Alright, so what we might want to start out with is that 9/22 can be multiplied by 0.5/0.5 to get 4.5/11. We know that 1/11=0.090909 and so on, and we can round that to 0.09. Multiplying that by 4, we get 0.36. Adding 1/2 of 0.09 to that (as 9*4.5 = 1*9+1*9+1*9+1*9+0.5*9), we get 0.405, which can be rounded to 0.41. If we wanted to get more specific, we could round to, for example, 0.09090909. Multiplying that by 4.5, we get 0.409090905, establishing a pattern of 0.4+a bunch of 09's repeating.
Feel free to ask further questions!