Simplify : `2 div (3)/(17) "of" (2 (3)/(4)+ 3(5)/(8)) + (2)/(5) div 2 (1)/(5) + (2)/(9)`

Answer

To simplify the expression \( 2 \div \frac{3}{17} \text{ of } \left(2 \frac{3}{4} + 3 \frac{5}{8}\right) + \frac{2}{5} \div 2 \frac{1}{5} + \frac{2}{9} \), we will follow the order of operations (BODMAS/BIDMAS).

### Step-by-Step Solution:

1. **Convert Mixed Numbers to Improper Fractions:**
- Convert \( 2 \frac{3}{4} \) to an improper fraction:
\[ 2 \frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4} \]
- Convert \( 3 \frac{5}{8} \) to an improper fraction:
\[ 3 \frac{5}{8} = \frac{3 \times 8 + 5}{8} = \frac{24 + 5}{8} = \frac{29}{8} \]
- Convert \( 2 \frac{1}{5} \) to an improper fraction:
\[ 2 \frac{1}{5} = \frac{2 \times 5 + 1}{5} = \frac{10 + 1}{5} = \frac{11}{5} \]

2. **Substitute Back into the Expression:**
\[ 2 \div \frac{3}{17} \text{ of } \left(\frac{11}{4} + \frac{29}{8}\right) + \frac{2}{5} \div \frac{11}{5} + \frac{2}{9} \]

3. **Calculate the Sum Inside the Parentheses:**
- Find a common denominator for \( \frac{11}{4} \) and \( \frac{29}{8} \):
- The LCM of 4 and 8 is 8.
- Convert \( \frac{11}{4} \) to eighths:
\[ \frac{11}{4} = \frac{11 \times 2}{4 \times 2} = \frac{22}{8} \]
- Now add:
\[ \frac{22}{8} + \frac{29}{8} = \frac{22 + 29}{8} = \frac{51}{8} \]

4. **Calculate the First Part of the Expression:**
- Now substitute back:
\[ 2 \div \frac{3}{17} \text{ of } \frac{51}{8} \]
- "of" means multiplication:
\[ 2 \div \frac{3}{17} \times \frac{51}{8} \]
- Dividing by a fraction is the same as multiplying by its reciprocal:
\[ 2 \times \frac{17}{3} \times \frac{51}{8} \]

5. **Calculate the Multiplication:**
- First calculate \( 2 \times \frac{17}{3} = \frac{34}{3} \):
\[ \frac{34}{3} \times \frac{51}{8} = \frac{34 \times 51}{3 \times 8} = \frac{1734}{24} \]

6. **Simplify \( \frac{1734}{24} \):**
- Divide both numerator and denominator by 6:
\[ \frac{1734 \div 6}{24 \div 6} = \frac{289}{4} \]

7. **Calculate the Second Part of the Expression:**
- Now calculate \( \frac{2}{5} \div \frac{11}{5} \):
\[ \frac{2}{5} \times \frac{5}{11} = \frac{2 \times 5}{5 \times 11} = \frac{2}{11} \]

8. **Combine All Parts:**
- Now combine:
\[ \frac{289}{4} + \frac{2}{11} + \frac{2}{9} \]
- Find a common denominator for \( 4, 11, \) and \( 9 \):
- The LCM is \( 396 \).
- Convert each fraction:
\[ \frac{289}{4} = \frac{289 \times 99}{396} = \frac{28511}{396} \]
\[ \frac{2}{11} = \frac{2 \times 36}{396} = \frac{72}{396} \]
\[ \frac{2}{9} = \frac{2 \times 44}{396} = \frac{88}{396} \]

9. **Add the Converted Fractions:**
\[ \frac{28511 + 72 + 88}{396} = \frac{28671}{396} \]

10. **Final Simplification:**
- Check if \( \frac{28671}{396} \) can be simplified further. The GCD is 3:
\[ \frac{28671 \div 3}{396 \div 3} = \frac{9557}{132} \]

### Final Answer:
The simplified expression is \( \frac{9557}{132} \).