Problem 63 - Algebra for College Students

Question

Solve each problem by setting up and solving an appropriate inequality. Suppose that Derwin shot rounds of \(82,84,78\), and 79 on the first four days of a golf tournament. What must he shoot on the fifth day of the tournament to average 80 or less for the five days?

Step-by-Step Solution

Answer

Derwin must shoot 77 or less on the fifth day to average 80 or less.

Step 1: Define the Problem

We need to find the score Derwin must shoot on the fifth day so that his average score over five rounds is 80 or less.

Step 2: Set up the Inequality

To have an average score of 80 or less, the total score after five rounds, divided by 5, should be less than or equal to 80. Let \( x \) be the score of the fifth round. The inequality is \( \frac{82 + 84 + 78 + 79 + x}{5} \leq 80 \).

Step 3: Simplify the Inequality

First, calculate the sum of the scores from the first four rounds: \( 82 + 84 + 78 + 79 = 323 \). Substitute this into the inequality: \( \frac{323 + x}{5} \leq 80 \).

Step 4: Solve for x

Multiply both sides of the inequality by 5 to eliminate the fraction: \( 323 + x \leq 400 \). Subtract 323 from both sides: \( x \leq 400 - 323 \).

Step 5: Calculate the Result

Perform the subtraction: \( x \leq 77 \). Thus, Derwin needs to shoot 77 or less on the fifth day to maintain an average score of 80 or less.

Key Concepts

Average Score Calculation

Calculating an average score involves taking the sum of all the individual scores and dividing by the number of scores. In this problem, we want to find out what score Derwin needs on the fifth day to achieve an average of 80 or less across five days of play. Here's how it works:

• Add up all the scores from the first few days, which are given as 82, 84, 78, and 79.
• Next, assume the fifth score as a variable, let's call it \( x \).
• To find the average, add these scores together and then divide by the number of scores, which is 5 in this scenario.

So, the average is calculated using the formula: \[\frac{82 + 84 + 78 + 79 + x}{5}\] This formula needs to result in 80 or less, as per the problem statement. Understanding this concept is essential for solving many average-based problems.

Problem Solving in Algebra

Algebra is a powerful tool for problem solving, helping us express relationships between numbers in terms of variables and equations. In this exercise, we have a situation where we need to use an inequality to find a solution. The thought process involves:

• Defining what exactly you're solving for— in this case, the score Derwin needs on the fifth day.
• Setting up an inequality that represents the problem situation— balancing the known scores and the unknown final score against the target average.
• Simplifying the inequality to eventually solve for the variable in question.

By following these systematic steps, you can solve a wide range of problems, from simple arithmetic to more complex algebraic equations. Always remember: identify what's given, what you need, and use algebra to bridge that gap.

Setting Up Equations

Setting up equations, or in this case inequalities, involves translating a word problem into a mathematical expression. This translation is crucial because it forms the foundation for solving the problem. Here's a breakdown of how to effectively set up equations:

• Identify what you know and what you need to find out. In this case, you know the scores for four days and you need to find the fifth score.
• Use a variable to represent the unknown. Here, we use \( x \) to represent the unknown score on the fifth day.
• Formulate an equation or inequality that encapsulates the entire problem. For this problem, it's capturing the condition that the average should not exceed 80.

The equation for Derwin’s problem starts as an inequality: \[\frac{323 + x}{5} \leq 80\]This formulation captures all the given information and allows us to solve for the unknown by identifying and manipulating the variable.