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

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In mathematics, a quotient is the result of a division operation. When we divide one number by another, the result is the quotient. In this article, we will explore which expression has a positive quotient among the given options.

Understanding Quotient

A quotient can be positive or negative, depending on the signs of the dividend and divisor. If both the dividend and divisor have the same sign (either both positive or both negative), the quotient will be positive. However, if the dividend and divisor have opposite signs, the quotient will be negative.

Analyzing the Options

Let's analyze each option to determine which one has a positive quotient.

Option A: −34−23\frac{-\frac{3}{4}}{-\frac{2}{3}}

To evaluate this expression, we need to follow the order of operations (PEMDAS). First, we will divide the fractions in the numerator and denominator.

−34−23=−3−2×43\frac{-\frac{3}{4}}{-\frac{2}{3}} = \frac{-3}{-2} \times \frac{4}{3}

Now, we can simplify the expression by canceling out common factors.

−3−2×43=12×43=46=23\frac{-3}{-2} \times \frac{4}{3} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}

Since the quotient is positive, option A has a positive quotient.

Option B: −18-\frac{1}{8}

This option is already in its simplest form, and we can see that the quotient is negative. Therefore, option B does not have a positive quotient.

Option C: 227−15\frac{2 \frac{2}{7}}{-\frac{1}{5}}

To evaluate this expression, we need to convert the mixed number to an improper fraction.

227=147+27=1672 \frac{2}{7} = \frac{14}{7} + \frac{2}{7} = \frac{16}{7}

Now, we can rewrite the expression as:

167−15\frac{\frac{16}{7}}{-\frac{1}{5}}

To divide fractions, we need to invert the divisor and multiply.

167−15=167×−51\frac{\frac{16}{7}}{-\frac{1}{5}} = \frac{16}{7} \times -\frac{5}{1}

Now, we can simplify the expression by multiplying the numerators and denominators.

167×−51=−807\frac{16}{7} \times -\frac{5}{1} = -\frac{80}{7}

Since the quotient is negative, option C does not have a positive quotient.

Option D: −653\frac{-6}{\frac{5}{3}}

To evaluate this expression, we need to follow the order of operations (PEMDAS). First, we will divide the fraction in the denominator.

−653=−6×35\frac{-6}{\frac{5}{3}} = -6 \times \frac{3}{5}

Now, we can simplify the expression by multiplying the numerators and denominators.

−6×35=−185-6 \times \frac{3}{5} = -\frac{18}{5}

Since the quotient is negative, option D does not have a positive quotient.

Conclusion

In conclusion, only option A has a positive quotient among the given options. The other options have negative quotients. When evaluating expressions with fractions, it's essential to follow the order of operations and simplify the expression to determine the sign of the quotient.

Key Takeaways

  • A quotient can be positive or negative, depending on the signs of the dividend and divisor.
  • If both the dividend and divisor have the same sign (either both positive or both negative), the quotient will be positive.
  • If the dividend and divisor have opposite signs, the quotient will be negative.
  • When evaluating expressions with fractions, it's essential to follow the order of operations and simplify the expression to determine the sign of the quotient.

Final Thoughts

In our previous article, we explored which expression has a positive quotient among the given options. Now, let's dive deeper into the world of quotients and answer some frequently asked questions.

Q: What is a quotient?

A: A quotient is the result of a division operation. When we divide one number by another, the result is the quotient.

Q: How do I determine the sign of the quotient?

A: To determine the sign of the quotient, you need to consider the signs of the dividend and divisor. If both the dividend and divisor have the same sign (either both positive or both negative), the quotient will be positive. If the dividend and divisor have opposite signs, the quotient will be negative.

Q: What is the order of operations for evaluating expressions with fractions?

A: The order of operations for evaluating expressions with fractions is:

  1. Follow the order of operations (PEMDAS)
  2. Simplify the expression
  3. Determine the sign of the quotient

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, you need to:

  1. Follow the order of operations (PEMDAS)
  2. Cancel out common factors
  3. Multiply or divide the fractions as needed

Q: What is the difference between a quotient and a product?

A: A quotient is the result of a division operation, while a product is the result of a multiplication operation. For example, 6 ÷ 2 = 3 (quotient) and 6 × 2 = 12 (product).

Q: Can a quotient be a fraction?

A: Yes, a quotient can be a fraction. For example, 1/2 ÷ 1/3 = 3/2.

Q: How do I evaluate an expression with multiple fractions?

A: To evaluate an expression with multiple fractions, you need to:

  1. Follow the order of operations (PEMDAS)
  2. Simplify the expression
  3. Determine the sign of the quotient

Q: What is the importance of understanding quotients in mathematics?

A: Understanding quotients is essential in mathematics because it helps you to:

  1. Evaluate expressions with fractions
  2. Determine the sign of the quotient
  3. Simplify complex expressions

Q: Can you provide examples of real-world applications of quotients?

A: Yes, here are some examples of real-world applications of quotients:

  1. Cooking: When you divide a recipe by a certain number of people, the result is the quotient.
  2. Finance: When you divide a loan by the number of years, the result is the quotient.
  3. Science: When you divide a measurement by a certain factor, the result is the quotient.

Conclusion

In conclusion, understanding quotients is essential in mathematics. By following the order of operations and simplifying expressions, you can determine the sign of the quotient and evaluate complex expressions. We hope this Q&A guide has helped you to better understand quotients and their applications.

Key Takeaways

  • A quotient is the result of a division operation.
  • The sign of the quotient depends on the signs of the dividend and divisor.
  • To simplify an expression with fractions, follow the order of operations (PEMDAS) and cancel out common factors.
  • Understanding quotients is essential in mathematics and has real-world applications.

Final Thoughts

In this article, we answered some frequently asked questions about quotients and provided examples of real-world applications. We hope this guide has helped you to better understand quotients and their importance in mathematics.