I was wrong. The problem is, you can't go to the least common denominator. So if your running average is represented by 10/5, a ratio would represent that as 2/1. Then if you wanted to average in a sixth quantity, its numerator would be added to 2, not 10, and your count would be back to 2 instead of updating to 6.
Also, ratios only work with ints. You couldn't use them to average floats. Well, you could try converting back to a ratio but then you'd have unpredictable numerators/denominators again.