[mr0077]

Four gallons = 512 ounces of liquid. If he wants to change out at least 90% of the old fluid, he can allow 10%, or 51.2 oz of old fluid left in the system. Each time he changes 1/4 of the fluid he leaves 3/4 (75% or 0.75) of whatever is in it. If he did this routine 8 times, he would have 512 x 0.75^8 of old fluid. 0.75^8 = 0.1001, roughly 10%, so 512 x 0.75^8 = 51.2578 oz, slightly more than 10% old fluid. Therefore he has to do this change 9 times to have 10% or less of the old fluid (he would have 7.51% old fluid and 92.49% fresh fluid, or 473.5566 oz of fresh fluid and 38.4434 oz of old fluid.

Why did I just do that?