Alright, let's solve the given expression step-by-step to understand how we arrive at the correct answer.
The expression we want to solve is:
[tex]\[
\left[\frac{4}{5}-\frac{2}{3}\left(\frac{4}{5}+\frac{1}{2}\right) \div \frac{1}{5}\right].
\][/tex]
### Step 1: Calculate the addition inside the parentheses
First, we need to add the fractions inside the parentheses:
[tex]\[
\frac{4}{5} + \frac{1}{2}.
\][/tex]
To add these fractions, we need a common denominator. The least common multiple of 5 and 2 is 10. Converting both fractions to have a denominator of 10:
[tex]\[
\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10},
\][/tex]
[tex]\[
\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}.
\][/tex]
Add the two fractions:
[tex]\[
\frac{8}{10} + \frac{5}{10} = \frac{13}{10}.
\][/tex]
### Step 2: Compute the multiplication inside the parentheses
Next, we need to multiply [tex]\(\frac{2}{3}\)[/tex] by [tex]\(\frac{13}{10}\)[/tex]:
[tex]\[
\frac{2}{3} \times \frac{13}{10} = \frac{2 \times 13}{3 \times 10} = \frac{26}{30} = \frac{13}{15}.
\][/tex]
### Step 3: Perform the division inside the expression
We now need to divide [tex]\(\frac{13}{15}\)[/tex] by [tex]\(\frac{1}{5}\)[/tex]. Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[
\frac{13}{15} \div \frac{1}{5} = \frac{13}{15} \times \frac{5}{1} = \frac{13 \times 5}{15 \times 1} = \frac{65}{15} = \frac{13}{3}.
\][/tex]
### Step 4: Subtract from [tex]\(\frac{4}{5}\)[/tex]
Finally, we subtract [tex]\(\frac{13}{3}\)[/tex] from [tex]\(\frac{4}{5}\)[/tex]:
We need a common denominator to perform the subtraction. The least common multiple of 5 and 3 is 15. Converting both fractions to have a denominator of 15:
[tex]\[
\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15},
\][/tex]
[tex]\[
\frac{13}{3} = \frac{13 \times 5}{3 \times 5} = \frac{65}{15}.
\][/tex]
Subtract the two fractions:
[tex]\[
\frac{12}{15} - \frac{65}{15} = \frac{12 - 65}{15} = \frac{-53}{15}.
\][/tex]
This simplifies to:
[tex]\[
-3.533333333333333.
\][/tex]
Thus, the final result of the given expression is:
[tex]\[
\boxed{-3.533333333333333}.
\][/tex]
