Mixed Number Division: 6 1/3 ÷ 1/6 Explained

Dec 7, 2025 by Alex Johnson 45 views

When tackling math problems involving fractions, especially mixed numbers, it's easy to feel a bit daunted. But don't worry! Breaking down the process step-by-step can make even complex-looking division problems, like $6 rac{1}{3} ext{ divided by } rac{1}{6}$ , much more manageable. This article will guide you through the exact steps to solve this specific problem and understand the underlying principles of dividing mixed numbers by fractions. We'll explore why each step is crucial and how it leads you to the correct answer. By the end, you'll not only know the solution but also feel more confident in your ability to handle similar fraction division scenarios. Let's dive in and demystify this mathematical operation together!

Understanding Mixed Numbers and Fraction Division

Before we jump into solving $6 rac{1}{3} ext{ divided by } rac{1}{6}$ , it's important to grasp the fundamentals of working with mixed numbers and the rules of fraction division. A mixed number, like $6 rac{1}{3}$ , is a whole number combined with a proper fraction. To perform operations like division, we first need to convert this mixed number into an improper fraction. An improper fraction has a numerator that is greater than or equal to its denominator. This conversion is key because it allows us to apply a consistent method for fraction division. The rule for dividing fractions is to invert the divisor (the second fraction) and multiply. So, instead of dividing by $rac{1}{6}$ , we'll multiply by its reciprocal, which is $rac{6}{1}$ . This process is often remembered by the phrase "Keep, Change, Flip." We keep the first fraction as it is, change the division sign to multiplication, and flip the second fraction. This mathematical maneuver is what allows us to solve division problems using multiplication, a skill we're typically more comfortable with. Understanding these two core concepts – converting mixed numbers and the invert-and-multiply rule for division – lays the groundwork for successfully solving our problem.

Step-by-Step Solution for $6 rac{1}{3} ext{ \div } rac{1}{6}$

Let's meticulously work through the problem $6 rac{1}{3} ext{ divided by } rac{1}{6}$ . Our first crucial step is to convert the mixed number $6 rac{1}{3}$ into an improper fraction. To do this, we multiply the whole number (6) by the denominator of the fraction (3) and then add the numerator (1). This gives us $6 imes 3 + 1 = 18 + 1 = 19$ . The denominator remains the same (3). So, $6 rac{1}{3}$ is equivalent to $rac{19}{3}$ . Now, our division problem transforms into $rac{19}{3} ext{ \div } rac{1}{6}$ . The next step, as per the rule of fraction division, is to invert the second fraction ( $rac{1}{6}$ ) and change the division to multiplication. The reciprocal of $rac{1}{6}$ is $rac{6}{1}$ . Therefore, our problem becomes $rac{19}{3} imes rac{6}{1}$ . Now, we multiply the numerators together and the denominators together: $19 imes 6 = 114$ and $3 imes 1 = 3$ . This results in the fraction $rac{114}{3}$ . The final step is to simplify this improper fraction by performing the division. $114 ext{ divided by } 3$ equals 38. So, the answer to $6 rac{1}{3} ext{ \div } rac{1}{6}$ is 38. This systematic approach ensures accuracy and clarity in solving mixed number division problems. Remember, converting to improper fractions and applying the "Keep, Change, Flip" rule are the keys to unlocking these solutions.

Why Does Inverting and Multiplying Work?

It's natural to wonder why we invert the divisor and multiply when dividing fractions. This method, often called the "invert and multiply" rule, is fundamental to fraction division and might seem a bit abstract at first. Let's break it down using a simpler example. Imagine we want to calculate $6 ext{ ÷ } 2$ . We know the answer is 3 because 2 goes into 6 three times. Now, let's think of 2 as a fraction: $rac{2}{1}$ . So, the problem is $6 ext{ \div } rac{2}{1}$ . If we apply the invert-and-multiply rule, we keep 6 (which is $rac{6}{1}$ ), change the division to multiplication, and flip $rac{2}{1}$ to $rac{1}{2}$ . So we get $rac{6}{1} imes rac{1}{2} = rac{6}{2} = 3$ . It works! The underlying principle is related to the concept of reciprocals and how they interact with multiplication. Dividing by a number is the same as multiplying by its reciprocal. For instance, dividing by 2 is the same as multiplying by $rac{1}{2}$ . When we divide fractions, say $rac{a}{b} ext{ \div } rac{c}{d}$ , we are essentially asking how many times $rac{c}{d}$ fits into $rac{a}{b}$ . By multiplying $rac{a}{b}$ by the reciprocal of $rac{c}{d}$ (which is $rac{d}{c}$ ), we are effectively scaling the problem in a way that allows us to find this relationship through multiplication. The equation $rac{a}{b} ext{ \div } rac{c}{d} = rac{a}{b} imes rac{d}{c}$ is derived from the property that any number multiplied by its reciprocal equals 1 (e.g., $rac{c}{d} imes rac{d}{c} = 1$ ). This ensures that the value of the division remains unchanged while transforming the operation into a multiplication problem that we can solve directly. This principle is crucial for understanding why the algorithm for fraction division is valid and consistently provides the correct results.

Connecting to the Multiple-Choice Options

Now that we've arrived at the solution for $6 rac{1}{3} ext{ \div } rac{1}{6}$ , let's see how it matches up with the provided multiple-choice options. We calculated the result to be 38. Let's look at the choices given:

A. 6 B. 9 C. 12 D. 19 E. 38

Our calculated answer, 38, directly corresponds to option E. This confirms that our step-by-step process, from converting the mixed number to an improper fraction to applying the invert-and-multiply rule, has led us to the correct solution. It's always a good practice to double-check your work, especially in timed tests or when precision is critical. Sometimes, a quick estimation can also help. For example, $6 rac{1}{3}$ is a bit more than 6. Dividing a number by a fraction less than 1 (like $rac{1}{6}$ ) will always result in a number larger than the original number. Since $rac{1}{6}$ is significantly less than 1, we expect a result much larger than 6. This quick check helps eliminate options A, B, and C right away. Options D (19) and E (38) are both larger than 6, but our detailed calculation points definitively to 38. This process of solving and then verifying against the options is a robust way to ensure mathematical accuracy and confidence in your answers. You've successfully navigated a mixed number division problem!

Conclusion: Mastering Mixed Number Division

In conclusion, solving the division problem $6 rac{1}{3} ext{ \div } rac{1}{6}$ involves a clear, two-step process. First, we convert the mixed number $6 rac{1}{3}$ into an improper fraction, $rac{19}{3}$ . Second, we apply the rule for dividing fractions: invert the divisor ( $rac{1}{6}$ becomes $rac{6}{1}$ ) and multiply. This transforms the problem into $rac{19}{3} imes rac{6}{1}$ , which simplifies to $rac{114}{3}$ . Finally, dividing 114 by 3 gives us the answer, 38. This methodical approach not only solves the problem but also builds a stronger understanding of fraction arithmetic. Mastering mixed number division is a valuable skill that opens doors to solving more complex mathematical challenges. Remember the key steps: convert mixed numbers to improper fractions and use the "invert and multiply" technique. With practice, these operations will become second nature. For further exploration into the fascinating world of fractions and arithmetic operations, you might find resources from Khan Academy to be incredibly helpful.