problem: The cost to carpet a floor is directly proportional to the area.  It costs $\$$105 to carpet a floor that is $14\times 5
  $ square feet.  How much will it cost to carpet a floor that is $16\times 13$ square feet? Express your answer in dollars.
solution: Let $C$ be the cost of carpeting a floor and $A$ be the area.  By the definition of a direct proportion, we know that $C=kA$, where $k$ is a constant.  Substituting $105$ for $C$ and $14\times 5=70$ for $A$, we can find that $k=3/2$.  Then, the cost to carpet a floor that is $16\times13$ square feet will be: \begin{align*}
  C&=kA\\
  &=(3/2)(16\times13)\\
  &=\boxed{312 \text{ dollars}}.
  \end{align*}