Problem 61 - Algebra for College Students

Question

Solve each problem by setting up and solving an appropriate inequality. Marsha bowled 142 and 170 in her first two games. What must she bowl in the third game to have an average of at least 160 for the three games?

Step-by-Step Solution

Answer

Marsha must bowl at least 168 in her third game.

Step 1: Understanding the Problem

We need to determine the score Marsha must achieve in her third game to have an overall average of at least 160 for three games. We know her scores for the first two games and need to find the lowest possible third game score that maintains the average.

Step 2: Define Variables

Let \( x \) represent the score Marsha needs in her third game.

Step 3: Set Up the Average Equation

The average score for three games can be found using the formula: \( \text{Average} = \frac{S_1 + S_2 + S_3}{3} \). Here, \( S_1 = 142 \), \( S_2 = 170 \), and \( S_3 = x \).

Step 4: Set Up the Inequality

We know the average should be at least 160, so we have the inequality: \[ \frac{142 + 170 + x}{3} \geq 160 \]

Step 5: Clear the Denominator

Multiply both sides of the inequality by 3 to eliminate the denominator: \[ 142 + 170 + x \geq 480 \]

Step 6: Combine Like Terms

Add the known scores: \( 142 + 170 = 312 \). This simplifies the inequality to: \[ 312 + x \geq 480 \]

Step 7: Solve for x

Subtract 312 from both sides to isolate \( x \): \[ x \geq 480 - 312 \] which simplifies to \( x \geq 168 \).

Key Concepts

Average Calculation

Calculating an average means finding the central value among a set of numbers.
This is done by summing all numbers and then dividing by the number of values. To put this into the context of a real-world problem, such as Marsha's bowling scores, you want to figure out what the average score needs to be.
For example:

• Add up all the scores you have: In this case, Marsha's two scores, 142 and 170.
• Consider what additional scores are required to achieve the target average: In Marsha's case, the target average is 160.
• Use the average formula: \( \text{Average} = \frac{\text{Total Scores}}{\text{Number of Scores}} \).

By understanding how each score contributes to the average, we can determine what is needed to meet a specific goal.

Problem Solving

Problem-solving in mathematics involves understanding what is being asked and systematically working through the solution.
The first step is breaking down the problem into understandable chunks like Marsha's bowling average. Let's illustrate:

• **Understand the problem**: We need a third score to maintain an average of at least 160.
• **Define what's needed**: Here, the score Marsha must achieve to meet the requirement.
• **Set up your inequality**: Using inequality to establish boundaries for possible solutions \( \frac{142 + 170 + x}{3} \geq 160 \).

Once you work through these steps, the unknown value becomes clearer, making it easier to find a solution systematically.

Mathematical Modeling

Mathematical modeling involves using mathematical language and expressions to represent real-world scenarios.
This means creating equations or inequalities from given situations. Consider Marsha's example:

• Model her bowling situation: Here, we know two scores and need to find the third.
• Translate the problem into an inequality: Use \( x \) to represent the unknown score and form an inequality \( 142 + 170 + x \geq 480 \).
• Solve the inequality to find a solution: This gives us the minimum score Marsha needs to ensure her average is 160.

Through modeling, these abstract concepts connect back to everyday life, aiding in solving practical problems with mathematical precision.