Party A and Party B run back and forth on the straight track AB.

-It takes 3 minutes for A to run from A to B, that is, the speed of A is AB/3;

It takes 4 minutes for B to run from A to B, which means that the speed of B is AB/4.

Suppose we meet for the first time, A runs for x minutes and B runs for x minutes. According to their speed, the equation can be listed as follows:

Distance a operation: AB = (AB/3) * x

B running distance: ab = (ab/4) * X.

Substituting AB into the equation gives:

(AB/3) * x = (AB/4) * x

After sorting and simplifying, you can get:

4x = 3x

x = 3

So, when they first met, A and B ran for 3 minutes.

This problem is actually a problem of the least common multiple. Party A and Party B set out at the same time, and when they met for the first time, they actually found the first coincidence point. Because the speeds of A and B are different, their meeting time is determined by their respective speeds. By calculating the least common multiple, we can find the time when they first met.

In this problem, the speed of A and B are AB/3 and AB/4 respectively, and they ran for 3 minutes when they first met. This method can help students better understand the concept of least common multiple and apply it to practical problems.

I hope this answer can help you understand the solution to this problem! If you have any other questions, please keep asking.