Example 1: A math teacher counts the final exam as 30% of the course grade. A student's average going into the final is 81. Devise a mathematical model that shows how this student's course grade is related to the final exam score.
Choose variables for this model
Since we want to obtain the course grade, let G represent the course grade. Since G will depend on the grade on the final exam, let F represent that grade.
Express how G depends on F.
Since we are given that F is 30% of G,
G depends on 0.30 * F.
How does G depend on the average going into the final exam?
The other 70% of G is from this grade:
G depends on 0.70 * 81
How are these two parts of G to be combined? Added, subtracted, multiplied, or divided?
They are parts of G and must be added.
Combine all of these results to express G in terms of the data and F.
G = 0.30 * F + 0.70 * 81
Determine the course grade if this student gets 90 on the final exam.
Now F is known, and we are solving for G:
G = 0.30 * 90 + 0.70 * 81 = 84
This student would get a course grade of 84.
Example 2: What grade on the final exam would this student have to earn to get a course grade of 86?
In this situation, G is known as 86, and we are solving for F.
G = 0.30 * F + 0.70 * 81
86 = 0.30 * F + 0.70 * 81
86 = 0.30 * F + 56.7
Subtract 56.7 from both sides of the equation.
86 - 56.7 = 0.30 * F + 56.7 - 56.7
29.3 = 0.30F
Divide both sides by 0.30
, or F ~ 98
The student needs a practically perfect performance on the final exam - a grade of 98.