The value of `(8/3 div3/5xx7/5)/(5/3div5/7xx8/9)div6` is

Options

A. `1/6`
B. `1/4`
C. `1/3`
D. `1/2`

Answer

To solve the expression \((\frac{8}{3} \div \frac{3}{5} \times \frac{7}{5}) \div (\frac{5}{3} \div \frac{5}{7} \times \frac{8}{9}) \div 6\), we will follow the order of operations (BODMAS/BIDMAS).

### Step-by-step Solution:

1. **Evaluate the first part of the expression:**
\[ \frac{8}{3} \div \frac{3}{5} \times \frac{7}{5} \]
To divide by a fraction, we multiply by its reciprocal:
\[ \frac{8}{3} \times \frac{5}{3} \times \frac{7}{5} \]

2. **Simplify the multiplication:**
\[ = \frac{8 \times 5 \times 7}{3 \times 3 \times 5} \]
The \(5\) in the numerator and denominator cancels out:
\[ = \frac{8 \times 7}{3 \times 3} = \frac{56}{9} \]

3. **Evaluate the second part of the expression:**
\[ \frac{5}{3} \div \frac{5}{7} \times \frac{8}{9} \]
Again, we multiply by the reciprocal:
\[ \frac{5}{3} \times \frac{7}{5} \times \frac{8}{9} \]
The \(5\) cancels out:
\[ = \frac{7 \times 8}{3 \times 9} = \frac{56}{27} \]

4. **Now substitute back into the main expression:**
\[ \frac{\frac{56}{9}}{\frac{56}{27}} \div 6 \]
To divide by a fraction, we multiply by its reciprocal:
\[ \frac{56}{9} \times \frac{27}{56} \]
The \(56\) cancels out:
\[ = \frac{27}{9} = 3 \]

5. **Now divide by 6:**
\[ 3 \div 6 = \frac{3}{6} = \frac{1}{2} \]

### Final Answer:
The value of the expression is \(\frac{1}{2}\).