Problem 51 - Elementary Algebra

Question

Set up an algebraic inequality and then solve it. Joe earned scores of \(72,85,\) and 75 on his first three algebra exams. What must he score on the fourth exam to average at least \(80 ?\)

Step-by-Step Solution

Answer

Joe must score at least 88 on the fourth exam.

Step 1: Set Up the Average Formula

The formula to calculate the average of Joe’s four exams is \( \text{average} = \frac{\text{Sum of Four Scores}}{4} \). Let \( x \) be the score Joe needs on the fourth exam. Thus, the sum of the scores is \( 72 + 85 + 75 + x \). We want the average to be at least 80, so we set up the inequality: \( \frac{72 + 85 + 75 + x}{4} \geq 80 \).

Step 2: Clear the Fraction

Multiply both sides of the inequality by 4 to eliminate the fraction: \( 72 + 85 + 75 + x \geq 320 \).

Step 3: Simplify the Expression

Combine the known scores: \( 232 + x \geq 320 \). Here, 232 is the sum of Joe's first three test scores.

Step 4: Solve for x

Subtract 232 from both sides of the inequality to isolate \( x \): \( x \geq 88 \).

Step 5: Interpret the Solution

Joe must score at least an 88 on his fourth exam to achieve an average of at least 80.

Key Concepts

Average Calculation

Calculating an average involves determining the "central" value among a group of numbers. In Joe's case, he's trying to find out what his exam scores average out to by taking the mean of four tests. To do this, we sum all the scores and then divide by the total number of exams, which is four in this scenario. For example, Joe's scores for the first three tests are 72, 85, and 75. You calculate the average of these scores by first adding them together:

• Sum of the first three scores = 72 + 85 + 75 = 232.

Next, if Joe scores x on the fourth exam, you add this to the previous total and divide by 4:

• Average = \( \frac{232 + x}{4} \)

This formula helps determine what score Joe needs to reach his desired average.

Exam Scores

Exam scores are the results of tests that students take to show what they've learned over a period of time. In Joe's situation, each score reflects his performance in an algebra exam. His scores of 72, 85, and 75 give a sense of how he's doing overall, but don't quite reach the target average of 80 yet. Exam scores can vary due to a number of factors, such as:

• Difficulty of the test
• Individual preparation and understanding
• External factors like stress or fatigue during the exam

Joe wants to ensure his average is at least 80. That means his performance on the fourth exam is crucial. The higher his score, the better his average will be.

Solving Inequalities

Solving inequalities is a lot like solving equations, but with a focus on discovering ranges of possible values rather than a single outcome. For Joe, the inequality tells us what score he needs to achieve a target average. The algebraic inequality in Joe's situation is given as:\[ \frac{72 + 85 + 75 + x}{4} \geq 80 \]To solve, follow these steps:

• First, clear the fraction by multiplying through by 4: \(72 + 85 + 75 + x \geq 320\)
• Combine like terms: \(232 + x \geq 320\)
• Subtract 232 from both sides: \(x \geq 88\)

This tells us that Joe needs at least an 88 on his final exam to hit the desired average of 80. Solving inequalities like this shows not just if a condition is met, but how much needs to change to meet a specific goal.