Si \( A=\left(3 \frac{1}{2}+\frac{2}{3}\right): \frac{15}{6}-\frac{2}{3} \)

To solve for \( A \), we first need to simplify the expression step by step. 1. **Convert mixed number to improper fraction**: \[ 3 \frac{1}{2} = 3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2} \] 2. **Add \( \frac{7}{2} \) and \( \frac{2}{3} \)**: To add these fractions, we need a common denominator. The least common multiple of 2 and 3 is 6. \[ \frac{7}{2} = \frac{7 \times 3}{2 \times 3} = \frac{21}{6} \] \[ \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \] Now we can add: \[ \frac{21}{6} + \frac{4}{6} = \frac{21 + 4}{6} = \frac{25}{6} \] 3. **Calculate \( \frac{15}{6} - \frac{2}{3} \)**: Again, we need a common denominator. We can convert \( \frac{2}{3} \) to sixths: \[ \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \] Now we can subtract: \[ \frac{15}{6} - \frac{4}{6} = \frac{15 - 4}{6} = \frac{11}{6} \] 4. **Now we have the expression**: \[ A = \frac{25}{6} : \frac{11}{6} \] Dividing by a fraction is the same as multiplying by its reciprocal: \[ A = \frac{25}{6} \times \frac{6}{11} \] 5. **Simplify**: The \( 6 \) in the numerator and denominator cancels out: \[ A = \frac{25}{11} \] Thus, the final result is: \[ \boxed{\frac{25}{11}} \]