Problem 100 - College Algebra with Modeling and Visualization

Question

Grades In order to receive an \(\mathrm{A}\) in a college course it is necessary to obtain an average of \(90 \%\) correct on three 1-hour exams of 100 points each and on one final exam of 200 points. If a student scores \(82,88,\) and 91 on the 1-hour exams, what is the minimum score that the person can receive on the final exam and still earn an A?

Step-by-Step Solution

Answer

The minimum score on the final exam is 189.

Step 1: Understand the Total Points

To achieve an A, the student needs an average of 90% on all exams combined. The total possible points are calculated by adding the points from the three 1-hour exams and the final exam: \[ 100 + 100 + 100 + 200 = 500 \] points in total. To get an average of 90%, the student needs at least 90% of these 500 points, which is \[ 0.9 \times 500 = 450 \] points.

Step 2: Sum Scores of the 1-Hour Exams

Compute the sum of the scores from the three 1-hour exams: \[ 82 + 88 + 91 = 261 \] points. These are the points already earned from the 1-hour exams.

Step 3: Calculate Minimum Final Exam Score

Determine how many additional points are needed from the final exam to reach the minimum total of 450 points required: \[ 450 - 261 = 189 \] points. This means the student needs to score at least 189 on the final exam to earn an A.

Key Concepts

Exam Scoring

When dealing with exam scores, it is crucial to understand how different exams contribute to the overall grade. For college courses, exams often have varying weights and point values, affecting the overall score significantly. In this particular exercise, we have three 1-hour exams, each worth 100 points, and one final exam worth 200 points.

A key step in solving the problem is identifying the total possible score across all exams. By doing this, you can determine the cumulative points needed to reach a certain average. Here, to secure an A grade, a student must achieve a 90% average across all exams, which equates to 450 out of a possible 500 cumulative points.

Keep these steps in mind when calculating exam scores:

• Determine the total points possible for all exams combined.
• Calculate the required percentage of that total for your target grade.
• Understand how individual exam scores contribute toward reaching that target percentage.

Grade Calculation

Grade calculation can often be compared to solving a puzzle where each exam score is a piece of the total grade picture. Especially in college, where multiple assessments contribute to the final mark, understanding this process is essential.

In this case, the calculation involves summing the scores from the multiple exams and determining what remains needed to reach the required average for an A. The student has already earned 261 points from the three 1-hour exams, so it is about finding how the score from the final can complete the needed 450 points.

The steps to calculate grades include:

• Sum total scores earned from completed assessments.
• Calculate the remaining number of points required to reach the desired total.
• Adjust scores in remaining assessments accordingly.

Mathematical Problem Solving

Mathematical problem-solving involves a systematic approach to reach a solution. In this case, achieving a desired exam score blends arithmetic calculations with logical pacing.

The process starts with understanding the problem statement and what is needed. Next, it involves mathematical operations such as addition and subtraction to calculate options and decisions. For our issue, we first determined the total needed points, then calculated the difference between those and already acquired points.

Effective problem-solving includes:

• Breaking down the problem into manageable parts.
• Performing relevant mathematical operations to piece together a solution.
• Continuously checking your work to ensure accuracy.