The value of `((1)/(2)" of " 1(1)/(2))div(3(1)/(2)-1(1)/(4)) "of" 1(1)/(2)-1(1)/(2)div2(1)/(4)+1(1)/(3)` is :

Options

A. `(2)/(9)`
B. `(8)/(9)`
C. `(2)/(3)`
D. `(4)/(3)`

Answer

To solve the expression `((1)/(2) of 1(1)/(2)) div (3(1)/(2) - 1(1)/(4)) of 1(1)/(2) - 1(1)/(2) div 2(1)/(4) + 1(1)/(3)`, we will follow the order of operations, often referred to as BODMAS/BIDMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction).

### Step-by-Step Solution:

1. **Convert Mixed Numbers to Improper Fractions:**
- \( 1(1/2) = \frac{3}{2} \)
- \( 2(1/4) = \frac{9}{4} \)
- \( 1(1/3) = \frac{4}{3} \)

The expression now looks like:
\[ \left(\frac{1}{2} \cdot \frac{3}{2}\right) \div \left(\frac{3}{2} - \frac{1}{4}\right) \cdot \frac{3}{2} - \frac{3}{2} \div \frac{9}{4} + \frac{4}{3} \]

2. **Calculate the First Part:**
- Calculate \( \frac{1}{2} \cdot \frac{3}{2} = \frac{3}{4} \)

Now the expression is:
\[ \frac{3}{4} \div \left(\frac{3}{2} - \frac{1}{4}\right) \cdot \frac{3}{2} - \frac{3}{2} \div \frac{9}{4} + \frac{4}{3} \]

3. **Calculate the Denominator of the First Division:**
- Find a common denominator for \( \frac{3}{2} - \frac{1}{4} \):
- The common denominator is 4.
- Convert \( \frac{3}{2} = \frac{6}{4} \)
- Now, \( \frac{6}{4} - \frac{1}{4} = \frac{5}{4} \)

The expression now looks like:
\[ \frac{3}{4} \div \frac{5}{4} \cdot \frac{3}{2} - \frac{3}{2} \div \frac{9}{4} + \frac{4}{3} \]

4. **Perform the Division:**
- \( \frac{3}{4} \div \frac{5}{4} = \frac{3}{4} \cdot \frac{4}{5} = \frac{3}{5} \)

The expression now simplifies to:
\[ \frac{3}{5} \cdot \frac{3}{2} - \frac{3}{2} \div \frac{9}{4} + \frac{4}{3} \]

5. **Calculate the Multiplication:**
- \( \frac{3}{5} \cdot \frac{3}{2} = \frac{9}{10} \)

The expression now looks like:
\[ \frac{9}{10} - \frac{3}{2} \div \frac{9}{4} + \frac{4}{3} \]

6. **Perform the Division:**
- \( \frac{3}{2} \div \frac{9}{4} = \frac{3}{2} \cdot \frac{4}{9} = \frac{12}{18} = \frac{2}{3} \)

The expression now simplifies to:
\[ \frac{9}{10} - \frac{2}{3} + \frac{4}{3} \]

7. **Combine the Terms:**
- Combine \( -\frac{2}{3} + \frac{4}{3} = \frac{2}{3} \)

The expression now looks like:
\[ \frac{9}{10} + \frac{2}{3} \]

8. **Find a Common Denominator:**
- The common denominator for \( 10 \) and \( 3 \) is \( 30 \).
- Convert \( \frac{9}{10} = \frac{27}{30} \)
- Convert \( \frac{2}{3} = \frac{20}{30} \)

Now, add the fractions:
\[ \frac{27}{30} + \frac{20}{30} = \frac{47}{30} \]

### Final Answer:
The value of the expression is \( \frac{47}{30} \).