Question

$$ \begin{aligned} &\text { Elena's math scores }\\\ &\begin{array}{|l|l|} \hline \text { Test 1 } & 78 \\ \hline \text { Test 2 } & 83 \\ \hline \text { Test 3 } & 81 \\ \hline \text { Midterm exam } & 85 \\ \hline \text { Test 4 } & 89 \\ \hline \text { Test 5 } & 94 \\ \hline \text { Final Exam } & 96 \\ \hline \end{array} \end{aligned} $$ If each test is \(10 \%\), the midterm is \(20 \%\), and the final exam is \(30 \%\) of her grade, what is Elena's weighted average? A. 88 B. 89 C. 90 D. 92

Short answer

A. 88.3

Step-by-step solution

- Understand the Weighting

Each test is worth 10%, the midterm exam is worth 20%, and the final exam is worth 30% of the total grade.

- Assign Weights to Scores

Multiply each test score by 0.10 (10%), the midterm score by 0.20 (20%), and the final exam score by 0.30 (30%).\[\begin{aligned} &\text{Weighted Test 1} = 78 \times 0.10 \ &\text{Weighted Test 2} = 83 \times 0.10 \ &\text{Weighted Test 3} = 81 \times 0.10 \ &\text{Weighted Midterm} = 85 \times 0.20 \ &\text{Weighted Test 4} = 89 \times 0.10 \ &\text{Weighted Test 5} = 94 \times 0.10 \ &\text{Weighted Final Exam} = 96 \times 0.30 \end{aligned}\]

- Calculate Individual Weighted Scores

\[\begin{aligned} &\text{Weighted Test 1} = 78 \times 0.10 = 7.8 \ &\text{Weighted Test 2} = 83 \times 0.10 = 8.3 \ &\text{Weighted Test 3} = 81 \times 0.10 = 8.1 \ &\text{Weighted Midterm} = 85 \times 0.20 = 17.0 \ &\text{Weighted Test 4} = 89 \times 0.10 = 8.9 \ &\text{Weighted Test 5} = 94 \times 0.10 = 9.4 \ &\text{Weighted Final Exam} = 96 \times 0.30 = 28.8 \end{aligned}\]

- Sum All Weighted Scores

Add all the individual weighted scores together:\[\begin{aligned} &7.8 + 8.3 + 8.1 + 17.0 + 8.9 + 9.4 + 28.8 \ = 88.3 \end{aligned}\]

- Determine the Answer

The total weighted average score is 88.3. Thus, the closest option is: A. 88

Key concepts

weighted average calculation

To calculate a weighted average, first understand each component’s weight in the total average. In this case, each test scores 10%, the midterm 20%, and the final exam 30% of the final grade. By multiplying each test score by its respective weight, you get the weighted scores. Then, sum these weights to find the total weighted average.

Here is a step-by-step approach:

• Multiply each test score by its weight (0.10 for Tests 1-5, 0.20 for the midterm, 0.30 for the final).
• Add all the results together.

For example, the weighted score for Test 1 is:
\(78 \times 0.10 = 7.8\)

Repeat this for each score and add them up. This will give you the weighted average.

percentages in grading

When working with percentages in grading, understanding the weight each component contributes to the final grade helps in ensuring fair assessment. Each percentage signifies a part of the whole grade, distributing the importance of various assessments.

In Elena's case:

• Each Test's weight is 10%.
• The Midterm is 20%.
• The Final Exam is 30%.

This means Tests 1-5 combined contribute 50% to her grade, while the midterm and the final exam contribute 20% and 30%, respectively. By multiplying the score with its respective percentage (converted into a decimal), you get the weighted contribution of each score.

For instance:

• Test 2: \(83 \times 0.10 = 8.3\)
• Midterm: \(85 \times 0.20 = 17.0\)
• Final: \(96 \times 0.30 = 28.8\)

math problem-solving steps

Breaking down math problems into clear, manageable steps is essential. Let's use Elena's problem as an example:

• First, recognize what you are solving: a weighted average.
• Identify the weights and scores given.
• Assign weights to each score by multiplying them by their respective percentage weights.
• Calculate each weighted score separately, which can be shown as: \( \text{Weighted Test 1} = 78 \times 0.10 = 7.8\)
• Sum up all these weighted scores.
• The sum provides the weighted average score.

The final step in problem-solving is verifying your work. Ensure all calculations are correctly executed and the sum is as expected. In Elena's case, we added 7.8, 8.3, 8.1, 17.0, 8.9, 9.4, and 28.8 to get 88.3, making the weighted average closest to 88.