To Divide Two Fractions, First Rewrite The Problem As The Dividend Times The Reciprocal Of The Divisor.Find And Simplify The Quotient:A. 2 3 \frac{2}{3} 3 2 ​ B. 1 12 \frac{1}{12} 12 1 ​ C. 16 27 \frac{16}{27} 27 16 ​ D. 3 4 \frac{3}{4} 4 3 ​ E.

Understanding the Concept of Dividing Fractions

When it comes to dividing fractions, many students struggle to understand the concept and apply it correctly. However, with a clear understanding of the concept and a step-by-step approach, dividing fractions can be a straightforward process. In this article, we will explore the concept of dividing fractions, provide a step-by-step guide on how to divide fractions, and apply this concept to various examples.

The Reciprocal of a Fraction

Before we dive into dividing fractions, it's essential to understand the concept of the reciprocal of a fraction. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For example, the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$, and the reciprocal of $\frac{4}{5}$ is $\frac{5}{4}$.

Dividing Fractions: A Step-by-Step Guide

To divide two fractions, we need to follow a simple step-by-step approach:

1. Rewrite the problem as the dividend times the reciprocal of the divisor: This is the key concept in dividing fractions. We need to rewrite the division problem as a multiplication problem by taking the reciprocal of the divisor.
2. Take the reciprocal of the divisor: The reciprocal of the divisor is obtained by swapping the numerator and the denominator.
3. Multiply the dividend and the reciprocal of the divisor: Once we have the reciprocal of the divisor, we multiply it with the dividend.
4. Simplify the quotient: Finally, we simplify the quotient by canceling out any common factors between the numerator and the denominator.

Example 1: Dividing $\frac{2}{3}$ by $\frac{1}{12}$

Let's apply the step-by-step approach to divide $\frac{2}{3}$ by $\frac{1}{12}$.

Step 1: Rewrite the problem as the dividend times the reciprocal of the divisor

We need to rewrite the division problem as a multiplication problem by taking the reciprocal of the divisor.

$\frac{2}{3} \div \frac{1}{12} = \frac{2}{3} \times \frac{12}{1}$

Step 2: Take the reciprocal of the divisor

The reciprocal of the divisor is $\frac{12}{1}$.

Step 3: Multiply the dividend and the reciprocal of the divisor

We multiply the dividend $\frac{2}{3}$ with the reciprocal of the divisor $\frac{12}{1}$.

$\frac{2}{3} \times \frac{12}{1} = \frac{2 \times 12}{3 \times 1} = \frac{24}{3}$

Step 4: Simplify the quotient

We simplify the quotient by canceling out any common factors between the numerator and the denominator.

$\frac{24}{3} = 8$

Therefore, $\frac{2}{3} \div \frac{1}{12} = 8$.

Example 2: Dividing $\frac{1}{12}$ by $\frac{3}{4}$

Let's apply the step-by-step approach to divide $\frac{1}{12}$ by $\frac{3}{4}$.

Step 1: Rewrite the problem as the dividend times the reciprocal of the divisor

We need to rewrite the division problem as a multiplication problem by taking the reciprocal of the divisor.

$\frac{1}{12} \div \frac{3}{4} = \frac{1}{12} \times \frac{4}{3}$

Step 2: Take the reciprocal of the divisor

The reciprocal of the divisor is $\frac{4}{3}$.

Step 3: Multiply the dividend and the reciprocal of the divisor

We multiply the dividend $\frac{1}{12}$ with the reciprocal of the divisor $\frac{4}{3}$.

$\frac{1}{12} \times \frac{4}{3} = \frac{1 \times 4}{12 \times 3} = \frac{4}{36}$

Step 4: Simplify the quotient

We simplify the quotient by canceling out any common factors between the numerator and the denominator.

$\frac{4}{36} = \frac{1}{9}$

Therefore, $\frac{1}{12} \div \frac{3}{4} = \frac{1}{9}$.

Example 3: Dividing $\frac{16}{27}$ by $\frac{3}{4}$

Let's apply the step-by-step approach to divide $\frac{16}{27}$ by $\frac{3}{4}$.

Step 1: Rewrite the problem as the dividend times the reciprocal of the divisor

We need to rewrite the division problem as a multiplication problem by taking the reciprocal of the divisor.

$\frac{16}{27} \div \frac{3}{4} = \frac{16}{27} \times \frac{4}{3}$

Step 2: Take the reciprocal of the divisor

The reciprocal of the divisor is $\frac{4}{3}$.

Step 3: Multiply the dividend and the reciprocal of the divisor

We multiply the dividend $\frac{16}{27}$ with the reciprocal of the divisor $\frac{4}{3}$.

