box can fit exactly 5 large cubes. It can also fit exactly 40 medium cubes orexactly 320 small cubes. A mix of large, medium and small cubes are put into sucha box. What is the greatest possible number of small cubes that can fit in the box?

To solve this problem, we need to understand that the volume of the box is constant and that the volume of the cubes is inversely proportional to the number of cubes that can fit in the box.

1. First, we need to find the volume of the box. We know that the box can fit exactly 5 large cubes, so the volume of the box is 5 times the volume of a large cube.

2. We also know that the box can fit exactly 40 medium cubes, so the volume of a medium cube is 1/40th of the volume of the box.

3. Similarly, the box can fit exactly 320 small cubes, so the volume of a small cube is 1/320th of the volume of the box.

4. If we want to fit the maximum number of small cubes in the box, we need to fill the box with the largest cubes possible, then fill the remaining space with medium cubes, and finally fill any remaining space with small cubes.

5. Since the volume of a large cube is 8 times the volume of a medium cube (because 40/5 = 8), and the volume of a medium cube is 8 times the volume of a small cube (because 320/40 = 8), we can fit 8 small cubes in the space of 1 medium cube, and 64 small cubes in the space of 1 large cube.

6. Therefore, if we fill the box with 4 large cubes and 1 medium cube, we can fit 4*64 + 8 = 264 small cubes in the box.

7. However, if we fill the box with 5 large cubes, we can fit 5*64 = 320 small cubes in the box.

8. Therefore, the greatest possible number of small cubes that can fit in the box is 320.