Maximizing Wrong Answers in a School Test: A Mathematical Insight
Understanding the intricacies of test scoring can be crucial for students aiming to maximize their scores under unique conditions. This article delves into a specific scenario where a student faces a 150-question test with a unique scoring system: 3 points for each correct answer, -1 point for each wrong answer, and -0.5 points for each unanswered question. The goal is to identify the maximum number of questions a student could answer wrongly while still achieving a score of 162.
Problem Setup
The problem at hand is to determine the number of wrong answers a student could have while maintaining a score of 162. To analyze this, we define the following variables:
nx represents the number of correct answers. ny represents the number of wrong answers. nz represents the number of unattempted questions.The total number of questions is 150 and the student's score is 162. Hence, the following equations govern the problem:
Total questions:
nx ny nz 150
nz 150 - nx - ny
Score:
3nx - ny - 0.5nz 162
Substituting the expression for nz into the score equation:
3nx - ny - 0.5(150 - nx - ny) 162
Simplifying the equation:
3nx - ny - 75 0.5nx 0.5ny 162
3.5nx - 0.5ny - 75 162
3.5nx - 0.5ny 237
Multiplying the equation by 2 to eliminate decimals:
7nx - ny 474
Solving the System of Equations
The two equations are:
nx ny nz 150
7nx - ny 474
Solving for ny in terms of nx:
ny 7nx - 474
Substituting ny into the first equation:
nx 7nx - 474 nz 150
8nx - 474 nz 150
nz 624 - 8nx
Ensuring nz must be non-negative:
624 - 8nx geq 0
624 geq 8nx
78 geq nx
Setting nx 78:
ny 7(78) - 474 546 - 474 72
nz 624 - 8(78) 624 - 624 0
This solution suggests:
Correct answers: 78 Wrong answers: 72 Unattempted questions: 0Maximizing Wrong Answers
To find the maximum number of wrong answers while keeping the score at 162, we need to lower nx and ensure nz remains non-negative. We continue this process to find the maximum:
If nx 77, ny 65, nz 8 If nx 76, ny 58, nz 16The maximum occurs when nz 0, which means:
nx ny 150 ny 7(150 - ny) - 474 1050 - 8ny 474 8ny 576 ny 72Therefore, the maximum number of wrong answers is 72.