Problem 9 - Precalculus Mathematics for Calculus

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; \(s=\) third test score

Step-by-Step Solution

Answer

The average is \(\frac{160 + s}{3}\).

Step 1: Write the Formula for Average

The average of three numbers is calculated by summing the numbers and dividing by three. For this problem, the formula for the average is: \[ \text{Average} = \frac{x_1 + x_2 + x_3}{3} \] where \(x_1\), \(x_2\), and \(x_3\) are the three test scores.

Step 2: Substitute the Known Values

Substitute the known test scores into the formula for \(x_1\) and \(x_2\). Given that the first two scores are 78 and 82, the expression becomes: \[ \text{Average} = \frac{78 + 82 + x_3}{3} \]

Step 3: Replace the Third Score with the Variable

The problem asks to express the quantity in terms of the third test score, which is given as \(s\). Thus, replace \(x_3\) with \(s\) to get: \[ \text{Average} = \frac{78 + 82 + s}{3} \]

Step 4: Simplify the Expression

Add 78 and 82 to simplify the fraction. This gives: \[ 78 + 82 = 160 \] which results in the expression: \[ \text{Average} = \frac{160 + s}{3} \]

Key Concepts

Understanding Test Scores

Test scores are a way to gauge a student's performance. Think of them as special numbers representing how well you did on a test. When you have multiple test scores, you might need to calculate an overall average to see how well you performed across those tests. Let's say you have three test scores. Adding them up and then dividing by the number of scores (which is three) gives you an average score. This average helps understand your overall performance, instead of focusing on just one test. It's a helpful way to see improvement or notice if there's a pattern in your scores.

What is an Algebraic Expression?

An algebraic expression is a combination of numbers, variables, and operations like addition or multiplication. They are mathematical phrases showing a value, like a sentence in language. In our exercise, we want to find the average with scores expressed as an algebraic expression. Variables, such as the letter 's' in our problem, represent unknown values or numbers that can change. Using algebraic expressions allows us to work with numbers flexibly, making adjustments without redoing calculations each time. It’s like using a recipe where some ingredients can be switched or measured out differently based on what's available.

Substitution in Formulas

Substitution in a formula means replacing variables with their actual values. This concept is like filling in the blanks of a story with specific characters. In our exercise, we start with a general formula for the average of three test scores. We know two of these scores (78 and 82), so we substitute them into our formula. This makes the expression more specific, leaving just the third score as a variable. Substitution helps in narrowing down unknowns and working step-by-step towards an answer. It’s a basic, but essential skill in algebra that simplifies complex problems.

Simplifying Expressions

Simplifying expressions means making them easier to work with, often by condensing numbers and operations. For the problem, after substituting the known values, you simplify by performing basic arithmetic operations. First, you add the two known scores. In this case, 78 and 82 become 160. This reduces the expression to \({160 + s}\/{3}\). Simplification helps recognize patterns or solutions faster and makes complicated calculations manageable. It’s similar to decluttering a room, where you remove what's unnecessary and create a neater, more functional space. This makes it easier to reach the intended result without confusion.