2 4/15 - (2-1 1/15) : 4/9 + 2/3

To solve the expression \(2 \frac{4}{15} - (2 - 1 \frac{1}{15}) : \frac{4}{9} + \frac{2}{3}\), let's follow the order of operations step by step.

1. Convert mixed numbers to improper fractions:

   • \(2 \frac{4}{15} = \frac{2 \times 15 + 4}{15} = \frac{30 + 4}{15} = \frac{34}{15}\)
   • \(1 \frac{1}{15} = \frac{1 \times 15 + 1}{15} = \frac{15 + 1}{15} = \frac{16}{15}\)

2. Evaluate the subtraction in the parentheses: \[ 2 - 1 \frac{1}{15} = 2 - \frac{16}{15} \] Convert 2 to an improper fraction: \[ 2 = \frac{2 \times 15}{15} = \frac{30}{15} \] Thus, \[ 2 - 1 \frac{1}{15} = \frac{30}{15} - \frac{16}{15} = \frac{30 - 16}{15} = \frac{14}{15} \]

3. Now substitute back into the expression: \[ \frac{34}{15} - \left(\frac{14}{15}\right) : \frac{4}{9} + \frac{2}{3} \]

4. Handle the division: Division by a fraction is the same as multiplying by its reciprocal. \[ \frac{14}{15} : \frac{4}{9} = \frac{14}{15} \cdot \frac{9}{4} = \frac{14 \cdot 9}{15 \cdot 4} = \frac{126}{60} \] Simplify \(\frac{126}{60}\):

   • The GCD of 126 and 60 is 6. \[ \frac{126 \div 6}{60 \div 6} = \frac{21}{10} \]

5. Now substitute back into the expression: \[ \frac{34}{15} - \frac{21}{10} + \frac{2}{3} \]

6. Find a common denominator for all three fractions (the least common multiple of 15, 10, and 3 is 30):

   • Convert \(\frac{34}{15}\): \[ \frac{34}{15} = \frac{34 \cdot 2}{15 \cdot 2} = \frac{68}{30} \]
   • Convert \(\frac{21}{10}\): \[ \frac{21}{10} = \frac{21 \cdot 3}{10 \cdot 3} = \frac{63}{30} \]
   • Convert \(\frac{2}{3}\): \[ \frac{2}{3} = \frac{2 \cdot 10}{3 \cdot 10} = \frac{20}{30} \]

7. Now the expression looks like this: \[ \frac{68}{30} - \frac{63}{30} + \frac{20}{30} \]

8. Combine the fractions: \[ \frac{68 - 63 + 20}{30} = \frac{5 + 20}{30} = \frac{25}{30} \]

9. Simplify \(\frac{25}{30}\): The GCD of 25 and 30 is 5. \[ \frac{25 \div 5}{30 \div 5} = \frac{5}{6} \]

Thus, the final answer is \(\frac{5}{6}\).