College Algebra, Exercise P, Exercise P.1, Section Exercise P.1, Problem 26

In many universitites students are given grade points for each credit unit according to the following scale:

$\begin{array}{|c|c|} \hline\\ A & 4 \text{ points}\\ \hline\\ B & 3 \text{ points}\\ \hline\\ C & 2 \text{ points}\\ \hline\\ D & 1 \text{ points}\\ \hline\\ F & 0 \text{ points}\\ \hline \end{array}$


For example, a grade of $A$ in a 3-unit course earns $4 \times 3 = 12$ grade points and a grade of $B$ in a 5-unit course earns $3 \times 5 = 15$ grade points. A student's grade point average (GPA) for these two courses is the total number of grade points earned divided by the number of units; in this case the GPA is $\displaystyle \frac{12+15}{8} = 3.375$.

a.) Find a formula for GPA of a student who earns grade of $A$ in a units of course work, $B$ in b units, $C$ in c units, $D$ in d units and $F$ in f units.
Let,
$A, B, C, D, E,$ and $F$ be the grade of students.
$a,b,c,d,e$ and $f$ be the number of units of course work.

$\displaystyle \text{GPA} = \frac{(A \times a) + (B \times b) + (C \times c) + (D \times d) + (F \times f)}{a + b + c + d + f}$ model

b.) Find the GPA of a student who has earned a grade of $A$ in two 3-unit courses, $B$ in one 4-unit course, and $C$ in three 3-unit courses.
Based from the model in part(a) and the given condition, we have...

$\begin{equation} \begin{aligned} \text{GPA} &= \frac{2(A \times a) + (B \times b) + 3 (C \times c)}{2a + b + 3c} && \text{model}\\ \\ \text{GPA} &= \frac{2(4 \times 3) + (3 \times 4) + 3(2 \times 3)}{2(3) + 4 + 3 (3)} && \text{Substitute } A=4, B=3, C= 2, a = 3, b = 4 \text{ and } c =3\\ \\ \text{GPA} &= \frac{24+12+18}{6+4+9} && \text{Simplify}\\ \\ \text{GPA} &= 2.842 \end{aligned} \end{equation}$