Certainly! Let's analyze which pairs of fractions from the list [tex]\(-\frac{1}{2}, \frac{4}{5}, -\frac{3}{8}\)[/tex] yield a negative quotient when divided. Remember, a quotient is negative when one fraction is positive, and the other is negative.
First, list all possible pairs and their quotients:
1. [tex]\(\left(-\frac{1}{2}\right) \div \left(\frac{4}{5}\right)\)[/tex]
2. [tex]\(\left(-\frac{1}{2}\right) \div \left(-\frac{3}{8}\right)\)[/tex]
3. [tex]\(\left(\frac{4}{5}\right) \div \left(-\frac{1}{2}\right)\)[/tex]
4. [tex]\(\left(\frac{4}{5}\right) \div \left(-\frac{3}{8}\right)\)[/tex]
5. [tex]\(\left(-\frac{3}{8}\right) \div \left(\frac{4}{5}\right)\)[/tex]
6. [tex]\(\left(-\frac{3}{8}\right) \div \left(-\frac{1}{2}\right)\)[/tex]
Next, evaluate the pairs to determine which ones have negative quotients:
- [tex]\(\left(-\frac{1}{2}\right) \div \left(\frac{4}{5}\right) = -\frac{1}{2} \times \frac{5}{4} = -\frac{5}{8}\)[/tex]
- [tex]\(\left(\frac{4}{5}\right) \div \left(-\frac{1}{2}\right) = \frac{4}{5} \times -2 = -\frac{8}{5}\)[/tex]
- [tex]\(\left(\frac{4}{5}\right) \div \left(-\frac{3}{8}\right) = \frac{4}{5} \times -\frac{8}{3} = -\frac{32}{15}\)[/tex]
- [tex]\(\left(-\frac{3}{8}\right) \div \left(\frac{4}{5}\right) = -\frac{3}{8} \times \frac{5}{4} = -\frac{15}{32}\)[/tex]
From the calculations above, the pairs that yield a negative quotient are:
1. [tex]\(\left(-\frac{1}{2}\right) \div \left(\frac{4}{5}\right)\)[/tex]
2. [tex]\(\left(\frac{4}{5}\right) \div \left(-\frac{1}{2}\right)\)[/tex]
3. [tex]\(\left(\frac{4}{5}\right) \div \left(-\frac{3}{8}\right)\)[/tex]
4. [tex]\(\left(-\frac{3}{8}\right) \div \left(\frac{4}{5}\right)\)[/tex]
Now, let's evaluate one of these pairs. We'll take the first one:
Evaluate [tex]\(\left(-\frac{1}{2}\right) \div \left(\frac{4}{5}\right)\)[/tex]:
- Calculate the reciprocal of [tex]\(\frac{4}{5}\)[/tex], which is [tex]\(\frac{5}{4}\)[/tex].
- Multiply by [tex]\(-\frac{1}{2}\)[/tex]: [tex]\(-\frac{1}{2} \times \frac{5}{4} = -\frac{5}{8}\)[/tex].
Therefore, the evaluated problem [tex]\(\left(-\frac{1}{2}\right) \div \left(\frac{4}{5}\right)\)[/tex] equals [tex]\(-\frac{5}{8}\)[/tex].
In all these problems, we notice that in order to obtain a negative quotient, one of the fractions must always be negative, and the other must always be positive. This is essential because the product or division of a positive and a negative number always results in a negative number.
Answer:
[tex]\[
\left(-\frac{1}{2}\right) \div \left(\frac{4}{5}\right), \quad \left(\frac{4}{5}\right) \div \left(-\frac{1}{2}\right), \quad \left(\frac{4}{5}\right) \div \left(-\frac{3}{8}\right), \quad \left(-\frac{3}{8}\right) \div \left(\frac{4}{5}\right)
\][/tex]
Evaluated problem: [tex]\(\left(-\frac{1}{2}\right) \div \left(\frac{4}{5}\right) = -\frac{5}{8}\)[/tex]
The number that must be in all of these problems is one fraction being negative, which ensures that the quotient is negative.
