The given mathematical expression is:

$5 \frac{4}{9} \div 1 \frac{2}{3} - \left( \frac{2}{3} - \frac{1}{5} \right)$

Let's break it down step by step:

1. Convert the mixed numbers to improper fractions:

• $5 \frac{4}{9} = \frac{5 \times 9 + 4}{9} = \frac{49}{9}$
• $1 \frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{5}{3}$

So the expression becomes:

$\frac{49}{9} \div \frac{5}{3} - \left( \frac{2}{3} - \frac{1}{5} \right)$

2. Perform the division:

To divide fractions, multiply by the reciprocal:

$\frac{49}{9} \div \frac{5}{3} = \frac{49}{9} \times \frac{3}{5} = \frac{49 \times 3}{9 \times 5} = \frac{147}{45}$

Now, simplify $\frac{147}{45}$ by dividing both numerator and denominator by 3:

$\frac{147 \div 3}{45 \div 3} = \frac{49}{15}$

3. Simplify the subtraction within the parentheses:

$\frac{2}{3} - \frac{1}{5}$

Find the common denominator (LCM of 3 and 5 is 15):

$\frac{2}{3} = \frac{10}{15}, \quad \frac{1}{5} = \frac{3}{15}$

Now subtract:

$\frac{10}{15} - \frac{3}{15} = \frac{7}{15}$

4. Final expression:

The expression now is:

$\frac{49}{15} - \frac{7}{15}$

Since the denominators are the same, subtract the numerators:

$\frac{49 - 7}{15} = \frac{42}{15}$

Simplify $\frac{42}{15}$ by dividing both the numerator and denominator by 3:

$\frac{42 \div 3}{15 \div 3} = \frac{14}{5}$

Final Answer:

The simplified result of the expression is $\frac{14}{5}$, which can also be written as $2 \frac{4}{5}$ in mixed form.

Would you like further clarification on any part of this solution?

Here are 5 related questions:

1. How do you multiply two fractions together?
2. What are the steps to find the least common denominator?
3. How do you convert improper fractions back into mixed numbers?
4. How does subtracting fractions differ from adding them?
5. Can you explain how to simplify a fraction?

Tip: When subtracting or adding fractions, always make sure to find a common denominator before performing the operation.