Since when it rained in the morning it was clear in the afternoon, and when it rained in the afternoon it was clear in the morning, one can assume that it did not rain for an entire day. Given that it rained on a total of 100 days, i.e. either in the morning or in the afternoon, then what we really need to find is the number of completely clear days. It was clear on 19 afternoons and on 95 mornings, but some of these mornings and afternoons fell on the same day. That is to say, it did not rain during every one of the 19 mornings during which it was clear in that afternoon nor did it rain during every one of the 95 afternoons during which it was clear in the morning. Let n be the number of clear days, then (19 - n) + (95 - n) = 100, or n = (19 + 95 - 100) / 2 = 7. Therefore, there were 107 days altogether (100 days on which it rained and 7 completely clear days). That ratio sounds about right.
And for those of you still wondering which came first, the chicken or the egg: the egg did. Eggs have been around for millions of years (like dinosaurs, for instance) while chickens have only been around for thousands. No one said it had to be a chicken egg. Groan.