(5) \( \frac{2+\frac{1}{\frac{3}{4}+\frac{3}{5}}}{\frac{2}{3}-\frac{2}{7}-\frac{4}{5}} \)

To simplify the expression \[ \frac{2+\frac{1}{\frac{3}{4}+\frac{3}{5}}}{\frac{2}{3}-\frac{2}{7}-\frac{4}{5}}, \] we will first simplify the numerator and the denominator separately. **Step 1: Simplify the numerator** The numerator is \[ 2 + \frac{1}{\frac{3}{4} + \frac{3}{5}}. \] First, we need to simplify \(\frac{3}{4} + \frac{3}{5}\). To do this, we find a common denominator, which is \(20\): \[ \frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}, \] \[ \frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20}. \] Now we can add these fractions: \[ \frac{3}{4} + \frac{3}{5} = \frac{15}{20} + \frac{12}{20} = \frac{27}{20}. \] Next, we substitute this back into the numerator: \[ 2 + \frac{1}{\frac{27}{20}} = 2 + \frac{20}{27}. \] To add these, we convert \(2\) into a fraction with a denominator of \(27\): \[ 2 = \frac{54}{27}. \] Now we can add: \[ \frac{54}{27} + \frac{20}{27} = \frac{54 + 20}{27} = \frac{74}{27}. \] So, the numerator simplifies to \[ \frac{74}{27}. \] **Step 2: Simplify the denominator** The denominator is \[ \frac{2}{3} - \frac{2}{7} - \frac{4}{5}. \] We will find a common denominator for these fractions. The least common multiple of \(3\), \(7\), and \(5\) is \(105\). Now we convert each fraction: \[ \frac{2}{3} = \frac{2 \times 35}{3 \times 35} = \frac{70}{105}, \] \[ \frac{2}{7} = \frac{2 \times 15}{7 \times 15} = \frac{30}{105}, \] \[ \frac{4}{5} = \frac{4 \times 21}{5 \times 21} = \frac{84}{105}. \] Now we can substitute these back into the denominator: \[ \frac{70}{105} - \frac{30}{105} - \frac{84}{105} = \frac{70 - 30 - 84}{105} = \frac{70 - 114}{105} = \frac{-44}{105}. \] **Step 3: Combine the results** Now we have: \[ \frac{\frac{74}{27}}{\frac{-44}{105}}. \] To divide by a fraction, we multiply by its reciprocal: \[ \frac{74}{27} \times \frac{105}{-44} = \frac{74 \times 105}{27 \times -44}. \] **Step 4: Simplify the fraction** Calculating the numerator: \[ 74 \times 105 = 7770. \] Calculating the denominator: \[ 27 \times -44 = -1188. \] Thus, we have: \[ \frac{7770}{-1188}. \] Now we simplify this fraction. We can find the greatest common divisor (GCD) of \(7770\) and \(1188\). Using the Euclidean algorithm, we find that the GCD is \(18\). Now we divide both the numerator and the denominator by \(18\): \[ \frac{7770 \div 18}{-1188 \div 18} = \frac{430}{-66} = -\frac{430}{66}. \] We can simplify \(-\frac{430}{66}\) further. The GCD of \(430\) and \(66\) is \(2\): \[ -\frac{430 \div 2}{66 \div 2} = -\frac{215}{33}. \] Thus, the final simplified result is \[ \boxed{-\frac{215}{33}}. \]