Problem 73 - Algebra for College Students

Question

Bonnie earned scores of 90 and 82 on her first two tests in English literature. What score must she make on her third test to keep an average of 84 or greater?

Step-by-Step Solution

Answer

Bonnie needs to score at least 80 on her third test.

Step 1: Understand the Problem

We need to find out what score Bonnie needs on her third test to have an average score of 84 or greater. She already has scores of 90 and 82 on her first two tests.

Step 2: Write the Formula for Average

The formula to calculate the average of three test scores is: \[ \text{Average} = \frac{\text{Score}_1 + \text{Score}_2 + \text{Score}_3}{3} \]

Step 3: Set Up the Inequality

Since we want the average to be 84 or greater, set up the inequality: \[ \frac{90 + 82 + x}{3} \geq 84 \]Here, \( x \) is the score Bonnie needs on her third test.

Step 4: Simplify the Inequality

Multiply both sides of the inequality by 3 to eliminate the denominator: \[ 90 + 82 + x \geq 252 \]

Step 5: Solve for \( x \)

Combine the constants on the left side: \[ 172 + x \geq 252 \]Subtract 172 from both sides to isolate \( x \): \[ x \geq 80 \]

Step 6: Conclusion

Bonnie must score at least 80 on her third test to maintain an average of 84 or greater.

Key Concepts

average calculation

Calculating an average is a very useful skill in many areas, especially in academics where scores matter. Let's break it down simply.

An average is found by taking the sum of all the values and dividing by the number of values. In the case of Bonnie's test scores, we need to find the average of three test scores, two of which are known. The formula is:
\[ \text{Average} = \frac{\text{Score}_1 + \text{Score}_2 + \text{Score}_3}{3} \]
This means we add the scores and divide by 3.
For instance, Bonnie's scores are: 90, 82, and a score we need to find (let's call it \(x\)). So our equation starts with:
\[ \frac{90 + 82 + x}{3} \]
From here, we will figure out what \(x\) needs to be to achieve Bonnie's desired average.

inequality solving

Inequality solving is key in ensuring that conditions such as minimum requirements are met. Let's work through Bonnie's problem with inequalities.

We want Bonnie's average score to be at least 84. This sets up an inequality:
\[ \frac{90 + 82 + x}{3} \geq 84 \]
To solve, we'll first eliminate the fraction. Multiply both sides of the inequality by 3:
\[ 90 + 82 + x \geq 252 \]
Combining the scores on the left side results in:
\[ 172 + x \geq 252 \]
To isolate \(x\), subtract 172 from both sides:
\[ x \geq 80 \]
This tells us that Bonnie needs to score at least 80 on her third test to meet her goal.

test scores

Test scores are often used to gauge a student's understanding and performance in a subject. They can also determine eligibility for certain academic standings or programs.

In Bonnie's case, her first two test scores were 90 and 82. These scores reflect what she has achieved so far.
When considering test scores for averages, each score contributes equally to the final average. Thus, each test score is critical.

Bonnie's aim is to maintain or surpass an average of 84. This requires more than just high scores; it requires consistent performance across all tests.

Understanding how individual test scores impact overall averages is key to managing and achieving academic goals.