A professor grades students on two tests, three quizzes, and a final examination. Each test counts the same as three quizzes and the final examination counts the same as two tests. Maegan has test scores of 85 and 97. Maegan's quiz scores are 85, 74, and 85. Her final examination score is 96. Use the weighted mean formula to find Maegan's average for the course. The professor rounds averages to the nearest integer.

To calculate Maegan's average using the weighted mean formula, we must first understand the weight assigned to each component of her grades:

- Each test counts the same as three quizzes.
- The final examination counts the same as two tests.

Let's break down the problem:

1. **Tests**: Maegan has two test scores, each with the weight of three quizzes:
   - First test score: 85
   - Second test score: 97
   - The weight of each test: 3 quizzes

2. **Quizzes**: Maegan has three quiz scores:
   - Quiz scores: 85, 74, and 85
   - Each quiz has a weight of 1.

3. **Final Exam**: Maegan's final exam score is 96.
   - The weight of the final exam is the same as two tests, which is equivalent to 6 quizzes (since each test is worth 3 quizzes).

### Step 1: Convert all scores to quiz equivalents.
We will express each component in terms of its quiz weight:

- Two test scores (weighted as 3 quizzes each): \(85 \times 3\) and \(97 \times 3\)
- Three quiz scores (each weighted as 1 quiz): 85, 74, and 85
- Final exam score (weighted as 6 quizzes): \(96 \times 6\)

### Step 2: Calculate the total weighted sum.
The weighted sum is calculated by multiplying each score by its respective weight and summing them all up.

\[
\text{Weighted sum} = (85 \times 3) + (97 \times 3) + 85 + 74 + 85 + (96 \times 6)
\]
\[
\text{Weighted sum} = 255 + 291 + 85 + 74 + 85 + 576 = 1366
\]

### Step 3: Calculate the total weight.
The total weight is the sum of the weights of all components:

- Tests: \(3 + 3 = 6\) quizzes
- Quizzes: \(1 + 1 + 1 = 3\) quizzes
- Final exam: 6 quizzes

Thus, the total weight is:
\[
6 + 3 + 6 = 15 \text{ quizzes}
\]

### Step 4: Calculate the weighted mean.
The weighted mean is the total weighted sum divided by the total weight:
\[
\text{Weighted mean} = \frac{1366}{15} = 91.07
\]

### Step 5: Round to the nearest integer.
The final result, rounded to the nearest integer, is:
\[
\boxed{91}
\]

Thus, Maegan's average for the course is **91**.

Would you like more details or have any questions? Here are 5 related questions to consider:

1. How does the professor’s grading scale affect the course's weighted average?
2. If Maegan improved her final exam score, how would that change her weighted average?
3. What is the impact of low quiz scores on the overall average?
4. How does rounding affect students' final grades in this method?
5. What if each test had a different weight? How would we adjust the calculation?

**Tip**: Always ensure the weights add up correctly to avoid errors in calculating weighted averages!

Learn how to calculate Maegan's average course grade using the weighted mean formula. This problem involves tests, quizzes, and a final exam, with each test weighted as three quizzes and the final exam weighted as two tests. Suitable for high school students in Grades 9-12.

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