166. A rectangular picture is inserted into a rectangular frame made of strips of the same width. The area of the picture is equal to the area of the frame. What is the ratio of the length to the width of the picture in the frame, if the ratio of the length to the width of the picture without the frame is $2: 3$?
Solution. Let $x$ be the length of the picture without the frame. Then $\frac{3 x}{2}$ is its width. If $l$ is the width of the strips from which the frame is made, then the area of the picture with the frame is $(x+2 l)\left(\frac{3 x}{2}+2 l\right)$. By equating this area to twice the area of the picture without the frame, i.e., to $3 x^{2}\left(2 \cdot x \cdot \frac{3 x}{2}=3 x^{2}\right)$, we arrive at the equation $3 x^{2}-10 x l=8 l^{2}$, from which it follows that $3\left(\frac{x}{l}\right)^{2}-10 \frac{x}{l}-8=0, \frac{x}{l}=4,(x+2 l):\left(\frac{3 x}{2}+\right.$ $+2 l)=3: 4$.

Answer: $3: 4$.