Applying the Invert and Multiply Rule

Oct 13, 2025 by recoveryshake.com 93 views

Introduction

In mathematics, division is a fundamental operation that involves sharing a quantity into equal parts. When dealing with fractions, division can be a bit more complex, but with the right approach, it can be simplified. In this article, we will explore the division expression equivalent to $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ and examine the options provided.

Understanding the Problem

The given expression is $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ . To simplify this expression, we need to understand the concept of division with fractions. When dividing a fraction by another fraction, we can invert the second fraction and multiply instead. This is known as the "invert and multiply" rule.

Invert and Multiply Rule

The invert and multiply rule states that to divide a fraction by another fraction, we can invert the second fraction and multiply instead. This rule can be represented as:

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

Applying the Invert and Multiply Rule

Now, let's apply the invert and multiply rule to the given expression:

\frac{4 \frac{1}{3}}{-\frac{5}{6}} = \frac{4 \frac{1}{3}}{1} \times \frac{1}{-\frac{5}{6}}

To simplify the expression, we need to convert the mixed number $4 \frac{1}{3}$ to an improper fraction. An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

Converting Mixed Numbers to Improper Fractions

A mixed number can be converted to an improper fraction by multiplying the whole number part by the denominator and then adding the numerator. The result is then written as a fraction with the new numerator and the same denominator.

4 \frac{1}{3} = \frac{(4 \times 3) + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3}

Now, let's substitute the improper fraction back into the expression:

\frac{4 \frac{1}{3}}{-\frac{5}{6}} = \frac{\frac{13}{3}}{1} \times \frac{1}{-\frac{5}{6}}

Simplifying the Expression

Now, let's simplify the expression by applying the invert and multiply rule:

\frac{\frac{13}{3}}{1} \times \frac{1}{-\frac{5}{6}} = \frac{13}{3} \times \frac{6}{-5}

To simplify the expression further, we can multiply the numerators and denominators:

\frac{13}{3} \times \frac{6}{-5} = \frac{13 \times 6}{3 \times -5} = \frac{78}{-15}

Reducing the Fraction

To reduce the fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator. The GCD of 78 and 15 is 3.

\frac{78}{-15} = \frac{78 \div 3}{-15 \div 3} = \frac{26}{-5}

Conclusion

In conclusion, the division expression equivalent to $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ is $\frac{26}{-5}$ . This can be represented as an option in the discussion category:

Option A: $\frac{13}{3} \div \left(-\frac{5}{6}\right)$

This option is incorrect because it does not represent the correct division expression.

Option B: $-\frac{5}{6} \div \frac{13}{3}$

This option is incorrect because it does not represent the correct division expression.

Option C: $\frac{13}{3} \div \frac{5}{6}$

This option is correct because it represents the correct division expression.

Option D:

This option is not provided.

Final Answer

The final answer is $\boxed{C}$ .

Discussion

The discussion category is mathematics, and the topic is division expression equivalency. The question is which division expression is equivalent to $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ . The correct answer is option C: $\frac{13}{3} \div \frac{5}{6}$ . This option represents the correct division expression, which is equivalent to the given expression.

Additional Information

The invert and multiply rule is a fundamental concept in mathematics that can be used to simplify division expressions. This rule states that to divide a fraction by another fraction, we can invert the second fraction and multiply instead. This rule can be represented as:

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

This rule can be applied to simplify division expressions, and it is an essential tool for solving mathematical problems.

References

[1] "Invert and Multiply Rule" by Math Open Reference
[2] "Division of Fractions" by Khan Academy
[3] "Mathematics for Dummies" by Mark Ryan

Conclusion

In conclusion, the division expression equivalent to $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ is $\frac{26}{-5}$ . This can be represented as option C: $\frac{13}{3} \div \frac{5}{6}$ . The invert and multiply rule is a fundamental concept in mathematics that can be used to simplify division expressions. This rule states that to divide a fraction by another fraction, we can invert the second fraction and multiply instead. This rule can be represented as:

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

Introduction

In our previous article, we explored the division expression equivalent to $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ . We applied the invert and multiply rule to simplify the expression and arrived at the final answer. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q&A Section

Q: What is the invert and multiply rule?

A: The invert and multiply rule is a fundamental concept in mathematics that states that to divide a fraction by another fraction, we can invert the second fraction and multiply instead. This rule can be represented as:

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

Q: How do I apply the invert and multiply rule?

A: To apply the invert and multiply rule, you need to follow these steps:

Invert the second fraction by flipping the numerator and denominator.
Multiply the first fraction by the inverted second fraction.

Q: What is the difference between division and multiplication?

A: Division and multiplication are two different operations in mathematics. Division involves sharing a quantity into equal parts, while multiplication involves combining quantities.

Q: Can I use the invert and multiply rule for all division expressions?

A: Yes, the invert and multiply rule can be used for all division expressions involving fractions. However, it's essential to remember that the rule only applies to division expressions involving fractions and not to division expressions involving whole numbers.

Q: How do I simplify a division expression using the invert and multiply rule?

A: To simplify a division expression using the invert and multiply rule, follow these steps:

Invert the second fraction by flipping the numerator and denominator.
Multiply the first fraction by the inverted second fraction.
Simplify the resulting expression by reducing the fraction to its simplest form.

Q: What is the final answer to the division expression $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ ?

A: The final answer to the division expression $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ is $\frac{26}{-5}$ .

Q: Can I use the invert and multiply rule for division expressions involving decimals?

A: Yes, the invert and multiply rule can be used for division expressions involving decimals. However, it's essential to remember that the rule only applies to division expressions involving fractions and not to division expressions involving whole numbers or decimals.

Q: How do I apply the invert and multiply rule for division expressions involving decimals?

A: To apply the invert and multiply rule for division expressions involving decimals, follow these steps:

Convert the decimal to a fraction by writing it as a fraction with a denominator of 10 or 100.
Invert the second fraction by flipping the numerator and denominator.
Multiply the first fraction by the inverted second fraction.
Simplify the resulting expression by reducing the fraction to its simplest form.

Conclusion

In conclusion, the invert and multiply rule is a fundamental concept in mathematics that can be used to simplify division expressions involving fractions. This rule states that to divide a fraction by another fraction, we can invert the second fraction and multiply instead. This rule can be represented as:

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

This rule can be applied to simplify division expressions, and it is an essential tool for solving mathematical problems. We hope this Q&A article has helped clarify any doubts and provided additional information on the topic.

Additional Resources

[1] "Invert and Multiply Rule" by Math Open Reference
[2] "Division of Fractions" by Khan Academy
[3] "Mathematics for Dummies" by Mark Ryan

Final Answer

The final answer to the division expression $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ is $\frac{26}{-5}$ .