1. **State the problem:** Find the next number in the sequence: $10 \frac{1}{9}$, $11 \frac{1}{2}$, $13 \frac{2}{7}$, $15 \frac{11}{14}$. 2. **Convert mixed numbers to improper fractions:** $10 \frac{1}{9} = \frac{91}{9}$, $11 \frac{1}{2} = \frac{23}{2}$, $13 \frac{2}{7} = \frac{93}{7}$, $15 \frac{11}{14} = \frac{221}{14}$. 3. **Find the differences between consecutive terms:** $\frac{23}{2} - \frac{91}{9} = \frac{23 \times 9}{18} - \frac{91 \times 2}{18} = \frac{207}{18} - \frac{182}{18} = \frac{25}{18}$ $\frac{93}{7} - \frac{23}{2} = \frac{93 \times 2}{14} - \frac{23 \times 7}{14} = \frac{186}{14} - \frac{161}{14} = \frac{25}{14}$ $\frac{221}{14} - \frac{93}{7} = \frac{221}{14} - \frac{186}{14} = \frac{35}{14} = \frac{5}{2}$. 4. **Analyze the pattern in differences:** The differences are $\frac{25}{18}$, $\frac{25}{14}$, and $\frac{5}{2} = \frac{35}{14}$. Notice the numerators: 25, 25, 35; denominators: 18, 14, 14. The denominators decrease from 18 to 14 and then stay at 14, numerators increase from 25 to 35. 5. **Predict the next difference:** Assuming the numerator increases by 10 again (from 25 to 35 to 45) and denominator stays 14, next difference is $\frac{45}{14}$. 6. **Calculate the next term:** Next term = last term + next difference $$\frac{221}{14} + \frac{45}{14} = \frac{266}{14} = \frac{133}{7} = 19 \frac{0}{7} = 19.$$ 7. **Conclusion:** The next number in the sequence is $19$, which corresponds to option C.