Identifying Expressions With Positive Quotients A Comprehensive Guide

In the realm of mathematics, understanding the nuances of division is crucial. One key aspect is determining the sign of the quotient, which is the result of a division operation. When dividing numbers, the signs of the dividend (the number being divided) and the divisor (the number dividing) play a significant role in determining whether the quotient is positive or negative. In this comprehensive exploration, we will delve into the intricacies of dividing fractions and mixed numbers, focusing on how to identify expressions with positive quotients. We will analyze several examples, breaking down the steps involved in each calculation and highlighting the rules governing the signs of quotients. By the end of this journey, you will have a solid grasp of how to confidently determine whether an expression involving division yields a positive result.

Understanding the Rules of Signs in Division

The foundation for determining the sign of a quotient lies in understanding the fundamental rules of signs in division. These rules are simple yet powerful:

• Positive ÷ Positive = Positive: When a positive number is divided by another positive number, the result is always positive.
• Negative ÷ Negative = Positive: When a negative number is divided by another negative number, the result is also positive.
• Positive ÷ Negative = Negative: When a positive number is divided by a negative number, the result is negative.
• Negative ÷ Positive = Negative: When a negative number is divided by a positive number, the result is negative.

These rules can be summarized concisely: dividing numbers with the same sign (both positive or both negative) results in a positive quotient, while dividing numbers with different signs (one positive and one negative) results in a negative quotient. This principle is paramount when working with fractions and mixed numbers, as we will see in the examples below. Mastering these rules is not just about getting the right answer; it’s about developing a deeper intuitive understanding of how numbers interact.

Analyzing the Expressions for Positive Quotients

Now, let's apply these rules to the given expressions and identify the one with a positive quotient. We will meticulously examine each expression, step by step, to ensure clarity and comprehension.

Expression 1: $\frac{-\frac{3}{4}}{-\frac{2}{3}}$

This expression involves dividing a negative fraction by another negative fraction. To perform this division, we multiply the first fraction by the reciprocal of the second fraction. Remember, the reciprocal of a fraction is obtained by swapping the numerator and the denominator.

So, we have:

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

When multiplying fractions, we multiply the numerators together and the denominators together. In this case, we also need to consider the signs.

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

Since we are dividing a negative number by a negative number, the result is positive, as confirmed by our calculation. The quotient $\frac{9}{8}$ is indeed positive. Understanding why this is positive is crucial: the two negatives cancel each other out, resulting in a positive outcome. This is a fundamental concept in dealing with signed numbers and fractions.

Expression 2: $\frac{-\frac{1}{8}}{3 \frac{1}{5}}$

In this expression, we are dividing a negative fraction by a positive mixed number. First, we need to convert the mixed number into an improper fraction. A mixed number consists of a whole number and a fraction. To convert it to an improper fraction, we multiply the whole number by the denominator of the fraction, add the numerator, and then place the result over the original denominator.

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

Now, we can rewrite the expression as:

$\frac{-\frac{1}{8}}{3 \frac{1}{5}} = \frac{-\frac{1}{8}}{\frac{16}{5}} = -\frac{1}{8} \div \frac{16}{5}$

To divide fractions, we multiply by the reciprocal of the divisor:

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

Here, we are dividing a negative number by a positive number, so the result is negative. The quotient $-\frac{5}{128}$ is negative. The key takeaway here is the interplay between the signs: a negative divided by a positive always yields a negative result. This principle is consistent across all forms of division, whether dealing with integers, fractions, or mixed numbers.

Expression 3: $\frac{2 \frac{2}{7}}{-\frac{1}{5}}$

This expression involves dividing a positive mixed number by a negative fraction. As before, we first convert the mixed number into an improper fraction.

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

Now, we can rewrite the expression as:

$\frac{2 \frac{2}{7}}{-\frac{1}{5}} = \frac{\frac{16}{7}}{-\frac{1}{5}} = \frac{16}{7} \div -\frac{1}{5}$

To divide fractions, we multiply by the reciprocal of the divisor:

$\frac{16}{7} \div -\frac{1}{5} = \frac{16}{7} \times -\frac{5}{1} = -\frac{16 \times 5}{7 \times 1} = -\frac{80}{7}$

In this case, we are dividing a positive number by a negative number, resulting in a negative quotient. The result $-\frac{80}{7}$ is negative. This reinforces the rule that different signs in division lead to a negative result. The negative sign is crucial; without it, the answer would be incorrect.

Expression 4: $\frac{-6}{\frac{5}{3}}$

This expression involves dividing a negative integer by a positive fraction. To perform this division, we can rewrite the integer as a fraction with a denominator of 1:

$\frac{-6}{\frac{5}{3}} = \frac{-\frac{6}{1}}{\frac{5}{3}} = -\frac{6}{1} \div \frac{5}{3}$

Now, we multiply by the reciprocal of the divisor:

$- \frac{6}{1} \div \frac{5}{3} = -\frac{6}{1} \times \frac{3}{5} = -\frac{6 \times 3}{1 \times 5} = -\frac{18}{5}$

Here, we are dividing a negative number by a positive number, so the result is negative. The quotient $-\frac{18}{5}$ is negative. This example reiterates the consistency of the sign rules in division, applicable even when integers and fractions are combined.

Identifying the Expression with a Positive Quotient

After analyzing all four expressions, we can clearly identify the one with a positive quotient. Expression 1, $\frac{-\frac{3}{4}}{-\frac{2}{3}}$, resulted in a positive quotient of $\frac{9}{8}$. The other expressions yielded negative quotients due to the division of numbers with different signs. This exercise demonstrates the importance of not only performing the calculations correctly but also understanding the underlying principles that govern the signs of quotients.

Conclusion: Mastering the Art of Division with Fractions and Mixed Numbers

In conclusion, determining whether an expression has a positive quotient hinges on a clear understanding of the rules of signs in division. Dividing numbers with the same sign (both positive or both negative) results in a positive quotient, while dividing numbers with different signs (one positive and one negative) results in a negative quotient. By meticulously applying these rules and following the steps for dividing fractions and mixed numbers, we can confidently identify expressions with positive quotients. This skill is not just a mathematical exercise; it’s a crucial foundation for more advanced concepts in algebra and beyond. The ability to quickly and accurately determine the sign of a quotient is a powerful tool in any mathematical endeavor. Remember, practice makes perfect, and the more you work with these concepts, the more intuitive they will become.

Therefore, the expression with a positive quotient is:

$\frac{-\frac{3}{4}}{-\frac{2}{3}}$