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To simplify the given expression, let's break down the steps involved. The expression is:

\[
3 \frac{4}{9} \div \left( 5 \frac{1}{3} - 2 \frac{3}{4} \right) + 5 \frac{9}{10}
\]

### Step 1: Convert Mixed Numbers to Improper Fractions
1. \( 3 \frac{4}{9} = \frac{27}{9} + \frac{4}{9} = \frac{31}{9} \)
2. \( 5 \frac{1}{3} = \frac{15}{3} + \frac{1}{3} = \frac{16}{3} \)
3. \( 2 \frac{3}{4} = \frac{8}{4} + \frac{3}{4} = \frac{11}{4} \)
4. \( 5 \frac{9}{10} = \frac{50}{10} + \frac{9}{10} = \frac{59}{10} \)

### Step 2: Simplify the Expression Inside the Parentheses
\[
\frac{16}{3} - \frac{11}{4}
\]

To subtract these fractions, find a common denominator:
- Common denominator of 3 and 4 is 12.
- Convert each fraction:
  \[
  \frac{16}{3} = \frac{64}{12}, \quad \frac{11}{4} = \frac{33}{12}
  \]

Subtract the fractions:
\[
\frac{64}{12} - \frac{33}{12} = \frac{31}{12}
\]

### Step 3: Divide the First Fraction by the Result of the Parentheses
\[
\frac{31}{9} \div \frac{31}{12}
\]

Division of fractions means multiplying by the reciprocal:
\[
\frac{31}{9} \times \frac{12}{31} = \frac{31 \times 12}{9 \times 31} = \frac{12}{9} = \frac{4}{3}
\]

### Step 4: Add the Result to the Last Fraction
\[
\frac{4}{3} + \frac{59}{10}
\]

Find a common denominator for 3 and 10, which is 30:
\[
\frac{4}{3} = \frac{40}{30}, \quad \frac{59}{10} = \frac{177}{30}
\]

Add these fractions:
\[
\frac{40}{30} + \frac{177}{30} = \frac{217}{30}
\]

### Final Answer
The simplified expression is:
\[
\frac{217}{30}
\]

Or, as a mixed number:
\[
7 \frac{7}{30}
\]

Would you like any further details or have any questions about the steps? Here are some related questions:

1. How do you convert mixed numbers to improper fractions?
2. What is the process to find a common denominator for fractions?
3. How do you multiply and divide fractions?
4. How can you simplify complex fractions more efficiently?
5. Can this result be expressed in decimal form?

**Tip:** Always perform operations inside parentheses first and convert mixed numbers to improper fractions for easier calculation.

Simplify 3 4/9 ÷ (5 1/3 - 2 3/4) + 5 9/10

Learn how to simplify the expression 3 4/9 ÷ (5 1/3 - 2 3/4) + 5 9/10 with step-by-step solutions. This covers key fraction operations, including converting mixed numbers to improper fractions and using common denominators.

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