Let's assume that both Miguel and Grace had x number of rare coins when they started collecting.
Miguel has been collecting 5 coins each week, and he now has 38 coins. This means that he has been collecting for (38 - x) / 5 weeks.
Grace has been collecting 3 coins each week, and she now has 24 coins. This means that she has been collecting for (24 - x) / 3 weeks.
Since Miguel and Grace started collecting at the same time, the number of weeks they have been collecting should be the same. Therefore, we can set up the equation:
(38 - x) / 5 = (24 - x) / 3
Simplifying this equation, we get:
3(38 - x) = 5(24 - x)
114 - 3x = 120 - 5x
2x = 6
x = 3
So, Miguel and Grace each had 3 rare coins when they started collecting. Therefore, they had a total of 3 + 3 = 6 rare coins together when they started collecting.
The correct answer is not provided in the options.