To solve the given expression step by step, we need to follow the order of operations carefully: parentheses first, then multiplication, subtraction, negation, and finally division.
The given expression is:
[tex]\[ -\left[\frac{4}{5} - \frac{2}{3} \left(\frac{4}{5} + \frac{1}{2}\right)\right] \div \frac{1}{5} \][/tex]
1. Calculate the expression inside the innermost parentheses:
[tex]\[ \frac{4}{5} + \frac{1}{2} \][/tex]
To add these fractions, find a common denominator, which is 10.
[tex]\[ \frac{4}{5} = \frac{8}{10} \][/tex]
[tex]\[ \frac{1}{2} = \frac{5}{10} \][/tex]
Therefore,
[tex]\[ \frac{4}{5} + \frac{1}{2} = \frac{8}{10} + \frac{5}{10} = \frac{13}{10} \][/tex]
2. Multiply this result by [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[ \frac{2}{3} \times \frac{13}{10} \][/tex]
Multiply the numerators and the denominators:
[tex]\[ \frac{2 \times 13}{3 \times 10} = \frac{26}{30} \][/tex]
Simplify the fraction:
[tex]\[ \frac{26}{30} = \frac{13}{15} \][/tex]
3. Subtract this product from [tex]\(\frac{4}{5}\)[/tex]:
[tex]\[ \frac{4}{5} - \frac{13}{15} \][/tex]
Find a common denominator, which is 15.
[tex]\[ \frac{4}{5} = \frac{12}{15} \][/tex]
Therefore,
[tex]\[ \frac{4}{5} - \frac{13}{15} = \frac{12}{15} - \frac{13}{15} = \frac{12 - 13}{15} = -\frac{1}{15} \][/tex]
4. Apply negation to this result:
[tex]\[ -\left(-\frac{1}{15}\right) = \frac{1}{15} \][/tex]
5. Divide by [tex]\(\frac{1}{5}\)[/tex]:
[tex]\[ \frac{1}{15} \div \frac{1}{5} \][/tex]
Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[ \frac{1}{15} \times \frac{5}{1} = \frac{1 \times 5}{15 \times 1} = \frac{5}{15} = \frac{1}{3} \][/tex]
The final result of the expression is [tex]\(\frac{1}{3}\)[/tex]. Hence, the correct answer is:
[tex]\[ \boxed{\frac{1}{3}} \][/tex]
Given the multiple-choice options:
A. [tex]\(-\frac{25}{3}\)[/tex]
B. [tex]\(-\frac{1}{3}\)[/tex]
C. [tex]\(\frac{1}{3}\)[/tex]
D. [tex]\(\frac{25}{3}\)[/tex]
The correct answer is:
[tex]\[ \boxed{\text{C. }\frac{1}{3}} \][/tex]
