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First of all, nothing is shown to the right, although the post promises it.
I understand your difficulties. In such case, you MUST provide an adequate wording description.
I strongly (strictly) suspect that these circles touch each other, having one (and only one) common point.
(Simply because I do not see any other alternative possibility).
If so (I am 129% confident it is so), then the solution can be easily done as follows.
The pirate' ship makes = of his circle per day.
It needs 8 days to make 3 circles, and it makes integer number of circles every 8 days (as a check, 8*12 = 96 = 32*3 miles.)
The merchant ship makes = of his circle per day.
It needs 6 days to make 1 circle, and it makes integer number of circles every 6 days (as a check, 6*8 = 48 = 48 miles.)
The two ships can meet each other at the touching point ONLY, and
for it, each ship should make integer number of their corresponding circles.
Therefore, we should find the Least Common Multiple of the integer numbers 8 and 6.
This LCM = LCM(8,6) = 24.
ANSWER. In 24 days.
CHECK. In 24 days, the pitate' ship will make = 9 full circles;
in 24 days, the merchant ship will make = 4 full circles.
Thus after 24 days each of them will complete the integer number of their corresponding circles
and, hence, they will meet each other again for the first time.
Solved.