Problem 87 - Algebra A Combined Function

Question

Eric Daly has scores of \(75,83,\) and 85 on his history tests. Use an inequality to find the scores he can make on his final exam to receive a \(\mathrm{B}\) in the class. The final exam counts as two tests, and a \(\mathrm{B}\) is received if the final course average is greater than or equal to 80 .

Step-by-Step Solution

Answer

Eric needs to score at least 79 on his final exam to maintain a B average.

Step 1: Understand the problem

The problem requires finding the possible scores Eric can earn on his final exam to maintain a B average. The B average means his overall average should be at least 80, and the final exam counts twice.

Step 2: Set up the total average formula

The average is calculated by dividing the total sum of scores by the number of tests. In this case, he has 3 test scores already, and the final counts as two, making a total of 5 test scores.

Step 3: Express the inequality

Let the score on the final exam be represented by \( x \). The inequality for his average can be expressed as: \[ \frac{75 + 83 + 85 + 2x}{5} \geq 80 \]

Step 4: Solve for \( x \)

Multiply both sides by 5 to clear the denominator: \[ 75 + 83 + 85 + 2x \geq 400 \] Combine the constants: \[ 243 + 2x \geq 400 \] Subtract 243 from both sides to isolate terms with \( x \): \[ 2x \geq 157 \] Finally, divide by 2: \[ x \geq 78.5 \]

Step 5: Interpret the result

Since the score on a test must be a whole number, Eric must score at least 79 on his final exam to keep a B average.

Key Concepts

Final Exam Score Calculation

To calculate what Eric needs on his final exam, we consider his past scores and how the final exam influences his overall grade. Eric's current scores are 75, 83, and 85. The key point to remember is that the final exam counts as two separate tests in his total average calculation.

With the final exam considered twice, his total number of scores becomes 5. This means that when we calculate the average, the final exam score will have a strong impact. Here's how you calculate the final test's contribution:

• Add up all scores, including twice the final exam score.
• Divide by the total number of scores, which in this case is 5.

Understanding this effect is crucial because it doubles the impact the final exam score has on the final grade. Calculating the contribution of the final uses the formula: \[ \frac{75 + 83 + 85 + 2x}{5} \geq 80 \] where \( x \) is the final exam score.

B Average Calculation

A 'B' average requires Eric’s total course average to be at least 80. Since the final exam counts as two tests, it's weighted heavily in achieving this target. To calculate whether he meets the 'B' threshold, we use Eric's previous test scores (75, 83, and 85) and project the impact of the final.

Steps to find out how he can achieve this average include:

• Add current test scores (75, 83, 85)
• Include the final exam score twice in the calculation
• Divide by the total number of test scores, which is 5

The calculation aims to ensure the result meets or exceeds 80, which is the bottom line for a 'B' grade. Solving this for \( x \), his final exam score, gives us the minimum he needs to reach or surpass that average. This involves understanding that averages smooth out individual scores, making consistent or impactful tests like the final significant in these calculations.

Test Score Inequality

Test score inequalities are used to set conditions for grades. They help determine what score is needed on a remaining test—here, the final exam—to achieve a desired average. The challenge is to write an inequality that represents the needed condition. In this case, Eric wants to maintain a 'B' average, translating mathematically to the inequality:\[ \frac{75 + 83 + 85 + 2x}{5} \geq 80 \]This inequality is solved step by step:

• Clear fractions by multiplying both sides by 5: \( 243 + 2x \geq 400 \)
• Simplify to isolate terms with \( x \): \( 2x \geq 157 \)
• Divide to find \( x \): \( x \geq 78.5 \)

In practice, because scores are whole numbers, Eric needs at least 79 to maintain his 'B'. Understanding these inequalities is crucial for setting and reaching academic goals, making them a powerful tool in tracking and achieving desired grades.