1. **State the problem:** Simplify the expression $$3 \frac{1}{8} \left( \frac{1}{5} - 12a \right) - 1 \frac{3}{4} \left( \frac{2}{5} a + \frac{5}{7} \right)$$. 2. **Convert mixed numbers to improper fractions:** $$3 \frac{1}{8} = \frac{25}{8}, \quad 1 \frac{3}{4} = \frac{7}{4}$$. 3. **Rewrite the expression:** $$\frac{25}{8} \left( \frac{1}{5} - 12a \right) - \frac{7}{4} \left( \frac{2}{5} a + \frac{5}{7} \right)$$. 4. **Distribute each term:** $$\frac{25}{8} \times \frac{1}{5} - \frac{25}{8} \times 12a - \frac{7}{4} \times \frac{2}{5} a - \frac{7}{4} \times \frac{5}{7}$$. 5. **Calculate each multiplication:** $$\frac{25}{8} \times \frac{1}{5} = \frac{25}{40} = \frac{5}{8}$$ $$\frac{25}{8} \times 12a = \frac{25 \times 12}{8} a = \frac{300}{8} a = 37.5a$$ $$\frac{7}{4} \times \frac{2}{5} a = \frac{14}{20} a = \frac{7}{10} a$$ $$\frac{7}{4} \times \frac{5}{7} = \frac{35}{28} = \frac{5}{4}$$. 6. **Rewrite the expression with simplified terms:** $$\frac{5}{8} - 37.5a - \frac{7}{10} a - \frac{5}{4}$$. 7. **Combine like terms:** Constants: $$\frac{5}{8} - \frac{5}{4} = \frac{5}{8} - \frac{10}{8} = -\frac{5}{8}$$ Variables: $$-37.5a - \frac{7}{10} a = -37.5a - 0.7a = -38.2a$$. 8. **Final simplified expression:** $$-\frac{5}{8} - 38.2a$$. **Answer:** $$-\frac{5}{8} - 38.2a$$