Will do math for chocolate

I don’t usually pay too much attention to the conversations in my office that are in German. My office mates both have many people coming in to talk to them and are both German, so speak in German with their German visitors. This particular conversation caught my attention primarily because it repeatedly used the work ‘Schokolade” one of the few German words I have no trouble with. Also there were four people involved. Eventually one of my office mates said, “Ask Dan, he is very interested in chocolate!”

So the conversation was about a simple demography puzzle, and my boss had offered a chocolate bar to whoever solved it first.
The problem is this: 100% of a population were alive at the beginning of a year. 60% were alive at the end of the year. Assuming that the mortality rate is constant throughout the year, what percentage of individuals were alive at the exact middle of the year?

Note that a constant mortality rate does not mean that the same absolute number of individuals die each day, but rather than the same proportion of those starting the day alive end the day dead.
Motivated by chocolate, and using one simple bit of algebra, I secured the Intense Orange Lindt dark chocolate bar.

So I can’t hang a chocolate bar in the comments section, but I can promise kudos if you can tell me what portion of individuals were alive after six months, and how you got that answer.