Solving Complex Examination Scoring Problems with Algebraic Equations

Examinations often present challenging problems that require a combination of logical reasoning and mathematical skills. One such problem involves determining the number of correct and incorrect answers in an exam where each correct answer is worth 2 marks and each incorrect answer deducts 1 mark. This article will demonstrate how to solve such a problem using algebraic equations.

Problem Statement and Equations

Suppose Ram answered 150 questions in an exam and received a total of 240 marks. The scoring is as follows:

Correct answer: 2 marks Incorrect answer: -1 mark

Let:

x represent the number of questions Ram answered correctly y represent the number of questions Ram answered incorrectly

We can set up the following equations based on the problem statement:

x y 150 2x - y 240

Solution Steps

Step 1: Express y in terms of x

Solving the first equation for y:

y 150 - x

Step 2: Substitute the expression for y into the second equation

Substitute y 150 - x into the second equation:

((2x - (150 - x)) 240)

This simplifies to:

((2x - 150 x) 240) ((3x - 150) 240)

Further simplification:

(3x - 150 240) (3x 390) (x 130)

Step 3: Calculate the value of y

Substitute x 130 back into the equation for y:

(y 150 - 130 20)

Conclusion

Therefore, Ram answered 130 questions correctly.

Alternative Approaches

While the primary method used above is algebraic, alternative methods also exist and can yield the same result. Here are some equivalent formulations:

Using Multiples: Consider the scenario where all answers are correct and see how many would be correct or incorrect (as some need to be adjusted). Solving Simultaneously: Use the equations directly and eliminate variables to simplify.

Understanding the Concept of Opportunity Loss

In the context of scoring systems, opportunity loss refers to the difference between the marks that could have been obtained if the answers were correct versus the marks actually obtained.

Example:

If 3 answers are correct and 2 answers are incorrect, the opportunity loss is 3 - 2 or 1 answer. Given the deductions, the equivalent marked loss is 3 (correct) - 2 (actual) 1 answer.

Final Remarks

Different methods can lead to the same solution, but the primary approach involves setting up a system of equations and solving them step-by-step. Understanding these methods can enhance your problem-solving skills and make exams less daunting.