (the top number) and the denominator (the bottom number). For instance, the reciprocal of $1/2$ is $2/1$, which is equal to 2. This principle is not just a mathematical trick; it stems from the fundamental properties of division and multiplication as inverse operations. When you divide a number by a fraction, you're essentially asking how many times that fraction fits into the number. Multiplying by the reciprocal provides a direct and efficient way to determine this. Understanding this concept is vital as it simplifies the division process, allowing you to transform a division problem into a more familiar multiplication problem. In the context of our example, $5/6 \\div 1/2$ becomes $5/6 \\times 2/1$, setting the stage for a straightforward calculation. This method ensures accuracy and efficiency in handling fraction division, a crucial skill in various mathematical contexts.

Step-by-Step Solution for 5/6 ÷ 1/2

To tackle the problem $5/6 \\div 1/2$, we'll break down the solution into manageable steps, ensuring clarity and understanding at each stage. First, as we've established, the division operation is transformed into multiplication by taking the reciprocal of the second fraction. This means $5/6 \\div 1/2$ becomes $5/6 \\times 2/1$. This transformation is the core of the solution, converting a potentially confusing division problem into a simple multiplication problem. Next, we perform the multiplication. To multiply fractions, we multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator. Thus, $5/6 \\times 2/1$ equals $(5 \\times 2) / (6 \\times 1)$, which simplifies to $10/6$. Now, we have the fraction $10/6$, but the job isn't done yet. The final step is crucial: reducing the fraction to its lowest terms. This involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by it. In this case, the GCD of 10 and 6 is 2. Dividing both 10 and 6 by 2, we get $5/3$. This is the fraction in its simplest form, the irreducible form of the answer.

Reducing Fractions to Lowest Terms: Why It Matters

Reducing fractions to their lowest terms is a fundamental aspect of fraction manipulation, not merely a cosmetic step. A fraction is said to be in its simplest form when the numerator and the denominator have no common factors other than 1. This process ensures that the fraction is represented in the most concise and easily understandable way. Why is this important? First, it promotes clarity. A simplified fraction is easier to grasp and compare with other fractions. For instance, $5/3$ is more immediately understandable than $10/6$. Second, it prevents unnecessary complexity in further calculations. Working with simplified fractions reduces the risk of errors and makes subsequent operations smoother. Third, it's often a requirement in mathematical contexts, such as standardized tests and academic assignments. Failing to reduce a fraction can sometimes lead to a loss of credit, even if the initial calculation was correct. The method for reducing fractions involves finding the greatest common divisor (GCD) of the numerator and denominator, as we did in the previous step. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. Dividing both parts of the fraction by their GCD results in the simplified form. This skill is indispensable for anyone working with fractions, ensuring accuracy and efficiency in mathematical problem-solving.

Identifying the Correct Answer

Now that we've solved the problem $5/6 \\div 1/2$ and arrived at the simplified fraction $5/3$, let's relate this back to the original question and the provided answer choices. The question asked us to perform the indicated operation and reduce the result to lowest terms, and it specified that both improper and mixed number forms should be considered if applicable. Our calculated answer, $5/3$, is an improper fraction because the numerator (5) is greater than the denominator (3). To determine the correct answer choice, we need to ensure that it matches our simplified improper fraction, $5/3$, and also consider its mixed number equivalent, reduced to its lowest terms. Converting $5/3$ to a mixed number involves dividing 5 by 3. The quotient is 1, and the remainder is 2. Thus, $5/3$ is equivalent to the mixed number $1 \\frac{2}{3}$. Now, let's examine the answer choices:

• A. $5/12$ - This is incorrect as it doesn't match our calculated result.
• B. $5/3$ - This is a correct answer, as it matches our simplified improper fraction.
• C. $16/12$ - This fraction can be simplified by dividing both numerator and denominator by their GCD, which is 4. $16/12$ simplifies to $4/3$, which does not match our answer of $5/3$.
• D. $12/3$ - This simplifies to 4, which is not our answer.
• E. None of the above - This would be incorrect since we've identified a correct answer.

Therefore, the correct answer is B. $5/3$. This exercise highlights the importance of not only performing the operation correctly but also simplifying the result and comparing it accurately with the provided options.

The final answer is B. 5/3