1. **State the problem:** Simplify the expression $3 \frac{1}{6} \div \frac{15}{2}$. 2. **Convert mixed number to improper fraction:** $$3 \frac{1}{6} = \frac{3 \times 6 + 1}{6} = \frac{19}{6}$$ 3. **Rewrite the division as multiplication by reciprocal:** $$\frac{19}{6} \div \frac{15}{2} = \frac{19}{6} \times \frac{2}{15}$$ 4. **Multiply the fractions:** $$\frac{19}{6} \times \frac{2}{15} = \frac{19 \times 2}{6 \times 15} = \frac{38}{90}$$ 5. **Simplify the fraction by dividing numerator and denominator by their greatest common divisor (GCD):** GCD of 38 and 90 is 2. $$\frac{\cancel{38}^{19}}{\cancel{90}^{45}} = \frac{19}{45}$$ 6. **Final answer:** $$\boxed{\frac{19}{45}}$$