Answer :
To solve the expression [tex]\( -\left[\frac{4}{5}-\frac{2}{3}\left(\frac{4}{5}+\frac{1}{2}\right)\right] \div \frac{1}{5} \)[/tex], let's proceed step-by-step:
### Step 1: Simplify the innermost parentheses
We start with the expression inside the parentheses:
[tex]\[ \frac{4}{5} + \frac{1}{2} \][/tex]
To add these fractions, we need a common denominator. The least common denominator (LCD) of 5 and 2 is 10. Convert both fractions:
[tex]\[ \frac{4}{5} = \frac{8}{10} \][/tex]
[tex]\[ \frac{1}{2} = \frac{5}{10} \][/tex]
Now add them:
[tex]\[ \frac{8}{10} + \frac{5}{10} = \frac{13}{10} \][/tex]
### Step 2: Multiply by [tex]\(\frac{2}{3}\)[/tex]
Next, multiply the 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 [tex]\(\frac{26}{30}\)[/tex] by dividing by their GCD which is 2:
[tex]\[ \frac{26 \div 2}{30 \div 2} = \frac{13}{15} \][/tex]
### Step 3: Subtract from [tex]\(\frac{4}{5}\)[/tex]
Now subtract [tex]\(\frac{13}{15}\)[/tex] from [tex]\(\frac{4}{5}\)[/tex].
First, we need a common denominator. The LCD of 5 and 15 is 15:
[tex]\[ \frac{4}{5} = \frac{12}{15} \][/tex]
So the expression becomes:
[tex]\[ \frac{12}{15} - \frac{13}{15} = \frac{12 - 13}{15} = \frac{-1}{15} \][/tex]
### Step 4: Negate the result
Now we negate the result of the subtraction:
[tex]\[ -\left(\frac{-1}{15}\right) = \frac{1}{15} \][/tex]
### Step 5: Divide by [tex]\(\frac{1}{5}\)[/tex]
Finally, we divide [tex]\(\frac{1}{15}\)[/tex] by [tex]\(\frac{1}{5}\)[/tex]:
[tex]\[ \frac{1}{15} \div \frac{1}{5} = \frac{1}{15} \times \frac{5}{1} = \frac{1 \times 5}{15 \times 1} = \frac{5}{15} \][/tex]
Simplify [tex]\(\frac{5}{15}\)[/tex] by dividing by 5:
[tex]\[ \frac{5 \div 5}{15 \div 5} = \frac{1}{3} \][/tex]
Therefore, the final result is:
[tex]\[ \boxed{\frac{1}{3}} \][/tex]
So, the correct answer is:
[tex]\[ C. \frac{1}{3} \][/tex]
### Step 1: Simplify the innermost parentheses
We start with the expression inside the parentheses:
[tex]\[ \frac{4}{5} + \frac{1}{2} \][/tex]
To add these fractions, we need a common denominator. The least common denominator (LCD) of 5 and 2 is 10. Convert both fractions:
[tex]\[ \frac{4}{5} = \frac{8}{10} \][/tex]
[tex]\[ \frac{1}{2} = \frac{5}{10} \][/tex]
Now add them:
[tex]\[ \frac{8}{10} + \frac{5}{10} = \frac{13}{10} \][/tex]
### Step 2: Multiply by [tex]\(\frac{2}{3}\)[/tex]
Next, multiply the 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 [tex]\(\frac{26}{30}\)[/tex] by dividing by their GCD which is 2:
[tex]\[ \frac{26 \div 2}{30 \div 2} = \frac{13}{15} \][/tex]
### Step 3: Subtract from [tex]\(\frac{4}{5}\)[/tex]
Now subtract [tex]\(\frac{13}{15}\)[/tex] from [tex]\(\frac{4}{5}\)[/tex].
First, we need a common denominator. The LCD of 5 and 15 is 15:
[tex]\[ \frac{4}{5} = \frac{12}{15} \][/tex]
So the expression becomes:
[tex]\[ \frac{12}{15} - \frac{13}{15} = \frac{12 - 13}{15} = \frac{-1}{15} \][/tex]
### Step 4: Negate the result
Now we negate the result of the subtraction:
[tex]\[ -\left(\frac{-1}{15}\right) = \frac{1}{15} \][/tex]
### Step 5: Divide by [tex]\(\frac{1}{5}\)[/tex]
Finally, we divide [tex]\(\frac{1}{15}\)[/tex] by [tex]\(\frac{1}{5}\)[/tex]:
[tex]\[ \frac{1}{15} \div \frac{1}{5} = \frac{1}{15} \times \frac{5}{1} = \frac{1 \times 5}{15 \times 1} = \frac{5}{15} \][/tex]
Simplify [tex]\(\frac{5}{15}\)[/tex] by dividing by 5:
[tex]\[ \frac{5 \div 5}{15 \div 5} = \frac{1}{3} \][/tex]
Therefore, the final result is:
[tex]\[ \boxed{\frac{1}{3}} \][/tex]
So, the correct answer is:
[tex]\[ C. \frac{1}{3} \][/tex]
