Simplify The Following Expression:${ 2 \frac{1}{5} \div 3 \frac{3}{4} }$

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Introduction

When dealing with mixed numbers, it's essential to understand how to simplify expressions that involve division. In this article, we will explore the process of simplifying the expression $2 \frac{1}{5} \div 3 \frac{3}{4}$. We will break down the steps involved in simplifying this expression and provide a clear understanding of the mathematical concepts involved.

Understanding Mixed Numbers

Before we dive into simplifying the expression, let's take a moment to understand what mixed numbers are. A mixed number is a combination of a whole number and a fraction. It's denoted by writing the whole number part followed by the fraction part. For example, $2 \frac{1}{5}$ is a mixed number that represents $2$ whole units and $\frac{1}{5}$ of a unit.

Converting Mixed Numbers to Improper Fractions

To simplify the expression $2 \frac{1}{5} \div 3 \frac{3}{4}$, 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 then add the numerator. The result is the new numerator, and the denominator remains the same.

For example, to convert $2 \frac{1}{5}$ to an improper fraction, we multiply $2$ by $5$ and add $1$. This gives us a new numerator of $11$. The denominator remains the same, so the improper fraction is $\frac{11}{5}$.

Converting the Second Mixed Number to an Improper Fraction

We also need to convert the second mixed number, $3 \frac{3}{4}$, to an improper fraction. Using the same process, we multiply $3$ by $4$ and add $3$. This gives us a new numerator of $15$. The denominator remains the same, so the improper fraction is $\frac{15}{4}$.

Simplifying the Expression

Now that we have converted both mixed numbers to improper fractions, we can simplify the expression. The expression $2 \frac{1}{5} \div 3 \frac{3}{4}$ is equivalent to $\frac{11}{5} \div \frac{15}{4}$. To divide fractions, we multiply the first fraction by the reciprocal of the second fraction. The reciprocal of $\frac{15}{4}$ is $\frac{4}{15}$.

Multiplying the Fractions

To simplify the expression, we multiply the first fraction by the reciprocal of the second fraction. This gives us $\frac{11}{5} \times \frac{4}{15}$. To multiply fractions, we multiply the numerators and multiply the denominators.

Simplifying the Result

Multiplying the numerators gives us $11 \times 4 = 44$. Multiplying the denominators gives us $5 \times 15 = 75$. Therefore, the result of multiplying the fractions is $\frac{44}{75}$.

Reducing the Fraction

The fraction $\frac{44}{75}$ can be reduced by dividing both the numerator and the denominator by their greatest common divisor. The greatest common divisor of $44$ and $75$ is $1$, so the fraction cannot be reduced further.

Conclusion

In conclusion, the expression $2 \frac{1}{5} \div 3 \frac{3}{4}$ can be simplified by converting the mixed numbers to improper fractions and then multiplying the fractions. The result is $\frac{44}{75}$, which cannot be reduced further.

Final Answer

The final answer to the expression $2 \frac{1}{5} \div 3 \frac{3}{4}$ is $\frac{44}{75}$.

Additional Tips and Tricks

• When dealing with mixed numbers, it's essential to understand how to convert them to improper fractions.
• To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and add the numerator.
• To divide fractions, multiply the first fraction by the reciprocal of the second fraction.
• To multiply fractions, multiply the numerators and multiply the denominators.
• To reduce a fraction, divide both the numerator and the denominator by their greatest common divisor.

Frequently Asked Questions

• Q: What is the difference between a mixed number and an improper fraction? A: A mixed number is a combination of a whole number and a fraction, while an improper fraction is a fraction where the numerator is greater than or equal to the denominator.
• Q: How do I convert a mixed number to an improper fraction? A: To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and add the numerator.
• Q: How do I divide fractions? A: To divide fractions, multiply the first fraction by the reciprocal of the second fraction.
• Q: How do I multiply fractions? A: To multiply fractions, multiply the numerators and multiply the denominators.
• Q: How do I reduce a fraction? A: To reduce a fraction, divide both the numerator and the denominator by their greatest common divisor.
  

