Problem 76 - Algebra for College Students

Question

To pass Algebra II requires an average of at least 70 on four tests. A student has scores of \(80,62,\) and \(73 .\) What possible scores on the fourth test would guarantee this student a passing score in the class?

Step-by-Step Solution

Answer

The student needs to score at least 65 on the fourth test.

Step 1: Determine the required average score

To pass Algebra II, the student needs an average score of at least 70 on four tests.

Step 2: Set up the average inequality

Let the score on the fourth test be denoted by the variable x. The average score of the four tests must be at least 70. Set up the inequality: \[\frac{80 + 62 + 73 + x}{4} \geq 70\]

Step 3: Simplify the inequality

Combine the known test scores and simplify the inequality: \[\frac{215 + x}{4} \geq 70\]

Step 4: Solve for x

Multiply both sides of the inequality by 4 to clear the denominator: \[215 + x \geq 280\] Subtract 215 from both sides to isolate x: \[x \geq 65\]

Step 5: Interpret the solution

The student must score at least 65 on the fourth test to have an average of at least 70 across all four tests.

Key Concepts

average score calculation

To understand how to calculate an average score, let's break it down. The average score is found by adding up all the individual scores and then dividing the total by the number of scores. For example, if we have four test scores: 80, 62, 73, and an unknown score x, we add them up to get a total.
Using our exercise, we have:
\[ 80 + 62 + 73 + x \ = 215 + x \]
Now, we need to find the average, so we divide this total by the number of tests, which is 4.
The equation for the average score of four tests is:
\[ \frac{215 + x}{4} \]
This average must be at least 70, leading us to the next step in solving the problem.

solving linear inequalities

Solving linear inequalities involves finding the values of a variable that make the inequality true. In this exercise, we need to solve for the score x on the fourth test to ensure an average of at least 70.
We start with the inequality:
\[ \frac{215 + x}{4} \ \text{is greater than or equal to } 70 \]
To clear the fraction, we multiply both sides by 4:
\[215 + x \ \text{greater than or equal to } 280 \]
Next, we isolate x by subtracting 215 from both sides:
\[ x \ \text{greater than or equal to 65} \]
This tells us that the student needs at least 65 on the fourth test to have an average score of 70 or higher.

algebraic expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. The expression we dealt with in this exercise was:
\[ \frac{80 + 62 + 73 + x}{4} \]
Here, 80, 62, and 73 are known scores, and x is the unknown score that we need to solve for.
Combining the known scores gives us a simplified algebraic expression:
\[ \frac{215 + x}{4} \ \text{is greater than or equal to } 70 \]
This simplification helps in solving the inequality and finding the required value for x. Identifying and manipulating such algebraic expressions is crucial in solving various algebra problems.