The value of `((8)/(3)div(3)/(5) xx(7)/(5))/((5)/(3) div(5)/(7) xx (8)/(9)) div 15` is

Options

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

Answer

To solve the expression \(\frac{\left(\frac{8}{3} \div \frac{3}{5} \times \frac{7}{5}\right)}{\left(\frac{5}{3} \div \frac{5}{7} \times \frac{8}{9}\right)} \div 15\), we will follow the order of operations (BODMAS/BIDMAS).

### Step-by-Step Solution:

1. **Rewrite the Expression**:
\[ \frac{\left(\frac{8}{3} \div \frac{3}{5} \times \frac{7}{5}\right)}{\left(\frac{5}{3} \div \frac{5}{7} \times \frac{8}{9}\right)} \div 15 \]

2. **Handle Division by Changing to Multiplication**:
- For division, we can multiply by the reciprocal.
- Thus, \(\frac{8}{3} \div \frac{3}{5} = \frac{8}{3} \times \frac{5}{3}\) and \(\frac{5}{3} \div \frac{5}{7} = \frac{5}{3} \times \frac{7}{5}\).

The expression now becomes:
\[ \frac{\left(\frac{8}{3} \times \frac{5}{3} \times \frac{7}{5}\right)}{\left(\frac{5}{3} \times \frac{7}{5} \times \frac{8}{9}\right)} \div 15 \]

3. **Simplify the Numerator**:
- The numerator is \(\frac{8}{3} \times \frac{5}{3} \times \frac{7}{5}\).
- Cancel \(5\) in the numerator and denominator:
\[ = \frac{8 \times 7}{3 \times 3} = \frac{56}{9} \]

4. **Simplify the Denominator**:
- The denominator is \(\frac{5}{3} \times \frac{7}{5} \times \frac{8}{9}\).
- Cancel \(5\) in the numerator and denominator:
\[ = \frac{7 \times 8}{3 \times 9} = \frac{56}{27} \]

5. **Combine the Results**:
- Now we have:
\[ \frac{\frac{56}{9}}{\frac{56}{27}} \div 15 \]
- This simplifies to:
\[ \frac{56}{9} \times \frac{27}{56} \]
- The \(56\) cancels out:
\[ = \frac{27}{9} = 3 \]

6. **Final Division**:
- Now we divide by \(15\):
\[ 3 \div 15 = \frac{3}{15} = \frac{1}{5} \]

### Final Answer:
\[ \frac{1}{5} \]