Introduction

In our previous article, we explored the process of simplifying the expression $2 \frac{1}{5} \div 3 \frac{3}{4}$. We broke down the steps involved in simplifying this expression and provided a clear understanding of the mathematical concepts involved. In this article, we will answer some of the most frequently asked questions related to simplifying expressions with mixed numbers.

Q&A

Q: What is the difference between a mixed number and an improper fraction?

A: A mixed number is a combination of a whole number and a fraction, while an improper fraction is a fraction where the numerator is greater than or equal to the denominator.

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

A: To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and add the numerator. For example, to convert $2 \frac{1}{5}$ to an improper fraction, we multiply $2$ by $5$ and add $1$. This gives us a new numerator of $11$. The denominator remains the same, so the improper fraction is $\frac{11}{5}$.

Q: How do I divide fractions?

A: To divide fractions, multiply the first fraction by the reciprocal of the second fraction. The reciprocal of $\frac{15}{4}$ is $\frac{4}{15}$. Therefore, to divide $\frac{11}{5}$ by $\frac{15}{4}$, we multiply $\frac{11}{5}$ by $\frac{4}{15}$.

Q: How do I multiply fractions?

A: To multiply fractions, multiply the numerators and multiply the denominators. For example, to multiply $\frac{11}{5}$ and $\frac{4}{15}$, we multiply the numerators $11$ and $4$ to get $44$, and multiply the denominators $5$ and $15$ to get $75$. Therefore, the result of multiplying the fractions is $\frac{44}{75}$.

Q: How do I reduce a fraction?

A: To reduce a fraction, divide both the numerator and the denominator by their greatest common divisor. The greatest common divisor of $44$ and $75$ is $1$, so the fraction $\frac{44}{75}$ cannot be reduced further.

Q: What is the difference between a proper fraction and an improper fraction?

A: A proper fraction is a fraction where the numerator is less than the denominator, while an improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Q: How do I add and subtract fractions?

A: To add and subtract fractions, we need to have the same denominator. We can then add or subtract the numerators and keep the same denominator.

Q: What is the order of operations for simplifying expressions with mixed numbers?

A: The order of operations for simplifying expressions with mixed numbers is:

1. Convert mixed numbers to improper fractions
2. Divide fractions by multiplying the first fraction by the reciprocal of the second fraction
3. Multiply fractions by multiplying the numerators and denominators
4. Reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor

Conclusion

In conclusion, simplifying expressions with mixed numbers requires a clear understanding of the mathematical concepts involved. By following the steps outlined in this article, you can simplify expressions with mixed numbers and arrive at the correct solution.

Final Tips and Tricks

• Always convert mixed numbers to improper fractions before simplifying expressions.
• To divide fractions, multiply the first fraction by the reciprocal of the second fraction.
• To multiply fractions, multiply the numerators and multiply the denominators.
• To reduce a fraction, divide both the numerator and the denominator by their greatest common divisor.
• Practice, practice, practice! The more you practice simplifying expressions with mixed numbers, the more comfortable you will become with the process.

Additional Resources

• For more information on simplifying expressions with mixed numbers, check out our previous article on the topic.
• For practice problems and exercises, try using online resources such as Khan Academy or Mathway.
• For a comprehensive guide to fractions, check out our article on the topic.

Frequently Asked Questions (FAQs)

• Q: What is the difference between a mixed number and an improper fraction? A: A mixed number is a combination of a whole number and a fraction, while an improper fraction is a fraction where the numerator is greater than or equal to the denominator.
• Q: How do I convert a mixed number to an improper fraction? A: To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and add the numerator.
• Q: How do I divide fractions? A: To divide fractions, multiply the first fraction by the reciprocal of the second fraction.
• Q: How do I multiply fractions? A: To multiply fractions, multiply the numerators and multiply the denominators.
• Q: How do I reduce a fraction? A: To reduce a fraction, divide both the numerator and the denominator by their greatest common divisor.