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Problem 599

AMC 12 late, AIME early

Geometry Difficulty 5.0 Multiple choice

18.58 - A 18 inch $\times$ 24 inch painting is framed in a wooden frame with the longer vertical sides. The width of the wood at the top and bottom of the frame is twice the width of the wood on the sides. If the area of the frame equals the area of the painting, then the ratio of the shorter to the longer side of the frame is

Pick one

Official solution

[Solution] Let the top and bottom width of the frame be $x$ , and the side width of the frame be $\frac{x}{2}$ . According to the problem, we have:
$\begin{array}{c} (2 x+24)(x+18)=2 \cdot 18 \cdot 24, \\ x^{2}+30 x-216=0 . \end{array}$

Solving for $x$ gives $x=6$ or -36 (discard).
Then the ratio of the frame sides is:
$(x+18):(2 x+24)=24: 36=2: 3 \text {. }$

Therefore, the answer is (C).

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.