1. **State the problem:** Solve the expression $\frac{1}{2} \times (6k+1) \times 3k$. 2. **Write the expression clearly:** $$\frac{1}{2} \times (6k+1) \times 3k$$ 3. **Multiply the constants first:** $$\frac{1}{2} \times 3k = \frac{3k}{2}$$ 4. **Rewrite the expression:** $$\frac{3k}{2} \times (6k+1)$$ 5. **Distribute $\frac{3k}{2}$ over $(6k+1)$:** $$\frac{3k}{2} \times 6k + \frac{3k}{2} \times 1 = \frac{3k \times 6k}{2} + \frac{3k}{2}$$ 6. **Multiply the terms:** $$\frac{18k^2}{2} + \frac{3k}{2}$$ 7. **Simplify the fraction $\frac{18k^2}{2}$ by canceling common factors:** $$\frac{\cancel{18}k^2}{\cancel{2}} = 9k^2$$ 8. **Final simplified expression:** $$9k^2 + \frac{3k}{2}$$ **Answer:** $9k^2 + \frac{3k}{2}$