Simplify: ${ 3 \frac{4}{5} \div 3 \frac{1}{3} }$

Introduction

Understanding Mixed Numbers and Division

In mathematics, mixed numbers are a combination of a whole number and a fraction. They are often used to represent quantities that are not whole. When it comes to division, we need to simplify the mixed numbers before performing the operation. In this article, we will explore how to simplify the expression ${ 3 \frac{4}{5} \div 3 \frac{1}{3} }$.

What are Mixed Numbers?

Mixed numbers are a combination of a whole number and a fraction. They are often used to represent quantities that are not whole. For example, $3 \frac{4}{5}$ is a mixed number that represents $3$ whole units and $\frac{4}{5}$ of a unit. Mixed numbers can be written in different ways, but they always have a whole number part and a fractional part.

Converting Mixed Numbers to Improper Fractions

To simplify the expression ${ 3 \frac{4}{5} \div 3 \frac{1}{3} }$, we need to convert the mixed numbers to improper fractions. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. To convert a mixed number to an improper fraction, we multiply the whole number part by the denominator and add the numerator. Then, we write the result as a fraction with the same denominator.

For example, to convert $3 \frac{4}{5}$ to an improper fraction, we multiply $3$ by $5$ and add $4$. This gives us $15 + 4 = 19$. So, $3 \frac{4}{5}$ is equal to $\frac{19}{5}$.

Converting the Second Mixed Number to an Improper Fraction

We also need to convert the second mixed number, $3 \frac{1}{3}$, to an improper fraction. To do this, we multiply $3$ by $3$ and add $1$. This gives us $9 + 1 = 10$. So, $3 \frac{1}{3}$ is equal to $\frac{10}{3}$.

Simplifying the Expression

Now that we have converted both mixed numbers to improper fractions, we can simplify the expression ${ 3 \frac{4}{5} \div 3 \frac{1}{3} }$. To do this, we need to divide the two fractions. When dividing fractions, we invert the second fraction and multiply. So, we have:

$$\frac{19}{5} \div \frac{10}{3} = \frac{19}{5} \times \frac{3}{10}$$

Multiplying the Fractions

To multiply the fractions, we multiply the numerators and multiply the denominators. This gives us:

$$\frac{19 \times 3}{5 \times 10} = \frac{57}{50}$$

Simplifying the Result

The result, $\frac{57}{50}$, is an improper fraction. We can simplify it by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of $57$ and $50$ is $1$, so we cannot simplify the fraction further.

Conclusion

In conclusion, to simplify the expression ${ 3 \frac{4}{5} \div 3 \frac{1}{3} }$, we need to convert the mixed numbers to improper fractions and then divide the fractions. The result is $\frac{57}{50}$, which is an improper fraction that cannot be simplified further.

Frequently Asked Questions

• What is a mixed number? A mixed number is a combination of a whole number and a fraction.
• How do I convert a mixed number to an improper fraction? To convert a mixed number to an improper fraction, you multiply the whole number part by the denominator and add the numerator.
• How do I divide fractions? To divide fractions, you invert the second fraction and multiply.

Final Thoughts

Simplifying mixed numbers and dividing fractions can be challenging, but with practice and patience, you can master these skills. Remember to convert mixed numbers to improper fractions and then divide the fractions to get the correct result. With this knowledge, you will be able to simplify complex expressions and solve problems with ease.

Additional Resources

• Mathematics textbooks: For a comprehensive understanding of mixed numbers and division, refer to a mathematics textbook.
• Online resources: Websites such as Khan Academy and Mathway offer interactive lessons and exercises to help you practice and improve your skills.
• Practice problems: Try solving practice problems to reinforce your understanding of mixed numbers and division.

References

• Mathematics textbooks: For a comprehensive understanding of mixed numbers and division, refer to a mathematics textbook.
• Online resources: Websites such as Khan Academy and Mathway offer interactive lessons and exercises to help you practice and improve your skills.

