Let's solve the given expression step by step.

The expression is:

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

Step 1: Simplify mixed numbers

1. $1 \frac{7}{8} = \frac{15}{8}$
2. $1 \frac{1}{3} = \frac{4}{3}$
3. $3 \frac{1}{5} = \frac{16}{5}$
4. $4 \frac{1}{2} = \frac{9}{2}$
5. $5 \frac{1}{3} = \frac{16}{3}$
6. $2 \frac{1}{2} = \frac{5}{2}$

Step 2: Perform operations

Now we substitute these into the expression:

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

Step 3: Simplify each part

1. $\frac{3}{5} \times \frac{15}{8} = \frac{45}{40} = \frac{9}{8}$
2. $\frac{9}{8} \div \frac{4}{3} = \frac{9}{8} \times \frac{3}{4} = \frac{27}{32}$
3. $\frac{27}{32} \times \frac{3}{16} = \frac{81}{512}$

Now for the second term inside the parenthesis:

$\frac{\frac{16}{5}}{\frac{9}{2} \times \frac{16}{3}} = \frac{\frac{16}{5}}{\frac{144}{6}} = \frac{16}{5} \times \frac{6}{144} = \frac{96}{720} = \frac{2}{15}$

Multiplying by $\frac{5}{2}$:

$\frac{2}{15} \times \frac{5}{2} = \frac{10}{30} = \frac{1}{3}$

Now the entire expression is:

$\frac{81}{512} - \frac{1}{3} + \frac{1}{2} + \frac{1}{8} \div \frac{1}{4}$

Next, handle the last term:

$\frac{1}{8} \div \frac{1}{4} = \frac{1}{8} \times 4 = \frac{1}{2}$

So the expression simplifies to:

$\frac{81}{512} - \frac{1}{3} + \frac{1}{2} + \frac{1}{2}$

Adding the $\frac{1}{2} + \frac{1}{2} = 1$:

$\frac{81}{512} - \frac{1}{3} + 1$

Step 4: Convert to common denominator

Convert $\frac{1}{3}$ to a fraction with denominator 512:

$\frac{1}{3} = \frac{170.67}{512} \approx \frac{171}{512}$

Now the expression is:

$\frac{81}{512} - \frac{171}{512} + \frac{512}{512}$

Simplify:

$\frac{81 - 171 + 512}{512} = \frac{422}{512} = \frac{211}{256}$

So the final answer is:

$\boxed{\frac{211}{256}}$

Would you like more details or have any questions?

Related Questions:

1. How do you simplify complex fractions step-by-step?
2. What are common mistakes to avoid when working with mixed numbers?
3. Can you explain the importance of finding a common denominator?
4. What are the benefits of converting mixed numbers into improper fractions?
5. How does the order of operations affect the outcome of such expressions?

Tip: Always convert mixed numbers to improper fractions before performing operations.