1. **State the problem:** We have two stacks of boxes, one with boxes 12 inches tall and the other with boxes 18 inches tall. We want to find the shortest height at which the two stacks are the same height. 2. **Set up the equation:** Let $x$ be the number of 12-inch boxes and $y$ be the number of 18-inch boxes. We want to find the smallest positive height where: $$12x = 18y$$ 3. **Simplify the equation:** Divide both sides by 6 to simplify: $$\cancel{6} \times 2x = \cancel{6} \times 3y \implies 2x = 3y$$ 4. **Find integer solutions:** We want the smallest positive integers $x$ and $y$ such that $2x = 3y$. This means $2x$ is a multiple of 3 and $3y$ is a multiple of 2. 5. **Find least common multiple (LCM):** The shortest height corresponds to the least common multiple of 12 and 18. 6. **Calculate LCM:** Prime factors: - 12 = $2^2 \times 3$ - 18 = $2 \times 3^2$ LCM takes the highest powers: $$LCM = 2^2 \times 3^2 = 4 \times 9 = 36$$ 7. **Answer:** The shortest height at which the two stacks are the same height is **36 inches**. This means: - Number of 12-inch boxes: $\frac{36}{12} = 3$ - Number of 18-inch boxes: $\frac{36}{18} = 2$