A painting $18$" X $24$" is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is:
$\text{(A) } 1:3\quad \text{(B) } 1:2\quad \text{(C) } 2:3\quad \text{(D) } 3:4\quad \text{(E) } 1:1$
Let the width of the frame on the sides to be $x$.
Then, the width of the frame on the top and bottom is $2x$.
The area of the frame is then $x\cdot 2x-18\cdot24$
Setting the area of the frame equal to the area of the picture,
\[(2x+18)(4x+24)-18\cdot24 = 18\cdot24\]
Solving,
\[8x^2+120x+432 = 864\]
\[x^2+15x-54=0\]
\[(x+18)(x-3)=0\Rightarrow x=3\]
Therefore, the ratio of the smaller side to the larger side is $\frac{18+2\cdot3}{24+4\cdot3}=\frac{24}{36}=\boxed{2:3 (C)}$.
~AOPS81619