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!

Learn how to solve the fraction expression (5/3 x 4 5/6) ÷ (1.75 + 5/3) with a step-by-step breakdown. This problem covers key fraction concepts such as multiplication, addition, and division of fractions, as well as mixed numbers. Suitable for students in Grades 6-8.

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