Problem 62 - Algebra for College Students

Question

Solve each problem by setting up and solving an appropriate inequality. Candace had scores of \(95,82,93\), and 84 on her first four exams of the semester. What score must she obtain on the fifth exam to have an average of 90 or better for the five exams?

Step-by-Step Solution

Answer

Candace needs a score of at least 96 on her fifth exam.

Step 1: Understand the Problem

To find Candace's required score on the fifth exam, we need her average score for all five exams to be at least 90. This average can be represented and solved using an inequality.

Step 2: Setup the Inequality

The average of the five exams can be expressed as \( \frac{95 + 82 + 93 + 84 + x}{5} =\) where \(x\) represents the score on the fifth exam. We want this average to be greater than or equal to 90, leading to the inequality: \( \frac{95 + 82 + 93 + 84 + x}{5} \geq 90 \).

Step 3: Simplify the Inequality

First, solve the left side: \( 95 + 82 + 93 + 84 + x = 354 + x \).Then, write the inequality:\[ \frac{354 + x}{5} \geq 90 \]

Step 4: Eliminate the Fraction

To eliminate the fraction, multiply both sides of the inequality by 5:\[ 354 + x \geq 450 \]

Step 5: Solve for x

Subtract 354 from both sides of the inequality:\[ x \geq 450 - 354 \]\[ x \geq 96 \]

Step 6: Verify the Solution

Check the calculation by substituting \(x = 96\) back into the average calculation:\( \frac{95 + 82 + 93 + 84 + 96}{5} = \frac{450}{5} = 90 \), confirming the solution is correct as the average is exactly 90.

Key Concepts

Average Score Calculation

Calculating an average score involves adding up all the scores and then dividing this sum by the number of scores. It's like sharing a total equally among all instances. To apply this concept to Candace's scores:

• Add up the scores of her first four exams: 95, 82, 93, and 84. The sum is 354.
• Include the unknown score of the fifth exam, represented by \( x \).
• The formula for the average becomes \( \frac{354 + x}{5} \).

We want the average to be at least 90, which means the equation \( \frac{354 + x}{5} \) must be equal to or greater than 90. By setting up this equation as an inequality, we can solve for \( x \) to find the score required on the fifth exam.

Solving Inequalities

Inequalities are mathematical expressions that denote that one value is larger or smaller than another. In solving inequalities, our goal is to determine the values that satisfy the inequality condition. For Candace’s problem, we have:

• An inequality \( \frac{354 + x}{5} \geq 90 \).
• The goal is to isolate the variable \( x \) to solve for what scores satisfy this condition.

To begin, multiply both sides by 5 to eliminate the fraction:\[ 354 + x \geq 450 \]Next, subtract 354 from both sides:\[ x \geq 96 \]This tells us that Candace needs to score at least a 96 on her next test to achieve an average of 90 or higher.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. In the context of Candace's scores, the expression \( 354 + x \) consists of a constant (354) and a variable (\( x \)). Here’s how we used it:

• Initially, the sum 354 is derived from the known scores.
• The unknown "\( x \)" represents the fifth exam score Candace aims for.

To solve for \( x \), make sure to work step by step:1. Start with the complete algebraic expression in the inequality.2. Manipulate this expression through legal algebraic operations like addition, subtraction, and multiplication.3. Simplify as needed to isolate \( x \) and solve the problem.The algebraic approach lets us model real-world problems and solve them effectively, as shown in how we handled Candace's exam scores to ensure she reaches her desired average.