Problem 75 - Algebra for College Students

Question

To earn a \(\mathrm{B}\) in an algebra course requires an average of at least 80 on five tests. A student has scores of \(75,91,82,\) and \(74 .\) What possible scores on the fifth test would guarantee this student a \(\mathrm{B}\) in the class?

Step-by-Step Solution

Answer

The student needs at least a 78 on the fifth test.

Step 1 - Determine the Required Total Score

To find the total score needed to average at least 80 over five tests, multiply the required average by the number of tests: \[ 80 \times 5 = 400. \] Thus, the student needs a total score of at least 400 points.

Step 2 - Calculate Current Score Total

Add the scores of the first four tests: \[ 75 + 91 + 82 + 74 = 322. \] The student has 322 points from the first four tests.

Step 3 - Determine the Required Fifth Test Score

Subtract the current total from the required total to find the minimum score needed on the fifth test: \[ 400 - 322 = 78. \] Therefore, the student must score at least 78 on the fifth test to achieve an average of at least 80.

Key Concepts

average score calculation

Calculating the average score is essential in many academic scenarios, including determining your grade for a course. An average is simply the sum of several numbers divided by the number of those numbers. In this exercise, the student needed an average score of at least 80 on five tests to get a B in the algebra course. To find out what their average score should be, they first needed to determine the total score required. Here’s how to handle this calculation step by step:

• Add up all the scores you have.
• Divide this total sum by the number of scores.

So, in our case, if the student already has scores of 75, 91, 82, and 74 from four tests, they would first add these scores to find the total. Then, divide the total by 4 to find the average score for these tests.

total score requirement

To achieve a certain average, it's essential to know the total score you need to hit. In the given problem, the student needed an average of at least 80 over five tests to earn a B grade. How do we find the total score required for this? It's simple:

• Multiply the desired average by the number of scores.

This formula works for any set of numbers where you want to achieve a specific average. For our specific problem, the desired average is 80, and there are five tests. So, multiply: 80 times 5 equals 400. Therefore, the student needs a total score of at least 400 points from the five tests combined to ensure they get a B.

minimum score determination

After understanding the total score requirement, the next step is to figure out the minimum score needed on any remaining tests. In this exercise, having already totaled the first four test scores, this student needed to determine what score was needed on the fifth test. To do this, follow these steps:

• Add the scores you already have.
• Subtract this sum from the total score requirement.

With test scores of 75, 91, 82, and 74, the student first adds them to get a current total of 322 points. Then, subtract 322 from the needed 400 points. The difference gives you the minimum score needed on the remaining test. So, 400 minus 322 equals 78. Therefore, the student must score at least 78 on the fifth test to average an 80 and earn their B grade.