Calculate the value of `2/5 + 2 4/9 div [(7 5/12 - 5 3/4) div 22 1/2 + 10 xx 5/18] - 4/5`

Answer

To solve the expression \( \frac{2}{5} + 2 \frac{4}{9} \div \left[ \left( 7 \frac{5}{12} - 5 \frac{3}{4} \right) \div 22 \frac{1}{2} + 10 \times \frac{5}{18} \right] - \frac{4}{5} \), we will follow the order of operations (BODMAS/BIDMAS).

### Step-by-Step Solution:

1. **Convert Mixed Numbers to Improper Fractions**:
- \( 2 \frac{4}{9} = \frac{2 \times 9 + 4}{9} = \frac{18 + 4}{9} = \frac{22}{9} \)
- \( 7 \frac{5}{12} = \frac{7 \times 12 + 5}{12} = \frac{84 + 5}{12} = \frac{89}{12} \)
- \( 5 \frac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{20 + 3}{4} = \frac{23}{4} \)
- \( 22 \frac{1}{2} = \frac{22 \times 2 + 1}{2} = \frac{44 + 1}{2} = \frac{45}{2} \)

2. **Calculate the Expression Inside the Bracket**:
- First, solve \( 7 \frac{5}{12} - 5 \frac{3}{4} \):
\[ \frac{89}{12} - \frac{23}{4} = \frac{89}{12} - \frac{69}{12} = \frac{20}{12} = \frac{5}{3} \]
- Now, substitute this back into the expression:
\[ \left( \frac{5}{3} \div \frac{45}{2} + 10 \times \frac{5}{18} \right) \]

3. **Calculate Each Part**:
- For \( \frac{5}{3} \div \frac{45}{2} \):
\[ \frac{5}{3} \times \frac{2}{45} = \frac{10}{135} = \frac{2}{27} \]
- For \( 10 \times \frac{5}{18} \):
\[ 10 \times \frac{5}{18} = \frac{50}{18} = \frac{25}{9} \]

4. **Combine the Results**:
- Now, add \( \frac{2}{27} + \frac{25}{9} \):
- Convert \( \frac{25}{9} \) to have a common denominator of 27:
\[ \frac{25}{9} = \frac{25 \times 3}{9 \times 3} = \frac{75}{27} \]
- Now add:
\[ \frac{2}{27} + \frac{75}{27} = \frac{77}{27} \]

5. **Substitute Back into the Main Expression**:
- Now replace the bracket in the original expression:
\[ \frac{2}{5} + \frac{22}{9} \div \frac{77}{27} - \frac{4}{5} \]

6. **Calculate the Division**:
- For \( \frac{22}{9} \div \frac{77}{27} \):
\[ \frac{22}{9} \times \frac{27}{77} = \frac{594}{693} = \frac{66}{77} = \frac{6}{7} \]

7. **Combine All Parts**:
- Now the expression is:
\[ \frac{2}{5} + \frac{6}{7} - \frac{4}{5} \]
- Combine \( \frac{2}{5} - \frac{4}{5} = -\frac{2}{5} \):
\[ -\frac{2}{5} + \frac{6}{7} \]

8. **Find a Common Denominator**:
- The common denominator for 5 and 7 is 35:
\[ -\frac{2}{5} = -\frac{14}{35}, \quad \frac{6}{7} = \frac{30}{35} \]
- Now combine:
\[ -\frac{14}{35} + \frac{30}{35} = \frac{16}{35} \]

### Final Answer:
\[ \frac{16}{35} \]