1. **State the problem:** Simplify the expression $$3\cdot\left(\frac{1}{2} + \frac{1}{3} - \frac{1}{6}\right) - 7\cdot\left(\frac{1}{4} + \frac{3}{4} - \frac{1}{8}\right) = \left(\frac{1}{7} \div \frac{1}{4}\right) - \left(\frac{3}{2} \cdot \frac{1}{2}\right)$$ 2. **Simplify inside the parentheses:** $$\frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$$ $$\frac{1}{4} + \frac{3}{4} - \frac{1}{8} = 1 - \frac{1}{8} = \frac{8}{8} - \frac{1}{8} = \frac{7}{8}$$ 3. **Substitute back:** $$3 \cdot \frac{2}{3} - 7 \cdot \frac{7}{8} = 3 \cdot \frac{2}{3} - 7 \cdot \frac{7}{8}$$ 4. **Multiply:** $$3 \cdot \frac{2}{3} = \cancel{3} \cdot \frac{2}{\cancel{3}} = 2$$ $$7 \cdot \frac{7}{8} = \frac{49}{8}$$ 5. **Calculate the first big term:** $$2 - \frac{49}{8} = \frac{16}{8} - \frac{49}{8} = -\frac{33}{8}$$ 6. **Simplify the right side:** Division: $$\frac{1}{7} \div \frac{1}{4} = \frac{1}{7} \cdot \frac{4}{1} = \frac{4}{7}$$ Multiplication: $$\frac{3}{2} \cdot \frac{1}{2} = \frac{3}{4}$$ 7. **Subtract:** $$\frac{4}{7} - \frac{3}{4} = \frac{16}{28} - \frac{21}{28} = -\frac{5}{28}$$ 8. **Final expression:** $$-\frac{33}{8} - \left(-\frac{5}{28}\right) = -\frac{33}{8} + \frac{5}{28}$$ 9. **Find common denominator and add:** Common denominator is 56: $$-\frac{33}{8} = -\frac{33 \cdot 7}{56} = -\frac{231}{56}$$ $$\frac{5}{28} = \frac{5 \cdot 2}{56} = \frac{10}{56}$$ $$-\frac{231}{56} + \frac{10}{56} = -\frac{221}{56}$$ **Final answer:** $$-\frac{221}{56}$$