1. **Stating the problem:** Multiply the three fractions $\frac{4}{5}$, $\frac{20}{9}$, and $\frac{18}{8}$.\n\n2. **Formula used:** To multiply fractions, multiply the numerators together and the denominators together: $$\frac{a}{b} \times \frac{c}{d} \times \frac{e}{f} = \frac{a \times c \times e}{b \times d \times f}.$$\n\n3. **Apply the formula:**\n$$\frac{4}{5} \times \frac{20}{9} \times \frac{18}{8} = \frac{4 \times 20 \times 18}{5 \times 9 \times 8}.$$\n\n4. **Calculate numerator and denominator:**\nNumerator: $4 \times 20 = 80$, then $80 \times 18 = 1440$.\nDenominator: $5 \times 9 = 45$, then $45 \times 8 = 360$.\nSo, $$\frac{1440}{360}.$$\n\n5. **Simplify the fraction:**\nDivide numerator and denominator by 360: $$\frac{\cancel{1440}^{4}}{\cancel{360}^{1}} = \frac{4}{1} = 4.$$\n\n6. **Final answer:** The product of the three fractions is $4$.