Miguel and Grace started collecting rare coins at the same time. Back then, they had the same number of rare coins. Miguel has been collecting 5 coins each week and he now has 38 coins. Grace has been collecting 3 coins each week and she now has 24 coins. How many rare coins did they have all together when they started collecting?
If Miguel has been collecting 5 coins each week and now has 38 coins, then he has collected 38-5=<<38-5=33>>33 coins since they started.
If Grace has been collecting 3 coins each week and now has 24 coins, then she has collected 24-3=<<24-3=21>>21 coins since they started.
Together, they have 33+21=<<33+21=54>>54 rare coins when they started collecting. Answer: \boxed{54}.
To find out how many rare coins Miguel and Grace had together when they started collecting, we can subtract the total number of coins Miguel has now from the total number of coins he has been collecting each week.
Miguel has been collecting 5 coins each week and has a total of 38 coins now.
So, he must have been collecting coins for (38 / 5) = 7.6 weeks.
Now, let's find out how many coins Miguel had when he started collecting. This can be done by subtracting the total number of coins he collected from the number of coins he has now.
Miguel's initial number of coins = Total number of coins now - Total number of coins collected
Miguel's initial number of coins = 38 - (5 x 7.6)
Miguel's initial number of coins = 38 - 38
Miguel's initial number of coins = 0 coins.
This means that Miguel did not have any rare coins when he started collecting.
Now, let's find out Grace's initial number of coins. We can use a similar approach as we did for Miguel. Grace has been collecting 3 coins each week and has a total of 24 coins now.
So, she must have been collecting coins for (24 / 3) = 8 weeks.
Grace's initial number of coins = Total number of coins now - Total number of coins collected
Grace's initial number of coins = 24 - (3 x 8)
Grace's initial number of coins = 24 - 24
Grace's initial number of coins = 0 coins.
This means that Grace also did not have any rare coins when she started collecting.
Therefore, when they started collecting rare coins, Miguel and Grace had a total of 0 coins together.