1. **State the problem:** Multiply the fractions $\frac{8}{9} \times \frac{5}{12} \times \frac{18}{35}$.\n\n2. **Formula and rules:** To multiply fractions, multiply the numerators together and the denominators together: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$\n\n3. **Multiply numerators and denominators:**\n$$\frac{8 \times 5 \times 18}{9 \times 12 \times 35}$$\n\n4. **Calculate numerator and denominator:**\n$$\frac{720}{3780}$$\n\n5. **Simplify the fraction:** Find the greatest common divisor (GCD) of 720 and 3780. The GCD is 180.\n\n6. **Divide numerator and denominator by 180:**\n$$\frac{\cancel{720}^{4}}{\cancel{3780}^{21}}$$\n\n7. **Final simplified answer:**\n$$\frac{4}{21}$$\n\nSo, the product of the fractions is $\frac{4}{21}$.