1. **State the problem:** We need to find the age $y$ of a child given the child's dosage $D=20$ mg and the adult dosage $A=40$ mg using Young's formula: $$D = \frac{yA}{y + 12}$$ 2. **Write down the formula:** $$20 = \frac{y \times 40}{y + 12}$$ 3. **Multiply both sides by $(y + 12)$ to eliminate the denominator:** $$20(y + 12) = 40y$$ 4. **Distribute 20 on the left side:** $$20y + 240 = 40y$$ 5. **Bring all terms involving $y$ to one side:** $$20y + 240 = 40y \implies 240 = 40y - 20y$$ 6. **Simplify the right side:** $$240 = 20y$$ 7. **Divide both sides by 20 to solve for $y$:** $$\frac{240}{\cancel{20}} = \frac{20y}{\cancel{20}} \implies 12 = y$$ 8. **Interpret the result:** The child is 12 years old. **Final answer:** $\boxed{12}$ years old. This corresponds to option D.