1. **State the problem:** We are given the sequence $10 \frac{1}{9}$, $11 \frac{1}{2}$, $13 \frac{2}{7}$, $15 \frac{11}{14}$ and asked to find the next number. 2. **Convert mixed numbers to improper fractions for easier analysis:** $$10 \frac{1}{9} = 10 + \frac{1}{9} = \frac{90}{9} + \frac{1}{9} = \frac{91}{9}$$ $$11 \frac{1}{2} = 11 + \frac{1}{2} = \frac{22}{2} + \frac{1}{2} = \frac{23}{2}$$ $$13 \frac{2}{7} = 13 + \frac{2}{7} = \frac{91}{7} + \frac{2}{7} = \frac{93}{7}$$ $$15 \frac{11}{14} = 15 + \frac{11}{14} = \frac{210}{14} + \frac{11}{14} = \frac{221}{14}$$ 3. **Look at the whole number parts:** 10, 11, 13, 15. The differences are +1, +2, +2. 4. **Look at the fractional parts:** $\frac{1}{9}$, $\frac{1}{2}$, $\frac{2}{7}$, $\frac{11}{14}$. 5. **Convert fractions to decimals to detect pattern:** $\frac{1}{9} \approx 0.111$, $\frac{1}{2} = 0.5$, $\frac{2}{7} \approx 0.2857$, $\frac{11}{14} \approx 0.7857$. 6. **Check if fractional parts follow a pattern:** - From $0.111$ to $0.5$ is an increase of about $0.389$. - From $0.5$ to $0.2857$ is a decrease of about $0.2143$. - From $0.2857$ to $0.7857$ is an increase of $0.5$. No simple arithmetic progression. 7. **Check the denominators:** 9, 2, 7, 14. Notice $14 = 2 \times 7$. 8. **Check the numerators:** 1, 1, 2, 11. No obvious pattern. 9. **Check the whole number plus fraction as decimals:** $10 \frac{1}{9} \approx 10.111$ $11 \frac{1}{2} = 11.5$ $13 \frac{2}{7} \approx 13.2857$ $15 \frac{11}{14} \approx 15.7857$ 10. **Look at the decimal parts: 0.111, 0.5, 0.2857, 0.7857.** Notice the last two decimals $0.2857$ and $0.7857$ share the same fractional part $0.7857 - 0.2857 = 0.5$. 11. **Try to find a pattern in the whole numbers:** 10, 11, 13, 15. Differences: +1, +2, +2. If the pattern continues, next difference might be +3, so next whole number is $15 + 3 = 18$. 12. **Check answer choices with whole number 18:** Only option B has 18 $\frac{25}{28}$. 13. **Check if $\frac{25}{28}$ fits the fractional pattern:** $\frac{25}{28} \approx 0.8929$. Compare to previous fraction $\frac{11}{14} = 0.7857$. Difference $0.8929 - 0.7857 = 0.1072$. No clear pattern but this is the only option with whole number 18. 14. **Conclusion:** The next number is likely $18 \frac{25}{28}$ (option B). **Final answer:** $\boxed{18 \frac{25}{28}}$