Agym membership at one gym costs $10 every month plus a onetime membership fee of $15, and a gym membership at another gym cost $4every month plus a onetime $40 membership fee. after about how many months will the gym memberships cost the same amount

Let us assume that the number of months after which
the gym memberships will cost the same = x
Then
15 + 10x = 40 + 4x
10x - 4x = 40 - 15
6x = 25
x = 25/6
  = 4 1/6
So the membership fee of both the gyms will become same after 4 1/6 months. You can also consider that as after 5 months the membership fees of both the gyms will become same. I hope the procedure is not very complicated for you to understand.