I think this is a really badly worded question.

If we let H mean "heavy traffic" and T mean "get there on time", then "half of the time there is heavy traffic" means that P(H)=0.5.

"In the past, there has been heavy traffic and you have made it to your appointment within 30 minutes 34% of the time." This is the wording of this question I don't like. To get the answer you say is correct, this has to be taken to mean P(H and T) = 0.34.

"You get out on the street and see that there is heavy traffic. What is the chance you will get to your appointment on time?"

So you are given that there is heavy traffic, so you need to find P(T l H).

So using P(T l H) = P(H and T) / P(T) you get 0.34/0.5 = 0.68.

As I said, the bit in the middle of the question is badly worded.