Nested Cuboids with 1/3 Ratio: Calculate Volume Relationship

To solve this problem, we'll calculate the volumes of both cuboids:

• Step 1: Determine the volume of the larger cuboid
• Step 2: Determine the volume of the smaller cuboid
• Step 3: Calculate how many smaller cuboids fit inside the larger cuboid

Now, let's work through each step:
Step 1: Let the dimensions of the larger cuboid be $(L, W, H)$. The volume is $V_{large} = L \times W \times H$.
Step 2: The dimensions of the smaller cuboid are $\left(\frac{L}{3}, \frac{W}{3}, \frac{H}{3}\right)$. Thus, the volume is $V_{small} = \frac{L}{3} \times \frac{W}{3} \times \frac{H}{3} = \frac{L \times W \times H}{27}$.
Step 3: To find the number of smaller cuboids that fit inside the larger cuboid, divide the larger volume by the smaller volume:
$\frac{V_{large}}{V_{small}} = \frac{L \times W \times H}{\frac{L \times W \times H}{27}} = 27$

Therefore, the number of times the smaller cuboid fits into the larger cuboid is 27.