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 -\frac{1}{5} − 5 1 ​ E. − 6 5 3 \frac{-6}{\frac{5}{3}} 3 5 ​ − 6 ​

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Introduction

When dealing with fractions and mixed numbers, it's essential to understand the concept of a positive quotient. A quotient is the result of a division operation, and in this case, we're looking for expressions that yield a positive quotient. In this article, we'll explore five different expressions and determine which one has a positive quotient.

Expression 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'll 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=11×43=43\frac{-3}{-2} \times \frac{4}{3} = \frac{1}{1} \times \frac{4}{3} = \frac{4}{3}

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

Expression B: −18315\frac{-\frac{1}{8}}{3 \frac{1}{5}}

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

315=1653 \frac{1}{5} = \frac{16}{5}

Now, we can rewrite the expression as:

−18165\frac{-\frac{1}{8}}{\frac{16}{5}}

To divide fractions, we'll multiply the first fraction by the reciprocal of the second fraction.

−18165=−18×516\frac{-\frac{1}{8}}{\frac{16}{5}} = -\frac{1}{8} \times \frac{5}{16}

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

−18×516=−5128-\frac{1}{8} \times \frac{5}{16} = -\frac{5}{128}

Since the quotient is negative, Expression B does not have a positive quotient.

Expression C: 2272 \frac{2}{7}

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

227=1672 \frac{2}{7} = \frac{16}{7}

Since the quotient is positive, Expression C has a positive quotient.

Expression D: −15-\frac{1}{5}

This expression is already in its simplest form. Since the quotient is negative, Expression D does not have a positive quotient.

Expression E: −653\frac{-6}{\frac{5}{3}}

To evaluate this expression, we need to divide the fraction by the reciprocal of the denominator.

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

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

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

Since the quotient is negative, Expression E does not have a positive quotient.

Conclusion

In conclusion, only two expressions have a positive quotient: Expression A and Expression C. Expression A is −34−23\frac{-\frac{3}{4}}{-\frac{2}{3}}, and Expression C is 2272 \frac{2}{7}. These expressions yield a positive quotient when evaluated.

Frequently Asked Questions

  • What is a positive quotient? A positive quotient is the result of a division operation that yields a positive value.
  • How do I evaluate expressions with fractions and mixed numbers? To evaluate expressions with fractions and mixed numbers, follow the order of operations (PEMDAS) and simplify the expression by canceling out common factors.
  • What is the difference between a positive and negative quotient? A positive quotient is a result of a division operation that yields a positive value, while a negative quotient is a result of a division operation that yields a negative value.

Final Thoughts

When dealing with fractions and mixed numbers, it's essential to understand the concept of a positive quotient. By following the order of operations and simplifying expressions, you can determine which expressions have a positive quotient. In this article, we explored five different expressions and determined which ones have a positive quotient.

Introduction

In our previous article, we explored five different expressions and determined which ones have a positive quotient. In this article, we'll answer some frequently asked questions about positive quotients.

Q: What is a positive quotient?

A: A positive quotient is the result of a division operation that yields a positive value.

Q: How do I evaluate expressions with fractions and mixed numbers?

A: To evaluate expressions with fractions and mixed numbers, follow the order of operations (PEMDAS) and simplify the expression by canceling out common factors.

Q: What is the difference between a positive and negative quotient?

A: A positive quotient is a result of a division operation that yields a positive value, while a negative quotient is a result of a division operation that yields a negative value.

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 I determine if a quotient is positive or negative?

A: To determine if a quotient is positive or negative, follow these steps:

  1. Evaluate the expression by simplifying the numerator and denominator.
  2. If the numerator and denominator have the same sign (both positive or both negative), the quotient is positive.
  3. If the numerator and denominator have different signs (one positive and one negative), the quotient is negative.

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 mixed number?

A: Yes, a quotient can be a mixed number. This occurs when the numerator is a multiple of the denominator plus a remainder.

Q: How do I simplify a quotient?

A: To simplify a quotient, follow these steps:

  1. Factor the numerator and denominator.
  2. Cancel out common factors.
  3. Simplify the resulting expression.

Q: Can a quotient be a decimal?

A: Yes, a quotient can be a decimal. This occurs when the numerator and denominator are both integers and the denominator is not a factor of the numerator.

Q: Can a quotient be a repeating decimal?

A: Yes, a quotient can be a repeating decimal. This occurs when the numerator and denominator are both integers and the denominator is not a factor of the numerator.

Conclusion

In conclusion, we've answered some frequently asked questions about positive quotients. By following the order of operations and simplifying expressions, you can determine which expressions have a positive quotient. We hope this article has been helpful in understanding positive quotients.

Final Thoughts

Positive quotients are an essential concept in mathematics, and understanding them can help you solve a wide range of problems. By following the steps outlined in this article, you can determine which expressions have a positive quotient and simplify complex expressions.