1. **Problema A:** Calcular $$\frac{2}{3} - \left[\frac{1}{6} + 2 - \left(\frac{3}{5} + \frac{2}{3}\right)\right] - \frac{1}{5} \cdot \left(5 \cdot \frac{1}{2} - \frac{7}{3}\right)$$ 2. **Problema B:** Calcular $$-\left[-3 - (6 - 9) \cdot 0.15\right]$$ 3. **Problema C:** Calcular $$\left(\frac{1}{3} + \frac{1}{2 - \frac{1}{5}}\right) : 3 + \frac{1}{3}$$ 4. **Problema D:** Calcular $$\left(2.5 - \frac{7}{6} : \frac{14}{15}\right) \cdot 0.64 + \frac{8}{9}$$ --- ### Solución A: 1. Resolver la suma dentro del paréntesis: $$\frac{3}{5} + \frac{2}{3} = \frac{9}{15} + \frac{10}{15} = \frac{19}{15}$$ 2. Resolver la expresión dentro del corchete: $$\frac{1}{6} + 2 - \frac{19}{15} = \frac{1}{6} + \frac{30}{15} - \frac{19}{15} = \frac{1}{6} + \frac{11}{15}$$ Convertimos $$\frac{1}{6}$$ a quinceavos: $$\frac{1}{6} = \frac{5}{30} = \frac{1}{6}$$ (mejor convertir a común denominador 30): $$\frac{1}{6} = \frac{5}{30}, \frac{11}{15} = \frac{22}{30}$$ Sumamos: $$\frac{5}{30} + \frac{22}{30} = \frac{27}{30} = \frac{9}{10}$$ 3. Ahora la expresión es: $$\frac{2}{3} - \frac{9}{10} - \frac{1}{5} \cdot \left(5 \cdot \frac{1}{2} - \frac{7}{3}\right)$$ 4. Resolver el paréntesis del producto: $$5 \cdot \frac{1}{2} = \frac{5}{2}$$ Entonces: $$\frac{5}{2} - \frac{7}{3} = \frac{15}{6} - \frac{14}{6} = \frac{1}{6}$$ 5. Multiplicamos: $$\frac{1}{5} \cdot \frac{1}{6} = \frac{1}{30}$$ 6. Finalmente: $$\frac{2}{3} - \frac{9}{10} - \frac{1}{30}$$ Convertimos a común denominador 30: $$\frac{2}{3} = \frac{20}{30}, \frac{9}{10} = \frac{27}{30}$$ Entonces: $$20/30 - 27/30 - 1/30 = \cancel{20/30} - \cancel{27/30} - 1/30 = -8/30 = -\frac{4}{15}$$ --- ### Solución B: 1. Resolver el paréntesis: $$6 - 9 = -3$$ 2. Multiplicar: $$-3 \cdot 0.15 = -0.45$$ 3. Resolver dentro del corchete: $$-3 - (-0.45) = -3 + 0.45 = -2.55$$ 4. Aplicar el signo negativo exterior: $$-(-2.55) = 2.55$$ --- ### Solución C: 1. Resolver el denominador dentro del segundo término: $$2 - \frac{1}{5} = \frac{10}{5} - \frac{1}{5} = \frac{9}{5}$$ 2. Invertir para la división: $$\frac{1}{\frac{9}{5}} = \frac{5}{9}$$ 3. Sumar con $$\frac{1}{3}$$: $$\frac{1}{3} + \frac{5}{9} = \frac{3}{9} + \frac{5}{9} = \frac{8}{9}$$ 4. Dividir por 3: $$\frac{8}{9} : 3 = \frac{8}{9} \cdot \frac{1}{3} = \frac{8}{27}$$ 5. Sumar $$\frac{1}{3} = \frac{9}{27}$$: $$\frac{8}{27} + \frac{9}{27} = \frac{17}{27}$$ --- ### Solución D: 1. Resolver la división dentro del paréntesis: $$\frac{7}{6} : \frac{14}{15} = \frac{7}{6} \cdot \frac{15}{14} = \frac{7 \cdot 15}{6 \cdot 14} = \frac{105}{84}$$ Simplificamos: $$\frac{105}{84} = \frac{\cancel{105}^{15 \cdot 7}}{\cancel{84}^{12 \cdot 7}} = \frac{15}{12} = \frac{5}{4}$$ 2. Restar: $$2.5 - \frac{5}{4} = \frac{5}{2} - \frac{5}{4} = \frac{10}{4} - \frac{5}{4} = \frac{5}{4}$$ 3. Multiplicar por 0.64: $$\frac{5}{4} \cdot 0.64 = 1.25 \cdot 0.64 = 0.8$$ 4. Sumar $$\frac{8}{9} \approx 0.8889$$: $$0.8 + 0.8889 = 1.6889 \approx \frac{152}{90} = \frac{76}{45}$$ --- **Respuestas finales:** A) $$-\frac{4}{15}$$ B) $$2.55$$ C) $$\frac{17}{27}$$ D) $$\frac{76}{45}$$