What is the value of `(3)/(7) div (9)/(21) +2-(4)/(3)+(1)/(2)" of "(12)/(5) xx (25)/(18) div (5)/(9)` ?

Options

A. A)2
B. B)4
C. C)`(17)/(3)`
D. D)`(14)/(3)`

Answer

To solve the expression \( \frac{3}{7} \div \frac{9}{21} + 2 - \frac{4}{3} + \frac{1}{2} \text{ of } \frac{12}{5} \times \frac{25}{18} \div \frac{5}{9} \), we will follow the order of operations (BODMAS/BIDMAS rules).

### Step-by-step Solution:

1. **Convert Division to Multiplication**:
\[ \frac{3}{7} \div \frac{9}{21} = \frac{3}{7} \times \frac{21}{9} \]
**Hint**: Remember that dividing by a fraction is the same as multiplying by its reciprocal.

2. **Simplify the Multiplication**:
\[ \frac{3 \times 21}{7 \times 9} = \frac{63}{63} = 1 \]
**Hint**: Look for common factors in the numerator and denominator to simplify.

3. **Rewrite the Expression**:
Now substitute back into the expression:
\[ 1 + 2 - \frac{4}{3} + \frac{1}{2} \text{ of } \frac{12}{5} \times \frac{25}{18} \div \frac{5}{9} \]

4. **Calculate the "of" Part**:
First, calculate \( \frac{1}{2} \text{ of } \frac{12}{5} \):
\[ \frac{1}{2} \times \frac{12}{5} = \frac{12}{10} = \frac{6}{5} \]

5. **Calculate the Remaining Multiplication and Division**:
Now calculate \( \frac{6}{5} \times \frac{25}{18} \div \frac{5}{9} \):
- First, calculate \( \frac{25}{18} \div \frac{5}{9} = \frac{25}{18} \times \frac{9}{5} = \frac{225}{90} = \frac{5}{2} \)
- Now multiply by \( \frac{6}{5} \):
\[ \frac{6}{5} \times \frac{5}{2} = \frac{6 \times 5}{5 \times 2} = \frac{6}{2} = 3 \]

6. **Combine All Parts**:
Now substitute back into the expression:
\[ 1 + 2 - \frac{4}{3} + 3 \]
Combine \( 1 + 2 + 3 = 6 \):
\[ 6 - \frac{4}{3} \]

7. **Convert to a Common Denominator**:
Convert \( 6 \) to a fraction:
\[ 6 = \frac{18}{3} \]
Now subtract:
\[ \frac{18}{3} - \frac{4}{3} = \frac{14}{3} \]

### Final Answer:
The value of the expression is \( \frac{14}{3} \).