Solving The Division Problem 5/8 Divided By 3/4

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Hey everyone! Let's dive into this math problem together and figure out the correct answer. We've got a division problem involving fractions, and it's crucial to understand the steps involved to get it right. We're tasked with solving $\left\lvert\, \frac{5}{8} \div \frac{3}{4}\right\rvert$, and we need to choose the correct answer from the options provided: A. $1 / 2$, B. $4 / 5$, C. $5 / 6$, and D. $3 / 8$. So, let's break it down step by step.

Understanding Fraction Division

When we talk about dividing fractions, it's not as straightforward as dividing whole numbers. The key here is to remember the phrase "keep, change, flip." This little mnemonic helps us remember the steps involved in dividing fractions. The first step involves keeping the first fraction exactly as it is. So, in our problem, $\frac{5}{8}$ stays the same. The second step is to change the division sign ($\div$) to a multiplication sign ($\times$). Finally, we flip the second fraction, which means we swap the numerator (the top number) and the denominator (the bottom number). So, $\frac{3}{4}$ becomes $\frac{4}{3}$. Now, we can rewrite the problem as a multiplication problem: $\frac{5}{8} \times \frac{4}{3}$.

Applying the Keep, Change, Flip Method

Okay, guys, let's apply the keep, change, flip method to our problem. We started with $\frac{5}{8} \div \frac{3}{4}$. Following the method, we keep the first fraction ($\frac{5}{8}$), change the division to multiplication, and flip the second fraction ($\frac{3}{4}$ becomes $\frac{4}{3}$). This gives us the new problem: $\frac{5}{8} \times \frac{4}{3}$. Now we're dealing with multiplication, which is much simpler. To multiply fractions, we simply multiply the numerators together and the denominators together. So, we multiply 5 by 4 to get the new numerator, and we multiply 8 by 3 to get the new denominator. This gives us $\frac{5 \times 4}{8 \times 3}$, which equals $\frac{20}{24}$. But hold on, we're not done yet! We need to simplify this fraction to its lowest terms.

Simplifying the Resulting Fraction

After multiplying the fractions, we ended up with $\frac{20}{24}$. Now, we need to simplify this fraction. Simplifying a fraction means reducing it to its lowest terms, which means finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by it. The GCD is the largest number that divides both numbers without leaving a remainder. In this case, we need to find the GCD of 20 and 24. The factors of 20 are 1, 2, 4, 5, 10, and 20. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The greatest common factor is 4. So, we divide both the numerator (20) and the denominator (24) by 4. This gives us $\frac{20 \div 4}{24 \div 4}$, which simplifies to $\frac{5}{6}$. So, the simplified answer to the division problem is $\frac{5}{6}$.

Step-by-Step Solution

To recap, let's go through the entire step-by-step solution to make sure we've got it crystal clear. First, we started with the problem $\left\lvert\, \frac{5}{8} \div \frac{3}{4}\right\rvert$. Then, we applied the keep, change, flip method, which transformed the problem into $\frac{5}{8} \times \frac{4}{3}$. Next, we multiplied the numerators and the denominators, resulting in $\frac{20}{24}$. Finally, we simplified the fraction $\frac{20}{24}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 4. This gave us the simplified fraction $\frac{5}{6}$. So, the final answer to the problem is $\frac{5}{6}$.

Visualizing Fraction Division

Sometimes, it helps to visualize what's happening when we divide fractions. Imagine you have $\frac{5}{8}$ of a pizza, and you want to divide it into slices that are $\frac{3}{4}$ of a whole pizza each. How many slices would you have? This is essentially what we're solving with this division problem. Thinking about it this way can make the concept of dividing fractions more intuitive. When we flip the second fraction and multiply, we're actually finding out how many $\frac{3}{4}$ portions are contained within $\frac{5}{8}$. The simplified answer, $\frac{5}{6}$, tells us that we can make slightly less than one whole $\frac{3}{4}$ portion from $\frac{5}{8}$ of the pizza.

Identifying the Correct Answer

Now that we've worked through the entire problem step by step, we can confidently identify the correct answer from the options provided. We started with $\left\lvert\, \frac{5}{8} \div \frac{3}{4}\right\rvert$, applied the keep, change, flip method, multiplied the fractions, and simplified the result. Our final answer was $\frac{5}{6}$. Looking back at the options, we have:

A. $1 / 2$ B. $4 / 5$ C. $5 / 6$ D. $3 / 8$

It's clear that the correct answer is C. $5 / 6$. We've successfully solved the division problem and found the solution by carefully following the rules of fraction division and simplification.

