Marilee currently has four test grades and her quiz average that counts as one test grade, plus her final exam, which counts as three test grades.
Let's summarize her current scores:
- Test grades: 92, 96, 90, 92 (these are 4 tests)
- Quiz average: 80 (counts as one test grade)
- Final exam: Let \( x \) be her score on the final exam (counts as three test grades).
First, we can calculate the current total of her test grades plus the quiz score:
\[ 92 + 96 + 90 + 92 + 80 = 450 \]
Next, we need to include the final exam, which counts 3 times, so her total test grades including the final exam will be:
\[ 450 + 3x \]
The overall number of test grades is now \( 4 + 1 + 3 = 8 \). To find the average, we set up the following equation to find the score required on the final exam to achieve an overall average of 90:
\[ \frac{450 + 3x}{8} \geq 90 \]
Now, multiply both sides by 8 to eliminate the fraction:
\[ 450 + 3x \geq 720 \]
Next, isolate \( 3x \) by subtracting 450 from both sides:
\[ 3x \geq 720 - 450 \]
This simplifies to:
\[ 3x \geq 270 \]
Now divide both sides by 3:
\[ x \geq 90 \]
Therefore, Marilee needs to score at least 90 on her final exam to achieve an overall average of at least 90 in the class.