Which Division Expression Is Equivalent To 4 1 3 − 5 6 \frac{4 \frac{1}{3}}{-\frac{5}{6}} − 6 5 ​ 4 3 1 ​ ​ ?A. \frac{13}{3} \div \left(-\frac{5}{6}\right ] B. − 5 6 ÷ 13 3 -\frac{5}{6} \div \frac{13}{3} − 6 5 ​ ÷ 3 13 ​ C. 13 3 ÷ − 5 6 \frac{13}{3} \div -\frac{5}{6} 3 13 ​ ÷ − 6 5 ​ D.

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Introduction

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In mathematics, division is a fundamental operation that involves splitting a quantity into equal parts. When dealing with fractions, division can be represented as a fraction of a fraction. In this article, we will explore the equivalence of a division expression involving fractions.

Understanding the Problem

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The given problem is to find the equivalent division expression for $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$. To solve this, we need to understand the concept of division as a fraction of a fraction. When we divide a fraction by another fraction, we can invert the second fraction and multiply instead.

Inverting the Second Fraction

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To make the division easier to understand, let's invert the second fraction, $-\frac{5}{6}$, and multiply instead. This gives us:

$$\frac{4 \frac{1}{3}}{-\frac{5}{6}} = \frac{4 \frac{1}{3}}{-\frac{5}{6}} \cdot \frac{6}{5}$$

Simplifying the Expression

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Now, let's simplify the expression by multiplying the numerators and denominators:

$$\frac{4 \frac{1}{3}}{-\frac{5}{6}} \cdot \frac{6}{5} = \frac{\left(4 \cdot 3 + 1\right)}{-\left(5 \cdot 6\right)} \cdot \frac{6}{5}$$

Evaluating the Numerator and Denominator

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Next, let's evaluate the numerator and denominator separately:

$$\left(4 \cdot 3 + 1\right) = 12 + 1 = 13$$

$$-\left(5 \cdot 6\right) = -30$$

Substituting the Values

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Now, let's substitute the values back into the expression:

$$\frac{\left(4 \cdot 3 + 1\right)}{-\left(5 \cdot 6\right)} \cdot \frac{6}{5} = \frac{13}{-30} \cdot \frac{6}{5}$$

Simplifying the Expression Further

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To simplify the expression further, let's multiply the numerators and denominators:

$$\frac{13}{-30} \cdot \frac{6}{5} = \frac{13 \cdot 6}{-30 \cdot 5}$$

Evaluating the Numerator and Denominator

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Next, let's evaluate the numerator and denominator separately:

$$13 \cdot 6 = 78$$

$$-30 \cdot 5 = -150$$

Substituting the Values

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Now, let's substitute the values back into the expression:

$$\frac{13 \cdot 6}{-30 \cdot 5} = \frac{78}{-150}$$

Simplifying the Expression Further

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To simplify the expression further, let's divide both the numerator and denominator by their greatest common divisor, which is 6:

$$\frac{78}{-150} = \frac{13}{-25}$$

Conclusion

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In conclusion, the equivalent division expression for $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ is $\frac{13}{-25}$.

Discussion

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The given problem is a classic example of a division expression involving fractions. By inverting the second fraction and multiplying instead, we can simplify the expression and find the equivalent division expression.

Answer

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The correct answer is:

$\boxed{C. \frac{13}{3} \div -\frac{5}{6}}$

Explanation

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The correct answer is C. $\frac{13}{3} \div -\frac{5}{6}$ because it is the equivalent division expression for $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$. The other options are incorrect because they do not represent the equivalent division expression.

Final Answer

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The final answer is $\boxed{C. \frac{13}{3} \div -\frac{5}{6}}$.

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Frequently Asked Questions

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Q: What is the equivalent division expression for $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$?

A: The equivalent division expression for $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ is $\frac{13}{3} \div -\frac{5}{6}$.

Q: How do I simplify a division expression involving fractions?

A: To simplify a division expression involving fractions, you can invert the second fraction and multiply instead.

Q: What is the greatest common divisor of 78 and -150?

A: The greatest common divisor of 78 and -150 is 6.

Q: How do I simplify a fraction by dividing both the numerator and denominator by their greatest common divisor?

A: To simplify a fraction by dividing both the numerator and denominator by their greatest common divisor, you can divide both numbers by the greatest common divisor.

Q: What is the equivalent division expression for $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ in terms of division?

A: The equivalent division expression for $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ in terms of division is $\frac{13}{3} \div -\frac{5}{6}$.

Q: How do I evaluate the numerator and denominator of a fraction?

A: To evaluate the numerator and denominator of a fraction, you can multiply the numbers together.

Q: What is the value of $\frac{13}{-25}$?

A: The value of $\frac{13}{-25}$ is -0.52.

Q: How do I simplify a fraction by dividing both the numerator and denominator by their greatest common divisor?

A: To simplify a fraction by dividing both the numerator and denominator by their greatest common divisor, you can divide both numbers by the greatest common divisor.

Q: What is the equivalent division expression for $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ in terms of multiplication?

A: The equivalent division expression for $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ in terms of multiplication is $\frac{13}{3} \cdot \frac{6}{5}$.

Additional Resources

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• Division Expression Equivalence
• Simplifying Division Expressions
• Evaluating Numerators and Denominators
• Simplifying Fractions

Conclusion

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In conclusion, the equivalent division expression for $\frac{4 \frac{1}{3}}{-\frac{5}{6}}$ is $\frac{13}{3} \div -\frac{5}{6}$. By inverting the second fraction and multiplying instead, we can simplify the expression and find the equivalent division expression.

Final Answer

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The final answer is $\boxed{C. \frac{13}{3} \div -\frac{5}{6}}$.