The value of ` 1 (2)/(3) +[16(1)/(2) - { 1 (1)/(10) (3(1)/(3) +5)}] 1/4 " of " 2 (1)/(2) xx 4 (4)/(5)` is :

Options

A. `1 (2)/(5)`
B. `1(4)/(5)`
C. `23(2)/(3)`
D. `3(1)/(5)`

Answer

To solve the expression \( 1 \frac{2}{3} + [16 \frac{1}{2} - \{ 1 \frac{1}{10} (3 \frac{1}{3} + 5)\}] \frac{1}{4} \text{ of } 2 \frac{1}{2} \times 4 \frac{4}{5} \), we will follow the order of operations (BODMAS/BIDMAS rules).

### Step-by-step Solution:

1. **Convert Mixed Numbers to Improper Fractions**:
- \( 1 \frac{2}{3} = \frac{5}{3} \)
- \( 16 \frac{1}{2} = \frac{33}{2} \)
- \( 1 \frac{1}{10} = \frac{11}{10} \)
- \( 3 \frac{1}{3} = \frac{10}{3} \)
- \( 2 \frac{1}{2} = \frac{5}{2} \)
- \( 4 \frac{4}{5} = \frac{24}{5} \)

So the expression becomes:
\[ \frac{5}{3} + \left[\frac{33}{2} - \left\{ \frac{11}{10} \left(\frac{10}{3} + 5\right) \right\}\right] \frac{1}{4} \text{ of } \frac{5}{2} \times \frac{24}{5} \]

2. **Calculate Inside the Braces**:
- First, calculate \( \frac{10}{3} + 5 = \frac{10}{3} + \frac{15}{3} = \frac{25}{3} \).
- Now, calculate \( \frac{11}{10} \times \frac{25}{3} = \frac{275}{30} = \frac{55}{6} \).

The expression now looks like:
\[ \frac{5}{3} + \left[\frac{33}{2} - \frac{55}{6}\right] \frac{1}{4} \text{ of } \frac{5}{2} \times \frac{24}{5} \]

3. **Calculate the Bracket**:
- Find a common denominator for \( \frac{33}{2} \) and \( \frac{55}{6} \). The LCM of 2 and 6 is 6.
- Convert \( \frac{33}{2} = \frac{99}{6} \).
- Now calculate \( \frac{99}{6} - \frac{55}{6} = \frac{44}{6} = \frac{22}{3} \).

The expression now is:
\[ \frac{5}{3} + \frac{22}{3} \times \frac{1}{4} \text{ of } \frac{5}{2} \times \frac{24}{5} \]

4. **Calculate the "of" part**:
- \( \frac{1}{4} \text{ of } \frac{5}{2} = \frac{1}{4} \times \frac{5}{2} = \frac{5}{8} \).
- Now calculate \( \frac{5}{8} \times \frac{24}{5} = \frac{24}{8} = 3 \).

The expression now simplifies to:
\[ \frac{5}{3} + \frac{22}{3} \times 3 \]

5. **Final Calculation**:
- Calculate \( \frac{22}{3} \times 3 = 22 \).
- Now combine \( \frac{5}{3} + 22 = \frac{5}{3} + \frac{66}{3} = \frac{71}{3} \).

6. **Convert to Mixed Number**:
- \( \frac{71}{3} = 23 \frac{2}{3} \).

### Final Answer:
The value of the expression is \( 23 \frac{2}{3} \).