During a certain period of days in Cucumberland recently it was observed that when it rained in the afternoon, it had been clear in the morning, and when it rained in the morning, it was clear in the afternoon. (In a given morning or afternoon, it is either raining or it is clear.) It rained on 100 days, and was clear on 19 afternoons and 95 mornings. How many days were there altogether?
We know there are 19 clear afternoons and 95 clear mornings. We also know there are 100 days on which it rained. The following equation should show if there are any days that are clear both in the morning and in the afternoon.
1/2 x (19 + 95 - 100) = 1/2 x 14 = 7.
Thus, there are 7 days that are clear in the morning and in the afternoon.
Proof: With 19 - 7 days of clear afternoon, we have 12 days of rain in the morning and none in the afternoon. With 95 - 7 we have 88 days of rain in the afternoon and clear in the morning. Therefore, we have 12 + 88 = 100 days in which it rained, as the problem requires.