Rewrite The Division Problem Using Improper Fractions.${ \frac{3 \frac{3}{4}}{-\frac{5}{8}} }$A. { -\frac{5}{8}+\frac{15}{4}$}$B. { \frac{15}{4}-\left(-\frac{5}{8}\right)$}$C. { \frac{15}{4}-\frac{5}{8}$}$D.

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Understanding the Problem

When dealing with division problems involving mixed numbers and fractions, it's essential to rewrite the problem using improper fractions. This allows us to perform the division operation more efficiently and accurately. In this article, we'll focus on rewriting the division problem 334−58\frac{3 \frac{3}{4}}{-\frac{5}{8}} using improper fractions.

Rewriting Mixed Numbers as Improper Fractions

To rewrite a mixed number as an improper fraction, we need to multiply the whole number part by the denominator and then add the numerator. The result will be the new numerator, and the denominator remains the same.

For example, let's rewrite the mixed number 3343 \frac{3}{4} as an improper fraction:

334=(3×4)+34=12+34=1543 \frac{3}{4} = \frac{(3 \times 4) + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4}

Rewriting the Division Problem

Now that we've rewritten the mixed number 3343 \frac{3}{4} as an improper fraction 154\frac{15}{4}, we can rewrite the division problem 334−58\frac{3 \frac{3}{4}}{-\frac{5}{8}} using improper fractions:

334−58=154−58\frac{3 \frac{3}{4}}{-\frac{5}{8}} = \frac{\frac{15}{4}}{-\frac{5}{8}}

Inverting the Divisor and Changing the Division Sign

When dividing fractions, we need to invert the divisor (i.e., flip the numerator and denominator) and change the division sign to a multiplication sign. This is a fundamental property of fractions and is essential for performing division operations.

In this case, we'll invert the divisor −58-\frac{5}{8} and change the division sign to a multiplication sign:

154−58=154×(−85)\frac{\frac{15}{4}}{-\frac{5}{8}} = \frac{15}{4} \times \left(-\frac{8}{5}\right)

Multiplying the Numerators and Denominators

Now that we've inverted the divisor and changed the division sign to a multiplication sign, we can multiply the numerators and denominators:

154×(−85)=15×(−8)4×5=−12020\frac{15}{4} \times \left(-\frac{8}{5}\right) = \frac{15 \times (-8)}{4 \times 5} = \frac{-120}{20}

Simplifying the Result

Finally, we can simplify the result by dividing both the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of -120 and 20 is 20.

−12020=−120÷2020÷20=−61=−6\frac{-120}{20} = -\frac{120 \div 20}{20 \div 20} = -\frac{6}{1} = -6

Conclusion

In this article, we've rewritten the division problem 334−58\frac{3 \frac{3}{4}}{-\frac{5}{8}} using improper fractions. We've also demonstrated the importance of inverting the divisor and changing the division sign to a multiplication sign when dividing fractions. By following these steps, we've arrived at the final answer of -6.

Answer Key

The correct answer is:

−6\boxed{-6}

Discussion

This problem requires a deep understanding of fractions and division operations. It's essential to remember that when dividing fractions, we need to invert the divisor and change the division sign to a multiplication sign. By following these steps, we can perform division operations more efficiently and accurately.

Additional Resources

For more information on fractions and division operations, please refer to the following resources:

Q: What is the rule for dividing fractions?

A: When dividing fractions, we need to invert the divisor (i.e., flip the numerator and denominator) and change the division sign to a multiplication sign.

Q: How do I invert the divisor?

A: To invert the divisor, we simply flip the numerator and denominator. For example, if we have the fraction ab\frac{a}{b}, the inverted fraction would be ba\frac{b}{a}.

Q: What is the difference between dividing fractions and multiplying fractions?

A: When dividing fractions, we invert the divisor and change the division sign to a multiplication sign. When multiplying fractions, we multiply the numerators and denominators directly.

Q: Can I simplify the result of a division problem involving fractions?

A: Yes, you can simplify the result of a division problem involving fractions by dividing both 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: There are several ways to find the GCD of two numbers, including:

  • Listing the factors of each number and finding the greatest common factor
  • Using the Euclidean algorithm
  • Using a calculator or online tool

Q: Can I use a calculator or online tool to simplify the result of a division problem involving fractions?

A: Yes, you can use a calculator or online tool to simplify the result of a division problem involving fractions. This can be especially helpful if you are unsure of the GCD or if the numbers are large.

Q: What are some common mistakes to avoid when dividing fractions?

A: Some common mistakes to avoid when dividing fractions include:

  • Failing to invert the divisor
  • Failing to change the division sign to a multiplication sign
  • Not simplifying the result by dividing both the numerator and denominator by their GCD

Q: How can I practice dividing fractions?

A: You can practice dividing fractions by working through examples and exercises, such as:

  • Dividing simple fractions (e.g., 12÷13\frac{1}{2} \div \frac{1}{3})
  • Dividing complex fractions (e.g., 34÷56\frac{3}{4} \div \frac{5}{6})
  • Dividing fractions with negative numbers (e.g., −12÷34-\frac{1}{2} \div \frac{3}{4})

Q: What are some real-world applications of dividing fractions?

A: Dividing fractions has many real-world applications, including:

  • Cooking and recipe scaling
  • Building and construction
  • Science and engineering
  • Finance and economics

Conclusion

Dividing fractions can seem intimidating at first, but with practice and patience, it becomes second nature. By following the rules and guidelines outlined in this article, you'll be able to divide fractions with confidence and accuracy. Remember to invert the divisor, change the division sign to a multiplication sign, and simplify the result by dividing both the numerator and denominator by their GCD. Happy calculating!