The value of `(2)/(3)div(3)/(1)" of "(4)/(9)-(4)/(5)xx1 (1)/(9)div(8)/(15)+(3)/(4)div(1)/(2)` is :

Options

A. A)`(49)/(12)`
B. B)`(17)/(9)`
C. C)`(29)/(6)`
D. D)`(1)/(3)`

Answer

To solve the expression \( \frac{2}{3} \div \frac{3}{1} \text{ of } \frac{4}{9} - \frac{4}{5} \times 1 \frac{1}{9} \div \frac{8}{15} + \frac{3}{4} \div \frac{1}{2} \), we will follow the order of operations (BODMAS/BIDMAS rules).

### Step-by-step Solution:

1. **Convert Mixed Numbers to Improper Fractions**:
\[ 1 \frac{1}{9} = \frac{10}{9} \]
So the expression becomes:
\[ \frac{2}{3} \div \frac{3}{1} \text{ of } \frac{4}{9} - \frac{4}{5} \times \frac{10}{9} \div \frac{8}{15} + \frac{3}{4} \div \frac{1}{2} \]

2. **Calculate "of" Operation**:
The "of" operation means multiplication:
\[ \frac{3}{1} \text{ of } \frac{4}{9} = \frac{3 \times 4}{1 \times 9} = \frac{12}{9} = \frac{4}{3} \]
Now the expression is:
\[ \frac{2}{3} \div \frac{4}{3} - \frac{4}{5} \times \frac{10}{9} \div \frac{8}{15} + \frac{3}{4} \div \frac{1}{2} \]

3. **Convert Division to Multiplication**:
\[ \frac{2}{3} \div \frac{4}{3} = \frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2} \]
Now the expression is:
\[ \frac{1}{2} - \frac{4}{5} \times \frac{10}{9} \div \frac{8}{15} + \frac{3}{4} \div \frac{1}{2} \]

4. **Calculate the Second Term**:
First, calculate \( \frac{4}{5} \times \frac{10}{9} \):
\[ \frac{4 \times 10}{5 \times 9} = \frac{40}{45} = \frac{8}{9} \]
Now convert the division:
\[ \frac{8}{9} \div \frac{8}{15} = \frac{8}{9} \times \frac{15}{8} = \frac{15}{9} = \frac{5}{3} \]
Now the expression is:
\[ \frac{1}{2} - \frac{5}{3} + \frac{3}{4} \div \frac{1}{2} \]

5. **Calculate the Third Term**:
\[ \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times 2 = \frac{3 \times 2}{4} = \frac{6}{4} = \frac{3}{2} \]
Now the expression is:
\[ \frac{1}{2} - \frac{5}{3} + \frac{3}{2} \]

6. **Combine the Terms**:
To combine these fractions, we need a common denominator. The least common multiple of 2 and 3 is 6.
- Convert \( \frac{1}{2} \) to sixths: \( \frac{1}{2} = \frac{3}{6} \)
- Convert \( \frac{5}{3} \) to sixths: \( \frac{5}{3} = \frac{10}{6} \)
- Convert \( \frac{3}{2} \) to sixths: \( \frac{3}{2} = \frac{9}{6} \)

Now we can combine:
\[ \frac{3}{6} - \frac{10}{6} + \frac{9}{6} = \frac{3 - 10 + 9}{6} = \frac{2}{6} = \frac{1}{3} \]

### Final Answer:
The value of the expression is \( \frac{1}{3} \).