Mastering Division With Fractions A Step-by-Step Solution For 15 ÷ 6 2/3

#H1 Listen Mastering Division with Fractions

Mathematics often presents challenges, and one common hurdle involves dividing by mixed fractions. This article breaks down the problem $15 \div 6 \frac{2}{3}$ step-by-step, clarifying the process and leading you to the correct answer. We'll explore the necessary transformations and calculations to confidently tackle such problems. Understanding these principles is crucial not just for academic success but also for practical real-world applications.

#H2 Understanding the Problem: $15 \div 6 \frac{2}{3}$

Our core problem is to solve $15 \div 6 \frac{2}{3}$. At first glance, dividing by a mixed fraction might seem daunting, but the key lies in converting the mixed fraction into an improper fraction. This conversion simplifies the division process, allowing us to perform the necessary calculations more efficiently. The ability to convert between mixed and improper fractions is a foundational skill in mathematics, essential for solving a wide range of problems. Before we dive into the solution, let's first understand the components of the problem. We have a whole number, 15, which we are dividing by a mixed fraction, $6 \frac{2}{3}$. Mixed fractions combine a whole number and a proper fraction, adding an extra layer of complexity to the division. By converting the mixed fraction into an improper fraction, we transform the problem into a simpler form involving the division of a whole number by a single fraction. This transformation involves multiplying the whole number part of the mixed fraction by its denominator and then adding the numerator. The result becomes the new numerator, while the denominator remains the same. This process is crucial for handling mixed fractions in mathematical operations, including addition, subtraction, multiplication, and division. In the following sections, we'll demonstrate this conversion and proceed with the division, step by step, to arrive at the correct answer. This methodical approach not only solves the problem at hand but also reinforces the underlying mathematical principles, making it easier to tackle similar challenges in the future. Remember, practice is key to mastering these concepts, so try working through additional examples to solidify your understanding.

#H2 Step 1 Converting the Mixed Fraction to an Improper Fraction

The initial step in solving $15 \div 6 \frac{2}{3}$ involves converting the mixed fraction, $6 \frac{2}{3}$, into an improper fraction. This conversion is crucial because it simplifies the division process. An improper fraction has a numerator that is greater than or equal to its denominator, making it easier to divide into a whole number. To convert $6 \frac{2}{3}$ into an improper fraction, we follow a specific procedure. First, we multiply the whole number part (6) by the denominator (3): $6 \times 3 = 18$. Next, we add the numerator (2) to this product: $18 + 2 = 20$. This sum becomes the new numerator of our improper fraction. The denominator remains the same, which is 3. Therefore, the improper fraction equivalent of $6 \frac{2}{3}$ is $\frac{20}{3}$. Now that we've successfully converted the mixed fraction into an improper fraction, our division problem is transformed into $15 \div \frac{20}{3}$. This form is much easier to handle because we can now directly apply the rules of fraction division. Understanding this conversion process is fundamental for working with fractions. It allows us to perform mathematical operations accurately and efficiently. The ability to convert between mixed and improper fractions is not only useful in arithmetic but also in algebra and other advanced mathematical fields. By mastering this technique, you'll be well-equipped to tackle more complex problems involving fractions. In the next step, we'll see how this conversion helps us to divide 15 by the improper fraction $\frac{20}{3}$, bringing us closer to the final solution. Remember, each step in mathematics builds upon the previous one, so a solid understanding of the basics is essential for progressing to more advanced concepts.

