Problem 130 - College Algebra

Question

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. To earn an A in a course, you must have a final average of at least \(90 \% .\) On the first four examinations, you have grades of \(86 \% .88 \% .92 \%,\) and \(84 \% .\) If the final examination counts as two grades, what must you get on the final to earn an A in the course?

Step-by-Step Solution

Answer

To get an A in the course, a score of at least 95% must be achieved on the final exam.

Step 1: Understanding the problem

On the first 4 exams, the grades are 86, 88, 92 and 84. However, the final exam is weighted, counting as two grades. We need to understand that we are looking for a final grade average of at least 90, inclusive of the final exam, where the final exam counts as two separate grades.

Step 2: Representation of all grades

Knowing that the final exam score counts twice, assume the score needed on the final exam is \(x\). The total grades can be represented as the sum of the four exams and two times the final exam, which equals to: \(86+88+92+84+2x\).

Step 3: Formulate and solve the inequality

To formulate the inequality, the total sum of grades (from step 2) should be divided by the total number of grades (6). This would be greater than or equal to 90. Formulating this results in: \(\frac{86+88+92+84+2x}{6} \geq 90\). Solving for \(x\) will provide the score that needs to be achieved on the final.

Step 4: Solving the inequality

The inequality simplifies to: \(350+2x \geq 540\). Subtract 350 from both sides of the inequality, resulting to: \(2x \geq 190\). Dividing both sides of the equation by 2 gives us: \(x \geq 95\). This means that to get an 'A' in the course, the final exam grade must be at least 95.

Key Concepts

Word Problems

Word problems are a common way to apply mathematical concepts to real-life scenarios. They require you to translate a written problem into a mathematical equation or inequality. This translation process involves identifying the quantities and relationships described in the text. For instance, when tackling a word problem about calculating the grade needed to achieve a certain average, you must first understand all the conditions given. In our example, the conditions involve using grades from previous exams and understanding how a final exam is weighted. A clear understanding of these relationships helps you set up the correct equations or inequalities to find the solution.

Final Average Calculation

Calculating the final average is an essential skill in evaluating one's performance in a course. To find the average, you sum up all the scores and then divide by the number of items. In our scenario, the student's goal is to have at least a 90% average. This means, mathematically, you have: \[ \text{Final Average} = \frac{\text{Sum of all grades}}{\text{Total number of grades}} \geq 90 \% \] Understanding the role of each exam and particularly how the final is treated differently (counts as two grades), is key to correctly calculating this average.

Weighted Grades

Weighted grades are an important aspect of many academic evaluations. Not all assessments carry the same weight, which means they contribute differently to the final grade. The final exam in this example has been given more importance and counts as two grades. This means its score needs to be considered twice in the average calculation. Why do weighted grades matter? They can significantly influence the overall score, meaning you need to perform better on these assessments to achieve your desired average. Knowing how to incorporate weights is crucial for accurate grade calculations.

Solving Inequalities

Solving inequalities is like solving equations, but with an added complexity: you are looking for a range of values rather than a single solution. In this example, the task is to find what final grade will ensure a total average of at least 90%. The inequality is set up as: \[ \frac{86 + 88 + 92 + 84 + 2x}{6} \geq 90 \] Through a series of operations, this is simplified to: \( 350 + 2x \geq 540 \). Further simplification yields \( x \geq 95 \). This indicates that the student must achieve a score of at least 95 on the final exam to meet their academic goal. Understanding and solving inequalities helps in determining ranges rather than absolute values, offering a broader look at possible solutions.