Problem 3 - Algebra and Trigonometry

Question

Express the given quantity in terms of the indicated variable. The average of three test scores if the first two scores are 78 and \(82 ; \quad s=\) third test score

Step-by-Step Solution

Answer

\( \frac{160 + s}{3} \)

Step 1: Understand the Problem

We need to find the formula for the average of three test scores given two scores, 78 and 82, and express it in terms of the third score, which we will call \( s \).

Step 2: Calculate the Sum of All Test Scores

To find the average of three scores, we first need the sum of all three scores. The formula for the sum is \( 78 + 82 + s \).

Step 3: Use the Average Formula

The average of three numbers is their sum divided by 3. So, the average test score is given by \( \frac{78 + 82 + s}{3} \).

Step 4: Simplify the Expression

First, calculate \( 78 + 82 = 160 \). Then, replace \( 78 + 82 \) in the average formula with \( 160 \), giving \( \frac{160 + s}{3} \). This is the expression for the average in terms of \( s \).

Key Concepts

Average of Numbers

The concept of an average, often called the mean, involves finding one value that represents a set of numbers. To calculate the average of numbers, you need to do the following:

• Add all the numbers together to get the total sum.
• Divide the sum by the number of values you added together.

This process helps in understanding the central tendency of a dataset. In our example of test scores, we are looking at three values: two given scores and one variable score that can change. Thus, the formula for their average is expressed as \[ \text{Average} = \frac{\text{Sum of all scores}}{3} \]where the sum consists of the given scores and the third variable, and the denominator reflects the count of scores.

Sum of Terms

The sum of terms in an algebraic expression involves adding up all parts of the expression, including constants and variables. In our example of test scores, the process begins with identifying each score as a term. The scores 78 and 82 are constants, and the score represented by the variable \( s \) makes up the final term.To proceed:

• Combine the constants: 78 and 82, so 78 + 82.
• Then add the variable term \( s \) to get a full expression: \( 78 + 82 + s \).

This step is crucial because it sets the stage for calculating the average of the test scores. Besides, efficiently managing each part of the sum can help avoid errors in subsequent calculations, such as when dividing to find the average.

Variable Manipulation

Variable manipulation is an essential skill in algebra that involves handling letters or symbols that represent numbers. In the given problem, \( s \) serves as a placeholder for the third test score, and it can vary as required.Some key tips for manipulating variables include:

• Understand the role each variable plays in the expression or equation.
• Perform operations such as addition or subtraction with both constants and variables.
• Simplify expressions where possible, which might involve combining like terms or solving for a particular variable.

For the test score problem, we use manipulation to arrive at a simplified expression for the average by replacing the sum of 78 and 82 with 160. Then we express this sum along with the variable \( s \) in a fraction, as in \[ \frac{160 + s}{3} \], giving us a neat formula to work with to calculate average scores with different values of \( s \).