10. Xiao Li and Xiao Zhang are running on a circular track at a uniform speed. They start at the same time and place. Xiao Li runs clockwise, completing a lap every 72 seconds; Xiao Zhang runs counterclockwise, completing a lap every 80 seconds. If a $\frac{1}{4}$ circular arc interval is marked on the track with the starting point as the center, then the duration for which both are running in the marked interval is $\qquad$ seconds.
$10.3,9,11,18$.
Let the start time be 0. Then the time intervals during which Xiao Li and Xiao Zhang run in the designated area are
$$
[0,9] \text{ and } [72k-9,72k+9]
$$

and $[0,10] \text{ and } [80k-10,80k+10]$,
where $k=1,2, \cdots$.
Obviously, $[0,9]$ is the common part of the two types of time intervals in the first 9 seconds.
In addition, consider
$$
[72k-9,72k+9] \text{ and } [80m-10,80m+10]
$$

the common interval, where $k, m$ are positive integers.
There are two cases.
(1) $72k=80m$, that is, Xiao Li and Xiao Zhang return to the starting point after running $k$ laps and $m$ laps respectively, and they both run in the designated area for 18 seconds.
(2) $72k \neq 80m$, for example,
$$
\begin{array}{l}
72k-9 \leqslant 80m-10 \leqslant 72k+9 \leqslant 80m+10 \\
\Leftrightarrow 1 \leqslant 80m-72k \leqslant 19 .
\end{array}
$$

Both run in the designated area for
$$
72k+9-(80m-10)=19-(80m-72k) \text{. }
$$

From equation (1), we know $80m-72k=8,16$.
Therefore, the duration for which both run in the designated area is
11 seconds or 3 seconds, and other cases similarly yield the same result.
In summary, the answers are $3,9,11,18$.