It rained exactly 11 days during Jane's trip.
When it rained, it rained in the morning or the afternoon but never both.
There are 13 afternoons when it did not rain.
There are 16 mornings when it did not rain.
How long was the trip?
The trip was 29 days long.
13 + 16 = 29.
The 11 days that it rained cannot be added to this total because:
If it rained in the afternoon, then the morning was clear and that day is already counted as a clear morning day.
If it rained in the morning, then the afternoon was clear and that day is already counted as a clear afternoon day.
Since there were less rainy days than either clear afternoon days or clear morning days, the rainy days could have all come from either pot or the rainy days could have come some from the clear morning pot and some from the clear afternoon pot.
Either way, the total was unchanged, and the total has to be 29.
Here's one way the days could look like:
You have to have 16 days that are clear in the morning and you have to have 13 days that are clear in the afternoon.
days 1 to 16 are clear in the morning (c).
days 17 to 29 are clear in the afternoon (c).
morning afternoon
1 c
2 c
3 c
4 c
5 c
6 c
7 c
8 c
9 c
10 c
11 c
12 c
13 c
14 c
15 c
16 c
17 c
18 c
19 c
20 c
21 c
22 c
23 c
24 c
25 c
26 c
27 c
28 c
29 c
Take any of those days and make them rainy in the afternoon or the morning depending on whether they are clear in the morning or afternoon.
You cannot add another day, because then you mess up the totals of clear mornings or clear afternoons.
You have to pick an existing day to make the other half of the day rainy.
There are many possible solutions. One of them would look like this:
morning afternoon
1 c r
2 c r
3 c r
4 c
5 c
6 c r
7 c r
8 c
9 c
10 c
11 c r
12 c
13 c r
14 c
15 c r
16 c
17 c
18 c
19 c
20 r c
21 c
22 c
23 c
24 r c
25 c
26 c
27 r c
28 r c
29 r c