Question:
Fake gold bricks are made by covering concrete cubes with gold paint, so the cost of the paint is proportional to their surface area while the cost of the concrete is proportional to their volume.  If a 1 inch cube costs $\$1.30$ to make while a 2 inch cube costs $\$6.80$, then how much would a 3 inch cube cost?

Answer:
Let $x$ be the cost per square inch of gold paint, and let $y$ be the cost per cubic inch of concrete.  Since a 1 inch cube has surface area 6 $\text{in}^2$ and volume 1 $\text{in}^3$ its total price will be $6x+y$ dollars.  Similarly, a 2 inch cube has surface area 24 $\text{in}^2$ and volume 8 $\text{in}^3$, so its total price will be $24x+8y$ dollars.  We are given that \begin{align*} 6x+y &=\$1.30 \\ 24x+8y&= \$6.80 \end{align*} Subtracting 4 times the first equation from the second gives $4y=\$1.60$, so $y=\$0.40$.  Hence $6x=\$0.90$, so $x=\$0.15$.  Since a 3 inch cube has surface area 54 $\text{in}^2$ and volume 27 $\text{in}^3$, its total price will be a total of $54(\$0.15)+27(\$0.40)=\boxed{\$18.90}$.