Question

a students course grades and their corresponding weights are given in the table.

\

$$\begin{tabular}{|l|l|l|} \\hline category & grade earned & weight by category \\\\ \\hline attendance & 95\\% & 10\\% \\\\ \\hline tests & 75\\% & 30\\% \\\\ \\hline homework & 100\\% & 20\\% \\\\ \\hline final exam & & 40\\% \\\\ \\hline \\end{tabular}$$

what is the minimum grade needed on the final exam to earn an overall grade of 80\\% in the class?

88\\%
70\\%
50\\%
28\\%

Explanation:

Identify given values and weights

We are given a table of course categories, grades earned, and their corresponding weights:

• Attendance: Grade = \(95\%\), Weight = \(10\%\)
• Tests: Grade = \(75\%\), Weight = \(30\%\)
• Homework: Grade = \(100\%\), Weight = \(20\%\)
• Final Exam: Grade = \(x\), Weight = \(40\%\)

The target overall grade is at least \(80\%\).

Set up the weighted average equation

The overall grade is calculated as the sum of the products of each category's grade and its corresponding weight:

$$ \text{Overall Grade} = (95 \times 0.10) + (75 \times 0.30) + (100 \times 0.20) + (x \times 0.40) $$

We set this overall grade to be at least \(80\%\):

$$ (95 \times 0.10) + (75 \times 0.30) + (100 \times 0.20) + 0.40x \ge 80 $$

Calculate the current earned points

Compute the contribution from the completed categories:

• Attendance contribution: \(95 \times 0.10 = 9.5\)
• Tests contribution: \(75 \times 0.30 = 22.5\)
• Homework contribution: \(100 \times 0.20 = 20.0\)

Sum these contributions:

$$ 9.5 + 22.5 + 20.0 = 52.0 $$

Solve for the required final exam grade

Substitute the sum back into the inequality:

$$ 52.0 + 0.40x \ge 80 $$

Subtract \(52.0\) from both sides:

$$ 0.40x \ge 28 $$

Divide by \(0.40\):

$$ x \ge \frac{28}{0.40} = 70 $$

Thus, the minimum grade needed on the final exam is \(70\%\).

Answer:

• (A) 88%
• (B) 70% (Correct answer)
• (C) 50%
• (D) 28%