The value of `(3)/(5) xx 1 (7)/(8) div 1 (1)/(3)` of ` (3)/( 16) - ( 3 (1)/(5) div 4 (1)/(2)" of " 5 (1)/(3) ) xx 2 (1)/(2) + (1)/(2) + (1)/(8) div (1)/(4)`

Options

A. `4 (1)/(8)`
B. `5 (1)/(6)`
C. `5 (5)/(6)`
D. `4 (1)/(3)`

Answer

To solve the expression given in the question, we will break it down step by step. The expression is:

\[ \frac{3}{5} \times 1 \frac{7}{8} \div 1 \frac{1}{3} \text{ of } \frac{3}{16} - \left(3 \frac{1}{5} \div 4 \frac{1}{2}\right) \text{ of } 5 \frac{1}{3} \times 2 \frac{1}{2} + \frac{1}{2} + \frac{1}{8} \div \frac{1}{4} \]

### Step 1: Convert Mixed Numbers to Improper Fractions

1. Convert \(1 \frac{7}{8}\) to an improper fraction:
\[ 1 \frac{7}{8} = \frac{8 \times 1 + 7}{8} = \frac{15}{8} \]

2. Convert \(1 \frac{1}{3}\) to an improper fraction:
\[ 1 \frac{1}{3} = \frac{3 \times 1 + 1}{3} = \frac{4}{3} \]

3. Convert \(3 \frac{1}{5}\) to an improper fraction:
\[ 3 \frac{1}{5} = \frac{5 \times 3 + 1}{5} = \frac{16}{5} \]

4. Convert \(4 \frac{1}{2}\) to an improper fraction:
\[ 4 \frac{1}{2} = \frac{2 \times 4 + 1}{2} = \frac{9}{2} \]

5. Convert \(5 \frac{1}{3}\) to an improper fraction:
\[ 5 \frac{1}{3} = \frac{3 \times 5 + 1}{3} = \frac{16}{3} \]

6. Convert \(2 \frac{1}{2}\) to an improper fraction:
\[ 2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2} \]

### Step 2: Substitute Back into the Expression

Now substituting back into the expression, we have:

\[ \frac{3}{5} \times \frac{15}{8} \div \frac{4}{3} \text{ of } \frac{3}{16} - \left(\frac{16}{5} \div \frac{9}{2}\right) \text{ of } \frac{16}{3} \times \frac{5}{2} + \frac{1}{2} + \frac{1}{8} \div \frac{1}{4} \]

### Step 3: Evaluate Each Part

1. **Evaluate \(\frac{3}{5} \times \frac{15}{8} \div \frac{4}{3}\)**:
- First, calculate \(\frac{3}{5} \times \frac{15}{8} = \frac{45}{40} = \frac{9}{8}\)
- Now divide by \(\frac{4}{3}\):
\[ \frac{9}{8} \div \frac{4}{3} = \frac{9}{8} \times \frac{3}{4} = \frac{27}{32} \]

2. **Evaluate \(\left(\frac{16}{5} \div \frac{9}{2}\right) \text{ of } \frac{16}{3} \times \frac{5}{2}\)**:
- First, calculate \(\frac{16}{5} \div \frac{9}{2} = \frac{16}{5} \times \frac{2}{9} = \frac{32}{45}\)
- Now calculate \(\frac{32}{45} \text{ of } \frac{16}{3} = \frac{32}{45} \times \frac{16}{3} = \frac{512}{135}\)
- Then multiply by \(\frac{5}{2}\):
\[ \frac{512}{135} \times \frac{5}{2} = \frac{2560}{270} = \frac{128}{27} \]

3. **Evaluate \(\frac{1}{2} + \frac{1}{8} \div \frac{1}{4}\)**:
- First, calculate \(\frac{1}{8} \div \frac{1}{4} = \frac{1}{8} \times 4 = \frac{1}{2}\)
- Now add \(\frac{1}{2} + \frac{1}{2} = 1\)

### Step 4: Combine All Parts

Now we combine everything:

\[ \frac{27}{32} - \frac{128}{27} + 1 \]

### Step 5: Find a Common Denominator and Simplify

1. Convert \(1\) to a fraction with a common denominator of \(32\) and \(27\):
- The LCM of \(32\) and \(27\) is \(864\).
- Convert \(\frac{27}{32}\) to \(\frac{729}{864}\) and \(\frac{128}{27}\) to \(\frac{4096}{864}\).
- Convert \(1\) to \(\frac{864}{864}\).

2. Now combine:
\[ \frac{729}{864} - \frac{4096}{864} + \frac{864}{864} = \frac{729 - 4096 + 864}{864} = \frac{-2503}{864} \]

### Final Answer

Thus, the final result is:

\[ \frac{-2503}{864} \]