Given a solid cuboid of dimensions $3 ext{ cm} imes 30 ext{ cm} imes 30 ext{ cm}$ and a cube of side $30 ext{ cm}$, compare:

**Quantity I:** Number of perfect cubes each of side $3 ext{ mm}$ that can be cut out of the cuboid.

**Quantity II:** Number of perfect cuboids of dimensions $3 ext{ mm} imes 30 ext{ mm} imes 30 ext{ mm}$ that can be cut out of the cube.

Question

Given a solid cuboid of dimensions $3 	ext{ cm} 	imes 30 	ext{ cm} 	imes 30 	ext{ cm}$ and a cube of side $30 	ext{ cm}$, compare:

**Quantity I:** Number of perfect cubes each of side $3 	ext{ mm}$ that can be cut out of the cuboid.

**Quantity II:** Number of perfect cuboids of dimensions $3 	ext{ mm} 	imes 30 	ext{ mm} 	imes 30 	ext{ mm}$ that can be cut out of the cube.

Accepted Answer

**Solution:**

1. Convert all dimensions to the same unit (millimeters):
   - $3 	ext{ cm} = 30 	ext{ mm}$
   - $30 	ext{ cm} = 300 	ext{ mm}$

2. **Quantity I:** Number of cubes of side $3 	ext{ mm}$ cut from cuboid $3 	ext{ cm} 	imes 30 	ext{ cm} 	imes 30 	ext{ cm}$

- Dimensions in mm: $30 	ext{ mm} 	imes 300 	ext{ mm} 	imes 300 	ext{ mm}$
   - Volume of cuboid = $30 	imes 300 	imes 300 = 2,700,000 	ext{ mm}^3$
   - Volume of one cube = $3 	imes 3 	imes 3 = 27 	ext{ mm}^3$
   - Number of cubes = $\frac{2,700,000}{27} = 100,000$

3. **Quantity II:** Number of cuboids of dimensions $3 	ext{ mm} 	imes 30 	ext{ mm} 	imes 30 	ext{ mm}$ cut from cube of side $30 	ext{ cm}$

- Cube side in mm: $300 	ext{ mm}$
   - Volume of cube = $300 	imes 300 	imes 300 = 27,000,000 	ext{ mm}^3$
   - Volume of one cuboid = $3 	imes 30 	imes 30 = 2,700 	ext{ mm}^3$
   - Number of cuboids = $\frac{27,000,000}{2,700} = 10,000$

4. Comparing the two quantities:
   - Quantity I = 100,000
   - Quantity II = 10,000

**Answer:** Quantity I > Quantity II

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