`5(5)/(6) + [2(2)/(3) - [3(3)/(4) (3(4)/(5) div 9(1)/(2))]]`

Options

A. 7
B. `22/3`
C. `(44)/(7)`
D. `(43)/(6)`

Answer

To solve the expression \( 5 \frac{5}{6} + \left[ 2 \frac{2}{3} - \left[ 3 \frac{3}{4} \left( \frac{3 \frac{4}{5}}{9 \frac{1}{2}} \right) \right] \right] \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \( 5 \frac{5}{6} \):
\[ 5 \frac{5}{6} = \frac{5 \times 6 + 5}{6} = \frac{30 + 5}{6} = \frac{35}{6} \]

2. Convert \( 2 \frac{2}{3} \):
\[ 2 \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3} \]

3. Convert \( 3 \frac{3}{4} \):
\[ 3 \frac{3}{4} = \frac{3 \times 4 + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4} \]

4. Convert \( 3 \frac{4}{5} \):
\[ 3 \frac{4}{5} = \frac{3 \times 5 + 4}{5} = \frac{15 + 4}{5} = \frac{19}{5} \]

5. Convert \( 9 \frac{1}{2} \):
\[ 9 \frac{1}{2} = \frac{9 \times 2 + 1}{2} = \frac{18 + 1}{2} = \frac{19}{2} \]

### Step 2: Rewrite the Expression
Now, we can rewrite the expression using the improper fractions:
\[ \frac{35}{6} + \left[ \frac{8}{3} - \left[ \frac{15}{4} \left( \frac{19}{5} \div \frac{19}{2} \right) \right] \right] \]

### Step 3: Solve the Division Inside the Brackets
To solve \( \frac{19}{5} \div \frac{19}{2} \), we multiply by the reciprocal:
\[ \frac{19}{5} \div \frac{19}{2} = \frac{19}{5} \times \frac{2}{19} = \frac{2}{5} \]

### Step 4: Substitute Back into the Expression
Now substitute back:
\[ \frac{35}{6} + \left[ \frac{8}{3} - \left[ \frac{15}{4} \times \frac{2}{5} \right] \right] \]

### Step 5: Solve the Multiplication
Now calculate \( \frac{15}{4} \times \frac{2}{5} \):
\[ \frac{15 \times 2}{4 \times 5} = \frac{30}{20} = \frac{3}{2} \]

### Step 6: Substitute and Simplify
Now substitute this back into the expression:
\[ \frac{35}{6} + \left[ \frac{8}{3} - \frac{3}{2} \right] \]

### Step 7: Find a Common Denominator
To subtract \( \frac{8}{3} - \frac{3}{2} \), we need a common denominator, which is 6:
\[ \frac{8}{3} = \frac{16}{6}, \quad \frac{3}{2} = \frac{9}{6} \]
So,
\[ \frac{8}{3} - \frac{3}{2} = \frac{16}{6} - \frac{9}{6} = \frac{7}{6} \]

### Step 8: Add the Fractions
Now add \( \frac{35}{6} + \frac{7}{6} \):
\[ \frac{35 + 7}{6} = \frac{42}{6} = 7 \]

### Final Answer
The final answer is:
\[ \boxed{7} \]