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

Answer

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

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

- \(1\frac{5}{6} = \frac{6 \times 1 + 5}{6} = \frac{11}{6}\)
- \(2\frac{2}{3} = \frac{3 \times 2 + 2}{3} = \frac{8}{3}\)
- \(3\frac{3}{4} = \frac{4 \times 3 + 3}{4} = \frac{15}{4}\)
- \(3\frac{4}{5} = \frac{5 \times 3 + 4}{5} = \frac{19}{5}\)
- \(9\frac{1}{2} = \frac{2 \times 9 + 1}{2} = \frac{19}{2}\)

Now, the expression becomes:
\[ \frac{11}{6} + \left[\frac{8}{3} - \left\{\frac{15}{4} \left(\frac{19}{5} - :\frac{19}{2}\right)\right\}\right] \]

### Step 2: Simplify the Division
Next, we simplify the division inside the curly braces:
\[ \frac{19}{5} - :\frac{19}{2} = \frac{19}{5} \div \frac{19}{2} = \frac{19}{5} \times \frac{2}{19} = \frac{2}{5} \]

Now, substitute this back into the expression:
\[ \frac{11}{6} + \left[\frac{8}{3} - \left\{\frac{15}{4} \times \frac{2}{5}\right\}\right] \]

### Step 3: Multiply the Fractions
Now, we calculate \(\frac{15}{4} \times \frac{2}{5}\):
\[ \frac{15 \times 2}{4 \times 5} = \frac{30}{20} = \frac{3}{2} \]

Now, substitute this back into the expression:
\[ \frac{11}{6} + \left[\frac{8}{3} - \frac{3}{2}\right] \]

### Step 4: Simplify Inside the Brackets
Now we need to simplify \(\frac{8}{3} - \frac{3}{2}\). To do this, 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 5: Add the Results
Now we add \(\frac{11}{6}\) and \(\frac{7}{6}\):
\[ \frac{11}{6} + \frac{7}{6} = \frac{18}{6} = 3 \]

### Final Answer
Thus, the value of the expression is \(3\).

---