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?

Without loss of generality we can let the sides of the original cube be $1$.
The larger of the two cuboids has four of its sides equal to say $x$. (The other $8$ sides are $1$.)  Then the smaller cuboid has four of its sides equal to $1-x$.
The surface area of the larger cuboid is $2+4x$. (Two $1\times 1$ faces, and four $x\times 1$ faces). Similarly, the surface area of the smaller cuboid is $2+4(1-x)=6-4x$.
We are told that 
$$\frac{2+4x}{6-4x}=\frac{7}{5}.$$
Solve. We get $x=\frac{2}{3}$, making $1-x=\frac{1}{3}$. So the volumes are in the ratio $2:1$.
