Defect Rate

Let X be the defect rate. If you make Y widgets, XY are defective, (1-X)Y are good. You screen out 0.8 XY and send all the good ones (1-X)Y plus the bad ones you didn't detect 0.2 XY to the customer. The bad ones are 20% of the total so:

.2 XY / (.2 XY + (1-X) Y) = 20%

The Y's cancel out so

.2X / (.2X + 1 - X) = .2
X/(1-.8X) = 1
1.8 X = 1
X = 55.5%