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: 0