Problem 52 - Introductory and Intermediate Algebra for College Students 4th

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

To achieve an average of \(80\%\), a fourth exam grade of \(87\%\) is required.

Step 1: Isolate the variable w

To solve the formula for \(w\), first start by multiplying both sides of the equation by 4, resulting in \(4A = x + y + z + w\). Afterwards, subtract \(x, y,\) and \(z\) from both sides, yielding the isolated equation: \(w = 4A - x - y - z.\)

Step 2: Substitute the values

Given that the first three exam grades are \(76\%, 78\%,\) and \(79\%\), it means that \(x=76, y=78,\) and \(z=79\), and the desired average is \(80\%\), i.e. \(A=80\). Substitute these values in to the isolated equation from Step 1, resulting in \(w = 4*80 - 76 - 78 -79\).

Step 3: Calculate the value of w

Now perform the calculation: \(w = 320 - 76 - 78 -79 = 87\). Therefore, you must get \(87\%\) on the fourth exam to have an average of \(80\%\).

Key Concepts

Average calculation

Calculating the average of a set of numbers is a fundamental concept in algebra. The average, or mean, is simply the sum of all the numbers divided by the count of the numbers. In the context of grades, you might often see it as a way to calculate the average score across exams or assignments. For example, if you have four exam grades labeled as \(x, y, z,\) and \(w,\) the average \(A\) is determined using the formula:

• \( A = \frac{x + y + z + w}{4} \)

This formula allows you to add up your scores and divide by the number of scores (which is 4 in this case) to find the mean grade. Knowing how to find the average is crucial because it gives you a sense of your overall performance compared to individual scores. It's also often used to set targets or understand the needed scores to reach a desired average.

Equation solving

Equation solving involves finding the value of a variable that makes the equation true. In the context of our exercise, we begin with the average formula \( A = \frac{x+y+z+w}{4} \). Here, the task is to determine the missing grade \(w\) that would result in a specific average. The steps involve algebraic manipulation:

• First, clear the fraction by multiplying both sides by 4, leading to \(4A = x + y + z + w\).
• Then, isolate \(w\) by subtracting \(x, y,\) and \(z\) from both sides, giving us \(w = 4A - x - y - z\).

By isolating \(w,\) we can substitute known values for \(A, x, y,\) and \(z,\) to solve for the unknown \(w\). This method is handy in problem-solving as it allows us to rearrange equations and solve for any single variable when the rest are known.

Grade determination

Determining what grade you need on an exam to achieve a desired average can help in goal-setting and planning your study efforts. In the example given, we are tasked with finding out what grade is needed on the fourth exam to reach an average of 80% when the first three grades are known: 76%, 78%, and 79%. By using the isolated formula \(w = 4A - x - y - z\), we substitute the known values:

• Desired average, \(A=80\%\)
• Grades: \(x=76, y=78, z=79\)

Substitute these into the equation: \(w = 4*80 - 76 - 78 - 79\). After performing these calculations, we find \(w = 87\). This means you need to score 87% on the fourth exam to achieve your target average. Understanding this process empowers you to work backwards from a target average to determine the outcomes you need on remaining assessments.