1. **State the problem:** Evaluate the expression without using a calculator or tables: $$\frac{3}{4} + \frac{12}{7} \div \frac{4}{7} \times \frac{7}{3} \times \frac{2}{3} \div \left[ \frac{10}{7} - \frac{21}{8} \right] \times \frac{2}{3}$$ 2. **Rewrite the expression clearly:** $$\frac{3}{4} + \left( \frac{12}{7} \div \frac{4}{7} \times \frac{7}{3} \times \frac{2}{3} \right) \div \left( \frac{10}{7} - \frac{21}{8} \right) \times \frac{2}{3}$$ 3. **Evaluate inside the parentheses step-by-step:** - Division of fractions: $$\frac{12}{7} \div \frac{4}{7} = \frac{12}{7} \times \frac{7}{4} = \frac{12 \times 7}{7 \times 4} = \frac{12}{4} = 3$$ - Multiply by $$\frac{7}{3}$$: $$3 \times \frac{7}{3} = \frac{3 \times 7}{3} = 7$$ - Multiply by $$\frac{2}{3}$$: $$7 \times \frac{2}{3} = \frac{14}{3}$$ 4. **Evaluate the denominator inside the brackets:** $$\frac{10}{7} - \frac{21}{8} = \frac{10 \times 8}{7 \times 8} - \frac{21 \times 7}{8 \times 7} = \frac{80}{56} - \frac{147}{56} = \frac{80 - 147}{56} = \frac{-67}{56}$$ 5. **Divide the numerator by the denominator:** $$\frac{14}{3} \div \left( \frac{-67}{56} \right) = \frac{14}{3} \times \frac{56}{-67} = \frac{14 \times 56}{3 \times (-67)} = \frac{784}{-201} = -\frac{784}{201}$$ 6. **Multiply by $$\frac{2}{3}$$:** $$-\frac{784}{201} \times \frac{2}{3} = -\frac{784 \times 2}{201 \times 3} = -\frac{1568}{603}$$ 7. **Add $$\frac{3}{4}$$ to the result:** Find common denominator for $$\frac{3}{4}$$ and $$-\frac{1568}{603}$$: - LCD of 4 and 603 is $$4 \times 603 = 2412$$ (since 4 and 603 are coprime) Convert fractions: $$\frac{3}{4} = \frac{3 \times 603}{4 \times 603} = \frac{1809}{2412}$$ $$-\frac{1568}{603} = -\frac{1568 \times 4}{603 \times 4} = -\frac{6272}{2412}$$ Add: $$\frac{1809}{2412} - \frac{6272}{2412} = \frac{1809 - 6272}{2412} = \frac{-4463}{2412}$$ 8. **Final answer:** $$\boxed{-\frac{4463}{2412}}$$ This fraction cannot be simplified further because 4463 and 2412 share no common factors other than 1.