#H2 Step 2 Dividing by a Fraction

Now that we have converted the mixed fraction into an improper fraction, our problem is now $15 \div \frac{20}{3}$. Dividing by a fraction might seem tricky, but it's actually quite straightforward. The key is to remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is simply the fraction flipped upside down. In other words, the numerator becomes the denominator, and the denominator becomes the numerator. So, the reciprocal of $\frac{20}{3}$ is $\frac{3}{20}$. Therefore, to solve $15 \div \frac{20}{3}$, we can rewrite it as a multiplication problem: $15 \times \frac{3}{20}$. To multiply a whole number by a fraction, we can think of the whole number as a fraction with a denominator of 1. So, 15 can be written as $\frac{15}{1}$. Now we have $\frac{15}{1} \times \frac{3}{20}$. To multiply fractions, we multiply the numerators together and the denominators together. So, $15 \times 3 = 45$ and $1 \times 20 = 20$. This gives us the fraction $\frac{45}{20}$. Before we consider this our final answer, we need to simplify the fraction. Simplifying a fraction means reducing it to its lowest terms. We look for a common factor between the numerator and the denominator and divide both by that factor. In this case, both 45 and 20 are divisible by 5. Dividing 45 by 5 gives us 9, and dividing 20 by 5 gives us 4. So, the simplified fraction is $\frac{9}{4}$. In the next step, we will convert this improper fraction back into a mixed fraction to match the format of the answer choices, completing our solution.

#H2 Step 3 Simplifying and Converting to a Mixed Fraction

After performing the division, we arrived at the improper fraction $\frac{9}{4}$. While this is a correct answer, the multiple-choice options are presented as mixed fractions, so we need to convert $\frac{9}{4}$ into a mixed fraction to match the format and identify the correct choice. Converting an improper fraction to a mixed fraction involves dividing the numerator by the denominator. The quotient becomes the whole number part of the mixed fraction, the remainder becomes the new numerator, and the denominator stays the same. When we divide 9 by 4, we get a quotient of 2 and a remainder of 1. This means that $\frac{9}{4}$ can be written as the mixed fraction $2 \frac{1}{4}$. The whole number part is 2, the new numerator is 1, and the denominator remains 4. Now that we have our answer in the form of a mixed fraction, we can confidently compare it to the given options and select the correct one. This final conversion step highlights the importance of being able to move between different forms of fractions – improper and mixed – to effectively solve problems and interpret results. Understanding these conversions not only helps in answering specific questions but also provides a deeper understanding of the nature of fractions themselves. By mastering these techniques, you'll be well-prepared to tackle a wide variety of mathematical problems involving fractions. In the next section, we will identify the correct answer from the given options and summarize the entire solution process, reinforcing the key steps and concepts we've covered.

#H2 Identifying the Correct Answer

Having solved the problem $15 \div 6 \frac{2}{3}$ and converted our result to the mixed fraction $2 \frac{1}{4}$, we can now confidently identify the correct answer from the given options. The options provided were:

A) 100 B) $2 \frac{3}{4}$ C) $100 \frac{1}{4}$ D) $2 \frac{1}{4}$

By comparing our solution, $2 \frac{1}{4}$, with the options, it's clear that D) $2 \frac{1}{4}$ is the correct answer. This confirms that our step-by-step process of converting the mixed fraction to an improper fraction, dividing, simplifying, and converting back to a mixed fraction was successful. Choosing the correct answer from a set of options requires not only solving the problem accurately but also paying attention to the format in which the answers are presented. In this case, recognizing that the options were in mixed fraction form prompted us to convert our improper fraction solution to a mixed fraction, ensuring a direct comparison and accurate selection. This attention to detail is crucial in mathematics and test-taking scenarios. The ability to confidently arrive at the correct answer and identify it among multiple choices demonstrates a strong understanding of the underlying mathematical concepts and problem-solving skills. In the final section, we will summarize the complete solution and emphasize the key takeaways from this exercise, reinforcing your understanding of fraction division and conversion techniques.

#H2 Summary of the Solution

Let's recap the steps we took to solve the problem $15 \div 6 \frac{2}{3}$. This problem, at first glance, might have seemed complex, but by breaking it down into manageable steps, we successfully arrived at the correct answer. First, we converted the mixed fraction $6 \frac{2}{3}$ into an improper fraction. We multiplied the whole number (6) by the denominator (3) to get 18, then added the numerator (2) to get 20. The denominator remained the same, resulting in the improper fraction $\frac{20}{3}$. Next, we tackled the division by a fraction. Dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of $\frac{20}{3}$ is $\frac{3}{20}$. So, we rewrote the problem as $15 \times \frac{3}{20}$. We then multiplied the fractions, treating 15 as $\frac{15}{1}$, which gave us $\frac{45}{20}$. We simplified the resulting fraction $\frac{45}{20}$ by dividing both the numerator and the denominator by their greatest common factor, which is 5. This simplification gave us the reduced fraction $\frac{9}{4}$. Finally, we converted the improper fraction $\frac{9}{4}$ back into a mixed fraction. We divided 9 by 4, which gave us a quotient of 2 and a remainder of 1. This resulted in the mixed fraction $2 \frac{1}{4}$. By following these steps, we not only solved the problem but also reinforced several important mathematical concepts, including converting between mixed and improper fractions, dividing by fractions, and simplifying fractions. These skills are fundamental to success in mathematics and have practical applications in everyday life. Mastering these techniques will enable you to confidently tackle similar problems in the future. Remember, practice is key to solidifying your understanding, so try working through additional examples to further develop your skills.

