To solve the expression [tex]\(18 \frac{1}{5}-\left(-22 \frac{2}{5}\right)-\left(-40 \frac{1}{5}\right)\)[/tex], let's break it down step-by-step:
1. Convert the mixed numbers to improper fractions:
- For [tex]\(18 \frac{1}{5}\)[/tex]:
[tex]\[
18 \frac{1}{5} = 18 + \frac{1}{5} = \frac{90}{5} + \frac{1}{5} = \frac{91}{5}
\][/tex]
- For [tex]\(-22 \frac{2}{5}\)[/tex]:
[tex]\[
-22 \frac{2}{5} = -22 + \frac{-2}{5} = \frac{-110}{5} + \frac{-2}{5} = \frac{-112}{5}
\][/tex]
- For [tex]\(-40 \frac{1}{5}\)[/tex]:
[tex]\[
-40 \frac{1}{5} = -40 + \frac{-1}{5} = \frac{-200}{5} + \frac{-1}{5} = \frac{-201}{5}
\][/tex]
2. Rewrite the expression with improper fractions:
[tex]\[
\frac{91}{5} - \left(\frac{-112}{5}\right) - \left(\frac{-201}{5}\right)
\][/tex]
3. Simplify the expression by removing the parentheses:
[tex]\[
\frac{91}{5} - \left(\frac{-112}{5}\right) - \left(\frac{-201}{5}\right) = \frac{91}{5} + \frac{112}{5} + \frac{201}{5}
\][/tex]
4. Combine the fractions (since the denominators are the same, combine the numerators):
[tex]\[
\frac{91 + 112 + 201}{5}
\][/tex]
5. Calculate the sum in the numerator:
[tex]\[
91 + 112 + 201 = 404
\][/tex]
6. Write the result as a single fraction:
[tex]\[
\frac{404}{5}
\][/tex]
7. Simplify the fraction to a mixed number or decimal form:
[tex]\[
\frac{404}{5} = 80.8
\][/tex]
The correct expression equivalent to [tex]\(18 \frac{1}{5}-\left(-22 \frac{2}{5}\right)-\left(-40 \frac{1}{5}\right)\)[/tex] is:
[tex]\[
18 \frac{1}{5} + 22 \frac{2}{5} + 40 \frac{1}{5}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
Note that we achieved 80.8 as the final result, confirming that the operations are consistent with our step-by-step breakdown and ensuring [tex]\(18 \frac{1}{5} + 22 \frac{2}{5} + 40 \frac{1}{5}\)[/tex] is indeed the correct equivalent expression.