Common Mistakes to Avoid

When working with fraction division, there are a few common mistakes that students often make. One of the most frequent errors is forgetting to flip the second fraction after changing the division sign to multiplication. It's crucial to remember the keep, change, flip method in the correct order. Another common mistake is failing to simplify the fraction after multiplying. Always make sure to reduce your fraction to its lowest terms to get the simplest and most accurate answer. Additionally, some students may struggle with finding the greatest common divisor (GCD) when simplifying fractions. Practicing finding the GCD of different pairs of numbers can help improve this skill. By being aware of these common mistakes, you can avoid them and ensure you solve fraction division problems correctly.

Conclusion

Alright, guys, we've successfully tackled the division problem $\left\lvert\, \frac{5}{8} \div \frac{3}{4}\right\rvert$! We remembered the keep, change, flip method, multiplied the fractions, and simplified the result to get the correct answer, which is $\frac{5}{6}$. So, the answer is C. $5 / 6$. I hope this step-by-step explanation has helped you understand fraction division better. Keep practicing, and you'll become a pro at dividing fractions in no time!

Understanding and Solving the Problem $\left\lvert\, \frac{5}{8} \div \frac{3}{4}\right\rvert$

In this comprehensive guide, we'll break down the division problem $\left\lvert\, \frac{5}{8} \div \frac{3}{4}\right\rvert$ step by step, ensuring you grasp the fundamental concepts of fraction division. We'll explore the methodology behind solving such problems, common pitfalls to avoid, and how to simplify your answers effectively. The options we're considering are: A. $1 / 2$, B. $4 / 5$, C. $5 / 6$, and D. $3 / 8$. Let's embark on this mathematical journey together and demystify fraction division!

The Core Principle Dividing Fractions Made Simple

When you encounter a division problem involving fractions, it's essential to remember the golden rule: you don't actually "divide" fractions in the traditional sense. Instead, you multiply by the reciprocal of the second fraction. This concept is often remembered through the mnemonic keep, change, flip. This method is the cornerstone of fraction division, and mastering it will enable you to solve a wide range of problems with ease. The "keep" refers to keeping the first fraction as it is. The "change" indicates changing the division operation to multiplication. The "flip" means taking the reciprocal of the second fraction. Understanding why this method works is crucial. Dividing by a fraction is the same as asking how many times the second fraction fits into the first. By multiplying by the reciprocal, we're essentially finding out that value in a mathematically efficient way.

Applying Keep, Change, Flip to $\frac{5}{8} \div \frac{3}{4}$

Let's apply the keep, change, flip method to our specific problem, $\frac{5}{8} \div \frac{3}{4}$. First, we keep the first fraction, which is $\frac{5}{8}$. Next, we change the division sign ($\div$) to a multiplication sign ($\times$). Finally, we flip the second fraction, $\frac{3}{4}$, to its reciprocal, which is $\frac{4}{3}$. This transformation gives us the new problem: $\frac{5}{8} \times \frac{4}{3}$. Now, instead of dividing fractions, we're dealing with a multiplication problem, which is generally more straightforward. Remember, the reciprocal of a fraction is simply the fraction turned upside down. The numerator becomes the denominator, and the denominator becomes the numerator. This step is crucial because it converts the division problem into a multiplication problem, which we know how to solve.

Multiplying the Fractions The Next Critical Step

Now that we've transformed our division problem into a multiplication problem ($\frac{5}{8} \times \frac{4}{3}$), the next step is to multiply the fractions. Multiplying fractions is a simple process: you multiply the numerators together and multiply the denominators together. So, in our case, we multiply 5 by 4 to get the new numerator, and we multiply 8 by 3 to get the new denominator. This gives us $\frac{5 \times 4}{8 \times 3}$, which equals $\frac{20}{24}$. At this point, we have a valid answer, but it's not in its simplest form. We need to simplify the fraction to make it easier to understand and compare with the answer choices. Remember, the goal is always to express fractions in their simplest form, which means reducing them to their lowest terms.

Simplifying Fractions Finding the Greatest Common Divisor

After multiplying the fractions, we obtained $\frac{20}{24}$. Simplifying a fraction means reducing it to its lowest terms. To do this, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both numbers without leaving a remainder. In our case, we need to find the GCD of 20 and 24. The factors of 20 are 1, 2, 4, 5, 10, and 20. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. By comparing these lists, we can see that the greatest common factor is 4. Therefore, we divide both the numerator (20) and the denominator (24) by 4. This gives us $\frac{20 \div 4}{24 \div 4}$, which simplifies to $\frac{5}{6}$. This simplified fraction represents the final answer to our division problem.

