The volume of a cube is four times the volume of a cuboid. If the sides of the cuboid are $32\text{cm}$, $8\text{cm}$ and $4\text{cm}$, then find the ratio of the total surface area of the cube to that of the cuboid.

Step 1: Calculate the volume and dimensions of the cube
The formula for the volume of a cuboid is $V_{\text{cuboid}}=l\times b\times h$.
Given dimensions of the cuboid: $l=32\text{cm}$, $b=8\text{cm}$, $h=4\text{cm}$.

$$V_{\text{cuboid}}=32\times 8\times 4=1024{\text{cm}}^3$$

The volume of the cube ($V_{\text{cube}}$) is four times that of the cuboid:

$$V_{\text{cube}}=4\times 1024=4096{\text{cm}}^3$$

Let the side of the cube be $a$. Then:

$$a^3=4096⟹a=\sqrt[3]{4096}=16\text{cm}$$

Step 2: Calculate the total surface area of both shapes

* Total Surface Area of the cube (${\text{TSA}}_{\text{cube}}$):

$${\text{TSA}}_{\text{cube}}=6a^2=6\times (16{)}^2=6\times 256=1536{\text{cm}}^2$$

* Total Surface Area of the cuboid (${\text{TSA}}_{\text{cuboid}}$):

$${\text{TSA}}_{\text{cuboid}}=2(lb+bh+lh)$$

$${\text{TSA}}_{\text{cuboid}}=2(32\times 8+8\times 4+4\times 32)=2(256+32+128)=2\times 416=832{\text{cm}}^2$$

Step 3: Find the required ratio

$$\text{Ratio}=\frac{{\text{TSA}}_{\text{cube}}}{{\text{TSA}}_{\text{cuboid}}}=\frac{1536}{832}$$

Dividing both terms by their greatest common divisor ($64$):

$$\text{Ratio}=\frac{1536÷64}{832÷64}=\frac{24}{13}=24:13$$