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A student's course grades and their corresponding weights are given in the table.

\begin{tabular}{|c|l|l|}
\hline Category & Grade Earned & Weight by Category \\
\hline Attendance & [tex]$100 \%$[/tex] & [tex]$10 \%$[/tex] \\
\hline Tests & [tex]$80 \%$[/tex] & [tex]$30 \%$[/tex] \\
\hline Homework & [tex]$95 \%$[/tex] & [tex]$20 \%$[/tex] \\
\hline Final Exam & & [tex]$40 \%$[/tex] \\
\hline
\end{tabular}

What is the minimum grade needed on the final exam to earn an overall grade of [tex]$83 \%$[/tex] in the class?

A. [tex]$30 \%$[/tex]
B. [tex]$57 \%$[/tex]
C. [tex]$75 \%$[/tex]
D. [tex]$90 \%$[/tex]


Sagot :

To determine the minimum grade needed on the final exam to achieve an overall grade of 83% in the class, we need to use the weighted average formula. Let's breakdown the calculations step-by-step:

### Step 1: Calculate the weighted contributions from Attendance, Tests, and Homework

1. Attendance:
- Grade Earned: 100%
- Weight: 10%
[tex]\[ \text{Weighted Attendance} = 100 \times 0.10 = 10.0 \][/tex]

2. Tests:
- Grade Earned: 80%
- Weight: 30%
[tex]\[ \text{Weighted Tests} = 80 \times 0.30 = 24.0 \][/tex]

3. Homework:
- Grade Earned: 95%
- Weight: 20%
[tex]\[ \text{Weighted Homework} = 95 \times 0.20 = 19.0 \][/tex]

### Step 2: Sum of the current weighted grades
[tex]\[ \text{Total Weighted Grades} = \text{Weighted Attendance} + \text{Weighted Tests} + \text{Weighted Homework} = 10.0 + 24.0 + 19.0 = 53.0 \][/tex]

### Step 3: Determine the total needed to reach the overall target
[tex]\[ \text{Overall Target} = 83 \][/tex]
[tex]\[ \text{Remaining Needed} = \text{Overall Target} - \text{Total Weighted Grades} = 83.0 - 53.0 = 30.0 \][/tex]

### Step 4: Calculate the required grade on the Final Exam

Since the final exam weight is 40%, we use the remaining needed divided by this weight:
[tex]\[ \text{Final Exam Grade Required} = \frac{\text{Remaining Needed}}{\text{Final Exam Weight}} = \frac{30.0}{0.40} = 75.0 \][/tex]

Thus, the minimum grade needed on the final exam to earn an overall grade of 83% in the class is [tex]\(75 \%\)[/tex].

### Answer:
[tex]\[ \boxed{75 \%} \][/tex]