Let's solve the given fraction step by step.
Given:
[tex]\[
\frac{-4}{5 \frac{2}{5}}
\][/tex]
Firstly, let's simplify the denominator [tex]\(5 \frac{2}{5}\)[/tex].
1. Convert the mixed number [tex]\(5 \frac{2}{5}\)[/tex] into an improper fraction:
[tex]\[
5 \frac{2}{5} = 5 + \frac{2}{5}
\][/tex]
2. Rewrite [tex]\(5\)[/tex] as a fraction:
[tex]\[
5 = \frac{5 \times 5}{5} = \frac{25}{5}
\][/tex]
3. Add [tex]\(\frac{25}{5}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex]:
[tex]\[
5 \frac{2}{5} = \frac{25}{5} + \frac{2}{5} = \frac{27}{5}
\][/tex]
So now, the original expression becomes:
[tex]\[
\frac{-4}{\frac{27}{5}}
\][/tex]
When dividing by a fraction, we multiply by its reciprocal:
[tex]\[
\frac{-4}{\frac{27}{5}} = -4 \times \frac{5}{27}
\][/tex]
Multiply the fractions:
[tex]\[
-4 \times \frac{5}{27} = -\frac{4 \times 5}{27} = -\frac{20}{27}
\][/tex]
So, the simplified value of the given expression is:
[tex]\[
-\frac{20}{27}
\][/tex]
Therefore, the correct answer from the provided options is:
[tex]\(-\frac{20}{27}\)[/tex].