#H1 Listen: Mastering Division with Fractions - Correct Answer and Step-by-Step Solution

Listen closely as we break down this mathematical challenge: $15 \div 6 \frac{2}{3} = ?$ This problem requires us to divide a whole number by a mixed fraction. Don't worry, we'll tackle it step by step. The options are:

A) 100 B) $2 \frac{3}{4}$ C) $100 \frac{1}{4}$ D) $2 \frac{1}{4}$

#H2 Step 1 The Crucial First Step: Converting the Mixed Fraction

The cornerstone of solving this problem is understanding how to handle a mixed fraction. Mixed fractions, like $6 \frac{2}{3}$, combine a whole number and a fraction. To make division easier, we need to transform this mixed fraction into an improper fraction. An improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number).

To convert $6 \frac{2}{3}$ into an improper fraction, we follow a simple procedure: Multiply the whole number (6) by the denominator (3): $6 \times 3 = 18$ Add the numerator (2) to the result: $18 + 2 = 20$ Keep the original denominator (3): This gives us the improper fraction $\frac{20}{3}$.

So, $6 \frac{2}{3}$ is equivalent to $\frac{20}{3}$. Now, our problem looks like this: $15 \div \frac{20}{3}$. This seemingly small change makes the division process much more manageable. The ability to convert between mixed and improper fractions is a fundamental skill in mathematics. It’s not just about following steps; it’s about understanding the underlying concept: representing the same quantity in different forms to simplify calculations. Without this conversion, dividing by a mixed fraction becomes significantly more complex. Think of it as translating a sentence into a language you understand better – the meaning remains the same, but the form makes it easier to process. Practicing this conversion with various mixed fractions will build your confidence and fluency in handling fractions. This foundational skill will serve you well in more advanced mathematical concepts and in everyday situations where you need to work with parts of a whole. Remember, the goal is not just to get the right answer but to understand why the steps work. This deeper understanding allows you to apply the same principles to different problems and to confidently tackle new mathematical challenges.

#H2 Step 2 Unlocking Division: Multiplication by the Reciprocal

The next key to solving our problem, $15 \div \frac{20}{3}$, lies in understanding how division interacts with fractions. Here's the golden rule: Dividing by a fraction is the same as multiplying by its reciprocal. But what exactly is a reciprocal? The reciprocal of a fraction is simply that fraction flipped upside down. The numerator becomes the denominator, and the denominator becomes the numerator. So, the reciprocal of $\frac{20}{3}$ is $\frac{3}{20}$. Now we can rewrite our division problem as a multiplication problem: $15 \times \frac{3}{20}$. This transformation significantly simplifies the calculation. Multiplying fractions is generally easier than dividing them, especially when dealing with whole numbers and fractions. To multiply a whole number by a fraction, we can think of the whole number as a fraction with a denominator of 1. In our case, 15 can be written as $\frac{15}{1}$. Therefore, our problem now looks like this: $\frac{15}{1} \times \frac{3}{20}$. To multiply fractions, we multiply the numerators together and the denominators together: Numerators: $15 \times 3 = 45$ Denominators: $1 \times 20 = 20$ This gives us the fraction $\frac{45}{20}$. However, we're not quite done yet. This fraction can be simplified, which we'll address in the next step. This principle of multiplying by the reciprocal is a cornerstone of fraction arithmetic. It allows us to elegantly handle division involving fractions, transforming a potentially complex operation into a straightforward multiplication. Understanding why this works involves delving into the concept of inverse operations. Division is the inverse operation of multiplication, and the reciprocal provides the multiplicative inverse for a fraction. Mastering this concept and practicing applying it to various problems will greatly enhance your ability to work with fractions confidently and efficiently.

