Let's break this problem into two parts:

Part 1: Best Course Grade Your Friend Can Earn

The final exam is worth 15% of the total course grade, and your friend currently has an 83% average from the rest of the course, which accounts for the remaining 85%.

To find the best course grade, we assume that your friend scores a perfect 100% on the final exam.

The overall course grade $G$ can be calculated using the following weighted average formula:

$G = (0.85 \times 83) + (0.15 \times 100)$

Part 2: Minimum Score Needed on the Final to Earn 75% for the Course

To find the minimum score your friend needs on the final to earn a 75% overall, we set the overall course grade $G$ equal to 75% and solve for the score on the final exam (denoted as $F$):

$G = (0.85 \times 83) + (0.15 \times F) = 75$

Now, I'll calculate both parts.### Part 1: Best Course Grade The best course grade your friend can earn, assuming they score 100% on the final, is 85.6%.

Part 2: Minimum Score for a 75% Overall

To achieve a final course grade of 75%, your friend would need to score at least 29.7% on the final exam.

Let me know if you want more details or have any questions!

Here are some related questions for further understanding:

1. What would the course grade be if your friend scores 80% on the final?
2. How does changing the weight of the final exam affect the required final exam score?
3. If the final exam were worth 20%, what score would be needed to maintain an 83% overall?
4. What is the lowest possible course grade if the final exam is failed with 0%?
5. How would the calculations change if the current average were 90%?

Tip: When calculating weighted averages, always ensure you correctly account for the different weights of each component.