Let's break down and solve the given expression step by step.

The expression is:
$\left( \frac{5}{3} \times 4 \frac{5}{6} \right) \div \left( 1.75 + \frac{5}{3} \right)$

Step 1: Convert the mixed fraction to an improper fraction.

Convert $4 \frac{5}{6}$ to an improper fraction: $4 \frac{5}{6} = \frac{29}{6}$

Step 2: Multiply the fractions.

Now multiply $\frac{5}{3} \times \frac{29}{6}$: $\frac{5}{3} \times \frac{29}{6} = \frac{5 \times 29}{3 \times 6} = \frac{145}{18}$

Step 3: Add the terms in the denominator.

Add $1.75$ and $\frac{5}{3}$. First, convert $1.75$ to a fraction: $1.75 = \frac{7}{4}$

Now add $\frac{7}{4}$ and $\frac{5}{3}$ by finding a common denominator: $\frac{7}{4} + \frac{5}{3} = \frac{21}{12} + \frac{20}{12} = \frac{41}{12}$

Step 4: Perform the division.

Now divide $\frac{145}{18}$ by $\frac{41}{12}$. To divide fractions, multiply the first fraction by the reciprocal of the second: $\frac{145}{18} \div \frac{41}{12} = \frac{145}{18} \times \frac{12}{41} = \frac{145 \times 12}{18 \times 41} = \frac{1740}{738} = \frac{290}{123}$

Step 5: Simplify the fraction.

The fraction $\frac{290}{123}$ is already in its simplest form.

Final Answer:

$\boxed{\frac{290}{123}}$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you multiply two fractions with mixed numbers?
2. How do you convert a decimal like 1.75 to a fraction?
3. Why do we multiply by the reciprocal when dividing fractions?
4. How do you simplify a complex fraction like $\frac{290}{123}$?
5. How do you find a common denominator for adding fractions?

Tip: When adding or subtracting fractions, always remember to find a common denominator before performing the operation!