Which Expression Has A Positive Quotient?A. − 3 4 − 2 3 \frac{-\frac{3}{4}}{-\frac{2}{3}} − 3 2 ​ − 4 3 ​ ​ B. − 1 8 3 1 5 \frac{-\frac{1}{8}}{3 \frac{1}{5}} 3 5 1 ​ − 8 1 ​ ​ C. 2 2 7 2 \frac{2}{7} 2 7 2 ​ D. − 1 5 ‾ \overline{-\frac{1}{5}} − 5 1 ​ ​ E. − 6 5 3 \frac{-6}{\frac{5}{3}} 3 5 ​ − 6 ​

Introduction

In mathematics, a quotient is the result of a division operation. When we divide two numbers, the quotient is the number that results from the division. In this article, we will explore which expression has a positive quotient. We will examine each option and determine whether the quotient is positive or negative.

Option A: $\frac{-\frac{3}{4}}{-\frac{2}{3}}$

To determine the quotient of this expression, we need to divide the numerator by the denominator. The numerator is $-\frac{3}{4}$ and the denominator is $-\frac{2}{3}$. When we divide a negative number by a negative number, the result is positive.

$\frac{-\frac{3}{4}}{-\frac{2}{3}} = \frac{-3}{4} \div \frac{-2}{3} = \frac{-3}{4} \times \frac{3}{-2} = \frac{-9}{-8} = \frac{9}{8}$

The quotient of this expression is $\frac{9}{8}$, which is a positive number.

Option B: $\frac{-\frac{1}{8}}{3 \frac{1}{5}}$

To determine the quotient of this expression, we need to divide the numerator by the denominator. The numerator is $-\frac{1}{8}$ and the denominator is $3 \frac{1}{5}$. We need to convert the mixed number to an improper fraction.

$3 \frac{1}{5} = \frac{3 \times 5}{5} + \frac{1}{5} = \frac{15}{5} + \frac{1}{5} = \frac{16}{5}$

Now we can divide the numerator by the denominator.

$\frac{-\frac{1}{8}}{\frac{16}{5}} = \frac{-1}{8} \div \frac{16}{5} = \frac{-1}{8} \times \frac{5}{16} = \frac{-5}{128}$

The quotient of this expression is $-\frac{5}{128}$, which is a negative number.

Option C: $2 \frac{2}{7}$

To determine the quotient of this expression, we need to convert the mixed number to an improper fraction.

$2 \frac{2}{7} = \frac{2 \times 7}{7} + \frac{2}{7} = \frac{14}{7} + \frac{2}{7} = \frac{16}{7}$

However, this expression is not in the form of a quotient. It is a mixed number, and we cannot determine the quotient without more information.

Option D: $\overline{-\frac{1}{5}}$

This expression is a repeating decimal, which can be written as a fraction.

$\overline{-\frac{1}{5}} = -\frac{1}{5}$

However, this expression is not in the form of a quotient. It is a fraction, and we cannot determine the quotient without more information.

Option E: $\frac{-6}{\frac{5}{3}}$

To determine the quotient of this expression, we need to divide the numerator by the denominator. The numerator is $-6$ and the denominator is $\frac{5}{3}$.

$\frac{-6}{\frac{5}{3}} = -6 \div \frac{5}{3} = -6 \times \frac{3}{5} = \frac{-18}{5}$

The quotient of this expression is $-\frac{18}{5}$, which is a negative number.

Conclusion

In conclusion, only one of the expressions has a positive quotient. The expression $\frac{-\frac{3}{4}}{-\frac{2}{3}}$ has a positive quotient of $\frac{9}{8}$. The other expressions do not have a positive quotient.

Frequently Asked Questions

• What is a quotient? A quotient is the result of a division operation.
• How do you determine the quotient of an expression? To determine the quotient of an expression, you need to divide the numerator by the denominator.
• What is a positive quotient? A positive quotient is a quotient that is greater than zero.
• What is a negative quotient? A negative quotient is a quotient that is less than zero.

Final Thoughts

In this article, we explored which expression has a positive quotient. We examined each option and determined whether the quotient is positive or negative. The expression $\frac{-\frac{3}{4}}{-\frac{2}{3}}$ has a positive quotient of $\frac{9}{8}$. The other expressions do not have a positive quotient.

Introduction

In our previous article, we explored which expression has a positive quotient. We examined each option and determined whether the quotient is positive or negative. In this article, we will provide a Q&A section to help you better understand the concept of a quotient and how to determine it.

Q&A

Q: What is a quotient?

A: A quotient is the result of a division operation. It is the number that results from dividing one number by another.

Q: How do you determine the quotient of an expression?

A: To determine the quotient of an expression, you need to divide the numerator by the denominator. The numerator is the number being divided, and the denominator is the number by which we are dividing.

Q: What is a positive quotient?

A: A positive quotient is a quotient that is greater than zero. It is a quotient that results from dividing a positive number by a positive number, or a negative number by a negative number.

Q: What is a negative quotient?

A: A negative quotient is a quotient that is less than zero. It is a quotient that results from dividing a positive number by a negative number, or a negative number by a positive number.

Q: How do you handle negative numbers when determining a quotient?

A: When determining a quotient, you need to handle negative numbers carefully. If both the numerator and denominator are negative, the quotient will be positive. If one of the numbers is negative and the other is positive, the quotient will be negative.

Q: Can a quotient be zero?

A: Yes, a quotient can be zero. This occurs when the numerator is zero and the denominator is not zero.

Q: Can a quotient be undefined?

A: Yes, a quotient can be undefined. This occurs when the denominator is zero and the numerator is not zero.

Q: How do you handle fractions when determining a quotient?

A: When determining a quotient, you need to handle fractions carefully. You can convert a fraction to a decimal or a percentage to make it easier to work with.

Q: Can a quotient be a fraction?

A: Yes, a quotient can be a fraction. This occurs when the numerator and denominator are both fractions.

Q: Can a quotient be a decimal?

A: Yes, a quotient can be a decimal. This occurs when the numerator and denominator are both decimals.

Q: Can a quotient be a percentage?

A: Yes, a quotient can be a percentage. This occurs when the numerator and denominator are both percentages.

Examples

Example 1: Determining a Quotient with Positive Numbers

Determine the quotient of 12 and 4.

$\frac{12}{4} = 3$

The quotient is 3, which is a positive number.

Example 2: Determining a Quotient with Negative Numbers

Determine the quotient of -12 and -4.

$\frac{-12}{-4} = 3$

The quotient is 3, which is a positive number.

Example 3: Determining a Quotient with a Fraction

Determine the quotient of $\frac{1}{2}$ and $\frac{1}{4}$.

$\frac{\frac{1}{2}}{\frac{1}{4}} = \frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = 2$

The quotient is 2, which is a positive number.

Conclusion

In conclusion, determining a quotient can be a complex process, but with practice and patience, you can become proficient in handling different types of numbers and expressions. Remember to handle negative numbers carefully and to convert fractions to decimals or percentages when necessary.

Frequently Asked Questions

• What is a quotient? A quotient is the result of a division operation.
• How do you determine the quotient of an expression? To determine the quotient of an expression, you need to divide the numerator by the denominator.
• What is a positive quotient? A positive quotient is a quotient that is greater than zero.
• What is a negative quotient? A negative quotient is a quotient that is less than zero.

Final Thoughts

In this article, we provided a Q&A section to help you better understand the concept of a quotient and how to determine it. We hope that this article has been helpful in clarifying any confusion you may have had about quotients. If you have any further questions or concerns, please don't hesitate to ask.