1. **State the problem:** Simplify the expression $$-3 \times \frac{3}{5} \cdot \left( 2 \times \frac{5}{6} - 1 \times \frac{5}{12} \right)$$. 2. **Recall the rules:** Multiplication and subtraction of fractions require finding common denominators and simplifying step-by-step. 3. **Calculate inside the parentheses first:** $$2 \times \frac{5}{6} = \frac{2 \times 5}{6} = \frac{10}{6}$$ $$1 \times \frac{5}{12} = \frac{5}{12}$$ 4. **Subtract the two fractions:** Find common denominator for $$\frac{10}{6}$$ and $$\frac{5}{12}$$, which is 12. $$\frac{10}{6} = \frac{10 \times 2}{6 \times 2} = \frac{20}{12}$$ So, $$\frac{20}{12} - \frac{5}{12} = \frac{20 - 5}{12} = \frac{15}{12}$$ 5. **Simplify $$\frac{15}{12}$$:** $$\frac{15}{12} = \frac{\cancel{15}}{\cancel{12}} = \frac{5}{4}$$ (dividing numerator and denominator by 3) 6. **Multiply the result by $$-3 \times \frac{3}{5}$$:** First multiply $$-3$$ and $$\frac{3}{5}$$: $$-3 \times \frac{3}{5} = \frac{-3 \times 3}{5} = \frac{-9}{5}$$ 7. **Now multiply $$\frac{-9}{5}$$ by $$\frac{5}{4}$$:** $$\frac{-9}{5} \times \frac{5}{4} = \frac{-9 \times 5}{5 \times 4} = \frac{-45}{20}$$ Cancel common factor 5: $$\frac{-\cancel{45}}{\cancel{20}} = \frac{-9}{4}$$ 8. **Final answer:** $$\boxed{\frac{-9}{4}}$$