3.48 A young man was returning home from vacation on a bicycle. At first, after traveling several kilometers, he spent one day more than half the number of days remaining after this until the end of his vacation. Now the young man has two options to travel the remaining distance to arrive home on time: to travel $h$ km more daily than originally planned, or to maintain the original daily travel distance, exceeding it only once - on the last day of the journey by $2 h$ km. How many days before the end of his vacation did the young man set out for home?
3.48 Let the segment $A B$ (Fig. 3.11) represent the entire journey of the young man and the number of days ($x$) it should take him to travel it at a rate of $v$ km per day. According to the problem, the time segment $A C$ is one unit more than half of the time segment $C B$; therefore,

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-112.jpg?height=169&width=484&top_left_y=706&top_left_x=352)

Fig. 3.11

$A C-1=\frac{C B}{2}$, or $x-C B-1=\frac{C B}{2}$,

from which $C B=\frac{2 x-2}{3}$ days. Let the distance represented by the segment $C B$ be taken as the unit. Then, according to the first option, $\frac{2 x-2}{3}(v+h)=1$, and according to the second, $\left(\frac{2 x-2}{3}-1\right) v+(v+2 h) \cdot 1=1$. By eliminating $v$ from these two equations, we find $x=4$.

Answer: $\quad$ in 4 days.