Nine, the circumference of a circular track is 400 meters. Person A and Person B start running clockwise along the circular track at the same time and place. A runs 52 meters per minute, and B runs 46 meters per minute. Both A and B rest for 1 minute after every 100 meters they run. When will A catch up with B?
Nine, let $x$ meters be the distance at which Yi is caught by Jia. Then
$$
\frac{x+400}{52}+\left(\frac{x+400}{100}\right)=\frac{x}{46}+\left[\frac{x}{100}\right) .
$$

where $[x]$ denotes the integer part of $x$.
Solving for $x$ gives $x=\frac{13984}{3}$.
The time when Yi is caught is:
$$
\frac{13984}{3 \times 46}+\left[\frac{13984}{300}\right]=147 \frac{1}{3} \text { (min). }
$$

Suppose Yi runs $100 n$ (where $n$ is a natural number) laps, and is then caught by Jia. Then $0 \leqslant\left(\frac{100 n+400}{52}+(a-1)+4\right)-\left(\frac{100 a}{46}-1-(n-1)\right)$ $\leqslant 1$.
Solving gives $42.62 \leqslant n \leqslant 46.6$,
Thus, when $n=43,44,45,46$, Yi can be caught by Jia.
When $n=43$, we have
$$
\frac{100 \times 43+400}{52}+42+4=136 \frac{5}{13} \text {. }
$$

And $\frac{100 \times 43}{46}+42=135.4782$.
Therefore, Jia can catch Yi after $136 \frac{5}{13}$ minutes.
When $n=44,45,46$, calculations show that Jia takes longer to catch Yi. $\square$
Comparing (1) and (2), we find that Jia needs $136 \frac{5}{13}$ minutes to catch Yi.