1. **State the problem:** We want to find how many $\frac{1}{3}$-inch cubes fit inside a rectangular prism with dimensions 3 inches (height), 2 inches (length), and $1 \frac{2}{3}$ inches (width). 2. **Formula and approach:** To find the total number of small cubes that fit, we calculate how many cubes fit along each dimension and then multiply these counts: $$\text{Total cubes} = \text{cubes along length} \times \text{cubes along width} \times \text{cubes along height}$$ 3. **Calculate cubes along each dimension:** - Length: $2$ inches divided by $\frac{1}{3}$ inch per cube: $$\frac{2}{\frac{1}{3}} = 2 \times 3 = 6$$ cubes - Width: $1 \frac{2}{3}$ inches is $\frac{5}{3}$ inches. Dividing by $\frac{1}{3}$ inch per cube: $$\frac{\frac{5}{3}}{\frac{1}{3}} = \frac{5}{3} \times 3 = 5$$ cubes - Height: 3 inches divided by $\frac{1}{3}$ inch per cube: $$\frac{3}{\frac{1}{3}} = 3 \times 3 = 9$$ cubes 4. **Calculate total number of cubes:** $$6 \times 5 \times 9 = 270$$ cubes 5. **Summary in table form:** - $\frac{1}{3}$-inch cubes across bottom layer = $6 \times 5 = 30$ - Number of layers (height) = 9 - Total number of $\frac{1}{3}$-inch cubes = 270 **Final answer:** $$\boxed{270}$$ cubes fit inside the rectangular prism.