Step-by-Step Recap The Complete Solution

To ensure clarity, let's recap the entire step-by-step solution. We started with the problem $\left\lvert\, \frac{5}{8} \div \frac{3}{4}\right\rvert$. Applying the keep, change, flip method, we transformed the division problem into a multiplication problem: $\frac{5}{8} \times \frac{4}{3}$. We then multiplied the numerators and the denominators, resulting in $\frac{20}{24}$. Finally, we simplified the fraction $\frac{20}{24}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 4, yielding the simplified fraction $\frac{5}{6}$. This step-by-step approach ensures accuracy and helps in understanding the process thoroughly. So, our final answer is $\frac{5}{6}$.

Visualizing Fraction Division Making It Concrete

Sometimes, visualizing the problem can make the concept of dividing fractions more accessible. Consider having $\frac{5}{8}$ of a pie and wanting to divide it into portions that are $\frac{3}{4}$ of a pie each. The question is, how many such portions can you make? This is precisely what our division problem is asking. Visualizing this scenario helps to connect the abstract mathematical concept to a real-world situation. When we flip the second fraction and multiply, we're essentially finding out how many $\frac{3}{4}$ portions are contained within $\frac{5}{8}$. The simplified answer, $\frac{5}{6}$, tells us that we can make slightly less than one whole $\frac{3}{4}$ portion from $\frac{5}{8}$ of the pie. This visual representation can enhance understanding and retention.

Identifying the Correct Option Selecting the Right Answer

Having solved the problem step by step, we are now in a position to confidently identify the correct answer from the options provided. We began with $\left\lvert\, \frac{5}{8} \div \frac{3}{4}\right\rvert$, employed the keep, change, flip method, multiplied the fractions, and simplified the result. Our final answer was $\frac{5}{6}$. Let's revisit the options:

A. $1 / 2$ B. $4 / 5$ C. $5 / 6$ D. $3 / 8$

Clearly, the correct answer is C. $5 / 6$. This methodical approach ensures we arrive at the right solution with confidence. We have successfully navigated the division problem and determined the correct answer by diligently following the rules of fraction division and simplification.

Avoiding Common Mistakes Ensuring Accuracy

When dealing with fraction division, certain common mistakes can lead to incorrect answers. One of the most prevalent errors is neglecting to flip the second fraction after changing the division sign to multiplication. It's crucial to adhere to the keep, change, flip method in the correct sequence. Another frequent mistake is failing to simplify the fraction after performing the multiplication. Always remember to reduce your fraction to its lowest terms to obtain the simplest and most accurate answer. Furthermore, some individuals may encounter difficulties in finding the greatest common divisor (GCD) when simplifying fractions. Regular practice in determining the GCD of various number pairs can significantly improve this skill. Being mindful of these common pitfalls will help you avoid them and solve fraction division problems accurately. By steering clear of these mistakes, you can enhance your problem-solving accuracy and confidence.

Conclusion Mastering Fraction Division

In summary, we've successfully addressed the division problem $\left\lvert\, \frac{5}{8} \div \frac{3}{4}\right\rvert$! We recalled the keep, change, flip method, multiplied the fractions, and simplified the result to arrive at the correct answer, which is $\frac{5}{6}$. Therefore, the answer is C. $5 / 6$. This comprehensive guide has equipped you with the knowledge and steps to confidently tackle fraction division problems. I trust that this step-by-step explanation has deepened your understanding of fraction division. Consistent practice will undoubtedly solidify your skills in this area. Keep up the great work, and you'll become proficient in dividing fractions in no time!

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This article provides a detailed, step-by-step guide on how to solve the fraction division problem $\left\lvert\, \frac{5}{8} \div \frac{3}{4}\right\rvert$. We will break down each step, explain the underlying concepts, and ensure you understand how to arrive at the correct answer. The answer options are A. $1 / 2$, B. $4 / 5$, C. $5 / 6$, and D. $3 / 8$. Let's dive into the solution!

1. Understanding the Basics of Fraction Division

Dividing fractions can seem tricky at first, but it becomes much simpler once you understand the core principle. The key is that dividing by a fraction is the same as multiplying by its reciprocal. This fundamental concept is crucial for solving any fraction division problem. The reciprocal of a fraction is obtained by swapping its numerator and denominator. For example, the reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$. So, instead of dividing by $\frac{3}{4}$, we will multiply by $\frac{4}{3}$. This method, often remembered as "keep, change, flip," is the foundation of fraction division. Let's break down each part of this method to ensure we understand it fully.

