1. **State the problem:** Calculate the value of the expression $21 \frac{1}{2} - 36 \frac{2}{7} - 16 \frac{1}{2} - 45 \frac{3}{7} + 10 \frac{5}{7}$.\n\n2. **Convert mixed numbers to improper fractions:**\n$21 \frac{1}{2} = \frac{43}{2}$\n$36 \frac{2}{7} = \frac{254}{7}$\n$16 \frac{1}{2} = \frac{33}{2}$\n$45 \frac{3}{7} = \frac{318}{7}$\n$10 \frac{5}{7} = \frac{75}{7}$\n\n3. **Rewrite the expression with improper fractions:**\n$$\frac{43}{2} - \frac{254}{7} - \frac{33}{2} - \frac{318}{7} + \frac{75}{7}$$\n\n4. **Find a common denominator:** The denominators are 2 and 7, so the common denominator is $14$.\n\n5. **Convert each fraction to denominator 14:**\n$\frac{43}{2} = \frac{43 \times 7}{14} = \frac{301}{14}$\n$\frac{254}{7} = \frac{254 \times 2}{14} = \frac{508}{14}$\n$\frac{33}{2} = \frac{33 \times 7}{14} = \frac{231}{14}$\n$\frac{318}{7} = \frac{318 \times 2}{14} = \frac{636}{14}$\n$\frac{75}{7} = \frac{75 \times 2}{14} = \frac{150}{14}$\n\n6. **Substitute back:**\n$$\frac{301}{14} - \frac{508}{14} - \frac{231}{14} - \frac{636}{14} + \frac{150}{14}$$\n\n7. **Combine all terms over common denominator:**\n$$\frac{301 - 508 - 231 - 636 + 150}{14} = \frac{(301 - 508) - 231 - 636 + 150}{14}$$\n$$= \frac{-207 - 231 - 636 + 150}{14} = \frac{-438 - 636 + 150}{14}$$\n$$= \frac{-1074 + 150}{14} = \frac{-924}{14}$$\n\n8. **Simplify the fraction:**\n$\frac{-924}{14} = -66$\n\n**Final answer:** $-66$