1. **State the problem:** Calculate the product of the three fractions $$\frac{4}{21} \times \frac{7}{6} \times \frac{9}{11}$$. 2. **Formula and rules:** When multiplying fractions, multiply the numerators together and the denominators together: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$ 3. **Apply the formula:** $$\frac{4}{21} \times \frac{7}{6} \times \frac{9}{11} = \frac{4 \times 7 \times 9}{21 \times 6 \times 11}$$ 4. **Calculate numerator and denominator:** Numerator: $4 \times 7 = 28$, then $28 \times 9 = 252$ Denominator: $21 \times 6 = 126$, then $126 \times 11 = 1386$ So the fraction is: $$\frac{252}{1386}$$ 5. **Simplify the fraction:** Find the greatest common divisor (GCD) of 252 and 1386. - Prime factors of 252: $2^2 \times 3^2 \times 7$ - Prime factors of 1386: $2 \times 3 \times 7 \times 11 \times 3$ Common factors: $2 \times 3 \times 7 = 42$ Divide numerator and denominator by 42: $$\frac{252 \div 42}{1386 \div 42} = \frac{6}{33}$$ 6. **Simplify further:** 6 and 33 share a common factor 3: $$\frac{6 \div 3}{33 \div 3} = \frac{2}{11}$$ **Final answer:** $$\frac{2}{11}$$