Team alpha had one fourth as many people as team beta. after 6 people quit team beta and 12 people were transferred to team alpha from team beta, both teams became equal. how many people were there originally on team alpha?

Let "a" and "b" represent the numbers of people on teams Alpha and Beta, respectively. The problem statement tells you
  a = (1/4)b
  b - 6 - 12 = a + 12

The first equation can be rewritten to give an expresion for b:
  b = 4a
This can be substituted into the second equation to get
  4a -18 = a +12
  3a = 30
  a = 10

There were originally 10 people on team Alpha.