To Divide Two Fractions, First Rewrite The Problem As The Dividend Times The Reciprocal Of The Divisor.Find And Simplify The Quotient 2 3 ÷ 8 9 \frac{2}{3} \div \frac{8}{9} 3 2 ​ ÷ 9 8 ​ .A. 1 12 \frac{1}{12} 12 1 ​ B. 16 27 \frac{16}{27} 27 16 ​ C.

Understanding the Concept of Dividing Fractions

Dividing fractions can be a challenging concept for many students, but it can be simplified by understanding the basic principles. When we divide two fractions, we are essentially finding the quotient of the two fractions. To do this, we need to rewrite the problem as the dividend times the reciprocal of the divisor. This concept is crucial in solving division problems involving fractions.

The Reciprocal of a Fraction

Before we dive into the concept of dividing fractions, let's first understand what the reciprocal of a fraction is. 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 $\frac{8}{9}$ is $\frac{9}{8}$.

Finding the Quotient of Two Fractions

Now that we understand the concept of the reciprocal of a fraction, let's find the quotient of two fractions. To do this, we need to rewrite the problem as the dividend times the reciprocal of the divisor. This can be represented as:

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

where $\frac{a}{b}$ is the dividend and $\frac{c}{d}$ is the divisor.

Simplifying the Quotient

Once we have rewritten the problem as the dividend times the reciprocal of the divisor, we can simplify the quotient by multiplying the numerators and denominators. This can be represented as:

$\frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

Finding and Simplifying the Quotient

Now that we have understood the concept of dividing fractions, let's find and simplify the quotient of $\frac{2}{3} \div \frac{8}{9}$. To do this, we need to rewrite the problem as the dividend times the reciprocal of the divisor.

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

Multiplying the Numerators and Denominators

Now that we have rewritten the problem as the dividend times the reciprocal of the divisor, we can simplify the quotient by multiplying the numerators and denominators.

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

Simplifying the Quotient

Now that we have multiplied the numerators and denominators, we can simplify the quotient by dividing the numerator and denominator by their greatest common divisor (GCD).

$\frac{2 \times 9}{3 \times 8} = \frac{18}{24}$

Simplifying the Quotient Further

Now that we have simplified the quotient, we can simplify it further by dividing the numerator and denominator by their GCD.

$\frac{18}{24} = \frac{3 \times 6}{4 \times 6}$

Canceling Out the Common Factors

Now that we have simplified the quotient, we can cancel out the common factors in the numerator and denominator.

$\frac{3 \times 6}{4 \times 6} = \frac{3}{4}$

Conclusion

In conclusion, to divide two fractions, we need to rewrite the problem as the dividend times the reciprocal of the divisor. This can be represented as:

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Once we have rewritten the problem as the dividend times the reciprocal of the divisor, we can simplify the quotient by multiplying the numerators and denominators. This can be represented as:

$\frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

By following these steps, we can find and simplify the quotient of two fractions.

Final Answer

The final answer is $\boxed{\frac{3}{4}}$.

Discussion

The discussion category for this problem is mathematics. This problem involves the concept of dividing fractions, which is a fundamental concept in mathematics. The problem requires the student to understand the concept of the reciprocal of a fraction and how to simplify the quotient by multiplying the numerators and denominators.

Related Problems

Some related problems to this problem include:

• Dividing fractions with like denominators
• Dividing fractions with unlike denominators
• Simplifying fractions
• Finding the reciprocal of a fraction

Conclusion

In conclusion, to divide two fractions, we need to rewrite the problem as the dividend times the reciprocal of the divisor. This can be represented as:

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Once we have rewritten the problem as the dividend times the reciprocal of the divisor, we can simplify the quotient by multiplying the numerators and denominators. This can be represented as:

$\frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

By following these steps, we can find and simplify the quotient of two fractions.

Final Answer

The final answer is $\boxed{\frac{3}{4}}$.

Discussion

The discussion category for this problem is mathematics. This problem involves the concept of dividing fractions, which is a fundamental concept in mathematics. The problem requires the student to understand the concept of the reciprocal of a fraction and how to simplify the quotient by multiplying the numerators and denominators.

Related Problems

Some related problems to this problem include:

• Dividing fractions with like denominators
• Dividing fractions with unlike denominators
• Simplifying fractions
• Finding the reciprocal of a fraction
  

Q&A: Dividing Fractions

Q: What is the concept of dividing fractions?

A: Dividing fractions is a mathematical operation that involves finding the quotient of two fractions. To do this, we need to rewrite the problem as the dividend times the reciprocal of the divisor.

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 $\frac{8}{9}$ is $\frac{9}{8}$.

Q: How do I rewrite the problem as the dividend times the reciprocal of the divisor?

A: To rewrite the problem as the dividend times the reciprocal of the divisor, we need to swap the dividend and the divisor, and then multiply the new dividend by the reciprocal of the new divisor. This can be represented as:

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Q: How do I simplify the quotient?

A: To simplify the quotient, we need to multiply the numerators and denominators, and then simplify the resulting fraction by dividing the numerator and denominator by their greatest common divisor (GCD).

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, we can use the Euclidean algorithm or list the factors of each number and find the largest common factor.

Q: Can I simplify the quotient further?

A: Yes, we can simplify the quotient further by canceling out any common factors in the numerator and denominator.

Q: What is the final answer to the problem $\frac{2}{3} \div \frac{8}{9}$?

A: The final answer to the problem $\frac{2}{3} \div \frac{8}{9}$ is $\frac{3}{4}$.

Q: What are some related problems to this problem?

A: Some related problems to this problem include:

• Dividing fractions with like denominators
• Dividing fractions with unlike denominators
• Simplifying fractions
• Finding the reciprocal of a fraction

Q: Why is it important to understand the concept of dividing fractions?

A: Understanding the concept of dividing fractions is important because it is a fundamental concept in mathematics that is used in many real-world applications, such as finance, science, and engineering.

Q: Can you provide some examples of dividing fractions?

A: Yes, here are some examples of dividing fractions:

• $\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}$
• $\frac{2}{3} \div \frac{5}{6} = \frac{2}{3} \times \frac{6}{5} = \frac{12}{15} = \frac{4}{5}$
• $\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$

Q: Can you provide some tips for dividing fractions?

A: Yes, here are some tips for dividing fractions:

• Make sure to rewrite the problem as the dividend times the reciprocal of the divisor.
• Multiply the numerators and denominators.
• Simplify the resulting fraction by dividing the numerator and denominator by their greatest common divisor (GCD).
• Cancel out any common factors in the numerator and denominator.
• Check your answer by multiplying the quotient by the divisor to make sure it equals the dividend.