The struggling college student's GPA Calculator

Jelly,  15  21 bytes (12 with no rounding)

+6 bytes for the strict formatting (almost certainly possible in less but it's bed time)

Oạ69.Ḟ×S×ȷ2÷⁹S¤RLDż”.

A full program taking the grades and the respective credit hours which prints the calculated GPA (Note: the rounding method is to floor, as allowed in the OP).

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With no rounding for 12 bytes:

Oạ69.Ḟ×S÷⁹S¤

How?

Oạ69.Ḟ×S×ȷ2÷⁹S¤RLDż”. - Link: list of characters, grades; list of number, creditHours
                      -                                   e.g. "AFBDC", [5, 2, 4, 1, 2]
O                     - cast to ordinals (vectorises)          [65,70,66,68,67]
  69.                 - literal 69.5
 ạ                    - absolute difference (vectorises)       [4.5,0.5,3.5,1.5,2.5]
     Ḟ                - floor (vectorises)                     [4,0,3,1,2]
      ×               - multiply by creditHours (vectorises)   [20,0,12,1,4]
       S              - sum                                    37
         ȷ2           - literal 100
        ×             - multiply                               3700
              ¤       - nilad followed by link(s) as a nilad:
            ⁹         -   chain's right argument, creditHours  [5, 2, 4, 1, 2]
             S        -   sum                                  14
           ÷          - division                               264.2857142857143
               R      - range                                  [1,2,3,...,264]
                L     - length                                 264
                 D    - digits                                 [2,6,4]
                   ”. - literal '.'
                  ż   - zip together                           [[2,'.'],6,4]
                      - implicit print (smashing)              2.64

Python 3, 66 bytes

-5 bytes thanks to Rod.

lambda g,c:'%.2f'%sum('FDCBA'.find(i)*j/sum(c)for i,j in zip(g,c))

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Perl 5, 57 53 + 2 (-an) = 59 55 bytes

$c+=$F[1];$\+=$F[1]*=!/F/&&69-ord}{printf'%.2f',$\/$c

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Edit: swapped the input around to save 4 bytes

Input format: line separated, credits followed by grade:

grade credits

Example:

A 3
B 4
A 1
A 1