Boxes that are 12 inches tall are boing stacked to boxes that are 18 inches tall What is the sho height at which the two stacks will be the same height?

To find the height at which the two stacks of boxes will be the same height, we can set up an equation based on the number of boxes in each stack. Let \( x \) be the number of 12-inch boxes and \( y \) be the number of 18-inch boxes. The height of the stack of 12-inch boxes is \( 12x \) inches, and the height of the stack of 18-inch boxes is \( 18y \) inches. We want to find the height at which these two stacks are equal: \[ 12x = 18y \] To simplify this equation, we can divide both sides by 6: \[ 2x = 3y \] Now, we can express \( y \) in terms of \( x \): \[ y = \frac{2}{3}x \] Now, we can substitute \( y \) back into the height equation to find the height of the stacks: \[ \text{Height} = 12x = 18y = 18\left(\frac{2}{3}x\right) = 12x \] This shows that the heights are equal for any integer values of \( x \) and \( y \) that satisfy the equation \( 2x = 3y \). To find specific heights, we can choose integer values for \( x \) and calculate \( y \): 1. If \( x = 3 \), then \( y = \frac{2}{3} \times 3 = 2 \). - Height = \( 12 \times 3 = 36 \) inches. - Height = \( 18 \times 2 = 36 \) inches. 2. If \( x = 6 \), then \( y = \frac{2}{3} \times 6 = 4 \). - Height = \( 12 \times 6 = 72 \) inches. - Height = \( 18 \times 4 = 72 \) inches. Thus, the stacks will be the same height at any multiple of 36 inches (36, 72, 108, etc.). The general solution for the height at which the two stacks will be the same is: \[ \text{Height} = 36n \text{ inches, where } n \text{ is a positive integer.} \]