`5 5/6 + [ 2 2/3 - { 3 2/3 - {3 3/4 (3 4/5 -: 9 1/2)} ]` is equal to:

Options

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

Answer

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

### Step 1: Convert Mixed Numbers to Improper Fractions
Convert all mixed numbers into improper fractions for easier calculations.

- \( 5 \frac{5}{6} = \frac{35}{6} \)
- \( 2 \frac{2}{3} = \frac{8}{3} \)
- \( 3 \frac{2}{3} = \frac{11}{3} \)
- \( 3 \frac{3}{4} = \frac{15}{4} \)
- \( 3 \frac{4}{5} = \frac{19}{5} \)

### Step 2: Solve the Innermost Bracket
Calculate \( 3 \frac{4}{5} - \frac{9}{2} \):

Convert \( \frac{9}{2} \) to a fraction with a denominator of 10:
- \( \frac{9}{2} = \frac{45}{10} \)

Convert \( 3 \frac{4}{5} \) to have the same denominator:
- \( 3 \frac{4}{5} = \frac{19}{5} = \frac{38}{10} \)

Now, subtract:
\[ \frac{38}{10} - \frac{45}{10} = \frac{-7}{10} \]

### Step 3: Multiply by \( 3 \frac{3}{4} \)
Now calculate \( 3 \frac{3}{4} \left( -\frac{7}{10} \right) \):
\[ \frac{15}{4} \times -\frac{7}{10} = -\frac{105}{40} = -\frac{21}{8} \]

### Step 4: Solve the Next Bracket
Now calculate \( 3 \frac{2}{3} - \left( -\frac{21}{8} \right) \):
\[ \frac{11}{3} + \frac{21}{8} \]

Find a common denominator (24):
- \( \frac{11}{3} = \frac{88}{24} \)
- \( \frac{21}{8} = \frac{63}{24} \)

Now add:
\[ \frac{88}{24} + \frac{63}{24} = \frac{151}{24} \]

### Step 5: Solve the Outer Bracket
Now calculate \( 2 \frac{2}{3} - \frac{151}{24} \):
\[ \frac{8}{3} - \frac{151}{24} \]

Convert \( \frac{8}{3} \) to have a common denominator (24):
- \( \frac{8}{3} = \frac{64}{24} \)

Now subtract:
\[ \frac{64}{24} - \frac{151}{24} = \frac{-87}{24} \]

### Step 6: Final Addition
Now add \( 5 \frac{5}{6} + \left( -\frac{87}{24} \right) \):
\[ \frac{35}{6} + \left( -\frac{87}{24} \right) \]

Convert \( \frac{35}{6} \) to have a common denominator (24):
- \( \frac{35}{6} = \frac{140}{24} \)

Now subtract:
\[ \frac{140}{24} - \frac{87}{24} = \frac{53}{24} \]

### Final Answer
Thus, the final answer is:
\[ \frac{53}{24} \]