About the Author

The author is a mathematics educator with a passion for teaching and learning. They have extensive experience in teaching mathematics to students of all ages and skill levels. The author is committed to providing high-quality content and resources to help students succeed in mathematics.

Q&A: Simplifying Mixed Numbers and Division

In our previous article, we explored how to simplify the expression ${ 3 \frac{4}{5} \div 3 \frac{1}{3} }$. In this article, we will answer some frequently asked questions about simplifying mixed numbers and division.

Q: What is a mixed number?

A: A mixed number is a combination of a whole number and a fraction. For example, $3 \frac{4}{5}$ is a mixed number that represents $3$ whole units and $\frac{4}{5}$ of a unit.

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 part by the denominator and add the numerator. For example, to convert $3 \frac{4}{5}$ to an improper fraction, you multiply $3$ by $5$ and add $4$. This gives you $15 + 4 = 19$. So, $3 \frac{4}{5}$ is equal to $\frac{19}{5}$.

Q: How do I divide fractions?

A: To divide fractions, you invert the second fraction and multiply. For example, to divide $\frac{19}{5}$ by $\frac{10}{3}$, you invert the second fraction and multiply. This gives you $\frac{19}{5} \times \frac{3}{10}$.

Q: What is the result of dividing $\frac{19}{5}$ by $\frac{10}{3}$?

A: The result of dividing $\frac{19}{5}$ by $\frac{10}{3}$ is $\frac{57}{50}$.

Q: Can I simplify the result further?

A: The result, $\frac{57}{50}$, is an improper fraction. We can simplify it by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of $57$ and $50$ is $1$, so we cannot simplify the fraction further.

Q: What are some common mistakes to avoid when simplifying mixed numbers and division?

A: Some common mistakes to avoid when simplifying mixed numbers and division include:

• Not converting mixed numbers to improper fractions before dividing
• Not inverting the second fraction when dividing fractions
• Not simplifying the result further when possible

Q: How can I practice simplifying mixed numbers and division?

A: You can practice simplifying mixed numbers and division by trying out different examples and exercises. You can also use online resources such as Khan Academy and Mathway to get additional practice and support.

Q: What are some real-world applications of simplifying mixed numbers and division?

A: Simplifying mixed numbers and division has many real-world applications, including:

• Cooking and measuring ingredients
• Building and construction
• Finance and accounting
• Science and engineering

Q: Can I use a calculator to simplify mixed numbers and division?

A: Yes, you can use a calculator to simplify mixed numbers and division. However, it's always a good idea to double-check your work and make sure you understand the steps involved in simplifying mixed numbers and division.

Q: How can I improve my skills in simplifying mixed numbers and division?

A: To improve your skills in simplifying mixed numbers and division, you can:

• Practice regularly and try out different examples and exercises
• Use online resources such as Khan Academy and Mathway to get additional practice and support
• Ask a teacher or tutor for help and guidance
• Review and practice regularly to reinforce your understanding of mixed numbers and division.

Conclusion

Simplifying mixed numbers and division can be challenging, but with practice and patience, you can master these skills. Remember to convert mixed numbers to improper fractions and then divide the fractions to get the correct result. With this knowledge, you will be able to simplify complex expressions and solve problems with ease.

Additional Resources

• Mathematics textbooks: For a comprehensive understanding of mixed numbers and division, refer to a mathematics textbook.
• Online resources: Websites such as Khan Academy and Mathway offer interactive lessons and exercises to help you practice and improve your skills.
• Practice problems: Try solving practice problems to reinforce your understanding of mixed numbers and division.

References

• Mathematics textbooks: For a comprehensive understanding of mixed numbers and division, refer to a mathematics textbook.
• Online resources: Websites such as Khan Academy and Mathway offer interactive lessons and exercises to help you practice and improve your skills.

About the Author

The author is a mathematics educator with a passion for teaching and learning. They have extensive experience in teaching mathematics to students of all ages and skill levels. The author is committed to providing high-quality content and resources to help students succeed in mathematics.