#H2 Step 3 The Final Touch: Simplifying and Converting Back

We've arrived at $\frac{45}{20}$, but to find the correct answer among the options, we need to simplify this fraction and potentially convert it back to a mixed fraction. Simplifying a fraction means reducing it to its lowest terms. We need to find the greatest common factor (GCF) of the numerator (45) and the denominator (20) and divide both by that factor. The GCF of 45 and 20 is 5. Dividing both the numerator and denominator by 5: $\frac{45 \div 5}{20 \div 5} = \frac{9}{4}$ Now we have the simplified improper fraction $\frac{9}{4}$. However, the answer choices are presented as mixed fractions, so we need to convert $\frac{9}{4}$ back into a mixed fraction. To convert an improper fraction to a mixed fraction, we divide the numerator (9) by the denominator (4): 9 divided by 4 is 2 with a remainder of 1. The quotient (2) becomes the whole number part of the mixed fraction. The remainder (1) becomes the numerator of the fractional part. The denominator (4) stays the same. Therefore, $\frac{9}{4}$ is equivalent to $2 \frac{1}{4}$. Now we have our final answer in the same format as the options, making it easy to identify the correct choice. This final step highlights the importance of being comfortable with both improper and mixed fractions and knowing how to convert between them. Different forms of fractions are useful in different situations, and being able to switch between them fluidly is a crucial mathematical skill. Furthermore, simplifying fractions is essential for presenting answers in their most concise and understandable form. It also reduces the risk of errors in subsequent calculations. By mastering these techniques, you not only solve the immediate problem but also strengthen your overall understanding of fractions and their properties. This deeper understanding will empower you to tackle more complex problems and to apply these concepts in various mathematical contexts.

#H2 The Solution Unveiled: Choosing the Right Option

We've successfully navigated the steps, and our final answer is $2 \frac{1}{4}$. Now, let's revisit the options:

A) 100 B) $2 \frac{3}{4}$ C) $100 \frac{1}{4}$ D) $2 \frac{1}{4}$

Comparing our solution to the options, it's clear that D) $2 \frac{1}{4}$ is the correct answer. Therefore, $15 \div 6 \frac{2}{3} = 2 \frac{1}{4}$.

#H2 Key Takeaways and Mastering Fraction Division

This problem demonstrates several crucial concepts for working with fractions: Converting mixed fractions to improper fractions: This is often the first step when dealing with mixed fractions in calculations. Dividing by a fraction is the same as multiplying by its reciprocal: This principle transforms a division problem into a multiplication problem. Simplifying fractions: Reducing fractions to their lowest terms is essential for presenting answers clearly and accurately. Converting improper fractions to mixed fractions: This allows you to express answers in a format that is often more intuitive and practical. By mastering these skills, you'll be well-equipped to tackle a wide range of problems involving fractions. Remember, practice is key to solidifying your understanding. Work through additional examples, and don't hesitate to review the steps whenever you encounter a challenging problem. Fractions are a fundamental concept in mathematics, and a strong foundation in fraction arithmetic will serve you well in more advanced topics. So, keep practicing, keep exploring, and keep mastering the world of fractions!

#H2 Practice Problems

To further solidify your understanding of dividing fractions, try these practice problems:

1. $10 \div 2 \frac{1}{2} = ?$
2. $12 \div 1 \frac{3}{5} = ?$
3. $8 \div 3 \frac{1}{3} = ?$

Work through each problem step-by-step, applying the techniques we've discussed. Check your answers against the solutions provided (you can easily calculate them using the same method). The more you practice, the more confident you'll become in your ability to divide fractions!

#H3 Question Keywords

• $15 \div 6 \frac{2}{3} = ?$
• Divide 15 by 6 2/3
• What is 15 divided by 6 and 2/3?
• Solution for 15 divided by 6 2/3
• How to divide 15 by a mixed fraction 6 2/3