Problem 52 - Introductory Algebra for College Students

Question

The average, or mean, \(A\), of four exam grades, \(x, y, z,\) and \(w,\) is given by the formula $$A=\frac{x+y+z+w}{4}$$ a. Solve the formula for \(w\) b. Use the formula in part (a) to solve this problem. On your first three exams, your grades are \(76 \%, 78 \%,\) and \(79 \%: x=76, y=78,\) and \(z=79 .\) What must you get on the fourth exam to have an average of \(80 \% ?\)

Step-by-Step Solution

Answer

So, to get an average of 80%, you must get 87% on the fourth exam.

Step 1: Solving for 'w'

We can do that by rearranging the mean formula. \(A=\frac{x+y+z+w}{4}\) can be rewritten in terms of 'w' after isolating it on one side of the equation: \(w = 4A - x - y - z\)

Step 2: Substituting known values

Given: \(A=80, x=76, y=78,\) and \(z=79 .\) Substitute these values into the equation: \(w = 4*80 - 76 - 78 - 79\)

Step 3: Calculating the value for 'w'

Perform the arithmetic calculations to find the value for 'w'. So, \(w = 320 - 76 - 78 - 79 = 87\)

Key Concepts

Solving Equations

When we talk about solving equations, we're essentially looking for the value of the variable that makes the equation true. In the context of the given exercise, we're focusing on the equation for the average (or mean) of four exam grades. The equation provided is:\[A=\frac{x+y+z+w}{4}\]Here, we're interested in solving for the grade "w" when we already have values for "A", "x", "y", and "z". To make this happen, we need to rearrange the equation to isolate "w" on one side. This skill of rearranging equations is essential, as it helps you visualize and achieve the desired outcome in calculations. Here's how it's done:1. Start by multiplying both sides of the equation by 4 to eliminate the denominator:\[4A = x+y+z+w\]2. Next, subtract "x", "y", and "z" from both sides to get:\[w = 4A - x - y - z\]This new equation is what you use to solve for "w" when you have all other values. Solving equations like this is about logical steps, each one carefully considered to move closer to the unknown.

Mean Formula

The mean, or average, helps us understand the central tendency of a set of numbers; in this case, exam grades. To compute the mean, you sum all values and then divide by the number of values. The mean formula for "n" numbers looks like this:\[\text{Mean} = \frac{\text{Sum of all items}}{\text{Count of items}}\]In our problem, the mean formula is:\[A = \frac{x+y+z+w}{4}\]This formula tells us the average of the four grades.

• Sum: Add together grades: "x", "y", "z", "w"
• Count: Divide by the number of grades, which is 4 in this case

By rearranging this formula, you can solve for any one of the variables if the other ones are known. This formula is not only applied in academic settings but also practically used in everyday life for calculating averages, from budgeting to understanding scores.

Exam Grades

Exam grades are a crucial part of assessing student performance. They're usually presented as percentages and represent how well a student has understood the material compared to a maximum possible score. Let's break down the role and importance of calculating an average grade for exams.

• Performance Tracking: By calculating the mean grade, students can see their overall performance across multiple exams rather than looking at each score individually.
• Goal Setting: Knowing the average required (like reaching 80% in our example) can help students aim for specific scores on upcoming exams to reach their desired cumulative grade.

In our exercise, the student needed to calculate what they must score on the fourth exam to achieve an overall average of 80%. Using the formula derived previously, they plugged in the known grades into the equation and found that they needed a score of 87% on the fourth exam to meet this goal. This showcases how practical math concepts can assist students in planning their study efforts more effectively.