Problem 52 - Elementary Algebra

Question

Set up an algebraic inequality and then solve it. Maurice earned \(4,7,\) and 9 points out of 10 on the first three quizzes. What must he score on the fourth quiz to average at least \(7 ?\)

Step-by-Step Solution

Answer

Maurice must score at least 8 on the fourth quiz.

Step 1: Find total points needed for average

To find the total points Maurice needs to average at least 7, multiply the average score by the total number of quizzes. Since Maurice wants an average of at least 7 over 4 quizzes, calculate it as follows: \[7 \times 4 = 28\]So, Maurice needs a total of 28 points for all four quizzes.

Step 2: Sum points already earned

Add the scores Maurice earned on the first three quizzes. These scores are 4, 7, and 9. \[4 + 7 + 9 = 20\]This means Maurice has already scored 20 points in total.

Step 3: Set up the inequality

Let the score Maurice needs on the fourth quiz be represented by the variable \( x \). Establish an inequality to reflect that the total points Maurice must have on all quizzes must be at least 28:\[20 + x \geq 28\]

Step 4: Solve the inequality

Solve the inequality for \( x \):\[20 + x \geq 28\]Subtract 20 from both sides:\[x \geq 8\]Thus, Maurice must score at least 8 on the fourth quiz.

Key Concepts

Average Calculation

Calculating an average is an essential skill in mathematics and beyond. To find an average, sum all the numbers you're working with, then divide by how many numbers there are. In Maurice's case, he needed to average 7 over four quizzes. Knowing how to calculate averages helps in determining how well someone is doing relative to their goals.
When Maurice needed a specific average, the process was:

• Add scores from all quizzes (with the unknown score for the fourth quiz).
• Divide the total by the number of quizzes, which was 4.
• Calculate to ensure the result is at least the desired average, which in this case was 7.

This concept extends to various scenarios, like computing class averages or even financial metrics, making it invaluable.

Problem-Solving Steps

Solving algebraic inequalities involves systematic steps, helping ensure logical and correct results. These steps guide you from understanding what is asked to reaching a conclusion:
1. **Identify What's Given**First, look at the specific numbers or constraints. Maurice's known quiz scores were 4, 7, and 9. His goal was to have an average score of 7.
2. **Translate to Mathematical Expressions**Maurice's requirement for the fourth quiz score translates into the inequality statement: \[20 + x \geq 28\]This inequality set the stage for solving the problem.
3. **Solve the Inequality**Subtract from both sides of the inequality until you isolate the unknown variable on one side. In Maurice's case:

• Subtract 20 from both sides, leaving \(x \geq 8\) as the simplified inequality.

Always verify that the solution meets the conditions laid out initially.

Educational Mathematics

Educational mathematics often goes beyond just getting to the right answer. It helps foster logical thinking, problem-solving skills, and the ability to apply concepts to real-world situations. In Maurice's problem, learning the steps to solve an inequality not only helped him find the score needed for his goal, it also showed him how to handle similar situations in the future. Here are some benefits of engaging with exercises like Maurice's quiz problem:

• It builds a foundation in basic algebraic skills which are crucial for advanced mathematics.
• Understanding averages and inequalities develops critical thinking.
• These exercises often model practical scenarios, such as budgeting or planning schedules, linking mathematics to everyday life.

By practicing such problems, students acquire a skill set that is applicable in many areas, be it academics or daily decision-making scenarios.