Inequality For Final Exam Score Minimum Multiple-Choice Questions Needed
To help Lorenzo secure that coveted "B" in his class, understanding how his final exam score is calculated is crucial. This article delves into the inequality that represents the relationship between the number of correct multiple-choice questions (x) and the total points needed to achieve at least 50 points. We'll break down the problem step-by-step, ensuring a clear understanding of the concepts involved.
Understanding the Problem
The core of the problem lies in translating the given information into a mathematical inequality. Let's recap the key points:
- Multiple-choice questions are worth 4 points each.
- Word problems are worth 8 points each (although the problem doesn't specify the number of word problems, we'll address this implicitly in the inequality).
- Lorenzo needs at least 50 points.
- x represents the number of correct multiple-choice questions.
The phrase "at least" is crucial here. It signifies that Lorenzo's score must be greater than or equal to 50. This translates directly into the "≥" symbol in our inequality.
Building the Inequality: Multiple-Choice Points
The first part of our inequality focuses on the points earned from multiple-choice questions. Since each correct answer is worth 4 points, the total points earned from x correct answers is simply 4 multiplied by x, or 4x. This forms the foundation of our inequality, representing the contribution of multiple-choice questions to Lorenzo's overall score.
To further clarify, let's consider a few examples:
- If Lorenzo answers 5 multiple-choice questions correctly, he earns 4 * 5 = 20 points.
- If he answers 10 correctly, he earns 4 * 10 = 40 points.
- If he answers 15 correctly, he earns 4 * 15 = 60 points.
This linear relationship between the number of correct answers and the points earned is fundamental to understanding the inequality. The more questions Lorenzo answers correctly, the more points he accumulates towards his goal of 50.
Incorporating Word Problem Points
While the problem doesn't explicitly state the number of word problems, we need to account for their contribution to Lorenzo's total score. Let's represent the number of word problems Lorenzo answers correctly as y. Since each word problem is worth 8 points, the total points earned from word problems is 8y. Now, let's delve deeper into how this component interacts with the multiple-choice questions in the overall inequality.
The Combined Impact
Lorenzo's total score is the sum of the points earned from multiple-choice questions and word problems. Therefore, his total score can be represented as 4x + 8y. Remember, Lorenzo needs at least 50 points, which means his total score must be greater than or equal to 50. This crucial requirement translates directly into the following inequality:
4x + 8y ≥ 50
This inequality encapsulates the relationship between the number of correct multiple-choice questions (x), the number of correct word problems (y), and the minimum score Lorenzo needs to achieve. It provides a mathematical framework for analyzing different scenarios and determining the combinations of x and y that will allow Lorenzo to reach his goal.
Simplifying the Inequality
Often, inequalities can be simplified to make them easier to understand and work with. In this case, we can divide both sides of the inequality 4x + 8y ≥ 50 by 2. This simplification maintains the integrity of the inequality while reducing the coefficients, resulting in a more manageable form:
2x + 4y ≥ 25
This simplified inequality is equivalent to the original and represents the same relationship between x, y, and the minimum score. It highlights that for every two multiple-choice questions answered correctly, it contributes the same amount of points as half a word problem, comparatively speaking. This can be a useful way to think about the trade-offs between answering multiple-choice questions versus word problems.
A Note on the Variable 'y'
It's crucial to recognize that the problem explicitly asks for an inequality representing x, the number of correct multiple-choice questions. While we've incorporated y (the number of correct word problems) into our initial inequality for a complete picture, we need to consider scenarios where we isolate x or make assumptions about y to arrive at a final answer that focuses solely on x. This might involve considering the worst-case scenario, where Lorenzo answers no word problems correctly (y = 0), or making assumptions about a fixed number of word problems.
The Inequality Representing x (with y = 0)
To directly address the problem's request for an inequality representing x, we need to consider a scenario where we can eliminate y from the equation. A common approach is to consider the worst-case scenario: What if Lorenzo answers no word problems correctly? In this case, y = 0, and our inequality simplifies significantly.
Substituting y = 0 into the inequality 4x + 8y ≥ 50, we get:
4x + 8(0) ≥ 50
4x ≥ 50
This inequality now solely represents the relationship between the number of correct multiple-choice questions (x) and the minimum score required. It provides a direct answer to the problem's question, focusing on the specific variable of interest.
Solving for x
To further refine our understanding, let's solve the inequality 4x ≥ 50 for x. To do this, we divide both sides of the inequality by 4:
x ≥ 50 / 4
x ≥ 12.5
Since Lorenzo cannot answer half a question correctly, we need to round up to the nearest whole number. This means Lorenzo needs to answer at least 13 multiple-choice questions correctly to score 50 points if he answers no word problems. This result provides a concrete, actionable understanding of the problem's solution.
Interpreting the Solution
The inequality x ≥ 12.5 (rounded up to 13) provides valuable insight. It tells us that if Lorenzo relies solely on multiple-choice questions, he needs to get at least 13 correct to achieve his desired score of 50. This serves as a crucial benchmark. If Lorenzo answers any word problems correctly, the number of multiple-choice questions he needs to answer correctly will decrease. This highlights the trade-off between the two types of questions and the flexibility Lorenzo has in achieving his goal.
Final Answer and Conclusion
Therefore, the inequality that represents x, the number of correct multiple-choice questions Lorenzo needs to answer to score at least 50 points, assuming he answers no word problems correctly, is:
4x ≥ 50
And the minimum number of multiple-choice questions he needs to answer correctly is 13.
This detailed explanation has broken down the problem step-by-step, from understanding the initial information to building and simplifying the inequality, and finally, interpreting the solution. By understanding these concepts, Lorenzo (and anyone facing similar problems) can effectively analyze and solve mathematical inequalities to achieve their desired outcomes.
This comprehensive analysis demonstrates the power of translating real-world scenarios into mathematical expressions. By understanding the underlying principles, students can confidently tackle similar problems and apply these skills in various contexts.
