1. **State the problem:** We need to determine which fraction among $\frac{3}{5}$, $\frac{6}{13}$, $\frac{5}{9}$, and $\frac{4}{7}$ is the largest. 2. **Convert each fraction to a decimal for easy comparison:** - $\frac{3}{5} = 0.6$ - $\frac{6}{13} \approx 0.4615$ - $\frac{5}{9} \approx 0.5556$ - $\frac{4}{7} \approx 0.5714$ 3. **Compare the decimal values:** - $0.6$ (from $\frac{3}{5}$) - $0.4615$ (from $\frac{6}{13}$) - $0.5556$ (from $\frac{5}{9}$) - $0.5714$ (from $\frac{4}{7}$) 4. **Identify the largest decimal:** $0.6$ is the largest among these values. 5. **Conclusion:** The largest fraction is $\frac{3}{5}$. **Final answer:** $\boxed{\frac{3}{5}}$