1. **State the problem:** Simplify the expression $$\frac{36 \pi}{5} \times \frac{8}{20} = \frac{1}{\pi} - \frac{1}{\pi} \times \frac{5}{8}.$$\n\n2. **Simplify the left side:** Multiply the fractions on the left side:\n$$\frac{36 \pi}{5} \times \frac{8}{20} = \frac{36 \pi \times 8}{5 \times 20} = \frac{288 \pi}{100}.$$\n\n3. **Simplify the fraction:**\n$$\frac{288 \pi}{100} = \frac{\cancel{288}^{{\div 4}} \pi}{\cancel{100}^{{\div 4}}} = \frac{72 \pi}{25}.$$\n\n4. **Simplify the right side:**\n$$\frac{1}{\pi} - \frac{1}{\pi} \times \frac{5}{8} = \frac{1}{\pi} - \frac{5}{8\pi}.$$\n\n5. **Find common denominator on the right side:**\n$$\frac{1}{\pi} = \frac{8}{8\pi},$$\nso\n$$\frac{8}{8\pi} - \frac{5}{8\pi} = \frac{8 - 5}{8\pi} = \frac{3}{8\pi}.$$\n\n6. **Final simplified expression:**\n$$\frac{72 \pi}{25} = \frac{3}{8 \pi}.$$\n\nThis is the simplified form of the given expression.