1.1 Keep, Change, Flip: The Golden Rule

The "keep, change, flip" method is your best friend when it comes to dividing fractions. It provides a clear and memorable way to approach these problems. Here's what each part means:

• Keep: Keep the first fraction exactly as it is.
• Change: Change the division sign ($\div$) to a multiplication sign ($\times$).
• Flip: Flip the second fraction, which means swapping its numerator and denominator.

This method transforms a division problem into a multiplication problem, which is much easier to handle. Let's see how this applies to our problem in the next step.

2. Applying Keep, Change, Flip to Our Problem

Our problem is $\left\lvert\, \frac{5}{8} \div \frac{3}{4}\right\rvert$. Let's apply the "keep, change, flip" method step by step:

• Keep: We keep the first fraction, $\frac{5}{8}$, as it is.
• Change: We change the division sign ($\div$) to a multiplication sign ($\times$).
• Flip: We flip the second fraction, $\frac{3}{4}$, to its reciprocal, which is $\frac{4}{3}$.

By applying these steps, we transform our problem into $\frac{5}{8} \times \frac{4}{3}$. This conversion is the key to solving the division problem. Now, we move on to the multiplication step.

3. Multiplying the Fractions

Multiplying fractions is straightforward: you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. This is a fundamental operation in fraction arithmetic. So, in our case, we have:

Numerator: $5 \times 4 = 20$ Denominator: $8 \times 3 = 24$

This gives us the fraction $\frac{20}{24}$. However, this is not the final answer because the fraction can be simplified. Simplifying fractions is an essential step in arriving at the correct solution. We need to reduce this fraction to its lowest terms.

4. Simplifying the Fraction

To simplify the fraction $\frac{20}{24}$, we need to find the greatest common divisor (GCD) of 20 and 24. The GCD is the largest number that divides both 20 and 24 without leaving a remainder. Finding the GCD is crucial for simplifying fractions effectively. Let's list the factors of 20 and 24:

Factors of 20: 1, 2, 4, 5, 10, 20 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

The greatest common divisor of 20 and 24 is 4. Now, we divide both the numerator and the denominator by 4:

$\frac{20 \div 4}{24 \div 4} = \frac{5}{6}$

So, the simplified fraction is $\frac{5}{6}$. This simplified fraction is our final answer to the division problem.

5. Identifying the Correct Answer Option

Now that we have our simplified answer, $\frac{5}{6}$, we can look at the answer options provided:

A. $1 / 2$ B. $4 / 5$ C. $5 / 6$ D. $3 / 8$

The correct answer is C. $5 / 6$. We have successfully solved the fraction division problem and identified the correct option. By following the steps of keep, change, flip, multiplying the fractions, and simplifying, we arrived at the accurate solution.

6. Common Mistakes to Avoid

When working with fraction division, it's important to be aware of common mistakes. Avoiding these errors will help you solve problems more accurately. Here are a few common pitfalls:

• Forgetting to Flip: The most common mistake is forgetting to flip the second fraction after changing the division sign to multiplication. Always remember the "keep, change, flip" order.
• Not Simplifying: Another frequent error is not simplifying the fraction after multiplying. Always reduce the fraction to its lowest terms.
• Incorrect Multiplication: Ensure you multiply the numerators together and the denominators together correctly.
• GCD Errors: Mistakes in finding the greatest common divisor can lead to incorrect simplification. Practice finding GCDs to improve this skill.

By being mindful of these potential errors, you can improve your accuracy in solving fraction division problems.

7. Visualizing Fraction Division

Visualizing fraction division can help make the concept more intuitive. Visual aids can enhance understanding and retention of mathematical concepts. Think of our problem $\frac{5}{8} \div \frac{3}{4}$ as asking: "How many $\frac{3}{4}$ portions are there in $\frac{5}{8}$?"

Imagine you have $\frac{5}{8}$ of a pie, and you want to divide it into slices that are $\frac{3}{4}$ of a whole pie each. You'll find that you can make less than one whole slice. This corresponds to our answer of $\frac{5}{6}$, which is slightly less than 1. Visualizing the problem in this way can reinforce your understanding of the division process.

Conclusion: Mastering Fraction Division

In this guide, we have walked through a detailed, step-by-step solution to the fraction division problem $\left\lvert\, \frac{5}{8} \div \frac{3}{4}\right\rvert$. We covered the basics of fraction division, the keep, change, flip method, multiplying fractions, simplifying fractions, identifying the correct answer, common mistakes to avoid, and visualizing the problem. By mastering these steps, you can confidently tackle any fraction division problem. The correct answer is C. $5 / 6$. We hope this guide has been helpful and has improved your understanding of fraction division. Keep practicing, and you'll become a fraction division pro!