1. **State the problem:** We need to find which fraction among $\frac{5}{9}$, $\frac{4}{7}$, $\frac{3}{5}$, $\frac{6}{11}$, and $\frac{13}{21}$ has the smallest value. 2. **Method:** To compare fractions, convert them to decimal equivalents or cross-multiply to compare values without converting to decimals. 3. **Calculate decimal equivalents:** - $\frac{5}{9} \approx 0.5556$ - $\frac{4}{7} \approx 0.5714$ - $\frac{3}{5} = 0.6$ - $\frac{6}{11} \approx 0.5455$ - $\frac{13}{21} \approx 0.6190$ 4. **Compare decimals:** Among these values, $0.5455$ (which is $\frac{6}{11}$) is the smallest. 5. **Conclusion:** The fraction with the smallest value is $\boxed{\frac{6}{11}}$.