Simplify `(10^(2) " of " ((1)/(5))^(3) div (1)/(4) xx 4 - (2)/(5) " of " 15)/((4)/(5)(5 div 5 " of " 12 + (1)/(6)))`

Options

A. `-(65)/(73)`
B. 34
C. `-(147)/(73)`
D. `-(78)/(73)`

Answer

To simplify the expression

\[ \frac{10^2 \text{ of } \left(\frac{1}{5}\right)^3 \div \frac{1}{4} \times 4 - \frac{2}{5} \text{ of } 15}{\left(\frac{4}{5}\right) \left(5 \div 5 \text{ of } 12 + \frac{1}{6}\right)} \]

we will follow these steps:

### Step 1: Simplify the Numerator

1. **Calculate \(10^2\)**:
\[ 10^2 = 100 \]

2. **Calculate \(\left(\frac{1}{5}\right)^3\)**:
\[ \left(\frac{1}{5}\right)^3 = \frac{1}{125} \]

3. **Multiply \(100\) by \(\frac{1}{125}\)**:
\[ 100 \times \frac{1}{125} = \frac{100}{125} = \frac{4}{5} \]

4. **Calculate \(\frac{2}{5} \text{ of } 15\)**:
\[ \frac{2}{5} \times 15 = \frac{30}{5} = 6 \]

5. **Combine the results in the numerator**:
\[ \frac{4}{5} \div \frac{1}{4} \times 4 - 6 \]
First, calculate \(\frac{4}{5} \div \frac{1}{4}\):
\[ \frac{4}{5} \div \frac{1}{4} = \frac{4}{5} \times 4 = \frac{16}{5} \]
Now, substitute back:
\[ \frac{16}{5} - 6 = \frac{16}{5} - \frac{30}{5} = \frac{16 - 30}{5} = \frac{-14}{5} \]

### Step 2: Simplify the Denominator

1. **Calculate \(5 \div 5 \text{ of } 12\)**:
\[ 5 \div 5 = 1 \quad \text{so} \quad 1 \text{ of } 12 = 12 \]

2. **Combine with \(\frac{1}{6}\)**:
\[ 12 + \frac{1}{6} = \frac{72}{6} + \frac{1}{6} = \frac{73}{6} \]

3. **Multiply by \(\frac{4}{5}\)**:
\[ \frac{4}{5} \times \frac{73}{6} = \frac{292}{30} = \frac{146}{15} \]

### Step 3: Combine the Numerator and Denominator

Now, we have:
\[ \frac{\frac{-14}{5}}{\frac{146}{15}} \]

To divide fractions, multiply by the reciprocal:
\[ \frac{-14}{5} \times \frac{15}{146} = \frac{-14 \times 15}{5 \times 146} \]

### Step 4: Simplify the Final Expression

1. **Calculate the numerator**:
\[ -14 \times 15 = -210 \]

2. **Calculate the denominator**:
\[ 5 \times 146 = 730 \]

3. **Final fraction**:
\[ \frac{-210}{730} \]

4. **Simplify**:
\[ \frac{-21}{73} \]

### Final Answer

The simplified expression is:
\[ \frac{-21}{73} \]