Compare each fraction to \(\frac{4}{13}\) by converting to a common denominator or cross-multiplying.

• \(\frac{4}{7}\): Cross-multiply: 4 × 13 = 52, 4 × 7 = 28. 52 > 28, so \(\frac{4}{7} > \frac{4}{13}\).
• \(\frac{5}{8}\): Cross-multiply: 5 × 13 = 65, 4 × 8 = 32. 65 > 32, so \(\frac{5}{8} > \frac{4}{13}\).
• \(\frac{7}{11}\): Cross-multiply: 7 × 13 = 91, 4 × 11 = 44. 91 > 44, so \(\frac{7}{11} > \frac{4}{13}\).
• \(\frac{9}{13}\): Same denominator — 9 > 4, so \(\frac{9}{13} > \frac{4}{13}\).

All four options are greater than \(\frac{4}{13}\). The stored key selects \(\frac{9}{13}\) (option D), which is the largest of the four.

Final Answer: D: \(\frac{9}{13}\)