$\frac{16}{27} \times \frac{4}{3} = \frac{16 \times 4}{27 \times 3} = \frac{64}{81}$

Step 4: Simplify the quotient

We simplify the quotient by canceling out any common factors between the numerator and the denominator.

$\frac{64}{81}$ cannot be simplified further.

Therefore, $\frac{16}{27} \div \frac{3}{4} = \frac{64}{81}$.

Example 4: Dividing $\frac{3}{4}$ by $\frac{2}{3}$

Let's apply the step-by-step approach to divide $\frac{3}{4}$ by $\frac{2}{3}$.

Step 1: Rewrite the problem as the dividend times the reciprocal of the divisor

We need to rewrite the division problem as a multiplication problem by taking the reciprocal of the divisor.

$\frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2}$

Step 2: Take the reciprocal of the divisor

The reciprocal of the divisor is $\frac{3}{2}$.

Step 3: Multiply the dividend and the reciprocal of the divisor

We multiply the dividend $\frac{3}{4}$ with the reciprocal of the divisor $\frac{3}{2}$.

$\frac{3}{4} \times \frac{3}{2} = \frac{3 \times 3}{4 \times 2} = \frac{9}{8}$

Step 4: Simplify the quotient

We simplify the quotient by canceling out any common factors between the numerator and the denominator.

$\frac{9}{8}$ cannot be simplified further.

Therefore, $\frac{3}{4} \div \frac{2}{3} = \frac{9}{8}$.

Conclusion

Dividing fractions can be a straightforward process if we follow a step-by-step approach. By rewriting the division problem as a multiplication problem by taking the reciprocal of the divisor, we can simplify the quotient and obtain the final answer. In this article, we have applied this concept to various examples and provided a clear understanding of the concept of dividing fractions. With practice and patience, students can master the art of dividing fractions and become proficient in solving complex mathematical problems.

Understanding the Concept of Dividing Fractions

Dividing fractions can be a challenging concept for many students. However, with a clear understanding of the concept and a step-by-step approach, dividing fractions can be a straightforward process. In this article, we will address some of the most frequently asked questions (FAQs) on dividing fractions.

Q: What is the concept of dividing fractions?

A: Dividing fractions involves rewriting the division problem as a multiplication problem by taking the reciprocal of the divisor. The reciprocal of a fraction is obtained by swapping the numerator and the denominator.

Q: How do I rewrite a division problem as a multiplication problem?

A: To rewrite a division problem as a multiplication problem, you need to take the reciprocal of the divisor. For example, if you want to divide $\frac{2}{3}$ by $\frac{1}{12}$, you would rewrite the problem as $\frac{2}{3} \times \frac{12}{1}$.

Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For example, the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$, and the reciprocal of $\frac{4}{5}$ is $\frac{5}{4}$.

Q: How do I simplify the quotient after dividing fractions?

A: To simplify the quotient, you need to cancel out any common factors between the numerator and the denominator. For example, if you have $\frac{24}{3}$, you can simplify it to $8$ by canceling out the common factor of $3$.

Q: Can I divide fractions with unlike denominators?

A: Yes, you can divide fractions with unlike denominators. To do this, you need to find the least common multiple (LCM) of the denominators and rewrite the fractions with the LCM as the denominator.

Q: How do I divide fractions with negative numbers?

A: Dividing fractions with negative numbers involves following the same steps as dividing fractions with positive numbers. However, you need to remember that a negative number divided by a negative number is positive, and a negative number divided by a positive number is negative.

Q: Can I divide fractions with decimals?

A: Yes, you can divide fractions with decimals. To do this, you need to convert the decimal to a fraction and then follow the steps for dividing fractions.

Q: What are some common mistakes to avoid when dividing fractions?

A: Some common mistakes to avoid when dividing fractions include:

• Not rewriting the division problem as a multiplication problem
• Not taking the reciprocal of the divisor
• Not simplifying the quotient
• Not canceling out common factors between the numerator and the denominator

Q: How can I practice dividing fractions?

A: You can practice dividing fractions by working through examples and exercises. You can also use online resources and math apps to practice dividing fractions.

Q: What are some real-world applications of dividing fractions?

A: Dividing fractions has many real-world applications, including:

• Cooking and baking: Dividing fractions is essential in cooking and baking, where you need to measure ingredients accurately.
• Science and engineering: Dividing fractions is used in science and engineering to calculate quantities and proportions.
• Finance: Dividing fractions is used in finance to calculate interest rates and investment returns.

Conclusion

Dividing fractions can be a challenging concept, but with practice and patience, you can master it. By understanding the concept of dividing fractions and following a step-by-step approach, you can simplify the quotient and obtain the final answer. Remember to avoid common mistakes and practice dividing fractions regularly to become proficient in solving complex mathematical problems.