Simplify `((3+3/2 -: 6 1/2 xx 1/3) of (4 1/3)) /((-8/3-:8))`

Options

A. 10
B. `-30/13`
C. `-10`
D. `40/3`

Answer

To simplify the expression `((3 + 3/2 -: 6 1/2 xx 1/3) of (4 1/3)) /((-8/3 -: 8))`, we will follow the steps outlined below:

### Step-by-Step Solution:

1. **Convert Mixed Numbers to Improper Fractions**:
- Convert \(6 \frac{1}{2}\) to an improper fraction:
\[ 6 \frac{1}{2} = \frac{13}{2} \]
- Convert \(4 \frac{1}{3}\) to an improper fraction:
\[ 4 \frac{1}{3} = \frac{13}{3} \]

2. **Rewrite the Expression**:
- The expression now looks like:
\[ \left(3 + \frac{3}{2} \div \left(\frac{13}{2} \times \frac{1}{3}\right)\right) \times \frac{13}{3} \]

3. **Calculate the Division in the Parentheses**:
- First, calculate \(\frac{13}{2} \times \frac{1}{3}\):
\[ \frac{13}{2} \times \frac{1}{3} = \frac{13 \times 1}{2 \times 3} = \frac{13}{6} \]
- Now, calculate \(\frac{3}{2} \div \frac{13}{6}\):
\[ \frac{3}{2} \div \frac{13}{6} = \frac{3}{2} \times \frac{6}{13} = \frac{3 \times 6}{2 \times 13} = \frac{18}{26} = \frac{9}{13} \]

4. **Add the Result to 3**:
- Now, add \(3\) to \(\frac{9}{13}\):
\[ 3 = \frac{39}{13} \quad \text{(convert 3 to a fraction with a common denominator)} \]
\[ \frac{39}{13} + \frac{9}{13} = \frac{48}{13} \]

5. **Multiply by \(\frac{13}{3}\)**:
- Now multiply \(\frac{48}{13}\) by \(\frac{13}{3}\):
\[ \frac{48}{13} \times \frac{13}{3} = \frac{48 \times 13}{13 \times 3} = \frac{48}{3} = 16 \]

6. **Calculate the Denominator**:
- Now, simplify the denominator \(-\frac{8}{3} \div 8\):
\[ -\frac{8}{3} \div 8 = -\frac{8}{3} \times \frac{1}{8} = -\frac{8}{24} = -\frac{1}{3} \]

7. **Final Division**:
- Now, divide the result from the numerator by the result from the denominator:
\[ \frac{16}{-\frac{1}{3}} = 16 \times -3 = -48 \]

### Final Answer:
The simplified value of the expression is \(-48\).