In a picture trading, the price of the frames is directly proportional to the value of the paintings inside them. To reduce the price difference between certain paintings, the dealer exchanges the frames of two paintings. In one case, a painting that cost five times as much as the other, after the frames were exchanged, only cost three times as much. How will the price ratio of the paintings titled "Winter Landscape" and "Village Road" change if, before the frames were exchanged, the "Winter Landscape" cost nine times as much as the "Village Road"?
Let the price of the first picture be $x \mathrm{Ft}$, the price of its frame be $y \mathrm{Ft}$, then the price of the second picture is $5 x$, and its frame is $5 y \mathrm{Ft}$. After the frames are exchanged, the first picture costs $x+5 y$, and the second picture costs $5 x+y$ Ft with the frame, and

$$
(x+5 y)=3(5 x+y)
$$

From this, $x=7 y$, which means that the price of the pictures is 7 times the price of their frames.

| Let now the |  | Winter Landscape | Village Ruins |
| :--- | :--- | :---: | :---: |
|  | price of the frame | $u \mathrm{Ft}$ | $v \mathrm{Ft}$ |
|  | then the price of the picture | $7 u \mathrm{Ft}$ | $7 v \mathrm{Ft}$ |

and according to the condition, $7 u+u=9(7 v+v)$, from which $u=9 v$. After the exchange, let the ratio of the total price of the two pictures be $\lambda$. Then

$$
7 u+v=\lambda(7 v+u)
$$

After substituting $u=9 v$, we get $64 v=\lambda 16 v$, from which $\lambda=4$ is the ratio of the prices of the pictures.

Tamás Kovács (Miskolc, Földes F. Gymnasium, 10th grade)