Applying the Concept to the Given Expression

Oct 13, 2025 by windailys.com 114 views

Understanding the Problem

To simplify the given expression without using a calculator, we need to first understand the concept of dividing fractions and mixed numbers. A mixed number is a combination of a whole number and a fraction, while a fraction represents a part of a whole. When we divide a mixed number by another mixed number, we need to convert both numbers into improper fractions and then perform the division.

Converting Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and then add the numerator. The result becomes the new numerator, while the denominator remains the same.

Example: ${ 1 \frac{2}{3} = \frac{(1 \times 3) + 2}{3} = \frac{5}{3} }$

${ 2 \frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{8}{3} }$

Dividing Fractions

When we divide one fraction by another, we can invert the second fraction (i.e., flip the numerator and denominator) and then multiply the two fractions.

Inverting the Second Fraction

To invert a fraction, we simply flip the numerator and denominator.

Example: ${ \frac{a}{b} = \frac{b}{a} }$

Multiplying Fractions

To multiply two fractions, we multiply the numerators and denominators separately.

Example: ${ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} }$

Applying the Concept to the Given Expression

Now that we have understood the concept of dividing fractions and mixed numbers, we can apply it to the given expression.

${ 1 \frac{2}{3} \div 2 \frac{2}{3} = \frac{5}{3} \div \frac{8}{3} }$

Inverting the Second Fraction

To invert the second fraction, we flip the numerator and denominator.

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

Multiplying Fractions

To multiply the two fractions, we multiply the numerators and denominators separately.

${ \frac{5}{3} \times \frac{3}{8} = \frac{5 \times 3}{3 \times 8} = \frac{15}{24} }$

Simplifying the Result

To simplify the result, we can divide both the numerator and denominator by their greatest common divisor (GCD).

Finding the GCD

The GCD of 15 and 24 is 3.

Dividing the Numerator and Denominator

To divide both the numerator and denominator by their GCD, we simply divide each number by 3.

${ \frac{15}{24} = \frac{15 \div 3}{24 \div 3} = \frac{5}{8} }$

Conclusion

In conclusion, we have successfully simplified the given expression without using a calculator. We first converted the mixed numbers to improper fractions, then divided the fractions by inverting the second fraction and multiplying the two fractions. Finally, we simplified the result by dividing both the numerator and denominator by their GCD.

Final Answer

The final answer is $\boxed{\frac{5}{8}}$ .

Frequently Asked Questions

Q: What is the concept of dividing fractions and mixed numbers?

A: The concept of dividing fractions and mixed numbers involves converting both numbers into improper fractions and then performing the division.

Q: How do we convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and then add the numerator. The result becomes the new numerator, while the denominator remains the same.

Q: What is the process of inverting a fraction?

A: The process of inverting a fraction involves flipping the numerator and denominator.

Q: How do we multiply fractions?

A: To multiply fractions, we multiply the numerators and denominators separately.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is $\boxed{\frac{5}{8}}$ .

Additional Resources

For more information on dividing fractions and mixed numbers, please refer to the following resources:

Note: The above resources are provided for additional information and are not affiliated with this article.

Understanding the Basics

Dividing fractions and mixed numbers can be a challenging concept, but with the right understanding and practice, it can become second nature. In this article, we will address some of the most frequently asked questions on this topic.

Q: What is the concept of dividing fractions and mixed numbers?

A: The concept of dividing fractions and mixed numbers involves converting both numbers into improper fractions and then performing the division.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, you multiply the whole number by the denominator and then add the numerator. The result becomes the new numerator, while the denominator remains the same.

Example:

${ 1 \frac{2}{3} = \frac{(1 \times 3) + 2}{3} = \frac{5}{3} }$

Q: What is the process of inverting a fraction?

A: The process of inverting a fraction involves flipping the numerator and denominator.

Example:

${ \frac{a}{b} = \frac{b}{a} }$

Q: How do I multiply fractions?

A: To multiply fractions, you multiply the numerators and denominators separately.

Example:

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

Q: What is the final answer to the given expression?

A: The final answer to the given expression is $\boxed{\frac{5}{8}}$ .

Q: Can I use a calculator to simplify the expression?

A: No, the problem specifically states that we should not use a calculator to simplify the expression.

Q: How do I simplify the result?

A: To simplify the result, you can divide both the numerator and denominator by their greatest common divisor (GCD).

Example:

${ \frac{15}{24} = \frac{15 \div 3}{24 \div 3} = \frac{5}{8} }$

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

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

Example:

The GCD of 15 and 24 is 3.

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

A: You can find the GCD of two numbers by listing the factors of each number and finding the largest common factor.

Example:

The factors of 15 are 1, 3, 5, and 15. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The largest common factor is 3.

Q: Can I use a calculator to find the GCD of two numbers?

A: No, the problem specifically states that we should not use a calculator to simplify the expression.

Q: What is the final answer to the given expression?

A: The final answer to the given expression is $\boxed{\frac{5}{8}}$ .

Q: Can I use a calculator to check my answer?

A: Yes, you can use a calculator to check your answer, but the problem specifically states that we should not use a calculator to simplify the expression.

Q: How do I know if my answer is correct?

A: You can check your answer by plugging it back into the original expression and simplifying it.

Example:

${ \frac{5}{8} \div \frac{8}{5} = \frac{5 \times 5}{8 \times 8} = \frac{25}{64} }$

Q: What is the final answer to the given expression?

A: The final answer to the given expression is $\boxed{\frac{5}{8}}$ .

Additional Resources

For more information on dividing fractions and mixed numbers, please refer to the following resources:

Note: The above resources are provided for additional information and are not affiliated with this article.