1. A bicycle tire, if installed on the front wheel, will wear out after traveling $5000 \mathrm{~km}$; if installed on the back wheel, it will wear out after traveling $3000 \mathrm{~km}$. If the front and back tires are swapped after traveling a certain distance, so that a pair of new tires wear out simultaneously, then the maximum distance that can be traveled is $\mathrm{km}$.
II. 1.3750.
Assume the total wear of each new tire when it is scrapped is $k$, then the wear per kilometer for a tire installed on the front wheel is $\frac{k}{5000}$, and the wear per kilometer for a tire installed on the rear wheel is $\frac{k}{3000}$. Let a pair of new tires travel $x \mathrm{~km}$ before swapping positions, and $y \mathrm{~km}$ after swapping positions. By setting up equations based on the total wear of a tire, we have
$$
\left\{\begin{array}{l}
\frac{k x}{5000}+\frac{k y}{3000}=k, \\
\frac{k y}{5000}+\frac{k x}{3000}=k .
\end{array}\right.
$$

Adding the two equations gives
$$
\frac{k(x+y)}{5000}+\frac{k(x+y)}{3000}=2 k \text {. }
$$

Thus, $x+y=\frac{2}{\frac{1}{5000}+\frac{1}{3000}}=3750(\mathrm{~km})$.