The diagram shows a solid metal cuboid - Edexcel - GCSE Maths - Question 9 - 2018 - Paper 3

From the equations, we can express:

1. From the first equation, we express y: $y = \frac{27}{x}$
2. Substituting this into the second equation: $\frac{27}{x} \times z = 45 \Rightarrow z = \frac{45x}{27} \Rightarrow z = \frac{5x}{3}$
3. Substitute y into the third equation: $x \times \frac{5x}{3} = 15 \Rightarrow \frac{5x^2}{3} = 15 \Rightarrow 5x^2 = 45 \Rightarrow x^2 = 9 \Rightarrow x = 3$

This gives us x = 3 cm.

Now substituting x back to get y and z:

1. Calculate y: $y = \frac{27}{3} = 9 \text{ cm}$
2. Calculate z: $z = \frac{5 \times 3}{3} = 5 \text{ cm}$

Thus, the dimensions of the cuboid are 3 cm, 9 cm, and 5 cm.

The volume V of the cuboid is given by: $V = x \times y \times z = 3 \times 9 \times 5 = 135 \text{ cm}^3$

Now, each cube has a side length of 2.5 cm. Thus, the volume of each cube is: $V_{cube} = (2.5)^3 = 15.625 \text{ cm}^3$

To find the greatest number of cubes possible, divide the cuboid volume by the cube volume: $\text{Number of cubes} = \frac{135}{15.625} \approx 8.64$

Since we cannot have a fraction of a cube, the maximum number of cubes is 8.