1. **State the problem:** Calculate the value of $\frac{1}{5} \div \left(5 \frac{1}{2} - 3 \frac{1}{2}\right)$. 2. **Convert mixed numbers to improper fractions:** $5 \frac{1}{2} = \frac{11}{2}$ and $3 \frac{1}{2} = \frac{7}{2}$. 3. **Subtract the fractions inside the parentheses:** $$\frac{11}{2} - \frac{7}{2} = \frac{11 - 7}{2} = \frac{4}{2} = 2.$$ 4. **Rewrite the division problem:** $$\frac{1}{5} \div 2.$$ 5. **Recall the division rule for fractions:** Dividing by a number is the same as multiplying by its reciprocal. So, $$\frac{1}{5} \div 2 = \frac{1}{5} \times \frac{1}{2}.$$ 6. **Multiply the fractions:** $$\frac{1}{5} \times \frac{1}{2} = \frac{1 \times 1}{5 \times 2} = \frac{1}{10}.$$ **Final answer:** $$\boxed{\frac{1}{10}}.$$