Q1. Simplify: 2 1 3 ÷ 1 1 6 − 3 4 2 \frac{1}{3} \div 1 \frac{1}{6} - \frac{3}{4} 2 3 1 ​ ÷ 1 6 1 ​ − 4 3 ​

Understanding the Problem

In this problem, we are required to simplify an expression involving mixed numbers and fractions. The expression is $2 \frac{1}{3} \div 1 \frac{1}{6} - \frac{3}{4}$. To simplify this expression, we need to follow the order of operations (PEMDAS) and perform the division and subtraction operations.

Converting Mixed Numbers to Improper Fractions

To simplify the expression, we need to convert the mixed numbers to improper fractions. We can do this by multiplying the whole number part by the denominator and then adding the numerator.

• $2 \frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}$
• $1 \frac{1}{6} = \frac{(1 \times 6) + 1}{6} = \frac{6 + 1}{6} = \frac{7}{6}$

Simplifying the Division Operation

Now that we have converted the mixed numbers to improper fractions, we can simplify the division operation.

• $\frac{7}{3} \div \frac{7}{6} = \frac{7}{3} \times \frac{6}{7} = \frac{7 \times 6}{3 \times 7} = \frac{42}{21} = 2$

Simplifying the Subtraction Operation

Now that we have simplified the division operation, we can simplify the subtraction operation.

• $2 - \frac{3}{4} = \frac{8}{4} - \frac{3}{4} = \frac{8 - 3}{4} = \frac{5}{4}$

Conclusion

In this problem, we simplified an expression involving mixed numbers and fractions. We converted the mixed numbers to improper fractions, simplified the division operation, and then simplified the subtraction operation. The final answer is $\frac{5}{4}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Convert the mixed numbers to improper fractions.
   • $2 \frac{1}{3} = \frac{7}{3}$
   • $1 \frac{1}{6} = \frac{7}{6}$
2. Simplify the division operation.
   • $\frac{7}{3} \div \frac{7}{6} = \frac{7}{3} \times \frac{6}{7} = \frac{42}{21} = 2$
3. Simplify the subtraction operation.
   • $2 - \frac{3}{4} = \frac{8}{4} - \frac{3}{4} = \frac{8 - 3}{4} = \frac{5}{4}$

Key Concepts

• Converting mixed numbers to improper fractions
• Simplifying division operations involving fractions
• Simplifying subtraction operations involving fractions

Practice Problems

Here are some practice problems to help you reinforce your understanding of the concepts:

1. Simplify the expression: $3 \frac{1}{2} \div 2 \frac{1}{4} - \frac{1}{3}$
2. Simplify the expression: $4 \frac{2}{3} \div 3 \frac{1}{2} - \frac{2}{5}$
3. Simplify the expression: $2 \frac{1}{6} \div 1 \frac{1}{3} - \frac{3}{8}$
   Q&A: Simplifying Expressions with Mixed Numbers and Fractions =============================================================

Q1: What is the difference between a mixed number and an improper fraction?

A1: A mixed number is a combination of a whole number and a proper fraction, while an improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Q2: How do I convert a mixed number to an improper fraction?

A2: To convert a mixed number to an improper fraction, 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.

Q3: What is the order of operations when simplifying expressions with mixed numbers and fractions?

A3: The order of operations is:

1. Convert mixed numbers to improper fractions
2. Simplify division operations
3. Simplify subtraction operations

Q4: How do I simplify a division operation involving fractions?

A4: To simplify a division operation involving fractions, invert the second fraction and multiply.

Q5: What is the difference between a proper fraction and an improper fraction?

A5: 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.

Q6: How do I simplify a subtraction operation involving fractions?

A6: To simplify a subtraction operation involving fractions, find a common denominator and subtract the numerators.

Q7: What is the final answer to the expression $2 \frac{1}{3} \div 1 \frac{1}{6} - \frac{3}{4}$?

A7: The final answer is $\frac{5}{4}$.

Q8: Can you provide more practice problems to help me reinforce my understanding of the concepts?

A8: Here are some additional practice problems:

1. Simplify the expression: $3 \frac{1}{2} \div 2 \frac{1}{4} - \frac{1}{3}$
2. Simplify the expression: $4 \frac{2}{3} \div 3 \frac{1}{2} - \frac{2}{5}$
3. Simplify the expression: $2 \frac{1}{6} \div 1 \frac{1}{3} - \frac{3}{8}$

Q9: What are some common mistakes to avoid when simplifying expressions with mixed numbers and fractions?

A9: Some common mistakes to avoid include:

• Forgetting to convert mixed numbers to improper fractions
• Not simplifying division operations correctly
• Not finding a common denominator when simplifying subtraction operations

Q10: How can I apply these concepts to real-world problems?

A10: You can apply these concepts to real-world problems by using mixed numbers and fractions to represent quantities and perform calculations. For example, you can use mixed numbers to represent measurements or fractions to represent probabilities.

Conclusion

In this article, we have covered the basics of simplifying expressions with mixed numbers and fractions. We have discussed the order of operations, converting mixed numbers to improper fractions, simplifying division and subtraction operations, and avoiding common mistakes. We have also provided practice problems and real-world applications to help you reinforce your understanding of the concepts.