1. **State the problem:** Add the mixed numbers $21 \frac{7}{10}$, $60 \frac{4}{5}$, and $50 \frac{5}{8}$.\n\n2. **Convert mixed numbers to improper fractions:**\n- $21 \frac{7}{10} = 21 + \frac{7}{10} = \frac{21 \times 10}{10} + \frac{7}{10} = \frac{210}{10} + \frac{7}{10} = \frac{217}{10}$\n- $60 \frac{4}{5} = 60 + \frac{4}{5} = \frac{60 \times 5}{5} + \frac{4}{5} = \frac{300}{5} + \frac{4}{5} = \frac{304}{5}$\n- $50 \frac{5}{8} = 50 + \frac{5}{8} = \frac{50 \times 8}{8} + \frac{5}{8} = \frac{400}{8} + \frac{5}{8} = \frac{405}{8}$\n\n3. **Find a common denominator to add the fractions:**\nThe denominators are 10, 5, and 8. The least common denominator (LCD) is 40.\n\n4. **Convert each fraction to have denominator 40:**\n- $\frac{217}{10} = \frac{217 \times 4}{10 \times 4} = \frac{868}{40}$\n- $\frac{304}{5} = \frac{304 \times 8}{5 \times 8} = \frac{2432}{40}$\n- $\frac{405}{8} = \frac{405 \times 5}{8 \times 5} = \frac{2025}{40}$\n\n5. **Add the fractions:**\n$$\frac{868}{40} + \frac{2432}{40} + \frac{2025}{40} = \frac{868 + 2432 + 2025}{40} = \frac{5325}{40}$$\n\n6. **Simplify the fraction:**\nDivide numerator and denominator by 5:\n$$\frac{5325 \div 5}{40 \div 5} = \frac{1065}{8}$$\n\n7. **Convert improper fraction back to mixed number:**\nDivide 1065 by 8:\n$$1065 \div 8 = 133 \text{ remainder } 1$$\nSo,\n$$\frac{1065}{8} = 133 \frac{1}{8}$$\n\n**Final answer:** $133 \frac{1}{8}$