Here are two cubes, A and B - Edexcel - GCSE Maths - Question 9 - 2022 - Paper 1

Density is defined as mass per unit volume:

$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

For cube A:

$\text{Density}_A = \frac{81 \text{ g}}{27 \text{ cm}^3} = 3 \text{ g/cm}^3$

For cube B, with a side length of 4 cm:

$V_B = 4^3 = 64 \text{ cm}^3$

Using the same density formula:

$\text{Density}_B = \frac{128 \text{ g}}{64 \text{ cm}^3} = 2 \text{ g/cm}^3$

Now, we can find the ratio of the densities of cube A to cube B:

$\text{Density Ratio} = \frac{\text{Density}_A}{\text{Density}_B} = \frac{3}{2}$

Thus, the answer in the form a : b is:

$3 : 2$