1. **State the problem:** Evaluate the expression $\left(6 - 2 \frac{4}{5}\right) \cdot 3 \frac{1}{8} - 1 \frac{3}{5} : \frac{1}{4} - 1 : \frac{1}{3}$. 2. **Convert mixed numbers to improper fractions:** $2 \frac{4}{5} = \frac{2 \times 5 + 4}{5} = \frac{14}{5}$ $3 \frac{1}{8} = \frac{3 \times 8 + 1}{8} = \frac{25}{8}$ $1 \frac{3}{5} = \frac{1 \times 5 + 3}{5} = \frac{8}{5}$ 3. **Rewrite the expression with improper fractions:** $$\left(6 - \frac{14}{5}\right) \cdot \frac{25}{8} - \frac{8}{5} : \frac{1}{4} - 1 : \frac{1}{3}$$ 4. **Convert whole numbers to fractions for subtraction and division:** $6 = \frac{6}{1}$, $1 = \frac{1}{1}$ 5. **Calculate inside the parentheses:** $$6 - \frac{14}{5} = \frac{6}{1} - \frac{14}{5} = \frac{6 \times 5}{5} - \frac{14}{5} = \frac{30}{5} - \frac{14}{5} = \frac{16}{5}$$ 6. **Multiply:** $$\frac{16}{5} \cdot \frac{25}{8} = \frac{16 \times 25}{5 \times 8} = \frac{400}{40}$$ 7. **Simplify the fraction by canceling common factors:** $$\frac{400}{40} = \frac{\cancel{400}^{10 \times 40}}{\cancel{40}^{1 \times 40}} = 10$$ 8. **Divide $\frac{8}{5}$ by $\frac{1}{4}$:** $$\frac{8}{5} : \frac{1}{4} = \frac{8}{5} \times \frac{4}{1} = \frac{32}{5}$$ 9. **Divide $1$ by $\frac{1}{3}$:** $$1 : \frac{1}{3} = \frac{1}{1} \times \frac{3}{1} = 3$$ 10. **Rewrite the expression with these results:** $$10 - \frac{32}{5} - 3$$ 11. **Convert whole numbers to fractions with denominator 5:** $$10 = \frac{50}{5}, \quad 3 = \frac{15}{5}$$ 12. **Perform the subtraction:** $$\frac{50}{5} - \frac{32}{5} - \frac{15}{5} = \frac{50 - 32 - 15}{5} = \frac{3}{5}$$ **Final answer:** $$\boxed{\frac{3}{5}}$$