The value of ` (9)/(15) " of " ((2)/(3) // (2)/(3) " of " (3)/(2)) ÷ ((3)/(4) xx (3)/(4) ÷ (3)/(4) " of " (4)/(3)) " of " ((5)/(4) ÷ (5)/(2) xx (2)/(5) " of " (4)/(5)) ` is :

Options

A. ` (18)/(125)`
B. ` (20)/(9)`
C. ` (4)/(25)`
D. ` (40)/(9)`

Answer

To solve the given expression step by step, we will break it down into manageable parts. The expression is:

\[ \frac{9}{15} \text{ of } \left( \frac{2}{3} \text{ of } \left( \frac{2}{3} \text{ of } \frac{3}{2} \right) \right) \div \left( \left( \frac{3}{4} \times \frac{3}{4} \div \left( \frac{3}{4} \text{ of } \frac{4}{3} \right) \right) \text{ of } \left( \frac{5}{4} \div \frac{5}{2} \times \left( \frac{2}{5} \text{ of } \frac{4}{5} \right) \right) \right) \]

### Step 1: Solve the first part
We first solve the innermost expression:

\[ \frac{2}{3} \text{ of } \left( \frac{2}{3} \text{ of } \frac{3}{2} \right) \]

Calculating the inner part:

\[ \frac{2}{3} \text{ of } \frac{3}{2} = \frac{2}{3} \times \frac{3}{2} = 1 \]

Now substituting back:

\[ \frac{2}{3} \text{ of } 1 = \frac{2}{3} \]

### Step 2: Solve the second part
Next, we calculate:

\[ \frac{3}{4} \times \frac{3}{4} \div \left( \frac{3}{4} \text{ of } \frac{4}{3} \right) \]

Calculating the "of" part:

\[ \frac{3}{4} \text{ of } \frac{4}{3} = \frac{3}{4} \times \frac{4}{3} = 1 \]

Now substituting back:

\[ \frac{3}{4} \times \frac{3}{4} \div 1 = \frac{9}{16} \]

### Step 3: Solve the third part
Now we calculate:

\[ \frac{5}{4} \div \frac{5}{2} \times \left( \frac{2}{5} \text{ of } \frac{4}{5} \right) \]

Calculating the "of" part:

\[ \frac{2}{5} \text{ of } \frac{4}{5} = \frac{2}{5} \times \frac{4}{5} = \frac{8}{25} \]

Now substituting back:

\[ \frac{5}{4} \div \frac{5}{2} = \frac{5}{4} \times \frac{2}{5} = \frac{2}{4} = \frac{1}{2} \]

Now combining:

\[ \frac{1}{2} \times \frac{8}{25} = \frac{8}{50} = \frac{4}{25} \]

### Step 4: Combine all parts
Now we combine everything:

\[ \frac{9}{15} \text{ of } \left( \frac{2}{3} \right) \div \left( \frac{9}{16} \text{ of } \left( \frac{4}{25} \right) \right) \]

Calculating the "of" part:

\[ \frac{9}{16} \text{ of } \frac{4}{25} = \frac{9}{16} \times \frac{4}{25} = \frac{36}{400} = \frac{9}{100} \]

Now substituting back:

\[ \frac{9}{15} \times \frac{2}{3} \div \frac{9}{100} \]

Calculating:

\[ \frac{9}{15} \times \frac{2}{3} = \frac{18}{45} = \frac{2}{5} \]

Now dividing:

\[ \frac{2}{5} \div \frac{9}{100} = \frac{2}{5} \times \frac{100}{9} = \frac{200}{45} = \frac{40}{9} \]

### Final Answer
Thus, the final value of the expression is:

\[ \frac{40}{9} \]