1. **State the problem:** Simplify the expression $8 \frac{1}{15} - \frac{5}{6} - \frac{9}{10}$. 2. **Convert mixed number to improper fraction:** $8 \frac{1}{15} = 8 + \frac{1}{15} = \frac{8 \times 15}{15} + \frac{1}{15} = \frac{120}{15} + \frac{1}{15} = \frac{121}{15}$. 3. **Rewrite the expression:** $\frac{121}{15} - \frac{5}{6} - \frac{9}{10}$. 4. **Find the least common denominator (LCD):** The denominators are 15, 6, and 10. - Prime factors: 15 = 3 \times 5, 6 = 2 \times 3, 10 = 2 \times 5. - LCD = 2 \times 3 \times 5 = 30. 5. **Convert each fraction to have denominator 30:** - $\frac{121}{15} = \frac{121 \times 2}{15 \times 2} = \frac{242}{30}$. - $\frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30}$. - $\frac{9}{10} = \frac{9 \times 3}{10 \times 3} = \frac{27}{30}$. 6. **Rewrite expression with common denominator:** $\frac{242}{30} - \frac{25}{30} - \frac{27}{30}$. 7. **Combine the fractions:** $\frac{242 - 25 - 27}{30} = \frac{190}{30}$. 8. **Simplify the fraction:** - Find the greatest common divisor (GCD) of 190 and 30, which is 10. - Cancel common factor: $\frac{\cancel{10} \times 19}{\cancel{10} \times 3} = \frac{19}{3}$. 9. **Convert improper fraction to mixed number:** - $\frac{19}{3} = 6 \frac{1}{3}$. **Final answer:** $6 \frac{1}{3}$