## 39. Memories of the Trip

During the trip, I saw Romanesque cathedrals, triumphal arches, waterfalls, and medieval castles. I photographed half of these tourist attractions. I saw three times as many cathedrals as triumphal arches, and as many medieval castles as waterfalls. One quarter of the photographs I took depict medieval castles. However, I only photographed every second castle, while all the triumphal arches I saw are in my photographs.

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-025.jpg?height=671&width=1333&top_left_y=624&top_left_x=356)

Please tell me, what fraction of the photographs of triumphal arches are among all the photographs I took during the trip?
39. Let $x$ be the number of photos of castles, $y$ be the number of photos of triumphal arches, $z$ be the number of photos of waterfalls, and $t$ be the number of photos of cathedrals. Out of all the sights seen, the following were not photographed: 0 triumphal arches, $(3 y - t)$ cathedrals, $x$ castles, and $(2 x - z)$ waterfalls.

We know that one quarter of all the photos are of castles:

$$
x = (t + x + y + z) / 4
$$

Half of all the sights have been photographed:

$$
t + x + y + z = (3 y + 2 x + 2 x + y) / 2 = 2 x + 2 y
$$

from which we get

$$
t + z = x + y
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-122.jpg?height=551&width=1114&top_left_y=1089&top_left_x=471)

From these relations, we obtain

$$
x = y \text{.}
$$

Therefore, the photos of triumphal arches among all the photos constitute

$$
y / (x + y + z + t) = 1 / 4
$$