If a camera lens is randomly selected from the general population of all camera lenses, what is the probability that it was made by the greens manufacturing company, given that it is defective?

The likelihood of selecting a lens that is defective is determined by two factors:
1) The likelihood of selecting one company over another
2) The percent chance of getting a defective lens from that company. 
Probability of selecting a Greens Lens: 60/100
Probability of selecting a Parsons Lens: 15/100
Probability of selecting a Ratten Lens: 25/100
So, from that list, we then multiply against the defect rate to determine the likelihood of selecting a company AND a defective lens:
Probability of selecting a defective Greens Lens=(3/5)*(1/25)=3/125
Probability of selecting a defective Parsons lens=(3/20)*(1/10)=3/200
Probability of selecting a defective Ratten Lens=(1/4)*(3/50)= 3/200
I cannot tell from the answer you posted whether or not you needed defective or non-defective rates, so here are the nondefective probabilities:
Probability of selecting a NON-defective Greens Lens=(3/5)*(11/20)=33/100
Probability of selecting a NON-defective Parsons lens=(3/20)*(1/20)=3/400
Probability of selecting a NON-defective Ratten Lens=(1/4)*(19/100)=19/400