Volume Ratio Problem: 5 Spheres (10 cm³) Inside 125 cm³ Cube

To solve this problem, let's follow these steps:

• Step 1: Calculate the total volume of the spheres.
• Step 2: Determine the volume remaining in the cube after inserting the spheres.
• Step 3: Calculate the ratio between the total volume of the spheres and the remaining volume of the cube.

Now, let's work through each step:

Step 1: The volume of each sphere is given as 10 cm$^3$, and there are 5 spheres. Thus, the total volume of the spheres is:
$5 \times 10 = 50$ cm$^3$.

Step 2: The volume of the cube is given as 125 cm$^3$. After inserting the spheres, the remaining volume of the cube is:
$125 - 50 = 75$ cm$^3$.

Step 3: Calculate the ratio of the total volume of the spheres to the remaining volume of the cube:
$\frac{50 \text{ cm}^3}{75 \text{ cm}^3} = \frac{2}{3}$.

Therefore, the ratio between the total volume of the spheres and the volume remaining in the cube is $\frac{2}{3}$.