1. **State the problem:** Calculate the product of the fractions $\frac{9}{10} \times \frac{2}{7} \times \frac{5}{6}$.\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. **Apply the formula:** Multiply the numerators: $9 \times 2 \times 5 = 90$.\nMultiply the denominators: $10 \times 7 \times 6 = 420$.\nSo, $$\frac{9}{10} \times \frac{2}{7} \times \frac{5}{6} = \frac{90}{420}.$$\n\n4. **Simplify the fraction:** Find the greatest common divisor (GCD) of 90 and 420.\n- Prime factors of 90: $2 \times 3^2 \times 5$.\n- Prime factors of 420: $2^2 \times 3 \times 5 \times 7$.\nThe common factors are $2 \times 3 \times 5 = 30$.\nDivide numerator and denominator by 30: $$\frac{90 \div 30}{420 \div 30} = \frac{3}{14}.$$\n\n5. **Final answer:** $$\frac{9}{10} \times \frac{2}{7} \times \frac{5}{6} = \frac{3}{14}.$$