## 28. Passenger Ship

During a long voyage of one passenger ship, it was noticed that at each dock a quarter of the passenger composition is renewed, that among the passengers leaving the ship, only one in ten boarded at the previous dock, and finally, that the ship is always fully loaded.

Determine what fraction of the passengers, at each moment when the ship is underway, have not boarded at any of the two previous docks?
28. Let the ship travel from the $n$-th pier to the ( $n+1$ )-th pier. It is known that:
1) a quarter of the total number of passengers appeared on the ship at the $n$-th pier;
2) among the passengers who left the ship at the $n$-th pier, every tenth boarded at the ( $n-1$ )-th pier;
3) a quarter of the total number of passengers boarded at the $(n-1)$-th pier.

Therefore, the fraction of passengers who boarded at the ( $n-1$ )-th or $n$-th pier, in the total number of passengers currently on the ship, is

$$
1 / 4 + 1 / 4 - 1 / 4 \cdot 1 / 10 = 19 / 40
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-116.jpg?height=540&width=876&top_left_y=564&top_left_x=590)

Therefore, the fraction of passengers who did not board the steamboat at any of the two previous piers is

$$
1 - 19 / 40 = 21 / 40 = 52.5\% \text{. }
$$