<s>[INST] A cube is divided into two cuboids A cube is divided into two cuboids. The surfaces of those cuboids are in the ratio $7: 5$. Calculate the ratio of the volumes.
How can I calculate this?
 [/INST] Suppose the original cube had side $a$. And let dimensions of the resulting cuboids be $a \times a \times b$ and $a \times a \times c$, where $a = b+c$.
The surface areas of the resulting cuboids is $2\left(a \cdot a + a \cdot b + a \cdot b\right) = 2 a(a+2 b)$ and $2a(a+2 c)$, and volumes $a^2 b$ and $a^2 c$ respectively.
It is known that:
$$
  \frac{7}{5} = \frac{a+2b}{a+2c} \quad b+c = a
$$
Solving this gives $b = \frac{2}{3} a$ and $c=\frac{a}{3}$, hence the ratio of volumes is $\frac{a^2 b}{a^2 c} = \frac{b}{c} = 2$.
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